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2. Then G'")p is continuous and
const !20-k k IG,u
if N = 2,
_Galul
2. Hence lim,1. G'"p = Gp uniformly on RN, and we conclude that u = Gµ is continuous. In view of (7.1) the integral in (7.2) is absolutely convergent.
9
7. The Smoothness of Potentials; The Poisson Equation
Round off the integrand in (7.2) when it exceeds k""` in absolute value to obtain a continuous finite-valued integrand and repeat the reasoning proving the continuity of Gp to show that the right side of (7.2) defines a continuous function of . This function must be Proof of (b). In view of (7.1) and the Holder condition satisfied by Jon D the integrand in (7.3) for n in a neighborhood of in D is majorized in absolute value by constl - nI-"+"; so the integral converges absolutely. Let /3 be a function from R+ into R+ satisfying the following conditions:
05fl51,
$eC"'(R+),
(0 ifOSr 0. Then au(k) a
J
I
a;
a t c.=
I
a) = uniformly on R". The function a) is in so lima_, C"'(R"). To prove (b) it is sufficient to show that, on D, lim, .0 a4,/a4'h' is the right side of (7.3) and that the convergence is uniform on every compact subset of D. To show this, write in the form
fRN
aft gj) [1(n) -I()]P
a'a'J)
iN
a
8G(, n)
I
att, [!fin) -n
+ a Jp" +
r"(dn)
rl (- ) IN*)
+1()
_aL)
-« nI
I_
('RN
J
aG(, ), (14
)]fl,
I
a nI
tit IJ
- nI
1"(dn)
(J) - ncn -171
)
I
__171
1"(dn)
and denote the four terms on the right by I, It, III, IV, respectively. Observe that each integrand vanishes for n outside B(4, a).
10
1.1. Introduction to the Mathematical Background of Classical Potential Theory
The difference between I and the integral on the right side of (7.3) is at most const Jo r"!, (dr) for a < I - BDI ; so when a - 0, the term I has the integral on the right side of (7.3) as a limit uniformly on compact subsets of D.
If i # j in II, the integral over a) vanishes because the integrand is odd in nt". If i = j in II, the integral becomes f(IX'
-(N - 2) J t e.Q)
-
(i
(7.4)
0+N(N-2)
f
JB(.a) I - ql
at
if N > 2 and a < I - aDl. The second integral does not depend on the choice of i, and so it has as value the average of its values for i = 1, ... , N. We conclude that the sum in (7.4) vanishes. The corresponding argument when N = 2 shows that the integral II also vanishes in this case. If of < I - ODl, 11111 < const at
f rp,' (r)/1(dr) 5 const a";
so when a 0, there is uniform convergence to 0 on compact subsets of D. To evaluate IV observe that when i # j this integral over a) vanishes
because the integrand is odd in i'" - nt". If i = j and if N > 2 then IV =
-fad) (N - 2)
n"h2Q, (1
nl) at
L4.0
The integral is the same for all i, and it is equal to the average of its values
for i = 1, ..., N. Hence
IV= -
aN
0
n"l)(N-2)Ja
,()1i(dr) f
I(') N
If N = 2 the corresponding evaluation of IV yields and the proof of the theorem is now complete. 0 Extension. If v is a signed measure on I8" for which the integral defining Gv converges absolutely and if the projection of v on some bounded open set D is determined by a bounded density g relative to IN, then (Gv)1oe Ct"(D)
because if f = g1D and du = fdIN, then Theorem 7 is applicable to Gp which differs on D from Gv by a harmonic function. The same argument shows that if g is continuous on D and satisfies a Holder condition there, then (Gv)1D(-C"'(D) and AGv = -i g there.
8. Harmonic Measure and the Riesz Decomposition
11
Extension of the Gauss Integral Theorem
Under the hypotheses of (b) the flux evaluation (1.3) yields, whenever B is a smooth domain with closure in D,
f D,Gpol"_1. p(B) _ - in"f AGItdl" _ - 1 n"
I
(7.5)
ae
This evaluation generalizes the Gauss Integral Theorem (Section 6) in that the support of the measure u is allowed to meet OB.
8. Harmonic Measure and the Riesz Decomposition Let D be an open bounded subset of R". Suppose that JI E C'2)(D), and suppose that Au is bounded on D. Define a signed measure y on D by du = - Au dlN/n,. Theorem 7 implies that A(u - Gµ) = 0, so that u = (u - Gµ) + G,u is the sum of a harmonic function and the restriction to D of the potential Gp. This fact will now be proved in a slightly different version and developed further.
If D is a smooth open subset of Rr, if c D, and if uEC12)(D), an application of Green's identity to the pair of functions [u, on the smooth open set D - B(i, S) yields
uI"-MW aatt.a)
-
=
(8.1)
J8D
q)
is bounded, the first integral on the left is majorized in absolute value by const 61log 61 if N = 2 and by const 62 if N > 2; so this integral Since
tends to 0 when b -+ 0. The second integral on the left is equal to - n'" L(u, , S)
for N Z 2 and therefore has limit -ir u(g) when 6 - 0. Thus when b - 0 in (8.1), we find
u() = 1
[G(f, q)D u JaD 1
(8.2)
G( ,q)Au(l)I"(thl)
12
1.1. Introduction to the Mathematical Background of Classical Potential Theory
This representation of u is another version of that obtained above because the first term in (8.2) defines a function harmonic on D and the second
term is the potential of the signed measure with density -Au/nN. This representation of u remains valid if is replaced by u(t, ), is harmonic on D. Suppose e Ct21(D) for each in D and that can be chosen to satisfy (i) and also (ii) on OD. In this case the restriction to D of the difference GD(, ) = G(, ) - u(, ) is called the Green function of D with pole t, and GD is called the Green where (i)
function of D. The function u(, ), if there is such a function, is uniquely determined because the difference between two such functions is harmonic on D with limit 0 at every boundary point and therefore vanishes identically
(maximum-minimum theorem for harmonic functions). Recapitulating, GD(, ) is to satisfy the following conditions: (i') (ii')
GD is defined on D x D and GD(, ) = + oo.
GD(, ) is harmonic on D - {}, and GD(, ) - G(, ) is harmonic on D if defined suitably at .
(iii') GD(, ) has limit 0 at every point of aD. (iv') GD(, ) - G(, ) if defined suitably at and, if defined as -G(, ) on aD, is in Ct2j(D).
An application of the harmonic function maximum-minimum theorem to GD(, ) on D - {} shows that GD s 0. We shall show in Section 11.1 that a ball has a Green function in this classical sense. The existence of a Green function satisfying conditions (i')-(iv') is a restriction on the smoothness of aD, but in Chapter VII we shall define a Green function much more
loosely, keeping (i') and (ii'), weakening (iii'), and dropping (iv'). It will be shown that every nonempty open subset of RN for N > 2 and every not-too-large nonempty open subset of R2 (for example, every nonempty bounded open set) has a unique Green function in the looser sense. The present discussion shows what led to the more general definition and will suggest theorems to be proved.
Suppose then that D is smooth and that GD exists satisfying (i')-(iv'). If v is a measure on D, the function GDV on D is called the Green potential of v. Using GD instead of G in (8.2) and defining GD(, ) as 0 on aD reduces (8.2) to u(S) =
n--
f'Du(I)DeGD(,)lN-10I) -
NJDGD(,J)Au(1)lN(d+l) (8.3)
In particular
I = - N J DGD(, )lN-1(d 1). D
(8.4)
8. Harmonic Measure and the Riesz Decomposition
Define the measure
13
of Borel subsets of aD by
A) _ -
JDGD(i.)lN_l(dn).
(8.5)
N
Then ND(., ) is a positive measure because the normal derivative in the integrand is negative, and aD) = 1. The measure is called the harmonic measure relative to . The representation (8.3) can now be written
GD),
u(0 = JaD f
dv = -AudlN
(8.6)
In our later general treatment we shall prove the following facts: (a) (b)
GD is symmetric and continuous on D x D(= + o0 on the diagonal).
The function A) is harmonic on D for every A, and more generally the function J8Df(r1)LD( an) is harmonic on D for every bounded Borel measurable function f on OD.
These facts will be trivially verifiable when D is a ball (see Section 11.1), the only case when they will be used explicitly before the general discussion.
We therefore proceed without proving facts (a) and (b) in the present discussion. If u is harmonic on D the second term in (8.6) drops out; so
u() = J u a weighted average of the values of u on OD. In particular, if D is a ball with center , this averaging reduces to the defining average property of harmonic functions. More generally, if Au 5 0 and u > 0, (8.6) exhibits u as the sum of a positive harmonic function and the Green potential of a positive measure. One of the principal aims of the general theory is to generalize this result (the Riesz decomposition, Section IV.8) by dropping the smoothness con-
ditions on u, D, and GD.
Chapter II
Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions
1. The Green Function of a Ball; The Poisson Integral B denote by ' the image of under inversion in B. That is, ' is on the ray from o through , and 1 - u 1 1 ' - X01 = 62.
Let B =
To simplify the notation take o = 0. Then GB, as defined by
forN=2
7) =
712-N
712-N
-
(=1712-N -
-
(1.1) 62-N ifS
= 0)
forN>2
with the understanding that
oo, satisfies items (i')-(iv') of
Section 1.8, so that harmonic measure for B is given by I,
D
K(7Ni1N-1(d7),
(1.2)
where 1N- 1 here refers to surface area on aB and SN-2S2 - 1412
(1.3)
Hence, according to Section 1.8, if u is harmonic on a neighthorbood of B,
u( )= 7r b N
f
oB
u(,)s2-1FFSI21N-I(d1)= nNSIN_, J u(7)K(7,t)IN-1(d7). (1.4) 17 - y l dB
The function K(7, ) is harmonic on RN - (7) because GB(-, 7) is, and K(7, ) is normalized to be I at the origin. The function GB is symmetric, positive (= + oo when the arguments are equal), and increases with S. Moreover
15
1. The Green Function of a Ball; The Poisson Integral
lim lira 6-00
if N = 2,
+ oo 17) _
q)
if N>2.
(1.5)
Since ry) is harmonic on B - {ry}, all the partial derivatives of this function are also harmonic there so that in particular the harmonic measure density in (1.2) defines a harmonic function of on B. It follows that if µ is a finite measure of Borel subsets of aB, the Poisson integral (PI) J
(dq),
(1.6)
a8
defines a harmonic function on B. It should cause no confusion if we write PI(B,f) instead of PI(B,p) when t(dry) = j(ry)IN_1(dry)l(nNS" '). In later
applications the ball may not have the origin as center, and the Poisson formula is modified accordingly. Moreover, if f is defined on a superset S of the boundary of a ball B, we shall write PI(B,f) instead of PI(B, f ae)
Theorem. If f is Lebesgue integrable on aB, u = PI (B, f) is harmonic on B and lim sup
lim sup
4-4
teB
(CeaB).
(1.7)
4-C
4EOB
Combining (1.7) with the corresponding inequality for inferior limits, it follows that u has limitf(C) at C if f is continuous at C.
We have already noted that u is harmonic on B. Inequality (1.7) is a special case of Theorem I of Appendix VII with X = aB and n -+ oo replaced by C. The key fact is that for ry outside a neighborhood of C and on aB
the integrand K(ry, ) is at most const (62 to 0 when
and this majorant tends
in B tends to C.
Integral of K. The fact that JB K(ry, )IN(dC) = 1N(B) will be needed below.
To derive this evaluation observe that the value a = a(q) of this integral does not in fact depend on ry; so a = L(a, D, b) =
fB P
I(B, 1)lN(dq) = IN(B).
Solution of the Dirichlet Problem for a Ball If f is a finite continuous function on the boundary of a ball B, the function u = PI(B,f) is harmonic on B and, according to Theorem 1 has limit f(C) at every boundary point C. There is only one harmonic function on B with
16
1.11. Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions
this boundary limit property because the difference between two such functions is harmonic on B with limit 0 at every boundary point and therefore vanishes identically in view of the harmonic function maximum minimum
theorem. The function u is thus the unique solution of the classical first boundary (Dirichlet) problem for harmonic functions on B. A generalized form of this problem for the relevant class of open subsets of R' is treated in Chapter VIII. If u is a finite-valued function defined and continuous on the closure of a ball B and harmonic on B, u = PI(B, u) on B because u is the unique solution of the Dirichlet problem for the boundary function ulde. This result weakens the conditions under which (1.3) was derived using the general theory in Section 1.8.
If u is any Borel measurable function on an open subset D of R', if B is a ball with closure in D, and if the restriction of u to 8B is IN_, integrable, we define (PI(B, u) tBu
on
B,
on D-B.
=1 u
(1.8)
If u is upper or lower semicontinuous, Theorem I implies that tBu has the same property.
2. Harnack's Inequality Let A be a compact subset of the open connected subset D of RN. There is a function (A, D)"c(A, D) such that if u is harmonic and strictly positive on D, u() < c(A, D) u(ri)
If
[(c, ry) e A X A].
b) c D, the representation u = S2 - a2 6N_2
- oo if
6) c D.
Hence finiteness of u at a point implies finiteness 1N almost everywhere in 6) when 6) c D, and therefore a covering argument shows that u is finite 1N almost everywhere on D and that u is locally 1N integrable. In particular A(u, , b) is finite in (c') even when x. We shall show in Section 6(a) that L(u, , 6) in (c) is finite even when ao, and we shall show in Section 6(c) that (c') can replace (c) in the definitiop of superharmonicity. If u, and u2 are superharmonic functions on D and if c, and c2 are positive
constants, the functions c,u,, c,u, + c2u2, and u, A u2 are superharmonic
4. Harmonic, Subharmonic, and Superharmonic Functions
19
on D. Since superharmonic functions are lower semicontinuous, the limit of an upward-directed family of superharmonic functions on D is the limit of an increasing sequence of functions in the family (Theorem 2 of Appendix VIII). The limit function u is superharmonic if condition (b) is satisfied.
A subiamonic function is defined as the negative of a superharmonic function. A function is harmonic if and only if it is both subharmonic and superharmonic. It is unfortunate that it is natural in pure potential theory to consider superharmonic rather than subharmonic functions but that the applications of potential theory are likely to involve subharmonic functions rather than superharmonic functions. Positive Integral Operations on Superharmonic Functions Let D be a nonempty open subset of R", and let R(D) be the class of Borel subsets of D. Let (up, ig a 1) be a family of superharmonic functions on D, indexed by a set 1. If (1, .F, A) is a finite measure space and if the function is measurable from (D x 1, R(D) x ,F) into (R,R(R)), then the function u' = f r up A(dd) satisfies the superharmonic function average inequality, L(u', 4, b) = L
\Jr
uQA(dfl), , 6) //
= f L(up, t, b)A(d$) S Jr
(I - aDI > b), if the double integral involved converges absolutely. Thus u' is superharmonic if the superharmonic function finiteness and lower semicontinuity conditions are satisfied. EXAMPLES. Let b be a strictly positive number so small that
-aDj >b) is not empty. Then if u is a superharmonic function on D the function u' = L(u, , b) on Da is a special case of the preceding integral operation, with u( + P), fl e aB(0, 6) and with A the normalized surface area of B(0,6). In this case an application of Fatou's lemma shows that u' is lower semicontinuous. We shall show in Section 6(a) that L(u, , 6) is finite valued, and it follows that L(u, , 6) is superharmonic on D. We leave to the reader the verification of the fact that if u is a superharmonic function on D, the functions A (u, , 6) and Adu are also special cases of the above integral operation. Since (from Section 2) A (u, , 6) is finite valued and continuous and Aau is infinitely differentiable, these two functions are superharmonic on D6.
For u superharmonic on D the three functions L(u, , 6), A (u, , b), and Aau are all majorized by u, and it will be shown in Section 6(f) that these
20
1.11. Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions
functions increase monotonely to u when S -+ 0, in the sense that when So > 0 and b < 60, there is monotone convergence on D60 to u when 6 -+ 0.
5. Minimum Theorem for Superharmonic Functions Theorem. Let u be superharmonic on an open subset D of R'. (a)
If D is connected and if u attains its infunum at a point , then u is identically constant.
(b)
The infimum of u is the limit of u along a sequence of points approaching OD.
(c)
If u has a lower semicontinuous extension to D u 3D, the infunum of the extension is attained on aD.
A typical implication of this theorem is the fact that a superharmonic function on D with a positive inferior limit at every boundary point must be positive. This theorem is the generalization for superharmonic functions of the harmonic function maximum minimum theorem (see Section 1.4). The proof of Theorem 5(a) is precisely the same as that given of Theorem 1.4(a) involving the infimum of u because that proof required only lower semicontinuity of u and the superharmonic function average inequality. Parts (b) and (c) follow easily from (a).
6. Application of the Operation TB This operation, defined in Section 1, will be used to derive various important properties of superharmonic functions. (a) If u is superharmonic on D and if B is a ball with closure in D, then u is IN _ 1 integrable on aB and r9u 5 u. In fact, since u is lower semicontinuous,
it is bounded below on iB, and there is an increasing sequence f of finite-
valued continuous functions on aB with limit uka. Then PI(B,f) is an increasing sequence of harmonic functions on B with limit PI(B, u), and an application of the superharmonic function minimum theorem to u - PI(B, on B shows that the difference is positive there. Hence (n - oo) u is IN _I integrable on aB and TBU 5 u. (b) In (a) if B = B(g, S) and if
u is harmonic on B.
so that the restriction to In fact, under the stated conditions, B of u - r,u is positive superharmonic, vanishes at , and therefore vanishes identically. (c) In the definition of superharmonicity in Section 4, (c) can be replaced
by (c'). We have already seen that superharmonic functions satisfy (c'). Conversely, if u satisfies Section 4(a), (b), (c') the reasoning leading to the superharmonic function minimum theorem and thereby to (a) above remains
6. Application of the Operation rB
valid, and then if B =
21
b), the inequality TBU() S u(i;) is condition
Section 4(c). (d) In the definition of a superharmonic function in Section 4 condition (c) [or equivalently (c')] there need be supposed true only for sufficiently small b, depending on , because the reasoning leading to (a) of the present section
used only this weakened condition, and (a) implies Section 4(c) when B= (e)
b).
In (a) the function rBu is superharmonic on D because according to
Section 1 this function is lower semicontinuous, and it is trivial that for i; in D the inequality z L(TBu, g, b) is valid for sufficiently small b. More generally, if B is a ball with closure in D, if u is superharmonic on D, and if v is a function defined and superharmonic on B, with v z TBU on B, then the function v' (equal on B to v A u and on D - B to u) lies between u and TBU from which it follows easily that v' is superharmonic on D. (f) If u is superharmonic on D and if e D, the functions b - L(u,, b), b i-+ A (u, ia, bland b i--+ Aau() are monotone decreasing for 0 < b < 1 - OD I, when b -+ 0. To prove the monotoneity of L(u, c, -) observe with limit
that if B; = B(c, b,) with b, < b2 < I - aD1, then the inequality TB2TB,u 5 TB, u reduces at 1; to L(u, , b2) < L(u, s, b,). Next apply the lower semicontinuity of u and Fatou's lemma to derive the inequality S lim L(u, l;, b) 5 a-0
and thereby complete the proof of (f) for L(u, c, ). The corresponding results for and follow from 1(2.2) and 1(2.4). The latter results imply that the relation u >- v or u = v, if satisifed IN almost everywhere on their domain of definition D by superharmonic functions u and v, is satisfied everywhere on D. The preceding results imply the truth of a slight strengthening of the lower semicontinuity property of a superharmonic function u, namely,
u(S) = liminfu(,) = lim infu(q), rC nee q*A
(6.1)
where A is an arbitrary iN null set. We shall see in Section XI.1 that the natural class of sets A making (6.1) true is the class of sets thin at in the sense of the fine topology.
Observation. If b > 0 and if 06 is a function on the interval [0, b], Borel measurable and 11 almost everywhere strictly positive, with 160 Cd(r) dr = 1,
then every superharmonic function u on an open set D obviously satisfies the inequality
u() z f
d
0
1e(r)L(u, , r) dr
(6.2)
22
111. Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions
when
b) c D. Now suppose that So > 0 and that ¢a is defined and
satisfies the conditions imposed on ie above, for 0 < b < bo. Then if u is a function on the open set D, satisfying the superharmonic function defining conditions (a) and (b) of Section 4, together with (6.2) for sufficiently small b, depending on , the function is superharmonic. This assertion has already been proved for 4a(r) = Nb-l r"-' , in which case the right-hand side of (6.2) reduces to A (u, , b), and the proof needs no change in the general case. EXAMPLE (The Fundamental Kernel and the Green Function of a Ball). If the is fundamental kernel G is defined on R" x R", by 1(5.2) the function and is superharmonic on R". In fact for each point harmonic on R" -
we have noted in Section 1.5 that this function is harmonic on R" Moreover this function is continuous on R", and in view of the local nature of superharmonicity proved in (d) above we need only observe, to prove that
the function is superharmonic on R", that the superharmonic function average inequality is trivially satisfied at . Similarly, if B is a ball, the Green function GB is defined on B x B by (1.1), and for each point t of B, GB(4, ) and superharmonic on D. is harmonic on B -
7. Characterization of Superharmonic Functions in Terms of Harmonic Functions It is important to characterize superharmonicity intrinsically, without the use of balls. Let u be a lower semicontinuous function from the open subset D of R" into ] - oo, + oo], not identically + oe on any open connected component of D. Consider the following property of u: if Do is a relatively compact open subset of D and if v is a function defined and harmonic on a
neighborhood of b, with u - v -: 0 on OD,, then u - v > 0 on Do. This condition is necessary and sufficient for u to be superharmonic and justifies the name "superharmonic." In fact a superharmonic function on D has this
property according to the superharmonic function minimum theorem. Conversely, if u has this property, let B = B(4,6) be a ball with closure in D. L(u, a:, b) and even, since u is lower semiIt is enough to prove that z L(f, , b) for every finite-valued continuous continuous, to prove that
function f defined on aB and majorized by u. For such a function f the function v = PI(B,f) is harmonic on B with a continuous extension to h obtained by setting v = f on B. The difference u - v is lower semicontinuous
on B and positive on aB, so that ifs > 0, then u - v z -E near 6B (for example, on aDo, for Do a slightly smaller concentric ball). The given condition implies that u - v z -E on Do and therefore u - v z -E on B, in L(f, 4, b) - E. Since E can be arbitrarily small, particular at ; that is, the desired inequality is true. Whenever harmonic functions can be defined, for example, on a Riemann surface by Laplace's equation, superharmonic functions can be defined using the intrinsic condition of this section.
23
9. Application of Jensen's Inequality
8. Differentiable Superharmonic Functions Theorem. If D is an open subset of R' and if u E V ')(D), then u is superharmonic
if and only if Au 5 0.
Since linear functions and products of two different coordinate functions are harmonic, the Taylor expansion of u about a point of D yields
L(u,, S) = u() +
z
2N
0(62),
(8.1)
so that Au :!g 0 when u is superharmonic. Conversely if Au :r. 0 define Then A(u - Ev) < 0 when e > 0, so that by (8.1) with u - Ev instead of u,
L(u - Ev, , S)
6, and (8.2) becomes the superharmonic function 1
inequality for u when c -' 0.
Approximation of a Superharmonic Function by Differentiable Superharmonic Functions
If u is a superharmonic function on the open subset D of R", the function Aau (see Section 1.2) is defined and infinitely differentiable on the set { a D: I - 3DI > S}. According to Section 4 this function is superharmonic and majorized by u, and according to Section 6(f), for each point of D b < 1 - aDl. and the function lima.0 Aau = u. This approximation result will be improved in Section I V.10.
9. Application of Jensen's Inequality This inequality implies that if 0 is a convex function
0[L(u, , S)] 5 L(O(u), , b),
(9.1)
whenever u is Lebesgue measurable on 6) and all integrals involved exist. This inequality has the following consequences: (a) (b)
If u is harmonic and ¢ is convex, O(u) is subharmonic.
If u is subharmonic and ¢ is convex and monotone increasing, ¢(u) is subharmonic.
Thus if u is harmonic or is positive and subharmonic, JuI" is subharmonic
24
1.11. Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions
when p,2! 1. If u is positive and superharmonic, u° is superharmonic when
0 2). In particular, u is harmonic if and only if u is a linear function of 0). (b) If u is a superharmonic [harmonic] function on D, the function '-+ L(u, 0.1 1) is also superharmonic [harmonic] on D, and therefore the function S i--+ L(u, 0, S) on ]a, b[ is a concave [linear] function of
-log 6ifN=2and of (2-NifN>2. Observe that u is a concave function of - log I I if and only if u is a concave function of log I
-
Proof of (a). Since we can replace u by AQu and then let o(-+ 0, we can suppose that f in (a) is infinitely differentiable. The following evaluation of Au makes (a) trivial.
Au(k) =f'(I0 +
N-I Ici
d2
P101
if N=2 an d s= - log IC I,
ds2
(N - 2)2 d2f(II) x Iy12N-2
if N > 2 an d s __1C12-N
( 10 . 1 )
.
ds2
Proof of (b). We prove (b) for u superharmonic; the proof for u harmonic is easier and is left to the reader. Since u is locally lower bounded we can assume in the proof of (b), at the expense of increasing a and decreasing b, that u is lower bounded on D. In addition suppose first that u is bounded on D. Since a space rotation around the origin preserves superharmonicity, integrating over all rotations (see the remark in Section 4 on positive integral operations on superharmonic functions) yields the fact that the function F-+ L(u, 0, ICI) is superharmonic on D, as asserted in (b). If u is not bounded
and n e Z', the function u A n is a bounded superharmonic function on D; so the function - L(u A n, 0,1 I) is superharmonic on D and when n co, we find that the function i-- L(u, 0, I I) is either superharmonic or identically
+ oo. The latter case is excluded because u is locally !N integrable so (by Fubini's theorem) L(u, 0, b) < + oo for !1 almost every 6 in ] a, b[.
25
!i. Examples
The Minimum Function of a Superharmonic Function
Let D be as in Theorem 10, let u be a superharmonic function on D, and define
m(b) = min {u(): ICI = b}. We now show that the function is superharmonic on D; that is, on ]a, b[ is a concave function of -logb if N = 2 the function 6 and of bZ-' if N> 2. First observe that the function is lower semicontinuous on D because u is. Next observe that if y is an arbitrary rotation of D about the origin, the rotated function is superharmonic on D, and the infimum of the class of all these rotated functions is the function Since an elementary argument shows that the infimum of any locally lower bounded family of superharmonic functions on an open set is superharmonic if this infimum is lower semicontinuous, we conclude that the function r m(l' ) is superharmonic, as asserted.
11. Examples (a) Suppose that N = 2 and let f be a not identically vanishing analytic
function on the connected open set D. The real and imaginary parts of f are harmonic because they satisfy Laplace's equation, alternatively because the Cauchy integral formula applied to a ball yields the harmonic function average property for f. Taking absolute values in this average relation we
conclude that If I is subharmonic and therefore (Section 9) that If I° is subharmonic when p z 1. Actually If I° is subharmonic when p > 0 because if I° = I f °I is the absolute value of an analytic function in a neighborhood of a nonzero of f, and it is trivial that the subharmonic function average property is satisfied at a zero off. Similar reasoning shows that log if I = Re (log f) is subharmonic when defined as - oo at a zero off and is harmonic on the nonzero set off. Since If 1n for p > 0 is a monotone increasing convex function of log If I, we have again that If I° is subharmonic. (b) If N = 2 and if f is analytic on B(0, b) and does not vanish identically, and if p > 0, the function If I° is subharmonic so l
zx
If(reie)I°&
(r 0) u is the limit of an increasing sequence of bounded positive harmonic functions.
Furthermore the map u i- f is a one-to-one, linear, order-preserving map from the class of D(uB_) harmonic functions on B onto the class
L'(aB,IN t), and D(lui) in (b2) are subharmonic functions; so L(l ul. 0, ) and L(b(Iul ), 0, ) are monotone increasing functions on ]0, b[. In (a) the signed measure M. is the zero measure if and only if u = 0, and M. z 0 if and only if u z 0. In (b) the functionJ vanishes IN_, almost everywhere if and only if u = 0, and f, z 0 IN_, almost everywhere if and only if u z 0. The functionf will be identified in Section 15 as the nontangential boundary limit function of u.
Nomenclature. The fact that a positive harmonic function on a ball has a
Poisson-Stielijes representation (al) with M. z 0 is usually called the Herglotz theorem, although Herglotz himself referred to Riesz. To avoid confusion this theorem will be called the Riesz-Herglotz theorem in this book. (al) . (a2) If M. = u, - u2 is a Jordan decomposition of M. into the difference between two (positive) measures, the equation
29
14. Classes of Harmonic Functions on a Ball
u = PI(B,u,) - PI(B,u2) expresses u as the difference between two positive harmonic functions. (a2) . (a3) If u = u, - u2 with u, positive and harmonic, the function Jul has the harmonic majorant u, + u2. (a3) (a4) If Jul has the harmonic majorant v and if O < r < S, then L(l u1, 0, r) < L(v, 0, r) = v(0).
(a4)
(a 1)
If 0 < r < S, the function
is harmonic on a
-4
neighborhood of B and so is given on B by a Poisson integral, (dq)
u (!4) I xNbN-'. N-'
u = PI(B, u,),
(14.2)
S
The total variation of p, is
J
ju (a
,-
dNdh)=L(lul,0,r),
)I'n
and therefore, in view of the hypothesis of (a4), there is a strictly monotone
increasing sequence r, with limit S for which the sequence u,, of signed measures has a vague limit, denote this limit by M. When r - S along this sequence in (14.2), the Poisson representation becomes u = PI (B, M.). To show that this representation of u on B determines M. uniquely, we show that any representation u = PI(B, u) with p a signed measure on aB implies that vague lim,.su, = u when u, is defined by (14.2). That is, we show (14.1):
limf fdu,=f '-a
ae
fdu
ae
whenever f is continuous on aB. Using the fact that J i7 - r/Sl = l - rq/Sl for and q on 3B, an interchange of orders of integration yields
Jfd= e
as
PI(Bf) C) u(dq).
(14.3)
By Theorem 1 the right-hand side has limit Jd8 fdu when r - S, as was to be proved. The relation between L`(uB_) harmonic functions on B and signed measures on aB is obviously linear and positivity preserving. If the space of signed measures on aB is ordered by setting u, 5 P2 when the difference u, - u2 is positive, the relation u . M. is order preserving. Since the space of signed measures on aB is a conditionally complete vector lattice (Appendix IV.7),
30
1.11. Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions
the space L' (µB_) of harmonic functions is also a conditionally complete vector lattice (in the order determined by pointwise inequality). The vector lattice isomorphism deduced here has an exact analog for harmonic functions on Greenian sets provided with their Martin boundaries (Section XII.9). (bl) =:-. (b2) Choose a uniform integrability test function 0 for which f eBO(I f I) d!N _, < + oc. Then by Jensen's inequality 4)(IuI)
= ()[IPI(B,.ff)I] 5 PI(B, 1(I f I)).
(14.4)
The last term is a harmonic majorant of 4)(IuJ). (b2) . (b3) Let v be a harmonic majorant of (D(Iul). Then
L(O(Iul),0,r) 5 L(v, 0, r) = v(0); so (b3) is true.
(b3) . (bl) Since (b3) is stronger than (a4), u = PI(B, M.), and we have proved above that the signed measure u, tends to M. (vague convergence) when r S. Now the uniform integrability described in (b3) is equivalent to uniform integrability of the family i < r < S} of functions on aB relative to the measure IN_,. Hence (Appendix V) there is a strictly monotone increasing sequence r, with limit S for which the sequence
q- u(r.nlb) has a weak limit function f on aB in the sense that lim
u dB
(rte 9(17)IN-i(dn) = J f(rl)9(q)IN-r(dt)
\
(14.5)
aB
for every bounded IN_, measurable function g on aB. This limit relation implies (take g continuous) that the sequence u,., already known to be vaguely convergent to M., is vaguely convergent to the signed measure fdlN_,/(7r,6N-'). That is, M. is absolutely continuous relative to IN_, and dM = fd1N-t/(7rNS"-'). Thus (bl) is true and f. is uniquely determined up to IN_, null sets because M. is uniquely determined. The map u I-+f has
the stated properties because the map u F- M. has the corresponding properties. (bl) (b4) Under (bl), if u > 0, u = PI(B,f) with f. z 0, according to what we have proved, and u = A n) represents u as the limit of an increasing sequence of positive bounded harmonic functions. (b4) (bl) If u is bounded positive and harmonic on B, condition (b2) is satisfied, so (b I) is satisfied. Moreover, if u, is an increasing sequence of positive bounded harmonic functions on B with limit u, the sequence f,4
15. The Fatoi. Boundary Limit Theorem
31
is an increasing sequence (up to 1N_, null sets) with limit some function f, and the equation u = PI(B, f ) becomes in the limit u = PI(B, f ). Observation. If u e D(µo_) in the theorem, we have proved that lim,..ap, _ M. (vague convergence of signed measures on OB), where p, is the signed measure defined by (14.2). This convergence, or alternatively, a slight variation of the convergence proof, shows that if v, is the signed measure on B supported by aB(0, r) and defined by
(r =
v,(dry) =
1q1),
N
then lim,.av, = M. (vague convergence of signed measures on B).
15. The Fatou Boundary Limit Theorem Let B = B(0, 6) in RN. A "Stolz domain" in B with vertex C on aB is defined as the intersection of B with an open cone of revolution with vertex 1, axis
of rotation the ray from C through the origin, half-angle < n/2. A "deleted nontangential neighborhood" of C is defined as a subset A of B with the property that if S is a Stolz domain with vertex C, then S r B(C, e) c A for sufficiently small s, depending on S. The class of deleted nontangential neighborhoods of C is closed under finite intersections. A function w on B is said to have radial limit q at C if lim,.,, w(rC) = q. The function is said to have nontangential limit q at C if lime., q whenever the approach to C is in a Stolz domain with vertex C, that is, if w has limit q at C along the filter of deleted nontangential neighborhoods of C. A signed measure p on the Borel subsets of OB can be considered as a
signed measure of Borel subsets of R', supported by aB. In particular, (N - I)-dimensional "area" on OB when so extended to R" will be denoted by C,v. If u and v are signed measures on R" supported by aB, the symmetric variational derivate dp/dv (see Appendix IV. 12) can thus be defined as the symmetric derivate for measures on R" or equivalently in the obvious way for measures on OB.
Theorem. If u e L' (p _) and if h is a strictly positive harmonic function on B then if Z e aB,
(nontang) Sim u() = M'(C)
(symmetric variational derivate)
(15.1)
whenever both the indicated derivate and the symmetric derivate dl'N/dM,,(C) exist.
32
1.11. Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions
According to this theorem the limit in (15.1) exists at M,, almost every boundary point C.
Observation. In the classical version of this theorem, as proved by Fatou, lN_1/(1rN6N-1) on sB. h =_ 1, so that M,, is a normalized Lebesgue measure In particular the theorem implies that if f is an IN_, measurable and integrable
function on aB then the harmonic function PI(B, f) has nontangential boundary limit f(C) at IN_, almost every point C of B.
Reduction of the Theorem to a Special Case
Suppose that it is known that u/h has radial limit 0 at a boundary point C whenever
(RI) uz0, (R2) The symmetric derivate exists and is 0 at (R3) The symmetric derivate dlN_1(dMa exists and is finite at C.
It will now be shown that then the theorem follows. In the first place, when u z 0 and u/h has radial limit 0 at C then u/h also has nontangential limit 0 there. In fact let S be a Stolz domain with vertex C and let S, be a second Stolz domain with vertex C but with a larger vertex angle than S. Choose p with 0 < p < 1 and let B,,[B1,,] be the ball with center pC, internally tangent
to aS[aS,]. According to the Harnack inequality for balls as applied to u and h in B,o there is a strictly positive constant c with the property that u( )
5
u(PC)
eB,,.
ch(PC)'
The constant c does not depend on p because the constant in Harnack's inequality is unaffected by a similarity transformation of I. It follows that u/h has limit 0 on approach to C in S and therefore on nontangential approach. In the second place we need only remark that if the derivate value in (15.1) is a, to reduce the general case to that of positive u and a = 0 we can replace
M. by IM. - aM,,j, that is, replace u by the corresponding vector lattice maximum (u - ah) Y (ah - u) furnished by Theorem 14. To finish the proof of the theorem, we prove that under (Rl)-(R3) the function u/h has radial limit 0 at C. Let a (9) and a,,(6) be, respectively, the M. and M,, measure of the closed spherical cap on aB cut off by the closed
cone of revolution with vertex the origin, axis the radius to , half-angle 0 > 0, and define a (0) = a,,(0) = 0. The functions a and a,, are monotone increasing on [0, n] and right continuous except perhaps at 0, and a,(6) > 0 when 0 > 0 in view of (R3). To simplify the notation take 6 = 1. Define I -s 2 K:(e) = nN [(I - s)2 + 4ssin'(0/2)]x12 1
(15.2)
33
16. Minimal Harmonic Functions
so that
u(se) = JK(O)dcc(8),
h(sC ) =
J0K3(0)d0
(15.3)
The function K, is a decreasing function on [0, n], o
I
liminfKs(e)_N-2 lt(dO) = Jo S-1
K.(a)
sin' (a/2)01-2
sin' (0/2)
(a > 0),
!1(d9) _ +op
(15.4)
and according to hypotheses (R2) and (R3), lim ato) = 0-0 ap(e)
ON-1
0,
lim
/
< +oo.
(15.5)
8-0 ap(0)
According to Appendix VII.3 with f
1, p = N - 1, and n -+ oo replaced
by s -+ 1, relations (15.4) and (15.5) imply that Gm,.t u(sC)/h(st) = 0, as was to be proved.
16. Minimal Harmonic Functions A harmonic function u on an open connected subset D of RN is called minimal if u is positive and if every harmonic function v on D satisfying the inequality 0 S v 5 u is a constant multiple of u. If u is minimal, the positive
multiples of u are also minimal. If to e D and if r is the class of positive harmonic functions on D with value I at 0, the class r is convex, and the minimal members of r are the extreme elements of this convex set. In fact, if u is minimal and in r and if u is not extremal, then u has a representation
u=put+qu2,
p+q=1,
0 2. In Section VII.1 the Green point, shows that
function Go will be defined for every Greenian set D. The property 0 remains true and in fact is very nearly a defining property of GD.
2. Generalization of Theorem I Theorem 1 is included in the following theorem but was proved separately because of the importance of its constructive proof. Theorem. If a class r of superharmonic functions on an open subset D of RN has a subharmonic minorant, then GM IF exists and is harmonic.
Let F. be the class of subharmonic minorants of F. The class ro contains u, v u2 with u t , u2 and is therefore directed upward in the order of pointwise
inequality, with limit u', a function majorized by r. We prove the theorem by showing that u' is harmonic. Let B be a ball with closure in D and suppose that v e ro. For every function u in F, v S tBV 5 r8u 5 u. Thus rBv a ro, and the supremum of ro is the same as that of {rBv: ve ro}. This class on B is an upward-directed family of harmonic functions majorized by r on B; so u' is harmonic on B (Harnack convergence theorem) and therefore on D. as was to be proved.
3. Fundamental Convergence Theorem (Preliminary Version)
37
3. Fundamental Convergence Theorem (Preliminary Version) If u is a function from a topological space into IR, we denote (Appendix VIII. 1) by u the lower semicontinuous smoothing of u, that is, the maximal lower semicontinuous minorant of u:
u( ) = u() A Iiminfu(q).
rc
(3.1)
In the following theorem the trivial inequality u 5 u is stated for completeness and to facilitate reference.
Theorem. Let r : { u a e 1) be a family of superharmonic functions on an open subset of 08N, locally uniformly bounded below, and define the lower u 5 u, envelope (infimum) u of r by lim nfu(q)
+
(3.2)
11-4
and
(a) (b) (c)
(d)
u is superharmonic. u = u on each open set on which u is superharmonic. = u IN abnost everywhere. There is a countable subset of f whose lower envelope has the same lower semiconlinuous smoothing u.
Observation (1). The hypotheses of the theorem are not effectively weakened by the added assumption that u. is directed downward because if all finite minima uQ A A u,k are adjoined to the family, the enlarged family of superharmonic functions is directed downward and has the same lower envelope u, and (d) is true for the original family if true for the enlarged
family. The added assumption justifies the nomenclature "convergence theorem." Observation (2). According to Choquet's topological lemma (Appendix VIII.3) Theorem 3(d) is true, and we can even assume that r is a decreasing sequence. Furthermore, since the theorem is local, it can be assumed that
u is bounded below on its domain, say u z c, and replacing ua by ua - c, it can be assumed that the members of f are positive. In accordance with these observations the following proof assumes that u, is a decreasing sequence of positive superharmonic functions. Apply 11(6.1) to find
u
lim inf u (q) Z lim inf u(q), R-C
7"C
38
so that
1.111. Infima of Families of Superharmonic Functions
is at least equal to the last term on the right, and therefore
(3.1) implies (3.2). Since u is superharmonic, A(u,,, , S) 5 u.(4),
(3.3)
A(u, ,, b) 5 A(u, , S) 5
(3.4)
so that
Moreover continuity of A(u, , S) (Section 1.2) implies that A (u,
, S) 5 A(u, , b) 5
u is the largest lower semicontinuous function majorized by u. Inequality (3.5) implies that u is superharmonic. Finally u = u IN almost everywhere on D because when S- 0 inequality (3.5) becomes u 5 u 5 u 5 u IN almost everywhere on D, by a standard derivation theorem.
Application to GMDf' Let r : { u,, or e 1) be a family of superharmonic functions on an open subset
D of R' and suppose that r has a subharmonic minorant. Then GMDr exists; so r is locally lower bounded. Define u = infu3. According to Theorem 3, the function u is superharmonic, and we now show that GMDI- _
GMDU. It is trivial that GMDr z GMDu and in the other direction GMDr is a continuous minorant of u and is therefore a minorant of u by definition of u; so GMDrS GMDu.
4. The Reduction Operation Let D be an open subset of RN coupled with a boundary OD provided by a metric compactification. Let A be a subset of D v 8D, let v be a positive superharmonic function on D, and let r be the class of positive superhar-
monic functions u on D majorizing v on A n D and near A n.3D in the sense that to each function u corresponds a neighborhood Ao of A n 8D in the compact space D u 8D such that {u z v) (A u A0) n D. If u, and u2 are in r, their minimum u, A u2 is also in r; so r is directed downward in the order of pointwise inequality. The reduction RA of v on A is defined
as the infimum, that is, the limit, of the directed set r. Since v c- r , this reduction is also the infimum and limit of the class of functions v A u with u in r, that is, of the class of positive superharmonic functions on D equal to v on A n D and near A n W. Thus R" 5 v, with equality on A n D. The
39
4. The Reduction Operation
set D is part of the context but is omitted from the notation. In most applications A e D. According to the Fundamental Convergence Theorem in the preceding section, the lower semicontinuous smoothed reduction R" is majorized by RA, is superharmonic, and coincides IN almost everywhere
on D with R. (According to the more refined version of this theorem in Section VI.1, the set {R"v > RA } is not only IN null but polar, a more re+V strictive characterization to be defined in Section V.1, and it will be proved in Section VIA that R" = RA on D - A.) The notation 8 VOA will sometimes be used instead of R".
Obviously R" and R increase when A or v increases. Moreover the reduction operation is subadditive in the sense that
B5
RB + RA + RB,
(4.1)
and the corresponding inequality (4.Ism) for smoothed reductions is also true. To prove (4.1) we observe that if u" [v"] majorizes u [v] on A n D and near A n aD and if u8 [v8] majorizes u [v] on B n D and near B n OD, then u" + u8 + v" + vs majorizes u + v on (A u B) n D and near (A u B) n aD. To prove (4.Ism), observe that the two sides are superharmonic functions and that the inequality is true 1N almost everywhere on D and therefore everywhere on D. If D - A is not empty and if B is a ball with closure in D - A, the function r8u is in F whenever u is in r and rsu < u; so R" on B is the limit of the downward-directed family of harmonic functions. Hence (by the Harnack convergence theorem) R" is harmonic on B and so is harmonic on D - A and equal to R" there. In particular, when A c OD, the reduction RA is harmonic on D and R = R Since v on A n D, it follows that R4 = RA = V on the interior of A n D. Hence, if A is open in D u aD, the smoothed reduction RA is a positive superharmonic function majorizing v on A n D and near A n OD and majorized by RA ; so R" = RA on D. More generally we shall show in Section 5(d) that R" = RA whenever A n D is open. Ex iipu (a). Let D be a ball and let A be a relatively compact subset of D. Then RA (and therefore also RA) has limit 0 at every ball boundary point. +V It is instructive to prove this result directly here, although the result also follows naturally at a later stage from the remark that when A is relatively compact in D, the function R" is (Section IV.8) the potential G0µ of a measure with compact support in the ball and as such (Section IV. I) has limit 0 at every ball boundary point. To prove the stated result, let be the center of D, let A 1 be an open neighborhood of A and be relatively compact in D, and let B be an open neighborhood of A 1 and be relatively compact in D. Then R"' = v on A; so R. 5 R. and R.", is harmonic on a V
1.111. Infima of Families of Superharmonic Functions
40
neighborhood of aB. Choose a constant c so large that on 8B and therefore on a neighborhood of OB. The function u equal to R. A- on B
on D - B is superharmonic on D, majorizes on D - B; so R' 5 u 5 near 8D, and therefore RA has limit 0 at every boundary point of D. A similar argument shows that when N > 2, D = R', and A is a relatively compact and equal to RA' A
R", and is a minorant of
subset of D, then RA has limit 0 at the point oo. EXAMPLE (b). Let D be a ball, let A = {} c D be a singleton, and let v =- 1. Then RA(s) = 1, but since majorizes v on A whenever e > 0,
it follows that R' = 0 on D - {} and the smoothed reduction RA vanishes identically.
EXAMPLE (c). Let D be a ball with center the origin, and let v be a positive harmonic function on D with Riesz-Herglotz representation (Section 1I.14)
v=J
K(,7,
(4.2)
2D
We shall now prove
RA = R = fA K(q, )Mv(dq)
(4.3)
for every Borel boundary subset A. The first equality in (4.3) is trivial because RA is harmonic. Denote the integral on the right by vA. If u is a positive superharmonic function on D that majorizes v near A and if A0 is a closed subset of A, the difference u - vAo is superharmonic on D and positive near Ao. Moreover the fact that lime{ 4) = 0 uniformly on Ao when C E 8D - Ao implies that vAa has boundary limit 0 on 8D - Ao. Hence u - vAo has a positive inferior limit at every boundary point of D, and the superharmonic function minimum theorem implies that u - vA0 Z 0; so RA z vAo. If A, is an open boundary superset of A, then limt_{ 0 uniformly on DD - A, when C E A ; so v - vA has boundary limit 0 at every
point of A. Hence, if e > 0 the function VA, + e majorizes v near A, and it follows that vA, 2 RA. Thus vAo 5 RA 5 vA,, and this inequality implies the desired equality (4.3).
EXAMPLE (d). Let D be a Greenian subset of R', let B be an open relatively compact subset of D, and let v be a positive superharmonic function on D, finite and continuous at each point of 8B. Then R°-B = GMBv on B. In fact,
on the one hand, R" is harmonic on B and is majorized by v on B; so 5 GMBv on B. On the other hand, if u is a positive superharmonic function on D, with u >- v on D - B, and if vo is a harmonic function on RD-B
B and is majorized by v on B, then u - vo is superharmonic on B and has a positive inferior limit at every point of 8B and so is positive on B, and it follows that R" 2 GMBv on B.
5. Reduction Properties
41
According to Section VIII.18, the hypothesis in Example (d) that v be finite valued and continuous at each point of BB is stronger than necessary. It is sufficient, for example, if v is finite valued on B. More specifically, according to Theorem VIII.18(c), in which V1 B) can be identified with R°_B R°-B, the evaluation = GMBv on B is true if and only if, in the ter-
minology to be developed below, the set of irregular boundary points of B is a null set for the Riesz measure associated with v.
5. Reduction Properties The following list of reduction properties will be useful, and expanded, in later chapters. The reductions are relative to a Greenian subset D of OB", and no restrictions not specifically stated are imposed on the positive superharmonic functions on D to be reduced or on the subsets of D u OD on which these functions are reduced. (a) R" in terms of reductions on subsets of D.
inf { R": A n 8D c B, B open in D u 8D}
= inf{R,"-"": A ndD c B, B open in DuaD}. These evaluations follow from the fact that if u is a superharmonic function on D majorizing v on A n D and near A n 3D, then there is a neighborhood B of A n aD so small that u also majorizes v near B n aD. The smoothed version of (5.1) is
RA =inf{Rv"B:An ODcB,Bopen inDu0D} = inf{Rv"' °: Ana D c B, B open in DuaD}.
(5. l sm)
To prove (5.Ism), observe that in view of (5.1) and the Fundamental Convergence Theorem the right sides are superharmonic functions majorized by R" and therefore by Ro , and the reverse inequalities are trivial. If A n OD is compact, the sets B in (5.1) and (5.1 sm) need only run through a decreasing sequence of relatively compact open neighborhoods of A n OD with that set as intersection. (b) If p is a positive superharmonic function on D with GMDp = 0, if A c OD, and if v is a positive superharmonic function on D, then R," P = R. In particular, RD = 0. This property follows from the inequality
R"SRS P:5. RA + RA
because R", a positive harmonic function majorized by p, must vanish identically.
42
1.111. Infima of Families of Superharmonic Functions
Observation. In Section IV.8 the positive superharmonic functions p with GMDp = 0 will be identified with the superharmonic potentials GDµ. (c) R" in terms of separate reductions on subsets of D and of 8D. Recall
(Section 4) that R"naD is harmonic. We now prove that if v' = v - R"11 then RA = RAIMD +Rd AnD RA, V
(5.2)
and also prove the corresponding inequality (5.2sm) for smoothed reductions. In particular, if GMDV = 0 then RV = RAID and R " = RAnD. In fact, if vt is a positive superharmonic function on D, majorizing v near A r) 6D and if v2 ij a positive superharmonic function on D, majorizing v' on A n D, then vt + v2 >_ R", and therefore RV naD + RV n°
RA,.
(5.3)
Conversely, if v3 is a positive superharmonic function on D, majorizing v on A n D and near A n 8D, the positive superharmonic function v3 - RAnaD majorizes v' on A n D; so RVnD s RV
-
R"naD
(5.4)
and this inequality combines with (5.3) to yield (5.2). Equality (5.2) implies
that the corresponding equality for smoothed reductions, an equality between two superharmonic functions, is true IN almost everywhere on D and therefore is true everywhere on D. (d) RV = R" whenever A n D is open. If A is open, this property was pointed out in gection 4, and the more general property follows from (5.2) and (5.2sm). (e) If v is finite valued and continuous at each point of An D, then
RV = inf {Rv : A c B, B open in D u 8D}.
(5.5)
In fact, if u is a positive superharmonic function on D which majorizes v on A n D and near A n 8D, then for n z I the function (1 + 1 /n)u majorizes v on a neighborhood of A n D and on the trace on D of a neighborhood of A n 8D and therefore majorizes RB for some open superset B of A.
6. A Smallness Property of Reductions on Compact Sets Theorem. If D is a Greenian subset of GIN, if A is a relatively compact subset of D, and if v is a positive superharmonic function on D, then GMDRA = 0.
By hypothesis there is a nonconstant positive superharmonic function u on D. It can be supposed that D is connected and, replacing u by utn if
7. The Natural Order Decomposition for Positive Superharmonic Functions
43
necessary, that u is not harmonic. Replacing u by u - GMDU if necessary, it can also be supposed that GMDU = 0. If A, is an open relatively compact subset of D containing A, then v, = RA, = R"1 is a positive superharmonic function on D, harmonic on D - A, , and is equal to v on A ; so v, and v have the same reduction on A. Replacing v by v, if necessary, it can be supposed that v is harmonic on D - A,. Thus, if B is an open relatively compact subset of D containing A,, the function v is harmonic on a neighborhood of OB. Choose c so large that cu > v on B. This inequality will then hold on a neighborhood of aB, so that (cu) A V = v on this neighborhood. Define u, = v on B and u, = (cu) A v on D - B to get a positive superharmonic function on D majorizing v on A and therefore majorizing R" on D. Hence
GMDRA 5 R" = R' 5 cu on D - B, and (by the subharmonic function maximum theorem) therefore GMDRA 5 cu on D. Since GMDU = 0, it follows that GMDRn = 0, as was to be proved.
7. The Natural (Pointwise) Order Decomposition for Positive Superharmonic Functions Theorem. If u, u, , and u2 are positive superharmonic functions on an open subset D of RN, with u 5 u, + u2, then there are positive superharmonic functions u1, u2 on D for which U
u2 5 u2,
u = ui + u2.
(7.1)
Let u, be the smoothed infimum (given by the Fundamental Convergence Theorem) of the class of functions v, positive and superharmonic on D and satisfying the inequality u 5 v + u2. Then u 5 u, + u2 IN almost everywhere on D and therefore everywhere on D. Let u2 be the smoothed infimum of the class of functions v, positive and superharmonic on D and satisfying the inequality u 5 u, + v. Then u:5 ui + u2 on D, and if v is a positive superharmonic function on D for which u:!5; ui + v [u 5 v + u'2], it follows that v z u2 [v ui]. In particular (choose v = u), it follows that u z u, and
u ,-a u2. To prove the theorem, it will be shown that u = u, + u2. Define (u - u2) as the indicated difference whenever +oo and define (u-u'2)()=
whenever
liminf
[u(q) - u2(q)]
u - u2 is thereby lower semicontinuous
at each infinity of u2. and we shall now prove that u - u2 is also lower
44
I.M. Infima of Families of Superharmonic Functions
semicontinuous at the other points of D. Let B be a ball with closure in D. Then
U=tgu+U - r9U 2. Whatever the dimensionality, if u is a finite measure on the ball B, the potentials GBp and Gu are superharmonic and differ on B by a harmonic function.
2. Examples Let p be a unit mass distributed uniformly on OB(0, S). Then -N+ 2
-N+2 2
on B(O,6),
RN
-
on B(0, S) t1l, GB(o.a,µ=Gp-a-N+2 onB(0,a)ifa>6.
(N > 2),
(2.1)
In fact the potential Gp must be a function of the distance I q I from the origin, must be superharmonic, and must be harmonic on RN - 5B(0, 6), and therefore (according to Section 11.10) must be a concave function of lq1-N+2, and
1.1 V. Potentials on Special Open Sets
48
must be a linear function of IqI-N+2 on each open component of R' aB(O, S). Finally, this potential is 6-1+2 at the origin and has limit 0 at oo. Hence the first two lines of (2.1) are correct. If a > S, the difference Gµ Ge(o.a)F1 is a function of IriI, defined and harmonic on B(0, a), with limit Gp = a-N+2 at the boundary. Hence (maximum-minimum theorem for harmonic functions) this difference is identically a-N+2 ; so the third line of (2. 1) is correct. Similarly on B(0, S),
log b
Gµ(q) _
on R2 - B(0, S)
- log I q
(N = 2),
(2.2)
G8j0,a,p = Gµ + loga on B(0,a) if a > S. If µ,a is a unit mass distributed uniformly on
S), then
L(G(ri, -), , S) = Gp,a(q),
and it follows from (2.1) and (2.2) that for N z 2 the function of
(2.3) q)
defined by (2.3) is continuous on fl8N x R'. If a > S and if e B(0, a - S), the difference G4u - Ge a) and is harmonic and defined and superharmonic on I I i - AI > a, Ia; - OBI > a). Define Aa(GBM) on equal to GBµ on I
a B: I - OBI S a) as GBµ to obtain an infinitely differentiable superharmonic function on B. Since GBµ differs from Gp on B by a harmonic (
function the results in the preceding paragraph yield the fact that A1(GBµ) =
GBµ1, where µ1, the same measure as above, has density -A(GBµa)/a4 = - A(Gµ1)/rrN relative to IN, and GBµa increases to the limit GBµ when a 10.
50
LIV Potentials on Special Open Sets
6. Uniqueness of the Measure Determining a Potential Theorem. Let D be a Greenian subset of 68"(N > 2) or D = 682, and let p and v be measures on D for which GDp and GDv are superharmonic. Suppose that A is an open subset of D and that there is a function h defined and harmonic on A for which GDp = GDv + h on A. Then the projections of p and v on A are identical.
The proof will be given for D = 68" and requires only trivial changes for D a ball but the remarks validating the proof for a general Greenian set are deferred to Section VII.8. It is sufficient to prove the result for A relatively compact in 68", and therefore it can be supposed, replacing u and v by their projections on a ball containing A if necessary, that p and v have compact supports. According to Section 5, the functions A,(Gp), A,(Gv) are infinitely differentiable potentials of measures p v, with respective densities -AA,(Gp)/n;v, -L A,(Gv)/ah relative to I Since these densities are equal on the set ; e A : - aAI > a), the projections of p, and v, on this set are identical, and since p, and v, have vague limits p and v when a -' 0, the projections of p and v on A are identical, as was to be proved. I
Extension of Theorem 6 to Charges
In the following extension of Theorem 6 we consider charges p on D (see Appendix IV.7) with the following property: p has positive and negative variations p' and -p-, respectively, with the property that GDp* and GDpare superharmonic. As usual p* and p- are finite on compact subsets of D, but if D * 682 both p* (D) and p- (D) may be + ac, sop need not be a signed measure. The potential GDp is defined as GDp* - GDp- on the subset of D on which at least one of the potentials GDp`, GDp- is finite. If v is a second such charge on D, if A is an open subset of D and if h is a function defined and harmonic on A, suppose that GDp = GDv + h on A in the sense that
GD(p* + v) = GD(p- + v*) + h on A. According to Theorem 6, if this condition is satisfied, it follows that p* + v =p + v* on subsets of A. Since the four measures involved here are finite valued on compact subsets of D, we conclude that p = v on the compact subsets of A, so these measures have identical projections on A. Thus we have extended Theorem 6 to cover potentials of charges as restricted above. In particular, if GDp is harmonic on the open subset A of D, so that we can take v as the zero charge, we conclude that A is p null. As a second particular case suppose that GDp* and GDp- are finite valued on an open subset A of D and that (GDp)IA is in the class C11(A). Let v be the charge on A yvith density - (LtGDp)/nN relative to /,. If AO is an open relatively compact subset of A and if vo is the projection of v on Ao then (Section 1.7) (GDVO)IA is in the class C'21(A), and A(GDp - GDvO) = 0 on AO; so GDp differs from GDvo on
A0 by a harmonic function and it follows that p = vO on subsets of A0 and
7. Riesz Measure Associated with a Supcrharmonic Function
51
so on subsets of A; that is, dp = -A(GDp)/7[N d1N on A. The hypothesis that
(GDµ)J4 E C" )(A) can be weakened, for example, to the condition GDp c C'2)(A) by applying the result just obtained to a smoothing A,(GDIz), but we shall not need this refinement.
7. Riesz Measure Associated with a Superharmonic Function Theorem. If u is a superharmonic function on an open subset D of RN(N z 2), there is a unique associated measure p on D with the property that if A is an open relatively compact subset of D, and if pe is the projection of u on A, there is a superharmonic function hA on D, harmonic on A, with
u=Gµ,,+hA
(7.1)
on D.
The uniqueness of µA is obvious from Theorem 6, and incidentally this uniqueness implies that the measure determining a potential is the measure associated with the potential. If the theorem is true with h,, supposed only defined and harmonic on A and satisfying (7.1) on A, this function can be extended to a superharmonic function on D which satisfies (7.1) on D. In fact, if B is an open superset of A, relatively compact in D, and if v is the projection of Pa on B - A, then h4 = Gv + hB on A. Thus h,, has a superharmonic extension Gv + hB to B, and when h,, is so extended, u = Gµ,, + h4 on B. In this way hA can be extended in step* to D. Finally, it is sufficient to prove that if A is a ball with closure in D, there is a measure µA defined on A and a harmonic function h,, defined on A for which u = Gµ,, + h4 on A. In fact,
if two such balls overlap the parts of the two associated measures on the intersection of the balls must be identical by Theorem 6, and the desired measure on D is obtained by piecing together the measures associated with the balls. To prove the italicized statement we first replace D by a subset if necessary to make u bounded from below on D, and adding a constant if necessary, we can even suppose that u is positive on D. Since only the restriction of u to A is relevant to the desired result, we can replace u on D by R1, a positive super-
harmonic function on D, equal to u on A, harmonic on D - A. The proof now proceeds as follows. Let B be a ball concentric with and larger than A, with closure in D. Choose a so small that a is strictly exceeded by the difference between the radii of A and B and by IB - aDI. The function A,u is superharmonic and infinitely differentiable on a neighborhood of B, equal to u, and harmonic on a neighborhood of 8B, and 6ma.,o AQu = u on D. According to 1(8.6), A,u = GBµa + PI(B, u)
(7.2)
52
I.IV. Potentials on Special Open Sets
on B, where µ, is the measure on B with density - AA,u/tr, relative to 1,'. The
value of µ,(B) can be found by applying the Gauss integral theorem (integrate over 8B), and since A,u = u on B, µ,(B) does not depend on a. Thus as 0 along a suitable sequence p,, has a vague limit measure µo supported by 4. Applying Theorem 4, we find that if PA is the projection of µo on A, then a
(G8µ0 + PI(B,u)
on B,
u= Gµ,t + h,,
on A,
where h4 is harmonic on A, and we have used the fact that GBµA and Gµ4 differ on A by a harmonic function. Observation. It follows from Theorem 6 that the map u F-' p from superharmonic functions into their associated measures is additive (u1 Fpi and u2 r+p2 imply that u1 + u2 µ2) and positive homogeneous implies that cu E-- cp for c a positive constant).
8. Riesz Decomposition Theorem Theorem. If D is a Greenian subset of RR'(N 2), if u is a superharmonie function on D with associated measure p, and if u has a subharmonic minorant, then
u=G°µ+GM°u.
(8.1)
In the most common application u is supposed positive, in which case the harmonic function GM°u is also positive. We defer the general Greenian case to Section VII.8 and treat here only special sets D. If D is a ball, let B be
a smaller concentric ball. In the notation of Theorem 7, u = Gµ5 + hi, = G°µB + h8, where h'S is superharmonic on D and harmonic on B. Now according to Section 111.1, the minorant GM°v depends additively on the superharmonic function v, and according to Section 3, this minorant vanishes if v is a potential. Hence GMnu = GM°h'B 5 h8. When B increases to D, we conclude that G°µ is superharmonic, that h8 decreases to a harmonic func-
tion h on D, and that u = G°µ + h. Then h = GM°u, and the proof is complete. The proof for R" when N > 2 is essentially the same. The following assertions about a positive superharmonic function u on D are easy consequences of the Riesz theorem. The function u is a potential GDP if and only if GM°u = 0, equivalently, when D = b), if and only if lim,.a L(u, , r) = lim,_a 0 (or when D = RN with N > 2, if and only if this limit is 0 when r -, + oo). (b) If u is majorized by a superharmonic potential u is itself a potential. (c) Special case of (b). If u is a superharmonic potential, then R' is also, for every choice of A; in particular, 0 for every choice of 8D. (a)
9. Counterpart for Superharmonic Functions on R2 of the Riesz Decomposition
53
(d) The smoothed reduction Ru" is a potential if A is a relatively compact subset of D (because GMDRA = 0 according to Theorem 111.6).
Observe that a superharmonic function u on D is the sum of a potential and a harmonic function if and only if u has a subharmonic minorant.
9. Counterpart for Superharmonic Functions on R2 of the Riesz Decomposition Let u be a superharmonic function on R' and let b be a strictly positive number. Then (Section II.4) the function L(u, , b) is superharmonic on R2. For fixed in R2 the function bH
L(u, , b) - L(u, , 1) = m(u,
- log b
,
b)
is a positive increasing function on ] 1, +oo[ in view of the superharmonic function average inequality and the fact (Section 11.10) that the function L(u, l;, ) is a concave function of log b. Hence (Section 11.4) the function m(u) = lim&-m m(u, , b) is either identically + oo or a positive superharmonic function, and in the latter case it is identically constant because (Section IL 13)
the plane is not Greenian. Theorem. (a) I fu is a superharmonic function on 682, then u is harmonic if and only if m(u) = 0. _ (b) If µ is a measure on R2 and if g(b) = µ(B(0. b)), then Gp is superharmonic on R2 if and only if
log6dg(b) < +oo. (c)
(9.1)
If µ is a measure on 682 satisfying (9.1) and if u = Gp, then µ(68Z) = m(u) < + oo,
li m [L(u, 0, b) + g(b) log b] = 0,
b
(9.2)
and (9.2) is also true iJ'g(b) is replaced by p(68'). (d) Let u be a superharmonic Junction on 682 with associated Riesz measure p and define gas in (b). If
lim inf [L(u, 0, b) + g(b) log b] < + oo,
(9.3)
6-OD
then Gµ is superharmonic and u = Gp + h, with h a harmonic function on 682 given by
54
l IV. Potentials on Special Open Sets
lim [L(u, , b) + µ(I82) log b].
a-m
Proof. (a)
(9.4)
If u is harmonic on 682 the function m(u, , b) vanishes identically;
so m(u) = 0. Conversely, if u is superharmonic on 12 and m(u) = 0, then since m(u, c, ) is a positive increasing function on the interval ] I, + 00 [, it follows that this function vanishes on the interval ; that is, L(u, t, ) is a constant function there. Since L(u, , ) is a decreasing concave function of log b for b > 0, it follows that L(u, , ) is identically constant; that is, u satisfies the harmonic function average equality and so is harmonic. (b) and (c) The condition (9.1) that Gµ be superharmonic was derived in Section 1 [see (1.4)] in a trivially different form. If (9.1) is satisfied, the in Section 2 or Section 1.5 yields evaluation of
L(G/, 0, b) = -µ(B(0, b)) log b -
log 1171 µ(d7) RZ-B(O.b) (9.5)
= -g(b)logb - f
logsdg(s).
>a.+ml
The function g is positive, monotone increasing, and right continuous, with limit µ(I82) at + oo. Thus (9.2) is true. Moreover (9.1) implies that lim [µ(682) - g(b)] log b = 0;
6-M
(9.6)
so g(b) can be replaced in (9.2) by µ(I82). (d) Under (d) let µ, be the projection of µ on B(0,6). According to Theorem 7, there is a superharmonic function h8 on 682, harmonic on B(0, b), such that u = Gµa + ha.
(9.7)
Now (9.5) with µ replaced by µa yields the equality L(Gµ6, 0, b) = - g(b -) log b. Furthermore the function r '-. L(ha, 0, r) is continuous and equal to ha(0) for r < b; so L(h6, 0, b) = ha(0). Hence
L(u, 0, b) = -g(b-) log b + ha(0).
(9.8)
For fixed q the function b i-. Gµa(q) is monotone decreasing on the interval ] 1 + f , 1, + co [, so the function b --. ha(p7) is increasing on this interval. ha = + 00 on 68Z or Hence (Harnack convergence theorem) either lima_. ha = h is a harmonic function on 682. In view of (9.8) the first case is excluded by (9.3). Hence in the limit (9.7) yields the fact that Gµ is superharmonic and u = Gp + h. Moreover from (9.8)
55
10. An Approximation Theorem
h(O) = lim [L(u, 0, b) + g(b) log b], 6-OD
and as noted in the proof of (c), the value g(b) can be replaced by p(R') in this limit relation. Thus (9.4) is true for 4 = 0 and therefore for all since rather than the origin can be chosen as the reference point in this discussion.
10. An Approximation Theorem The following theorem is an example of the application of the reduction and related operations. Theorem. Every superharmonic function u on an open subset D of 68" is the limit of an increasing sequence u, of superharmonic junctions with the following properties :
(a) u is upper bounded. (b) infpu, = infDu. (c) (d)
u is infinitely differentiable.
[At the possible sacrifice of (a)] if u is harmonic outside a compact subset A of D, then u,, = u outside a compact neighborhood A. of A with
A A.
a A of D, we can choose u, to satisfy (b)-(d) as follows. Choose a sequence a, in 68+ satisfying
J!A - BDI > ao>at> ,
R-M
I - AI < The function u is and define u = A,Mu on the set B,, = superharmonic and is equal to u near BB,,. Define u = u on D - B. to obtain a sequence u, satisfying (b)-(d), with A = To find a sequence u, satisfying (a)-(c) for general superharmonic u suppose first that infD u = 0 and let D. be an increasing sequence of open relatively compact subsets of D with union D. Define v = R°na positive superharmonic function on D, bounded by n, equal to u A n on D,,, and harmonic on D - D,,. Now proceed as suggested by the method used in the
preceding paragraph. Define at, = ID - cD1/3, except that a = 1/n if D = 68", and define u,, = Aa,,(v,. ) on the set B. = {g: 1 - D.1 < 2a.). The function u is superharmonic, majorized by v,,, and equal to v,, near BB,,. Define u = v on D - B. to obtain a sequence u, satisfying (a)-(c). If infD u = ft > - oo, find a sequence u, satisfying (a)-(c) by adding f to each member of a sequence satisfying these conditions for the function u - IJ.
56
1. IV. Potentials on Special Open Sets
If u is not lower bounded, modify the approximation procedure as follows. Choose D. as above and also to satisfy D. c D.+1, and define v. by
v. = inf {v: v superharmonic on D, v
u n n on Den v(U aD2j11 n
(10.1)
JJ)
so that the lower semicontinuous smoothing + is majorized by u A n, coincides with u A n on D2, and (Fundamental Convergence Theorem) is superharmonic on D and harmonic on U. (D21 2 - D2). Define at. as a decreasing sequence of strictly positive numbers satisfying a. < IaD2n+2 - D2n+1I A IaD2n+1 - D2nl,
lima.=0, n-00
and define u. by AQn+n
on Den+1,
A,k+n
on Dzk+1 - D2k_1 for k > n.
_
(10.2)
The sequence u, has the desired properties (a)-(c). Specialization to Positive Superharmonic Functions and Potentials I f D in Theorem 10 is Greenian and if u is a positive superharmonic function on D, each approximation u. can be chosen to satisfy (a)-(c) and also to be the potential of a measure with compact support. If u is itself the potential of a measure with compact support A, each approximation u. can be chosen to satisfy (a)-(d) with un the potential of a measure supported by A. [notation of (d)]. It is convenient to suppose first that u is the potential of a measure with compact support A. Then u is harmonic on D - A. Furthermore u is bounded outside each neighborhood of A because the function (, 17) is a bounded function for q restricted to A and restricted to the complement of a neighborhood of A. (This property of the Green function is trivial for D special and will be proved in the general case.) The first paragraph of the proof of Theorem 10 furnishes a sequence u. with the properties (a)-(d); un is a potential because [Section 8(b)] u. is majorized by the potential u; the Riesz measure associated with u,, is supported by A. because u. is harmonic on D - A.. Thus the assertion for u when u is the potential of a measure with compact support in D is true. For u superharmonic and positive the proof of Theorem 10 for the case inf u = 0 provides a sequence u, with the desired properties. In fact each function u. is a potential because by compact-
ness of D. the function v. = R°
is a potential [Section 8(d)], and the
positive superharmonic function u. is a potential because it is majorized by vn .
Chapter V
Polar Sets and Their Applications
1. Definition A polar subset of RN is a set to each point of which corresponds an open neighborhood of the point that carries a superharmonic function equal to + oo at each point of the set in the neighborhood. An inner polar set is a set whose compact subsets are polar. It will be shown in Section VI.2 that an analytic inner polar set is polar. If a set is (inner) polar its Kelvin transforms are also. In particular, the set of infinities of a superharmonic function is a polar subset of its domain. Conversely, it will be shown (Theorem 2) that a polar set is always a subset of the set of infinities of a single superharmonic function defined on RN.
The polar sets are the negligible sets of classical potential theory. An assumption about points of R' true except for the points of an [inner] polar set is said to be true [inner] quasi everywhere. A subset of an [inner] is superpolar set is [inner] polar. A singleton is polar because
harmonic on RN and equal to + oo at . Although the point ao is not in RN, that point is considered a Euclidean boundary point of every unbounded set. In a context allowing oo in the domain of harmonic and superharmonic
functions, this point is polar for N = 2 but not for N > 2. Since a superharmonic function on an open subset of R' is IN integrable on every closed ball in its domain, and since every polar set A can be covered by a countable number of open sets, each carrying a positive superharmonic
function with value + cc on the part of A in its domain, a polar set has IN measure 0. It follows that an IN measurable inner polar set also has IN measure 0. If u and v are superharmonic functions on an open subset of RN and if u = v inner quasi everywhere, or if u >_ v inner quasi everywhere, then the same relation holds IN almost everywhere and therefore [Section 11.6(f)] everywhere.
I.V. Polar Sets and Their Applications
58
2. Superharmonic Functions Associated with a Polar Set Theorem. If A is a polar subset of R', there is a function superharmonic on and identically + oo on A. This function can be chosen to be the potential Gp of a measure u with µ(I8") finite and to be finite at any preassigned point
ofpN-A. To prove the theorem suppose that e I8" - A and apply the Lindelof covering theorem to cover A by balls Bo, B,, ... so small that l: is not in any ball closure and that to each ball Bk corresponds a function uk defined and superharmonic on an open neighborhood of Bk and identically + oo on B. n A. Let µk be the projection on Bk of the Riesz measure associated with uk ; choose a strictly positive constant ck so small that ck uk(Bk) < 2-k, 2-k, and if N = 2, that ck J, log IhI 2-k. The superthat harmonic potential G Eo ckpk is + oo on A and finite at .
Observation (a). Since the set of infinities of a superharmonic function v is the G6 set no {v > n}, every polar set is a subset of a G6 polar set.
Observation (b). Since a superharmonic function is 1,-, integrable on every ball boundary in its domain, a polar set meets a ball boundary in an 'N_, null set.
Observation (c). The complement of a closed polar subset A of RN is connected. To see this, let B be an open connected component of R' - A, let u be a superharmonic function on RN, identically + ao on A, and define v = + oo on B and v = u on R' - B. The function v satisfies the conditions for a function to be superharmonic on I8" except for the finiteness condition. Hence v is either identically + o0 or superharmonic. Both alternatives are impossible unless R' - A is connected.
Observation (d). If D is an open subset of R", the set I8N - D is polar if and only if the finite part a°D = I8" n aD of the boundary is polar, and then a°D = I8" - D. In fact, if a°D is polar, its complement is connected and everywhere dense and so is equal to D, and therefore R" - D = a°D is polar. Conversely, if R" - D is polar, the set D is everywhere dense; so a°D = R" - D is polar. Extension. If D is a Greenian subset of R' and if A is a polar subset of D, then there is a positive function superharmonic on D and identically + oo on A. This function can be chosen to be the potential GDµ of a measure p with u(D) finite and to be f inite at any preassigned point of D - A.
If D is special, the proof of Theorem 2 for N > 2 with RN replaced by D and G replaced by G. is valid in the present context. This same proof will be valid in the case of general Greenian D once (Section VIII) GD has been defined.
59
4. Properties of Polar Sets
3. Countable Unions of Polar Sets Theorem. A countable union of polar sets is polar. In particular, an inner polar F, set is polar.
In fact, if AO, A 1, ... are polar, if e IR" - Ua Ak, and if µk is a measure Gµk = +oo on Ak and (if N = 2) with f; log ryI µk(dry) < 2-k, then the superharmonic potential G(Eo µk) is
on R' with
+ooonU0Ak. Since singletons are polar, this theorem implies that countable sets are polar. For example, suppose that A is a countable dense subset of P", and let u be a superharmonic function on R", equal to + oo on A. The set of infinities of u is a polar dense GG set and therefore is not countable. It will be shown in Section VI.2 that an analytic inner polar set is polar. According to Theorem 3, if u and v are superharmonic functions on an open subset of R" and if v S u inner quasi everywhere on an F, set A, then this inequality must hold quasi everywhere on A because the set
{v>u}nA=U{v>r}n{rZu}nA
(r rational)
is an F. inner polar set and is therefore polar.
4. Properties of Polar Sets Theorem. The following conditions on a subset A of a connected Greenian set D are equivalent: (a)
(b)
A is polar.
There is a superharmonic potential u = GDµ with µ(D) < + oo and
u=+ooonA. (c) (d)
If u is superharmonic and strictly positive on D, R' =0. There is a strictly positive superharmonic function u on D for which has a zero.
In the following argument D is special. We defer the general Greenian case to Section VII.8. Proof. (a) (b) Has already been proved for D = R". The proof for a ball is similar. (b) . (c) Let u be a positive superharmonic function on D, identically + oo on A. Then (for n z 1) u/n z u on A ; so 5 u/n, and therefore 0 at every point where u is finite. Hence the positive superharmonic
I. V. Polar Sets and Their Applications
60
function R" has a zero and accordingly must vanish identically (superharmonic function minimum theorem). (c) . (d) The result follows because R' = (d) (a) The result follows because if
IN almost everywhere.
0, there is a function
v positive superharmonic on D, Z u on A, 5 2-" at , so that Eo v is superharmonic on D, identically + oo on A.
5. Extension of a Superharmonic Function Theorem. Let D be an open connected subset of RN, let A be a polar subset of D, and let u be an extended real-valued function on D - A satisfying the following conditions: - oo < u 5 + oo ; u ; + oo ; u is locally lower bounded; u is lower semicontinuous; if e D - A, then u(l;) Z L(u, , 6) for sufficiently small b, depending on . Then u has a unique superharmonic extension u' to D, determined on A by
liminfu(p).
(5.1)
n-4
In this theorem when 6) c D, the function u is defined IN_, almost everywhere on 6), and u is IN_, measurable and lower bounded on
this set; so L(u, , 6) is well defined. In the simplest case A is relatively closed in D, so that the hypotheses make u superharmonic on D - A, but these hypotheses allow A to be dense in D. A superharmonic extension u' of u satisfies (5.1) because of the strengthening of the lower semicontinuity
property of superharmonic functions in 11(6.1). If the theorem is true locally, it is true as stated; so we can assume that D is a ball, that u is not identically + oo, and (Section 4) that there is a positive superharmonic function v on D, identically + oc on A but finite at some point of D - A at which u is finite. When e > 0, the function u + ev, if defined as +oo on A. is superharmonic on D, and according to the Fundamental Convergence Theorem (Section 111.3), the function uo = lim,..o (u + ev) has lower semicontinuous smoothing uo superharmonic on D, equal IN almost everywhere on D to uo. If e D - A, the given inequality z L(u, , 6) implies [see 1(2.2)] that u(g) >- A(u, , 6) (for sufficiently small 6). Applying lower semicontinuity of u, we find that lim infb_o A (u, , 6) >-
lim A(u, , 6)
6-0
Hence
( e D - A).
(5.2)
Moreover u = uo = uo 1N almost everywhere on D. Therefore
lim A(uo, , 6) = uo() &-0
so uo is the desired superharmonic extension of u.
c D - A);
(5.3)
5. Extension of a Superharmonic Function
61
Special Case
If in the theorem u is bounded and if
L(u, , b) for
in D - A and
sufficiently small b depending on 4, then u has a unique harmonic extension
to D. In fact, under these stronger hypotheses if u' is the bounded superharmonic extension of u provided by Theorem 5 then when ED - A and S is sufficiently small the function u' is positive and superharmonic on S), vanishes at 4, and therefore vanishes identically on b). Thus u must be continuous on D - A, and Theorem 9 can now be applied to -u to provide a subharmonic extension u" of u to D. The function u' - u" vanishes identically on D because this difference is superharmonic on D and coincides !N almost everywhere with the harmonic function 0; so u' is the desired harmonic extension of u. Application to Analytic Functions
If A is a polar subset of the open subset D of R2, if A is closed in D, and if f is a bounded analytic function on D - A, the real and imaginary parts off have harmonic extensions to D according to the preceding paragraph. The extension to D off obtained in this way is analytic because the CauchyRiemann equations are satisfied. If A is a singleton this result reduces to Cauchy's classical theorem on isolated singularities of bounded analytic functions. Application to Isolated Singularities of Harmonic Functions Let u be a function defined and harmonic on an open deleted neighborhood B of a point i; and satisfying
lim
n-4
if N > 2.
Then Theorem 5 implies that there is a constant c such that the function
ifN=2, ifN>2 has a harmonic extension to B u
We give the proof for N = 2. According
-
to (5.4) there is a constant ct for which the function u + ct log I is positive on a deleted neighborhood of ; so this harmonic function has a superharmonic extension to B u { }. Since the Riesz measure associated I
62
1. V. Polar Sets and Their Applications
with the extension must be supported by there is a constant c such that the function u + clog I J has a superharmonic extension u' to B u { } and that the Riesz measure associated with u' vanishes identically. Hence u' is harmonic. Observe that by way of inversion in a sphere of center the result just obtained implies that if v is a function defined and harmonic on a deleted open neighborhood of the point co of 68" and if
-
lim inf V01) > - 00 v-m logInI
if N = 2,
liminfv(,)> -oo q-m
ifN> 2,
(5.5)
then there is a constant c such that the function
ifN=2, (v-c)1-1'-2 ifN>2 has a finite limit at the point co. Application to a Generalized Liouville Theorem (N = 2)
According to Section 11.2 a lower-bounded harmonic function on 68^` for N a 2 must be identically constant. The preceding application of Theorem 5 implies when N = 2 that a harmonic function on R2 which satisfies (5.5) must be identically constant because if c is chosen so that v - c log I I has a finite limit at the point co, then v must have a limit a at the point 00, finite or infinite according as c = 0 or c # 0. In view of the maximum-minimum theorem for harmonic functions a is finite and v =_ a. Application to the Greatest Subharmonic Minorants of Superharmonic Functions Let D be an open subset of 68^' and suppose that u is a superharmonic function
on D for which GMDu exists. Let A be a subset of D, closed relative to D, and define Do = D - A. Then GMD (uiDo) z GMDu on Do. We now show that there is equality if A is polar and is null for the Riesz measure associated with u. We can assume, replacing u by u - GMDu if necessary, that GMDu = 0, and it is then sufficient to prove that if h is a function defined and harmonic
on Do, with 0 5 h S u, it follows that h = 0. According to Theorem 5, such a function h has a positive superharmonic extension h, from Do to D and similarly u - h has a positive superharmonic extension v from Do to D. Since u = h, + v on Do, it follows that u = h, + v on D and therefore A must be null for the Riesz measure associated with ht. Hence h, is positive and
7. Superharmonic Function Minimum Theorem (Extension of Theorem 11. 5)
63
harmonic on D and is majorized there by u so h, = 0, and therefore h = 0, as was to be proved.
6. Greenian Sets in Ii2 as the Complements of Nonpolar Sets Theorem. A nonempty open subset D of 682 is Greenian if and only if 682 - D is not polar, equivalently, if and only if R2 n OD is not polar.
If R2 - D is polar (equivalently, according to Section 2, if 682 n aD is polar) D is not Greenian because a positive superharmonic function on D can be extended to be a positive superharmonic function on R2 and therefore (Theorem 11.13) is a constant function. Conversely, suppose that R2 n aD is not polar. If D is bounded, it was noted in Section 11. 13 that D is Greenian.
If D is unbounded, let A be the intersection of 682 - D with a closed disk so large that A is not polar. Let B be a disk containing A and define v on
682-Aby RI on B - A (reduction relative to B), 0
l
onR2-B.
In view of the fact that v is harmonic and positive on B - A with limit 0 at every boundary point of B [Section 111.4, Example (a)], the function v is subharmonic; so 1 - v is positive and superharmonic on D but not identi-
cally constant there because v = 0 on D - B but (Theorem 4) v > 0 on D n B. Hence D is Greenian.
7. Superharmonic Function Minimum Theorem (Extension of Theorem 11.5) The obvious application of the following theorem to harmonic functions is left to the reader.
Theorem. Let D be a Greenian subset of RN and let u be a lower bounded superharmonic function on D. Suppose that there is a constant c such that Um inf,,_tu(i) z c at quasi every finite point C of aD, and also at i; = oo if N> 2 and D is unbounded. Then u Z c.
Observation (a). We are only excluding trivia by the hypothesis that D be Greenian because if N > 2, all nonempty open subsets of R" are Greenian and if N = 2, a lower-bounded superharmonic function on a non-Greenian
open set is identically constant. (Moreover, according to Theorem 6, a
I.V. Polar Sets and Their Applications
64
condition on quasi every finite boundary point of a non-Greenian open subset of R2 is necessarily satisfied vacuously.)
Observation (b). Since the set A of finite boundary points of D at which u has inferior limit < c is a countable union of compact sets,
A= U )CeiD:ICIsn,liminfu(ry)Sc-I , n R=1
9'C
the set A is polar if it is inner polar. Thus Theorem 7 is true if "quasi every" is replaced by "inner quasi every."
To prove the theorem, observe that according to Theorem 2 if N > 2 there is a positive superharmonic function v on R", identically + 00 on A ; if N = 2 and D is bounded, there is a positive superharmonic function v on a ball containing D, identically + oo on A. In either case if e > 0 the function (u + sv),D is superharmonic on D with inferior limit z c at every point of OD, including oo if N > 2 and D is unbounded. Hence u + vv z c on D by the superharmonic function minimum theorem of Section 11.5, and therefore u z c quasi everywhere on D and so everywhere on D. If N = 2 and D is unbounded, we can suppose that lim inf,.,, u(PI) z c because the plane can be inverted in a circle with center a finite boundary point of D not in A, so that the transformed superharmonic function on the image of D has this property. Thus if c' < c, there is a disk B so large that u z c' on D - B. The part of the theorem already proved yields the inequality u z c' on D n B from which it follows that u z c on D, as was to be proved. Application to Analytic Function Theory
Let f be a bounded complex-valued function defined and analytic on a Greenian subset D of the plane; that is, the complement of D is nonpolar. The function If I is subharmonic; so it follows from Theorem 7 applied to - If I that if lim sup,.-=I f(z')I 5 c at quasi every finite boundary point of D, then
If I 0. Under the hypothesis of compactness of A, the restriction (Gp)ID is superharmonic and is the sum of GDu and a function harmonic on D (Section IV.1); so it follows that under the hypotheses of Theorem 8, the function (Gp)IA is continuous at and is finite there, and it is sufficient to prove that Gp is continuous at . Suppose first that N > 2. If pt8 is the implies that finiteness of 0 and projection of i on
that limj,0Gp 0. Furthermore, since G(µ - p4') is continuous on S), the function (Gp4l)IA must be continuous at . If e > 0 and if S is sufficiently small, depending on e, Gµ480) < e. For such a value of b continuity of (Gp46 )IA at implies that (Gp J)IA < e in some neighborhood of , and hence this inequality is true for all smaller values of S. It follows that lim
sup
Gµ4b(C') = 0.
(8.1)
6-0 4'e An (C,b)
If C is a point of D, let C' be a point of A at minimum distance from C so that if n e A,
!z'-nI51C'-CI+IC -n1521C-nl. The oscillation of Gµ at
(8.2)
can be majorized as follows:
5 sup
Gpl"(C)
Ce B((.a(2)
-5;2'-'
(8.3)
sup
Gp'6(C')
Cs B(4.a/2)
S) n A when C e S/2) because I - CI < 6/2; so the right side of (8.3) has limit 0 when S - 0, by (8.1). That is, Gµ is continuous at , as was to be proved. When N = 2, the only change needed in the preceding argument is that in (8.3) it should be supposed that S < I to ensure positivity of the potentials involved, and the last term in (8.3) should be replaced by Now C' e
sup
G, ({') +
S)) log 2,
CE B({.8/2)
which tends to 0 with S. Observation. If in the theorem it is supposed that (GDµ)IA is finite valued and continuous, it follows that GDu is finite valued and continuous on D. If D is special open and if p has compact support in D, the function GDp is
66
I.V. Polar Sets and Their Applications
then necessarily bounded because this potential has limit 0 at every boundary
point of D. For arbitrary Greenian D the boundedness of a continuous finite-valued potential GDp for p of compact support in D follows from the boundedness properties of GD to be proved in Section VII.5.
9. Approximation of a Potential by Continuous Potentials Theorem. Let D be either 682 or a Greenian subset of R' (N z 2) and let p be a measure on D. Suppose that GDµ is superharmonic and that GDP < + oo on some support A of D. There is then a sequence p. of measures on D for
which µ is supported by a compact subset of A, GDµ is finite valued and continuous (bounded if D o 682), and p =
p.; so GDP = L°. o GDµfl.
Recall our convention that "supported by A" means that A is p measurable and that the complement of A is p null. The generality of the statement of the theorem is convenient for reference, but there is no loss of generality in supposing that A is a Borel set because there is always a Borel support of p that is a subset of A. Apply Lusin's theorem to find a sequence A. of
disjoint compact subsets of A with the property that p(A - U; o 0 and that (GDp)JA. is bounded and continuous. If p is the projection of p on A,,, the continuous function (GDP)IAM is the sum of the restrictions to A.
of the lower semicontinuous functions GDµ and GD(p - pa). Hence these restrictions are continuous, and therefore (by the observation in Section 8)
GDp is bounded and continuous on D unless D = R2, in which case GDp = Gp is at least finite valued and continuous. Corollary. If B is a compact (but see the extension below) nonpolar subset of 68', there is a measure v on R' supported by B for which if N = 2, the function Gv is finite valued, continuous, and not identically 0 and if N 2, the function GDv is strictly positive, continuous, and bounded whenever D is a connected Greenian superset of'B. If D is a Greenian superset of B, the smoothed reduction (relative to D) RB is the potential of a measure p supported by B [Section IV.8(d)], and according to Theorem 9, there is a measure v supported by B with GDv as described. In particular, if N = 2, the potential Gv is then finite valued and continuous on 682.
Extension of the Corollary to Analytic Sets B
It will be proved in Section VI.2 that a nonpolar analytic subset of R" has a nonpolar compact subset. This fact will imply that the above Corollary is true if B is an analytic nonpolar set.
67
10. The Domination Principle
10. The Domination Principle The following theorem will be referred to as the domination principle, a designation sometimes used more restrictively.
Theorem. Let D be a Greenian subset of R', let p be a measure on D, and let v be a positive superharmonic function of D. Then each of the following three conditions implies that GDµ 5 v: (a) GDP < + co p almost everywhere and GDp 5 v inner quasi everywhere on some Borel support of p. (b) The inequalities GDp < +oo and GDp 5 v are true p almost everywhere.
(c)
Polar sets are p null and GDP 5 v p almost everywhere.
It is not obvious that conditions (a)-(c) are equivalent, and we shall not need this equivalence in the proof of Theorem 10, but according to Theorem 11, both conditions (a) and (b) imply that polar sets are p null and thereby that (a)-(c) are equivalent. It is a defect of Theorem 10 that polar sets must be y null, but it will be shown in Section XI.23 that GDp 5 v if lim sup G°p(q) 5 1,
n-t
GDph) 2.) It follows (Theorem 7)
that v ;-> u on D - A, so v z u inner quasi everywhere on D and hence (Section 1) everywhere on D, as was to be proved.
Special Hypotheses on r
If v in Theorem 10 is a constant function, the theorem is sometimes called the principle of the maximum. If v is the sum of a constant function and a potential, the theorem is sometimes called the complete principle of the maximum.
The Domination Principle for Potentials Gp When N = 2 If N = 2, if Gp is the superharmonic potential of a measure y on R2, and if there is a constant c such that Gp < c p almost everywhere then the method of proof of Theorem 10 shows that Gp < con R2.Observe that if c is replaced by a harmonic function h in this hypothesis then any further hypothesis on h leading to the conclusion that Gp S h on R2 would imply (according to Section IV.9) that lim infi,i_m oo and that therefore (Section 5) h is identically constant.
11. The Infinity Set of a Potential and the Riesz Measure Theorem. If D is a Greenian subset of RN, if A is a polar subset of D, and if u = GDp is the potential of a measure on D, then u = + ao p almost everywhere on A.
(The phraseology of the theorem requires that p be a completed measure.) We can assume that D is connected. If u is not superharmonic then u - + co,
so the theorem becomes trivial. If u is superharmonic the conclusion of the theorem can be restated in the following form: if u is finite valued on A then p(A) = 0. It is this assertion that will be proved. The triviality of the extension from special to Greenian sets will be explained in Section VII.8. Suppose then that D is special and that u = GDp is finite valued on the polar set A. Since the theorem is local it can be assumed that A is relatively compact
in D. It can also be assumed that A is a Borel set because there is a polar Ga superset of A and the intersection of this superset with the set of finiteness
of u is a Borel set. If A. = { eA: u :!g n} and if p is the projection of y on A then an application of the domination principle to the pair up = GDp,,, v =- n shows that u < n on D. Let GDv be a superharmonic potential identi-
11. The Infinity Set of a Potential and the Riesz Measure
69
cally + oc on A, with v(D) < + oo. Apply Fubini's theorem and the symmetry of GD to derive
nv(D) z f GDp,dv= f D
,JD
It follows that µ(A,) = 0; so µ(A) = 0, as was to be proved. Application to Gu When N = 2 If N = 2, if p is a measure on R 2, and if u = Gp is superharmonic, the conclusion of the theorem remains true. It is sufficient to prove that u = + 00 p almost everywhere on any bounded polar set A. Let D be a disk containing A and let µD be the projection of p on D. According to Theorem 11, GDPD = + oo µD almost everywhere on A. Since GDµD - Gp is harmonic on D if defined suitably on the set of common infinities of GDPD and Gp, it follows that Gp = + oo p almost everywhere on A, as asserted.
Chapter VI
The Fundamental Convergence Theorem and the Reduction Operation
1. The Fundamental Convergence Theorem Theorem. Let r : {u a e I } be a family of superharmonic functions defined on an open subset of R', locally uniformly bounded below, and define the lower envelope u by
u
inf
u u
u u
u
a countable subset of r whose lower envelope has the same lower semicontinuous smoothing u.
Conversely, if A is a polar subset of a Greenian subset D of RM, there is a decreasing sequence v, of positive superharmonic functions on D with limit v such that v > v on A.
The direct part of the present theorem is identical with Theorem 111.3 except that Theorem IIi.3(c) allows a larger exceptional set than Theorem 1(c). Thus there remains only the proof of Theorem l (c) and of the converse part of Theorem 1. Since Theorem 1(c) is a local assertion it can be assumed in its proof that the functions are defined on a ball D, and in view of the discussion in Section 111.3 it can be assumed that r is a decreasing sequence of positive superharmonic functions on D. The limit function need only be analyzed on a strictly smaller concentric ball B, with u replaced by (reduction relative to D). This reduction, equal to its lower semicontinuous smoothing because B is open, is a superharmonic potential Gals. (Section IV.8) and is equal to u on B, so the replacement is legitimate. The measure µ is supported by B. On D - B the sequence u, is a locally uniformly convergent sequence of harmonic functions; so the sequences of partial deriva-
tives are also locally uniformly convergent on D - B (Theorem 11.3).
2. Inner Polar versus Polar Sets
71
Evaluation of µ (D) by the Gauss Integral Theorem shows that µ,(D) is a convergent sequence. Going to a subsequence if necessary, it can be assumed
that the sequence p, is vaguely convergent to a measure p supported by B. If c is a strictly positive constant, the function GD A c is continuous and lim U. = u Z lim f
Ac
I
°-'m JD
GD(-, 1) A c p(dq), (1.2)
JD
so that u Z GDµ. Let v be a measure having compact support in D, with GDv finite valued and continuous. Then
f. GDP.dv =
`GDvdµ < +oo. 0
The equality
f udv = lim f
X-°D,JD
D
lim
`GDvdµ = f
4~00 JD
GDµdv,
GDVdµ =
D
(1.3)
D
combined with the inequality u Z GDµ, implies that u = GDµ at v almost every
point of D. Since # is the maximal lower semicontinuous minorant of u, it follows that u z u 2t GDµ with equality v almost everywhere. Thus, if A = (u > it has now been shown that v(A) = 0 whenever GDv is finite valued and continuous. In view of Corollary V.9 this fact means that every compact subset of A is polar; that is, the set A is a Bore] inner polar set. Theorem 1(c) follows from the next theorem, whose proof uses the partial result just obtained. Conversely, suppose that A is a polar subset of the Greenian subset D of R', and let r be the class of positive superharmonic functions on D equal at least to I on A. The infimum of the class is (reduction relative to D) RA, and (Theorem VI.4) RA m 0. According to Theorem 1(d), there is a sequence in r whose lower semicontinuous smoothed infimum vanishes identically. If V},
v is the minimum of the first n members of this sequence, v, has the properties stated in the converse of Theorem 1.
2. Inner Polar versus Polar Sets Theorem. An analytic inner polar subset of R' is polar.
It is obviously sufficient to prove the theorem for bounded sets so we shall consider subsets of a Greenian set D, say a ball. Reductions below are relative to D. The proof will be in several steps. Some of the preliminary
results proved below are more general than needed; this is to avoid later repetition when the results will be strengthened.
72
1.V1. The Fundamental Convergence Theorem and the Reduction Operation
(a)
If A is a compact subset of D and if v is a finite-valued positive contin-
uous superharmonic function on D, then R" = v quasi everywhere on A. In fact, according to the part of Theorem I actually proved in Section 1, the set ao
An{RA 0 so R ,D(). So there must be equality; that is, R" = R" on D - A when A c D. If A n 8D is not empty the expressions 111(5.2) and III(5.2sm) for reductions
R
and smoothed reductions in terms of those on A n D and A n 8D separately yield the equality of RA and R " on D - A. c3
Proof of (c). Let be in D - (A1 u A2), and (Theorem V.2) let u, be a positive superharmonic function on D, identically + oo on the polar set (A 1 - A 2) u (A 2 - A 1) but finite at . If u is a positive superharmonic function on D, majorizing v on A, n D and near A, n 8D, then for every e > 0 the function u + eu1 majorizes v on A2 n D and near A2 n D. Hence and by symmetry the reverse Z RA ,2(& so RA,, (t) Z RA u(g) + inequality is true so RA, = RA z on D - (A, Q A2). It then follows that the superharmonic functions R"' and R"z are equal quasi everywhere on D and +V +V therefore everywhere on D.
Proof of (d). The first line of (d) and the equality of the two infima are immediate consequences of (b) and (c).
Proof of (e). Under the hypotheses of (e) the reduction R increases with n. Define u = Ro n, and observe that u is superharmonic and that u = v quasi everywhere on A because un = v quasi everywhere on A. It follows from (d) that u z RA , and since the reverse inequality is trivial,
(3.lsm) is true. Equality (3.1) is trivial on A and is also true on D - A because on that set R"n = R" according to (b). In proving the +v, and Rv = R" +v second assertion of (e) we shall suppose that D is connected to avoid irrevant notational complexity. Assume then that v = v for all n, choose a point in D, and let u be a positive superharmonic function on D, majorizing v near A. n 8D, with un(c) 5 R fr
2-".
The function vk=u+(un_R"^^dD),
n=k
k>0,
(4.1)
78
I. V1 The Fundamental Convergence Theorem and the Reduction Operation
is positive and superharmonic on D because the sum is the limit of an increasing sequence of positive superharmonic functions and is finite at . Moreover vk Z u Z v quasi everywhere on A n D because RIM = v quasi everywhere on A. n D, and a° + (un - Ro" a°) z U.
vk z RV
when n z k ; so vk majorizes v near A n 8D, and therefore vk z R. A. When k oo, we find that u z RA quasi everywhere on D and therefore everywhere on D. Since the reverse inequality is trivial, (3.1sm) is true, and as in the first case of (e), it follows that (3.1) is true.
Proof of (f). Suppose first that there are only two summands in (3.2); so the equalities R", + RV,
Ru
(4.2)
and (4.2sm) are to be proved. By subadditivity (4.2sm) is true for "S"; so (natural order decomposition) there are positive superharmonic functions u1, u2 on D, satisfying RA+v
(4.3)
=ut+u2,
If A is an open subset of D, then R" = v, on A and R" = v, + v2 on A +v, +v, +vy so u; = v, on A, and it follows that u; 2, RBA. So (4.2sm) is true. If now A is i
still an open subset of D and if v, and v2 are supposed finite valued and continuous, equation 111(5.5) gives the value of a reduction on an arbitrary set in terms of reductions on open supersets, and it follows from (4.2sm) [the same as (4.2) for open sets] that (4.2) is true for an arbitrary subset A of D. Since a positive superharmonic function on D is the limit of an increasing sequence of finite-valued positive continuous superharmonic functions, (e) implies that (4.2) is true for an arbitrary subset A of D with no restriction on v v2, and then (4.2sm) must also be true with this generality because the two sides of (4.2sm) are superharmonic functions and are equal quasi everywhere on D. The evaluations in Section 111.5 of reductions and their smoothings in terms of reductions and their smoothings on subsets of D show that (4.2) and (4.2sm) are true with no restrictions on the sets or functions. Thus (3.2) and (3.2sm) are true for two and therefore any finite number of summands. If there are infinitely many summands, write r" for R'. Then k
A=
Rv
n
o
A Rvk +RAn
79
4. Proofs of the Reduction Properties
and since R" 5 F+ , vR, it follows that (3.2sm) is true quasi everywhere and therefore everywhere on D. Equation (3.2) is then true on D - A, and this equation is trivial on A r D.
Proof of (g). Define h = GMDv and define u = monic function. Obviously
R
a positive har
05 1ini RV _t,=u-R;. N_ QD The difference u - Rw is a positive harmonic minorant of v - h for all n, and therefore u - R,", 5 v - h ; so u = R; by definition of h. Finally, u = RA because R ,, is a positive harmonic minorant of h and so vanishes identically+
Proof of (h). In view of the trivial fact that the smoothed successive reducand BvO", it is tions of v on A and B in either order lie between sufficient to prove idempotence, that is, to prove that DBvO"DA = Qv0A. The proof will be carried through in several steps.
(h,) If A c D, the desired idempotence is a consequence of (d) because the condition on u in (d) is unchanged if v is replaced by OvOA. (h2) If A c 8D, the following argument yields idempotence. Use
Choquet's topological lemma to find a decreasing sequence v, of positive superharmonic functions on D, each majorizing v near A, with sequence limit O v D". Replacing v by V. A v if necessary, it can be supposed that v = v near A so that QV JA. It now follows
from (g) that
Jim vJA
n-m
(h3)
(4.4)
= D DvV"D".
For arbitrary A, DOVOADAnD =
DOVOADAndD =
OVOAnD
ODOAnJD
(4.5)
These inequalities follow, respectively, from OVBAnD =
OOVOADA'D
DOVOAnDDAnD 5
5
OVOAnD, (4.6)
OVOAn#D = D DVBAndDDAndD < D OVOADAndD 5 OVOAnW
Finally, to prove (h), apply 111(5.2) and (f) to write D0vO"D" in the form DOVOADAIW+D
D OVOADA=
OVOAD"nD
-
DOVOADAnODAnD
80
1 V1. The Fundamental Convergence Theorem and the Reduction Operation
and then apply (h,) and 111 (5.2) to show that the right-hand side is OvOA
Proof of (i). Observe that quasi everywhere on (A u B) n D, V + v' =
jv0A + llvrr
so that v OA,H =
p
OvOA DAv. +
(4.7)
= pupA + Ov 8.
Thus (3.4sm) is true, so (3.4) is true off A u B, and the latter equation is trivial on (A u B) n D. Proof of (j). We have derived a weakened version of (j) in Section 2. Instead
of going on from this result, we observe that (3.4sm) implies strong subadditivity of R* because if v' is defined as in Section 3(i), it follows that v' = v quasi everywhere on A n B; so RAUB +u
RAnB = RAn B,
Z
+V'
+v
and therefore (3.4sm) implies RA.B + +p
+p
RA +p + RH, +O
(4.8)
the strong subadditivity inequality. This inequality is trivial on A U B for the unsmoothed reductions and is true for these reductions elsewhere on D because it is true for the smoothed reductions. That is, both R;, and R, are strongly subadditive. Countable strong subadditivity follows from strong subadditivity by an application of (e).
Proof of (k). (This proof involves the domination principle and thereby the Green function, so at this stage the arguments are relevant only for special open sets D, but the extension of the domination principle to all Greenian sets will be seen to be trivial once GD has been defined for every Greenian set D in Chapter VII.) We shall not use in the following proof the partial result derived in the course of proving Theorem 2 that if v is continuous as well as finite valued,
is a Choquet capacity on D relative
to the class of compact subsets of D. To prove (k) in the present context except observe that all the capacity properties have been verified for the property that lim R' n
= R-
(4.9)
4. Proofs of the Reduction Properties
81
whenever A. is a decreasing sequence of compact subsets of D u aD with R" The function u is superharmonic, and intersection A. Define u = u = v quasi everywhere on A n D. Proof of (4.9) when A. c D. In this case u and R" A are finite-valued potentials on D (Section IV.8) whose associated Riesz measures are supported by A, and both potentials are equal to v quasi everywhere on A.
According to the domination principle these potentials are therefore identical. Since R". is harmonic on D - A., the function u is harmonic on D - A, so u = u on that set. Since R4 = R"- on D - A. and RA = R" on D - A, equation (4.9) is true on D - A, and this equation is trivially satisfied on A.
Proof of (4.9) when v is a potential. If B is an open neighborhood of A n OD the sequence A. - B is a decreasing sequence of compact subsets of
D with limit A - B; so by what we have just shown, lim, RA.-B =
RA-B
By subadditivity of R;,, RBm-B
+ RAY^B 2t RBA-B + RA m^B Z lim R" R-OD
(k 5 m).
(4.10)
Hence RA-B + RB Z lim R
(4.11)
and as B shrinks to A n OD, the left side becomes RAnD + RAID in view of III(5.1). The second reduction vanishes identically because v is a potential, so we find that RA
RBn° z lim RAE, 4-m
and the reverse inequality is trivial. Proof of (4.9) in the general case. In view of the Riesz decomposition of a positive superharmonic function v and of the reduction additivity property Section 3(f), it is sufficient to prove (4.9) separately for v a potential and for v harmonic. The proof for v a potential was just given. The proof for v
finite valued and continuous, and A c D, was given in Section 2 under part (d) of the proof of Theorem 2, and the proof is valid for A a compact subset of D u 8D. Thus (4.9) is true for v harmonic. The proof of (4.9) and thereby that of (k) is now complete. Proof of (1). Case 1. If v is finite valued, (3.5) follows from (k) since analytic subsets of D u 8D are capacitable for the Choquet capacity Case 2. If A c D but if v is not necessarily finite valued, apply Section 3(e) to derive
82
1 VI. The Fundamental Convergence Theorem and the Reduction Operation
R,A = supRA'. = sup (sup Rl,,n :Fc A, Fcompact} nzo
n2
= sup {R,',,,: F c A, Fcompact, n > 0) = sup {Rp : F c A, Fcompact}; so (3.5) is true. Case 3. In the general case write uF for Rp ^ d ° and observe that according to Section 111.5(c),
OF =uF - Rv^ F=VF-RV^°+RvF°.
(4.12)
From now on F is to be a compact subset of A. The class of these sets ordered by inclusion is directed so each term on the right in (4.12) defines a directed
set whose limit we now evaluate. Let h be the harmonic component of the Riesz decomposition of v. According to Section II1.5, uF = RndD; so by Case I above lim tF- RA^8° = RA^a° = VAh v Ft Since the class of sets F includes the compact subsets of A r D, Case 2 above
shows that limFt R1,D = RV"'. The same argument shows that if F' is a compact subset of A, then lim RfnD z lim RFnD uF. = RAnD VF. Ft Ft VF
(4.13)
Now Fi- uF is an upward-directed set of harmonic functions with limit vA ; so vA is the limit of an increasing sequence {vFn, n c- Z+ }, and therefore letting F' in (4.13) run through this sequence, we find from Section 3(e) that
limFt RFD > R0 . There must be equality because the reverse inequality is trivial, and (4.12) now yields limRF=vA-RA^D+RAD=R"
Ft
so (3.5) is true. Equation (3.5sm) is true on D - A because equations (3.5)
and (3.5sm) are identical on D - A, and this equality on D - A implies (3.5sm) on A n D by the following argument. If e A, the left side of (3.5sm) is unchanged according to Section 3(c) when A is replaced by A The right side is also unchanged in view of Section 3(c) and Section 3(e) [replace Fin (3.5sm) by F less each of a sequence of balls of center and radii tending to 0]. Thus (3.5sm) is reduced to (3.5).
Proof of (m). Assertion (i) has already been proved in Section III(e). To prove (ii) observe that if v is finite valued on D, then in view of the properties
(J) and (k) the restriction of the set function to the class of compact subsets of D v OD is a topological precapacity. This topological precapacity
4. Proofs of the Reduction Properties
83
generates a Choquet capacity
on D v 3D relative to the paving of
compact subsets of D v OD, as described in Appendix 11.8. More specifically, 1(g, ) = on the class of compact sets, and when A is open
A) = sup
F): Fc A, Fcompact}
= sup (R','(& Fc A, Fcompact}.
(4.14)
Hence A) = RA(s) when A is open, in view of Section 3(e) (since A is a countable union of compact sets) or Section 3(l). Finally, for arbitrary A,
A) = inf
B: B D A, B open)
= inf {RB(A): B c A, B open} = RA(s)).
If A is an capacitable set, in particular, if A is analytic, (4.14) is true and combined with (3.5) yields A) = R1.1(4) and thereby yields (3.6). According to property (a), each reduction on the right in (3.6) is equal to its smoothing; so an application of the Fundamental Convergence Theorem
o
to (3.6) yields (3.7).
Proof of (n). If c = supo1u - vj, so that v s u + c, we find that Ro 5
c,
and interchanging u and v yields the other half of (3.8). This argument applied to smoothed reductions yields (3.8sm). o
Proof of (ot). Obviously ue 5 v on D, and therefore h' + v8 5 v" on A. Hence h; + veA 5 v"; so hABA + VBABA 5 UBA, .... It follows that (3.9) is true. To prove (3.10), observe first that vB 5 v" and V" B:5 v; so vB 5 v A v".
Finally, take the smoothed reduction onto B of both sides of (3.9) to find that (3.10) is true.
o
Proof of (02). To prove (3.11) and (3.12), observe first that hAB 5 he 5 v/b and that vA 5 ahA. Hence hABA 5 h8A 5 b
5
(b) hA
(4.16)
from which it follows that hABAB 5 (a/b)hAB. Iterate and sum to derive (3.11)
and (3.12) with v instead of u A (bh). Relations (3.11) and (3.12) are true as written because if v is replaced by V A (bh), the sets A and B are unchanged. To prove (3.13) and (3.14), observe that the inequality between the second and fourth terms in (4.16) implies
hBABA 5 ()hAaA S ()h.
84
I V1 The Fundamental Convergence Theorem and the Reduction Operation
Iterate and sum to show that the left side of (3.14) is at most bh5A/(b - a). Since (4.16) implies that bhBA < v and bheA 5 ah, (3.13) and (3.14) are true. The reader is invited to derive (3.12) and (3.14) from (ot).
5. Reductions and Capacities Let D be a Greenian subset of R', provided with a boundary SD by a metric compactification, let t be a point of D, and let v be a positive finite-valued superharmonic function on D. Relation between the Capacity R,; and Topological Precapacities
According to Section 3(k), the set function
is a Choquet capacity on
D u 8D relative to the class r of compact subsets of D u 3D. This fact to r is a combined with Section 3(j) implies that the restriction of topological precapacity. Let
be the Choquet capacity on D U 8D rela-
tive to r generated by this topological precapacity. We have seen in the with equality on the course of proving Section 3(m) that 5 -)-capacitable sets and on the class of all boundary subsets, and class of that there is equality for all sets if v is continuous as well as finite valued. Finiteness of v Is Necessary in Section 3(k)
Let D be a ball of radius 1, let be the center of D, for n > 1, let A. be a closed concentric ball of radius I/n, and define v = Then A. is a decreasing sequence of compact subsets of D with intersection A = { }, and RA. = RA is false because RA = v for all n even though RA vanishes except at . Thus the finiteness of v is necessary in Section 3(k). Observe that with D, A, v, as just defined, (3.6) is false because the left-hand side of (3.6) is a function vanishing except at whereas the right-hand side is v. Is Not a Choquet Capacity on D Relative to the Class ro of Compact Subsets of D Choose v continuous and choose a compact subset A of D containing in such a way that RA If A. is a decreasing sequence of compact RA (g) for all neighborhoods of A with intersection A, then RAn(c) = n, and therefore even if v is finite valued and+vcontinuous, the +v set function is not a Choquet is not a topological precapacity on r'o and capacity on D relative to f0.
Chapter VII
Green Functions
1. Definition of the Green Function GD Let D be a nonempty open subset of R. If N > 2 or if N = 2 and D is bounded, the function is lower bounded for each point of D; so exists (Section 111. 1). If N = 2, if D is unbounded, and if
has a subharmonic minorant on D for some exists for every
in D, then the minorant
in D. In fact
is bounded below
outside each neighborhood of , and G(b', ) is bounded below on each compact neighborhood of so that if
exists,
zc+
Zc+
for some constant c depending on ' and . If exists for in D, define
by
G(4, ) -
(1.1)
The function GD on D x D is called the Green function of D, and
is
called the Green function of D with pole . The latter function is.positive with superharmonic, harmonic on D 0. It will be shown in Section 7 that D has a Green function if and only if D is Greenian. All we have proved so far is that D has a Green function if N > 2 or if N = 2 and if D is bounded. If N = 2 and if D is not everywhere dense in IB2, the Green function GD exists because if E D and if i; is an inner point of RZ - D, the restriction to D of G(e;, ) has G(C, ) - c as a harmonic minorant on D for sufficiently large c. If D is not connected, it follows that
each open component of D has a Green function, and it is trivial to verify that Go exists and that is defined for in the open component Do of D by
GD('
- (GD
0
)
0
on Do
on D - D0.
I.VII. Green Functions
86
The existence of the Green function GB of an open set B implies the existence of GD whenever D is a nonempty open subset of B and also implies that Go < GB on D x D. Moreover G can be obtained from GB just as GD is obtained from G:
To see this, note that if e D, then
GMDG(, ) (1.3)
= GQ,') - GMD[GB(, ) + GMBG(, )], and by linearity of the GM operator (Section 111. 1)
GD(, ) = G(, ) - GMDGB(, ) - GMBG(,'),
(1.4)
which yields (1.2). In particular, if B - D is polar then GD = G. on D x D in view of the corresponding result for greatest subharmonic minorants in Section V.5. Conversely, if D is an open subset of R" for which Go exists and if B is an open superset of D with B - D polar, then GB also exists. In fact, if E D, the function GMDG(, ) is locally upper bounded relative to B and so (Section V.5) has a subharmonic extension to B, and this extension is a subharmonic minorant of G(, ) on B; so GMBG(, ) exists. Evaluation of GD by Solving the First Boundary Value Problem
If D is an open subset of R", bounded if N = 2, and if for in D there is a harmonic function u(, ) on D with limit G(g, n) at each point n of BD (limit 0 at the point oo if N > 2 and D is unbounded), then an application of the superharmonic function minimum theorem and the harmonic function maximum-minimum theorem shows that u(, ) = GMDG(, ), so that
GD(, ) = G(, ') - u(, ').
(1.5)
Thus the Green function of a smooth open set as defined in Section 1.8 and evaluated for a ball in Section 11. 1 is the Green function in the present sense, and GD = G when N > 2 and D = R". We shall discuss the solution of the first boundary value problem for harmonic functions on Greenian subsets of R" in Chapter VIII. Let u(, ) be the solution in the generalized sense of Chapter VIII, that is, the "PWB" (Perron-Wiener-Brelot) solution, of this first boundary value problem on a Greenian set D for the boundary function
87
2. Extremal Property of GI)
ff = 91 D; if D is unbounded, define f4(oo) as - oo or 0 according as N = 2 or N > 2. According to Theorem VIII.18, the evaluation (1.5) is correct with this interpretation of u if N > 2 or if N = 2 and D is bounded. If N = 2 and D is unbounded, (1.5) is correct (Theorem VIII. 19) if and only if D does not certain too much of a neighborhood of the point oo in a sense made precise in the statement of Theorem VIII. 19.
2. Extremal Property of GD Theorem. Let D be an open subset of RN and let then
be a point of D. If GD exists,
inf {u z 0: u = G(i, ) + h with h defined and harmonic on D}
= inf {u z 0: u =
h with h defined and (2.1)
superharmonic on D}
= inf {u z 0: u superharmonic on D, u z on some neighborhood of }
= inf {u Z 0: u superharmonic on D, lim inf -c
u(g) q)
z 1 I.
In the following proof we denote the jth class on the right in (2.1) by r,. These classes are not empty if Go exists, but if GD does not exist, these classes
are empty because they all involve the existence of positive nonconstant superharmonic functions on D, and we shall see in Section 7 that Go exists if and only if D is Greenian. Proof for r2. If GD exists and if uE 1-2, then -h 5 on and therefore S u. On the other hand,
D; so -h 5 r2.
Proof that r, v r3 u r4 c r2. If GD exists and if u e r4, let p be the Riesz measure associated with u. If c < 1, the function u is positive and superharmonic on an open deleted neighborhood B of . Hence (Theorem V.5) u has a superharmonic extension to B v { 1; sop { } z c, and therefore p { } z l ; so u e r2 . Thus r4 c r2 . The inclusions r, c r2 and r3 c r4 are trivial. o Proof for r, . r3, r4. If u e IF, v r3 v r4, then we have just shown that u C- 1-2., and we have already seen it follows that u z Conversely, e
r,nr2nr4and (1
every e>0.
88
I.VII. Green Functions
3. Boundedness Properties of GD Theorem. Let D be an open subset of R" with a Green function GD, and let be a point of D.
(b)
If B is an open neighborhood of s in D, relatively compact in D, and if v is a positive superharmonic function defined on an open superset v of D - B, with v z GD(, ) on a neighborhood of on D - B. RBD,t , = R"Die = GD(i;, ) (reductions relative to D) whenever B is
(c)
a neighborhood of If v is a strictly positive superharmonic function on D, then outside
(a)
each neighborhood of , GD(tf , ) < const v. In particular (v a 1), GD(,, ) is bounded outside each neighborhood of , and if , is a point in the same open connected component of D as , then outside each
neighborhood of , GD(, ) 5 const GD(,, ). (d)
Let B, be a compact subset of D and let BZ be D less a neighborhood of B, . Then GD is bounded on Bt X B2.
Proof of (a).
Define
v, =
on B, `GD(, ) on D - B. Sl GD(, ) n v
The function v, is superharmonic on D, and G(1;,)-v,, defined as 0 at , is a subharmonic minorant of G(l:, ) and is therefore majorized by GMDG(, ); that is, v, Z GD(, ) on D - B. Proof of (b).
The function RB
is a positive superharmonic minorant
of GD(t, ) on D, equal to GD(, ) on a neighborhood of 4, and therefore by (a) must majorize GD(, ) outside this neighborhood. Hence these two functions are identical. Since the unsmoothed reduction lies between GD(, ) and the smoothed reduction, the three functions are identical on D.
Proof of (c) and (d).
In (c) we can suppose that the neighborhood in
question is open and relatively compact in D. Then if v is a strictly positive
superharmonic function on D, the function cv majorizes GD(, ) on a neighborhood of 8B for sufficiently large c; so (a) can be applied to yield (c). The particular cases of (c) follow trivially, and (d) follows from the first of these cases by an application of the Heine-Borel theorem. The Set D.: {ri: GD(, ry) > a)
(Note. S is fixed throughout the discussion.) If a > 0, the set D. is an open subset of D containing . No open connected component of D. not containing
89
3. Boundedness Properties of Go
is relatively compact in D. but D,, is itself connected and relatively compact in D if a is sufficiently large. In fact, if B is a relatively compact in D open connected component of D. not containing , then the function is harmonic on B, with boundary limit a at every boundary point of B, and so must be identically a on B, an impossibility. On the other hand, if /3 is the supremum of on the complement of a compact neighborhood of in D and if a > /3, then the set D. is relatively compact in the interior of this neighborhood and so also relatively compact in D and is connected by what was just proved.
Observation (N = 2). According to Theorem 3, the function GD(, ) = G(, ) - u(, ) is bounded outside each neighborhood of . It will now be shown that if
and 2 are in D and if c is an arbitrary constant, the minorants
GMD[G(,, ) A G(2, )],
GMD[G(l;,, ) A c]
(3.1)
exist and that the differences
G(1, ) A G(2,) - GMD[G(1,, ) A G(2, )], G(1, ) A c - GMD[G(1, ) n c]
(3.2)
are bounded outside any neighborhood of , and 2, outside any neighborhood of ,, respectively. These facts imply that if >q is a finite boundary point of D, the minorants in (3.1) are bounded on the trace on D of a compact neighborhood of q in 882, a fact needed later.
Since G(,, -) - G(2, ) is bounded outside any neighborhood of , and 2 and since GD(,, ) is bounded outside any neighborhood of i, the dif-
ference u(1,) - u(2, ) is bounded outside any neighborhood A of , and 2. Choose a compact neighborhood A of these two points, so that 1u( 1, ) - u(2, )1 5 b on D - A, where b depends on A, , , and b2. This inequality must hold on all of D in view of the maximum theorem for subharmonic functions. Then
G(1, ) A G(2, ) 2 GMD[u(1, ) A u(2, )] 2 u(,, ) - b;
(3.3)
so the first minorant in (3.1) exists. The harmonic function u(,, ) is bounded above by c, = sup,,D G(, , ) < + oc ; so
G(1, ) A c, 2 u(S,, ) A c, = u(, , and it follows that for any constant c,
G(c,,)Ac2G(y,,)Ac,-IC -c,I 2u(,,)-Ic-c,1;
(3.4)
90
I.VII. Green Functions
so the second minorant in (3.1) exists. Finally, the first difference in (3.2) is at most b = GD(bl, ) + b,
(3.5)
and similarly the second is at most const; so these differences are bounded outside each neighborhood of ,, 2 or each neighborhood of ,, respectively.
4. Further Properties of GD As already remarked we shall prove in Section 7 that the open sets with Green functions are the Greenian sets. In the following theorem the boundary involved is the Euclidean boundary.
Theorem. If D is an open subset of Il" for which there is a Green function, the Green function GD has the following properties:
(a) (b)
Go is continuous and symmetric on D x D (= + oo on the diagonal).
For quasi every finite point S of 6D and also for C = oo if N > 2 and D is unbounded, lim,._t
(c) (c 1)
(c2) (c3)
rl) = 0 for every
in D.
If e D, the function GD(c, ) has a positive extension GD
to
(F8", uniquely characterized by the following properties: Go 0 on R' - D and at quasi every finite point of 8D. GD is subharmonic on 68" - {ld}. If c is a finite boundary point of D,
GD (,
= lim sup Dan;
q);
if N > 2 and if D is unbounded, lim,,-,o Go
(4.1)
ri) = 0.
Observation (1). If D is connected, the finite points ( of 8D for which C) = 0 on D, equivalently, by lower semicontinuity of subharmonic functions the boundary points { for which lim,, GD(c, ti) = 0 for all in D, will be seen in Section VIII.14 to be the finite regular boundary points of D relative to the Dirichlet problem for harmonic functions on D. GD
Observation (2). We shall prove (b) as a consequence of (c). Alternatively it is possible to prove (b) after the Dirichlet problem for harmonic functions on D has been treated, by showing that quasi every finite boundary point of D is regular and that the regular boundary points are the points at which
has limit 0. The extension G() can then be obtained by an application of the extension Theorem V.5 to -
on R' - {l;}.
4. Further Properties of Go
91
Proof of (a). We have seen in Section III.1 that if GD exists, then GMDG(4, ) can be obtained as follows. A sequence B, of balls is chosen with closures in D and with the property that each point of D has a neighborhood which lies in B. for infinitely many values of n. If T aJ is the operator II(1.8) but applied here to functions on R', and if for 4 e D we define
t8.... TB.G(4, ), then
n e Z* } and
u,(
,
n e Z') are decreasing sequences of super-
harmonic functions on R" and D, respectively. The limit of the second We shall deal with the first sequence later. Since G is sequence is continuous on D x D, the functions uo, ut, ... are successively continuous, so u is upper semicontinuous. Let o be any point of D and choose B. in such a way that o e Bo and that there is an open neighborhood B e Bo of o which is a subset either of B. or of D - B for each n. A glance at the q) shows that for each point ry of D the function formula for is harmonic on B, and the same reasoning shows that ut ry), U2(',17), .. . are harmonic on B. It follows that u(-, j1) is harmonic on B, and since u is harmonic on D, is independent of the choice of B,, the function a minorant of q). By definition of u it follows that q) 5 u(17, -), and reversing arguments in this inequality yields the symmetry of u on D x D. Hence GD is symmetric on D x D. To prove continuity of GD, we prove that u is continuous. Let B' and B" be balls with closures in D, and express u on B' x B" using the Poisson integral on the boundary of each ball. If tLe balls either are the same or have disjoint closures, this representation of u shows that u is continuous on B' x B". Thus u is continuous on D x D. C3
n e Z) is a decreasing sequence Proof of (b) and (c). The sequence on R" - D. Moreover of superharmonic functions on R", all equal to
the sequence is locally uniformly bounded below on R" - 8D. In fact u on R" - D. We now show that z this sequence is uniformly bounded below on a neighborhood of each finite boundary point of D. This fact is trivial if N > 2 because u;, is then positive and is trivial if N = 2 and D is bounded because a lower bound of on a neighborhood of D is also a lower bound of on the neighborhood. If N = 2 and D is unbounded, the function is bounded on D in a neighborhood of OD (Theorem 3), say 2- c there; so Z
GD(,') Z G(:, ) - c
on D near OD, and G(4, ) on R2 - D; so the sequence {u;(4, ), NE Z') is locally lower bounded on a neighborhood of each finite point of 6D, as asserted. Thus {u;,(4, ), n e Z+ } is locally uniformly lower bounded
92
I.VII. Green Functions
on R'. It now follows from the Fundamental Convergence Theorem that the lim,,..m is equal quasi everywhere on R" to a superfunction harmonic function u More specifically,
u
u(, ) = G(, )
on D,
on R' - D and at quasi every finite point of OD.
u satisfies the conditions in (c) and The function Go is obviously uniquely determined by these conditions; that is, (c) is true. By the upper semicontinuity of subharmonic functions the positive function has limit 0 at GD is continuous at every zero; in particular, quasi every finite point of BD. If for some point , of D the function GD(.,, ) has limit 0 at C for all in has limit 0 at a boundary point C, then the same open connected component of D as , because (Theorem 3) outside is valid. const each neighborhood of the inequality Since the number of open connected components of D is countable, (b) is true. If N > 2 and D is unbounded, the function Go(C, ) has limit 0 at If C is a finite boundary point of D and N z 2, ao because GD then GD (
, C) = lim sup GD
PI)
n_t
(4.2)
This limit relation remains is subharmonic on R" because correct if I tends to C along the nonzero set of GD(C, ) and also [see 11(6. 1)]
if q avoids the polar set of points I' of BD for which (4. l) is true. o
of) # 0; that is,
Observation (3). The method of proof of (b) can be applied to show that the function differences in (3.2) also have limit 0 at quasi every finite
point of D.
5. The Potential GDp of a Measure p Let D be an open subset of R" for which GD exists. If p is a measure on D, the potential GDp is defined by
GDp() = JD
(5.1)
The following facts have been derived for D special but are true in general, and the proofs for special D are applicable. A potential GDp is either super-
93
5. The Potential GDµ of a Measure p
harmonic or identically + oo on each open connected component of D and, if superharmonic, is harmonic on everyp null open subset of D. Moreover GMDGDP = 0 if GDp is superharmonic. In particular, GDµ is superharmonic if p(D) < + cc. If p has compact support in D, Theorem 3(c) implies that GDµ is bounded outside each neighborhood of this support. Moreover in this case GDµ has limit 0 at every Euclidean boundary point C at which lim,r-{ q) = 0 for every because one can integrate to the limit in (5.1) when p has compact support. It follows (Section 4) that when p has compact support in D, the potential GDµ has limit 0 at quasi every finite Euclidean boundary point of D and also the point oo if D is unbounded and N > 2. Since a measure p on D can be chosen, supported by a countable dense subset of D, making GDp a superharmonic function + oo on a dense subset of D, the hypothesis of compact support for p was necessary in the preceding sentence.
If p has compact support in D, the potential GDp differs from Gp on D by a harmonic function. In fact, if GDp() = JA
f
q)p(drl).
(5.2)
A
The second integral defines a harmonic function on D because the function
is Borel measurable, has the harmonic function average property, and is finite valued on a dense set. Observe that what we have proved implies that if A is a compact subset of R', if p is a measure on R" supported by A, and if D, and D2 are open supersets of A whose Green functions exist, then GD, p and GD2µ differ on D, n DZ by a harmonic function.
In order to complete the proof of the Riesz Decomposition Theorem (Section IV.8) by proving it for open sets D for which GD exists-it will be seen in Section 7 that these sets are the Greenian sets-we need only follow the steps of the proof for D special. According to (5.2), the conclusion
of Theorem IV.7 can be stated in the form : u = GDPA + hA, where hA is superharmonic on D and harmonic on A. This representation of u was the basis for the proof of the Riesz Decomposition Theorem for special open
sets, and the continuation of that proof needs no change in the present more general context. Note again that according to that theorem, a positive superharmonic function u on D is a potential GDµ if and only if GMDU = 0. In view of(5.2) the Gauss Integral Theorem (Section 1.6) and also Theorem 1.7 on the smoothness of the potentials Gp of measures given by densities are valid for the more general potentials GDp. Application to the Function D f-- GD
Let D2 be a Greenian subset of R", let D, be a nonempty relatively compact open subset of D2, and let p be a measure on D2 with compact support in
I.VII. Green Functions
94
D, . Then GD p = Go, p + GMD, GD=µ on D, . The minorant is [Section III Example (d) y the restriction to D, of OGD:BD,-D, (reduction relative to D2).
This smoothed reduction must be a potential on D, , where u, is a measure supported by 8D, .
GD
6. Increasing Sequences of Open Sets and the Corresponding Green Function Sequences Theorem. Let D. be a monotone sequence of open subsets of l and suppose that GD. exists for all n. (a) (b)
with limit D,
If D. T D then the increasing sequence GD has limit GD if GD exists and has limit + oo otherwise. If D. I D and if D is open and not empty then the decreasing sequence GD has limit GD.
Proof of (a). To avoid trivia suppose that D is connected. If
of D and if u
then u
is a point on D,,, and
therefore (Theorem 11.3) the limit u(l;, ) of the sequence {u n > 0} on D is either identically - oc or harmonic, and it is easy to check that in the latter case GMDG(t, ). Conversely, if exists, the sequence { u n Z 0} is locally bounded below; so the limit function is harmonic on D. Since G(1;,) - GMDG(l;, ), assertion (a) follows.
Proof of (b). Define GD = lim,,.. GD, on D x D. For in D the function is harmonic on D The inequality GDS GD 5 GDS implies that is continuous with value + oo at and that the difference is a positive harmonic function on D if defined suitably
at g. This difference is bounded [because GDS(, ) is bounded outside a neighborhood of 1;] and has limit 0 at quasi every Euclidean boundary point of D, including the point oo if N > 2 and D is unbounded [because G0 has this boundary limit property]; so this difference vanishes identically according to the extended maximum-minimum theorem for harmonic functions in Section V.7.
7. The Existence of GD versus the Greenian Character of D Theorem. An open subset D of l ' is Greenian if and only if GD exists.
Since all open nonempty subsets of lE for N > 2 and all bounded nonempty open or not connected open subsets of R2 are both Greenian and have Green functions, only unbounded open connected subsets of l 2 re-
9. Approximation Lemma
95
main to be considered. If a set has a Green function, the set is trivially Greenian. Conversely, suppose that D is an open unbounded Greenian subset of l2, so that there is a positive nonconstant superharmonic function u on D, with associated nonnull Riesz measure p. Let D. be the part of D in B(0, n) and let p be the projection of p on D. According to Theorem 6,
unless D has a Green function, lim, Go. = + oo, but this limit relation is impossible because (Riesz decomposition) on D..
(7.1)
8. From Special to Greenian Sets Many theorems have been stated for Greenian sets D but proved only for special sets. The justification for this awkward procedure is that the properties of GD make the proofs already given for special D valid whenever D is Greenian, that is, whenever Go exists. Some of these extensions to Greenian D have already been checked in Sections 4 and 5. The rest are equally easy to check.
9. Approximation Lemma Lemma. If D is a Greenian subset of R', if 0 is a finite-valued continuous function on D, with compact support B, and if B, is a compact neighborhood of B in D, there is a sequence u, - v, of differences of finite-valued continuous potentials whose associated Riesz measures are supported by Bt, with u _
v,, on D - Bt and lima.,, (u -
uniformly on D.
For n sufficiently large, say n -- no, the function Atm4 is defined on a neighborhood of Bt and vanishes off Bt. For n this large define Atm4) = 0 on D - Bt where this function is not already defined. Then {Atm¢, n z no is a sequence of infinitely differentiable functions converging uniformly
to 0 on D. Define u
as the potential of the measure with density -((DA1m4) A 0)/niv [((AA1m4) v 0)/ni ]. Then Au o - (u - v.) = h is a
function harmonic on D, and v - u = h on D - Bt. The inequalities v z h and u z -h are valid on D - Bt and therefore (superharmonic function minimum theorem) also on D; so h = 0 since GMDV = GMDU _ 0. That is, Atm¢ = u - v,,, and the lemma follows. Application to the Ordering of Measures
Suppose that p and v are finite measures on a Greenian set D, with the property that whenever u and v are finite-valued positive continuous superharmonic functions on D, with u Z v, then
96
I.VII Green Functions
f (u - v) dp < f (u - v) dv.
(9.1)
D
D
We now show that it follows that µ S v. It is sufficient to show that if 0 is a positive continuous function on D with compact support then
j4)d 0. Then there is a function u on D, in the upper PWB" class for f, such thatforj >- 1, Hf is harmonic on D.,,
5
(2.1)
ifHf
ooon
(A dual assertion is true for lower PWB" solutions.) To see this, observe that by definition of upper PWB" classes there is a function uj in the upper PWB" class on D for f such that (2.1) is satisfied by uj. On each set D,, define
u = uj and define u = + oo on each remaining component of D. Internal Resolutivity
It is trivial that the one-point boundary of a Greenian set D is universally resolutive and that the PWB" solutions are the constant functions. At the other extreme, a boundary of D will be called internally h-resolutive if every bounded h-harmonic function is the PWB" solution of some boundary function, and a boundary will be called universally internally resolutive if the boundary is internally h-resolutive for every strictly positive harmonic h.
Observe that the characterization "h-resolutive" is not applied to any proper subset of a boundary.
EXAMPLE (c) (The Classical Solution). If f is a finite-valued continuous boundary function for which the traditional first boundary value problem has a solution, that is, there is an h-harmonic function u on D with boundary limit function f, then f is h-resolutive, and H' = u because u is in both upper and lower PWB" classes; so u = Hf = Hf. Thus [see Example (b) above] the Euclidean boundary of a ball is resolutive. This ball boundary will be shown to be universally resolutive in Section 9 and will be shown to be universally internally resolutive in Section IX.12. h-Regularity of Boundary Points
A boundary point i; of D will be called h-regular, or regular when h = 1, if whenever f is a finite-valued continuous boundary function
2. The PWB Method
103
lim HH(7) = urn Hf(ry) =f(0. 4
If every boundary point is h-regular, the boundary will be called h-regular, or regular when h = 1. An h-regular boundary is h-resolutive because whenever f is a finite-valued continuous boundary function Hf = Hf in view of the h-harmonic maximum-minimum theorem. For each such function f the PWB" solution H f" for an h-regular boundary is the solution of the traditional
Dirichlet problem. Conversely, if this problem has a solution for every finite-valued continuous f, the boundary is h-resolutive and h-regular.
Elementary Properties of Upper and Lower Solutions
It is trivial that fl! f = -Hg, that H f and H f increase with f, that H f = cH f and H f = cff when c is a positive constant, and that Hf+. = Hf + c and Hf+, = Hf + c when c is an arbitrary constant. Moreover, if e > 0 and if f and g are finite-valued boundary functions whose upper and lower solutions are finite-valued, and if If - g < e, then lH f - Ha : e and IHf` - H, 'l e because
Hf - e=Hf, SHa 2 and D is unbounded. (These conditions are convenient to verify but are more stringent than necessary.) The function u1 - u'2 is superharmonic with limit inferior !f(C) at every boundary point C of D. This difference is therefore lower bounded and is in the upper PWB class for f; so u1 >- u2 + Hf, and therefore u1 >- u2 + Hf. Interchanging u1 with u2 yields the inequality u2 z u1 - Hf and we conclude that Hf = Hf = u, - u2, that is, (i) is true. In Section VI 1.4 the Fundamental Convergence Theorem was used to show that the function GD(, ) = G(, )
- GMDG(, ) has limit 0 at quasi every finite point of 3D. Under the hypotheses of (ii) the same reasoning shows that u, - u,' has limit 0 at quasi every finite point of 8D and therefore that Hf has limit f(C) at quasi every finite boundary point C. To prove that Hf has limit f(oo) at oo when N > 2 and D is unbounded, we apply a barrier argument, to be developed more generally in Section 14. Define u() _ I12-" for in D, choose b > f(oo), choose a neighborhood of oo so small that f 5 b in the neighborhood, and
then choose n so large that b + nu 2:f on 8D outside this neighborhood. The restriction to D of the function b + nu is in the upper PWB class on D for f and has limit bat oo. Hence lim sup4-. Hf() 5 b; so this limit superior is at most f(oo). Since the same reasoning is applicable to -f, the function Hf has limit f(oo) at oo, as was to be proved.
Special case (b'). Let D be bounded and let f be the restriction to 8D of a polynomial u. Then u can be written as the difference between two polynomials superharmonic on a heighborhood of D: u(C) = [u(C) - c1C12] - (- cIC12)
(3.1)
for sufficiently large c. The assertions (b)(i) and (b)(ii) are therefore applicable to polynomial boundary functions when D is bounded. Special case (b"). According to Section VII.3, the hypotheses of (b) (i) and (b) (ii) are satisfied by u1= G(,, ) A c1jD for , in D and - oo < ci 5 + oo. Special case (b"). If N > 2 or if N = 2 and D is bounded, the hypotheses of (b)(i) and (b)(ii) are satisfied by u, = G(s, )JD' u2 0, for each point in D.
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I. VIII. The Dirichlet Problem for Relative Harmonic Functions
4. Continuous Boundary Functions on the Euclidean Boundary (h = 1) Theorem. The Euclidean boundary of a Greenian set D is resolutive. Moreover quasi every finite point of this boundary is regular, and oo is a regular Euclidean boundary point whenever D is unbounded and N > 2.
This theorem, together with the fact that the PWB method yields the classical solution whenever there is one, justifies this method. In the following, C(OD) denotes the class of finite-valued continuous functions on OD,
metrized by the supremum norm. In view of the fact (Section 2) that the limit f of a uniformly convergent sequence f of finite-valued resolutive boundary functions is resolutive and that Hf is uniformly convergent to Hf on D, it is sufficient to prove that there is a countable dense subset r: iga,aaI} of C(aD) with the property that for each index value a, the function ga is resolutive, and H,, has limit ga(C) at quasi every finite boundary
point C, as well as at i; = oo when N > 2 and D is unbounded. In fact the Euclidean boundary is then resolutive, and if A. is the exceptional polar
boundary subset for ga, the finite boundary points not in UaeiAa are regular.
In view of Section 3, Example (b'), when D is bounded F can be taken as any countable dense subset of C(aD) consisting of restrictions to aD of polynomials. The following proof is applicable to both unbounded and bounded sets D. If N > 2, let t, be the class of positive superharmonic functions on Ol;" which are finite valued and continuous with finite limit at oo. Define u(oo) for u in Ti as the limit of u at oo. Then IF, contains the positive constant functions and u A v, and au + by are in IF, if u and v are and if a and b are positive constants. Let F. be the class of differences u, - u2 with ui in 1-1, so that 1'2 is a vector lattice in the order determined by pointwise inequality. The set 173 of restrictions to OD of the members of F2 is a vector lattice of finite continuous boundary functions, which contains the constant functions and separates DD because c A -)Ian (defined as 0 at oo) is in 1-3 for an arbitrary positive constant c and an arbitrary point . The set 173 is dense in C(aD) (Stone-Weierstrass theorem), and in view of Section 3, Example (b) any countable subset of l'3 dense in C(OD) can serve as the desired set F.
If N = 2, let o be a point of R' and define Ti as the class of functions u satisfying the following two conditions: (a) u is a finite-valued continuous superharmonic function on R2. (b) There is a strictly positive constant a and a constant fi, both depending on u, such that u is identically fi outside some bounded set which may depend on u.
4. Continuous Boundary Functions on the Euclidean Boundary (h a I)
107
When u is in r,, the function cu is also in r, when c is a strictly positive constant, and u + c is in r, when c is an arbitrary constant. When u, and u2 are in r, , the functions u, + u2 and u, n u2 are in r,. Let r2 be the class of differences u, - u2 for which u, and u2 are in r, and have the same multiplier a in (b). Then u, - u2 is constant outside some bounded set. Define u, - u2 at oo as this constant value. The set r2 is a lattice in the order determined by pointwise inequality. If u is in this lattice, cu and u + c are also in the lattice, for every constant c. If I'3 is the class of restrictions to aD of the members of r2, T3 is a lattice of finite continuous boundary
functions which contains cf and f + c with f Moreover r3 separates aD. To see this, let , and 2 be distinct points of aD, and let B be a ball containing
o and either , or t;2, say ,, but not the other. The boundary function
f=
IA
18
is then in r3. If f has the same value at t, as at 2 decrease the radius of B to increase f at , but not at b2. Thus r3 separates aD and therefore (StoneWeierstrass theorem) is dense in C(aD). In view of Section 3 Example (b) any countable subset of r3 dense in C(0D) will serve as the desired set r.
Partial Generalization to PWB" Solutions
Suppose that D is Greenian, that h is a strictly positive harmonic function on D and that B is an open relatively compact subset of D. If f is a finitevalued continuous function on DB, a function u is in the upper [lower] PWB" class on B for the boundary function f if and only if the function uh on B is in the upper [lower] PWB class on B for the boundary function Jh18B It follows that f is h-resolutive, with Hi = Hf"/h, and that a boundary point of B is h-regular if and only if the point is regular. Thus quasi every boundary point of B is h-regular.
This partial generalization of Theorem 4 will be completed in Section 8 where it will be shown that if D is given an h-resolutive boundary by a metric compactification, if B is an open subset of D, and if M is the boundary of B relative to D v aD, then aB is h-resolutive. This fact can be deduced now
when h = I and OD is the one-point boundary provided by the Alexandrov compactification of D. In fact, in this specialization if f is a finite continuous function on tB, let f be the function on the Euclidean boundary a'B of B defined as! on OD n 8'B and (if B is not relatively compact in D) as f(aB n aD)
on the rest of a'B. The function f is finite and continuous and therefore resolutive on B for a'B. Moreover the upper and lower PWB classes on B for f on a'B are the same as those classes for f on B. Hencef is resolutive, as was to be proved.
108
I. Vill. The Dirichlet Problem for Relative Harmonic Functions
5. h-Harmonic Measure Null Sets Let D be a Greenian subset of R' provided with a boundary by a metric compactification, and let A be a boundary subset. If the upper solution H;A "R; = R" /h = RA/h) vanishes identically on D, the set A will be called an h-harmonic measure null set, or harmonic measure null set if h =_ 1.
In the following discussion D,, D2, ... are the open connected components of D and j is a point of Dj, held fast throughout. The proofs are given for infinitely many components and the simplifications to be made when there are only finitely many are left to the reader. (a) A countable union A = Uo A. of h-harmonic measure null sets is h-harmonic measure null. If e > 0 and if (Section 2) u" is an h-superharmonic function in the upper PWB" class on D for the boundary function 'Aft, with e2-'-' for ail j, then the function v = J:', u" is in the upper PWB" class for the boundary function IA, and H;A = 0 because 5 v(lj) < e for all j. (b) A boundary subset A is h-harmonic measure null if and only if there is a positive h-superharmonic function u on D with limit + oo at every point of A.
If A is h-harmonic measure null, define A. = A and observe that then the positive h-superharmonic function v in (a) has limit +oo at every point of A.
Conversely, if a function u as described in (b) exists, then for every e > 0 the function ru is in the upper PWB" class for the boundary function IA; so H A = 0 except possibly on the polar set of infinities of u, and so H; = 0 on D. (c)
If f is a positive boundary function and if Hf = 0, then the set
If > 0} is h-harmonic measure null. If n >_ I and if f is the indicator function on BD of the set { f > 1/n}, then
0=Nf z
H"n"= Hn"";
so the set { f > I /n} is h-harmonic measure null. Hence Ul f > 1 /n} _ { f > 0) is h-harmonic measure null. (d) If f is a boundary function for which H f < + oo, then the set A = { f = + oo } is h-harmonic measure null.
Choose (Section 2) an h-superharmonic function u on D in the upper PWB" class for the boundary function f. The function u is positive and has limit + oo at every point of A so A is h-harmonic measure null according to (b). (e) Euclidean boundary, h = 1. A polar subset A of OD is a harmonic measure null set. If N > 2, let u be a positive superharmonic function on R", identically + oo on A. Then v = ulD has limit + oo at every point of A ; so according to
S. h-Harmonic Measure Null Sets
109
(b), the set A is h-harmonic measure null. If N = 2 and if C is a finite boundary
point of D, choose a ball B of center r; so small that 682 - (D u B) is not polar, that is, so that D' = D v B is Greenian. Then if A n B is not empty, there is a positive superharmonic function u on D', identically + 00 on A n B; so UID has limit + oo at every point of A n B. Thus according to (b),
A is locally harmonic measure null, and it follows that A is a countable union of harmonic measure null sets and is therefore itself harmonic measure null. The converse of (e) is false : a harmonic measure null subset of a Euclidean
boundary aD need not be polar. For example, if D is a ball in R' it will be seen in Section 9 that the harmonic measure null subsets of the Euclidean boundary are the IN_, null boundary subsets, and it is not difficult to find examples of IN_, null subsets of a sphere which are not polar. The following is however a near converse to (e). (f) Let D be an arbitrary nonempty open subset of R', and let A be a nonpolar proper subset of D, closed relative to D. Provide D with a boundary
by a metric compactification, and let a(D - A) be the boundary of the Greenian set D - A in this compactification. Then the set A n a(D - A) is not harmonic measure null relative to D - A. In fact, if A n a(D - A) is harmonic measure null, then by (b) there is a positive superharmonic function u on D - A with limit + oo at each point of A n a(D - A). If u is extended to D by defining u = + co on A, the resulting function is superharmonic on D with value + co on A ; so A is polar, contrary to hypothesis.
h-Harmonic Measure Null Sets and the Kelvin Transformation We use here the notation of the discussion in Section 2 of the PWB method
and the Kelvin transformation. It is clear from that discussion that an h-harmonic measure null subset of aD is transformed under an inversion into an h,-harmonic measure null subset of OD'. We shall use the following additional fact when h = 1. If OD and aD' are Euclidean boundaries and if A is a harmonic measure null subset of aD, then ¢(A) is a harmonic measure
null subset of OD', under the additional hypothesis when N > 2 that all points of cb(A) are finite. This fact follows from the criterion for h-harmonic
measure null sets in (b) because if v is a positive superharmonic function on D with limit + oo at every point of A, then the Kelvin transform of v is a positive superharmonic function on D' with limit + oo at every finite point of ¢(A), every point of O(A) if N = 2. EXAMPLE (a) (Euclidean Boundary, h =- 1). If N Z 2 and if C is a finite boundary point of D, the singleton (C) is polar and therefore is harmonic measure null. If n = 2, the singleton { oo } is harmonic measure null for every
unbounded Greenian set D because the point oo can be made finite by an inversion of the plane relative to a circle centered at another boundary point
110
I.VIII. The Dinchlet Problem for Relative Harmonic Functions
and we have just seen that such a transformation of the plane preserves the harmonic measure null property. If N> 2, the singleton {oo} may not be a harmonic measure null set for an unbounded Greenian set. For example, if D = R' and if u is in the upper PWB class on RN for the boundary function 11,,), that is, if u is superharmonic and lower bounded on RN, with inferior limit z I at oo, then u >- 1 by the superharmonic function minimum theorem. EXAMPLE (b) (Euclidean Boundary). A polar subset of 8D need not be an h-harmonic measure null set for all h. For example, let D = B(0, b), and let
h be a minimal harmonic function on D corresponding to the boundary point C (Section 1I.16), say I
1z
h()= 6, - SIN IS -
Let g be the indicator function of A = OD Since the function e/h is in the upper PWBh class on D for g whenever c > 0, the set A is h-harmonic measure null; so is not, because aD is not an h-harmonic measure null set.
6. Properties of PWB' Solutions The following properties of PWBh solution will be needed in addition to those derived in Section 2. Proofs are given in the next section. The functions
f and g below are boundary functions. (a)
If f = g up to an h-harmonic measure null set, then H f = He and
Hh=Hh (b) Hf < + oo if and only if H f n < + oo.
If Hi < + oo, if h." < + oo, and if f + g is defined arbitrarily on the h-harmonic measure null set, where f or g is - oo and the other is +oo, then Hf+g :!g Hf + Ho. (d) The class of h-resolutive boundary functions contains the constant (c)
functions and is a vector lattice (that is, the class is linear and contains f v g and f A g when it contains f and g). Moreover, if f and g are h-resolutive, then Hj,, 6 = LM ,(H f v Hg),
H.hf+sB = aHJ + PH,, h
(e)
h
h
A Hg).
If f, is a monotone increasing sequence of boundary functions with limit f and if fl fo > - oo, then H! = lim,,..,p H H. Moreover H f = H j if each function f. is h-resolutive.
111
7. Proofs for Section 6
(f) If f is an h-resolutive boundary function, there are Borel measurable h-resolutive boundary functions f, and f2 such that f, 5 f 5 f2, H f = Hf = H fl,, and f, =f = f2 up to an h-harmonic measure null set. In particular, if A is a boundary subset with h-resolutive boundary indicator function f = 1A, there are Borel boundary subsets A, and A 2 such that the functions f, = 1,,, andf2 = 1A, are h-resolutive,
that A, c A c A 2, that H f, = H f' = H f,, and that A2 - A, is h-harmonic measure null. (g) If f is h-resolutive, if Do is an open subset of D, and if fo is defined
on (Do as f on OD, n 8D and as Hr' on D n BD,, then f0 is h(h)
resolutive for Do and the PWB' solution on Do forfo is the restriction of H4 to Do. For every subset A of 8D,
H" = inf {Hi8: 8D
B
A, B open in aD}, (6.1)
H A= sup {H;8: B c A, B compact}.
of D the set function A t--. H is a Choquet capacity relative to the class of compact boundary subsets.
For each point
h-Resolutive Boundaries. Property (e) and its dual for decreasing sequences
imply that if 8D is h-resolutive, all bounded Borel measurable boundary functions are h-resolutive. This result will be extended in Section 8.
7. Proofs for Section 6 Since the proofs do not depend on the choice of h they will be given for h I to simplify notation. The point is that the assertions in Section 6 are valid for the PWB method in a very general context. Proof of (a). Let A be the harmonic measure null set in question. It is enough to prove the second equality, and (by symmetry) it is even enough to prove
that Hf 5 H. If v is in the upper PWB class for g and if v, is a positive superharmonic function on D with limit + oo at every point of A, then v + ev, is in the upper PWB class for f whenever e > 0; so Hf 5 Hg quasi everywhere on D and therefore everywhere on D. o Proof of (b). If Hf < + oo, there is a lower-bounded superharmonic function v on D. say v z c, in the upper PWB class for f. Then v + Icl is in the upper
PWB class for f v 0; so Hf
Hf Hf z lim Hf. = Hf so there is equality throught, and the proof is complete.
Proof of (f). Suppose first that f is an arbitrary boundary function. Choose a point Sk in each open connected component of D, and let v, be a decreasing sequence of members of the upper PWB class for f with HM.-. V,(4) = for all k. There is such a sequence v, because the upper PWB class for f is directed downward. Define gq(C) = lim info..., v (ry) for i e BD. Then g, is a decreasing sequence of lower semicontinuous boundary functions
with Borel measurable limit function f2 z f; so H1 2 Hf, and there is equality on the set ,. Since an upper PWB solution for f is either harmonic or identically + oo or identically - o0 on each connected open component of D. we conclude that Hf, = Hf. A dual argument (or apply this result to
-,f) yields a Borel measureable boundary function f, with f, 5 f and Hf. = Hf. In particular, if f is PWB resolutive,
Hf=Hf, SHf. SHfSHf, SHfz=Hf; so f, and f2 are resolutive, and the equation Hf2 - Hf, = 0 together with the inequality f, 5 f 5 f2 implies (Section 5) that f, =f = f2 up to a harmonic measure null set, as was to be proved. If f = 1,, is the resolutive indicator function of a set A, we can replace f2 [ f,] by the indicator function of the set A 2 = { f2 = 1) [A, = (f, = 1) ] to obtain the second part of (f).
Proof of (g). If v is in the upper PWB class on D for f, the restriction of v to Do is in the upper PWB class on Do for fo. Hence the upper PWB solution Offfo on Do for fo satisfies the inequality oHfa 5 Hf on Do. Similarly, oHfo Z Hf on Do so (g) is true.
Proof of (h). Since H,A = I - H,aD_A, only the first equality in (6.1) need be proved. A function v on D is in the upper PWB class for lA if and only if on each open connected component of D the function v is either identically
1 14
I VI11 The Dirichlet Problem for Relative Harmonic Functions
+ oc or positive superharmonic with lim inf,,...z v(i) z I when e A. For such a function v, 0 < a < 1 implies that lim in some neighbor-
hood B of A relative to aD, so that v/a is in the upper PWB class for 1, and H, a < v/a. Since a can be chosen arbitrarily close to 1, the infimum in (6.1) is at most fit A* The inequality in the other direction is trivial. The set function -. H,A( ) = R; is a Choquet capacity of boundary subsets relative to the class of compact boundary subsets according to Section VI.3(k). [In the present context we have already proved all the desired properties of this set function except that if A is a decreasing sequence of H,A^ = H,A, and compact boundary subsets with intersection A, then this fact is elementary in view of the first equation in (6.1).]
8. h-Harmonic Measure Let D be a Greenian subset of R', coupled with a boundary OD provided by a metric compactification of D, and let h be a strictly positive harmonic function on D. In view of Sections 6(d) and 6(e) the class of boundary subsets whose indicator boundary functions are h-resolutive is a a algebra, and if A) = H;A for each set A in this a algebra, the set function we define µ"D(S. ) is a probability measure for each point of D. The sets of this a algebra will be called the uD measurable sets, pD will be called h-harmonic measure, and µ'W, ) will be called h-harmonic measure relative to . The h
will be omitted when h = 1. Observe that a boundary subset A is an hharmonic measure null set in the sense of Section 5 if and only if the set is A) -= 0, equivalently, if and only if the function pD measurable and A) has a zero in each open connected component of D. According to Section 6(f), a boundary subset A is pD measurable if and only if there are Bore] µD measurable boundary subsets A, and A2 such that A, c A c A2 and A2 - A, is µ"D null. Thus for i; in D, 4(S, ) is the completion of the to the µD measurable Borel boundary subsets if D is restriction of connected. If f is a boundary function, we write for JDf(,)4(', dn) A) can also be when this integral is defined. Under this convention written If f is defined on a superset of aD, the notation 4D(-,f) is fj.D). It is trivial that for f a linear combination of to be interpreted as f) is h-harmonic. indicator functions of µ"D measurable sets, the function Since a bounded uD measurable boundary function is the limit of a uniformly
f) is convergent sequence of such linear combinations, the function h-harmonic if f is bounded and µD measurable. If f is a pD measurable boundary function,
A(',IfI)= lim11D(',I fI n n) < +x, N--W
115
8. h-Harmonic Measure
and (Section 11.3) on each open connected component of D the limit is either
identically + oo or is h-harmonic. If this limit is h-harmonic on D, the function f will be called µD integrable; the class of such functions f will be denoted by L'(AD). Then feL'(4D) if and only if f v 0 and f n 0 are in this class and, if so, .U'(-,f) = 4D(-,f V 0) + 4D(-,f A 0).
Let Do, D,, ... be the open connected components of D, and let aDk be the boundary of Dk relative to D u aD. Let h be a strictly positive superharmonic function on D, and denote by hk the restriction of h to Dk. Let f be a function on aD, and denote by f the restriction off to aDk. Then f is an h-resolutive boundary function for D if and only iff is h-resolutive for Dk for all k; f is µD measurable if and only iffk is µ"Dk measurable for all k; f e L' (,uD) if and only if f e L' (p'k for all k. When A is a µD measurable subset of aD, the function p" k(-, A r aDk) is the restriction of 4(-, A) to D. Theorem. If D is a Greenian subset of R" coupled with a boundary aD provided
by a metric compactification of D and if h is a strictly positive harmonic function on D, then a boundary function f is h-resolutive if and only if f e L' (µ4D), equivalently, if and only if f is µD measurable with both H f and H f finite valued,
and then H1 = µD(-,f ).
(8.1)
Proof That h-Resolutive Boundary Functions Are µD Measurable
Let Co be the class of continuous functions from R into R with limit 0 at ± oo and let f be a finite-valued h-resolutive boundary function. The class of functions 0 in Co for which ¢(f) is h-resolutive is a vector lattice which is closed under uniform convergence and which separates points of R
because the class includes the function xf- (1 - Ix - n j) v 0 for n e Z'. Hence (Weierstrass approximation theorem) the class is Co. Since the class of functions from R into R for which i(f) is h-resolutive is a class closed under bounded monotone convergence [Section 6(e)] and includes Co, this class includes every bounded Borel measurable function. When ' is the
indicator function of an interval in R, we find that f is µp measurable. Finally, if f is an arbitrary h-resolutive boundary function, redefine f as 0 at its infinities to find an h-resolutive boundary function differing from f on a set of h-harmonic measure 0 and thereby to conclude that f is µD measurable. Proof of Theorem 8. If f is the indicator function of a µ4D measurable boundary
subset, then f is h-resolutive, and (8.1) becomes the definition of µ4D. More generally (8.1) is therefore true if f is a finite linear combination of indicator
116
1. VIII. The Dirichlet Problem for Relative Harmonic Functions
functions of uD measurable boundary subsets, that is, if f is an h-resolutive boundary function taking on only finitely many values, all finite. Denote this class of boundary functions by IF. Apply the dominated convergence
theorem and Section 6(e) to show that if f is a positive and h-resolutive [µD measurable and integrable] boundary function, then f, as the limit up to an h-harmonic measure null set of an increasing sequence of positive functions in r, is in L`(,uD) [is h-resolutive], and (8.1) is true. This result applied to f v 0 and (-f) v 0 shows that an arbitrary boundary function f is h-resolutive if and only if it is in L' (,uD) and that then (8.1) is true. Finally we prove that if the boundary function f is µD measurable, with Hf and Hf, finite valued, then f e L' (µD). According to Section 6(b), Hf o < + oo ; so in view of Section 6(e) and what we have just proved,
µD('>fv0)=JimAD(',(fv0)nn)=limHf
o)
,,=Hfvo
Hence f v 0eL'(p'D). Similarly f A 0eLI(jD); so feL1(µD), as was to be proved.
h-Regularity in Terms of h-Harmonic Measure
If aD is h-resolutive, the condition for h-regularity of a boundary point becomes
limµD(C, ') = 8 (vague convergence of measures on aD), where 8t is the probability measure supported by {C); equivalently, t is h-regular if and only if lim
A) = 1
t--4
whenever A is the trace on aD of a neighborhood of C. EXAMPLE (a) (Relation between uB and µg for B Relatively Compact in D).
If D is Greenian, if h is strictly positive and harmonic on D, and if B is an open relatively compact subset of D, it was shown in Section 4 that aB is h-resolutive with Hf = Hfh/h. It follows that d7) =
µ8(i,
a B).
(8.2)
EXAMPLE (b) (Extension of Example (a) to Non-relatively Compact B). Let D be Greenian, provided with a boundary OD by a metric compactification,
and let B be an open subset of D, with boundary aB relative to D u aD. Let h be a strictly positive harmonic function on D. Then the class of h-
117
8. h-Harmonic Measure
resolutive boundary functions on aB includes the indicator functions of the Borel subsets of D n aB, and for such a subset A and point in B the value A) does not depend on the choice of aD. Furthermore (8.2) is true for q c- D n aB when pB is defined using the Euclidean boundary of B. To prove these assertions, let A be a compact subset of D n aB, define! as the indicator
function of A on 8B, and define f' as the indicator function of A on the Euclidean boundary of B. A function u is in the upper PWB' class on B for f on aB if and only if uh is in the upper PWB class on B for f 'h on the Euclidean
boundary. In view of the fact that Euclidean boundaries are resolutive (Theorem 4), it follows that Hf = Hf"lh, where the notation refers to Dirichlet solutions on B. If e > 0 and if uh is in the lower PWB class on B forf'h on the Euclidean boundary, then u - s/h is in the lower PWB" class on B for f on 8B. It follows that Hf z Hf,,/h. This conclusion implies the truth of the assertions made above.
ExA.M1 a (b) (Continued). An assertion made in Section 4 can now be strengthened and proved. Suppose in Example (b) that OD is h-resolutive. Then aB is h-resolutive for B, and if A is a Borel subset of OD,
A) =
A n eB) + f
µD(17,
d17)
( a B).
(8.3)
nae
In view of Example (b) and Section 6(h) it is sufficient to prove that for A a compact subset of aD the function 1,,,,W on 8B is an h-resolutive boundary function for B and that (8.3) is true. Observe that if v2 [u2] is in the upper [lower] PWB" class on D for the boundary function 1 A on OD and if v, [u, ] is in the lower [upper] PWB" class on B for the boundary function defined as A) on D n 8B and as 0 elsewhere on 6B, the difference v2 - v, [u2 - u on B is in the upper [lower] PWB" class for the boundary function IA,,', on B. It follows that 1 A,,,1B is an h-resolutive boundary function on aB and that (8.3) is true. See Section 3.11.3(d) for the simple (after the necessary foundations have been laid) probabilistic derivation of (8.3).
Application to the h-Superharmonic Function Inequality Let u be an h-superharmonic function on D, and let B be an open relatively compact subset of D. Then the restriction of u to B is in the upper PWBh class for the boundary function uIaB; so u >u). A more delicate result is the following. Suppose that u is superharmonic and lower bounded on D, and define a lower semicontinuous boundary function f on aD by
f(o = lim inf u().
118
I.VIII. The Dirichlet Problem for Relative Harmonic Functions
Then if OD is h-resolutive, we show that f is h-resolutive and that u > P'D'(-,f).
(8.4)
In fact, for n e Z+ the boundary function f n n is h-resolutive [Section 6(d)], and u is in the upper PWB" class for this boundary function; so
u z Hf n= PD(-, f A n) and (8.4) follows. The function f is h-resolutive because it is in L1(p"D).
EXAMPLE (c). Let D, be a Greenian subset of I8", let D be an open subset of D, ,and let be a point of D. We consider PWB" solutions on Do = D with 8Do the Euclidean boundary and h = GD Let f be the function
lies on OD, We show that f is a PWB" resolutive boundary function and has the PWB" solution
GD, (
on Do.
(8.5)
,
In particular, if D = D, it follows that µ""D,,(-, { }) = I on Do. Let u be the restriction to Do of the function on the right in (8.5). Then u is h-harmonic
and is in the upper PWB" class on Do for the boundary function f The has boundary limit 0 at quasi every finite point of Green function aD and at the point oo if N > 2 and D is unbounded (Theorem VII.4), and therefore at quasi every finite point of aDo = aD v and at oo if N > 2 and D is unbounded. The polar exceptional set A is harmonic measure null (Section 5), so according to Section 5, there is a positive subharmonic function w on D with limit + oo at every point of A. Hence if e > 0, the h-subharmonic function e(l + w)
on D. is bounded above because (Section l) GD < GD,, and this h-subharmonic function has limit superior :5f at every point of aDo. This function is therefore in the lower PWB" class for f; so (e -. 0) the lower PWB" solution for f is at least u. Hence u is the PWB" solution for f, as stated in (8.5).
9. h-Resolutive Boundaries Section 6 implies that a boundary is h-resolutive if and only if the Bore] boundary subsets have h-resolutive indicator functions, equivalently, if and only if the compact boundary subsets have h-resolutive indicator functions.
9. h-Resolutive Boundaries
119
The following theorem gives a useful criterion for this h-resolutivity condition. Theorem. A boundary of a Greenian set is h-resolutive if and only if the set function A -. R' /h is additive on the class of compact boundary subsets. The equality HA = RA/h was pointed out in Section 2. If 8D is h-resolutive
A i- H, =
A) for A compact. To complete the proof of the theorem, it will be shown that conversely if there is the stated additivity, then the boundary function 'A is h-resolutive whenever A is compact. Let B be a compact subset of 8D - A. By hypothesis " Hh,A + HIB = H1A.e.
and when B increases to 8D - A through a sequence of compact sets, this equation becomes, in view of Section 6(e),
H,A+ H 10D-A =
1
so that H A = H H. That is, IA is h-resolutive, as was to be proved. (A Observation. For an arbitrary boundary the set function A F-. H compact) is subadditive; so there is additivity for all l; in D if and only if there is additivity for a point in each open connected component of D.
Application to Balls The Euclidean boundary of a ball B is universally resolutive. In fact, if h is a strictly positive harmonic function on B and if M" is the Riesz-Herglotz measure for h, then (Section III.4)
"Ri A = h" = HiA = f K(C, A
for every Borel boundary subset A. Hence the set function in Theorem 9 is additive. so the Euclidean ball boundary is universally resolutive. Moreover equation (9.1) implies that PhW,dC) = K(C,
)
h(
(9.2)
)
so that Hf = PI(B, fdM")/h. In particular (h - 1), the class of harmonic measure null boundary subsets is the class of IN_, null boundary subsets; the class of resolutive boundary functions, that is, the class L' (µe), is the class of IN_, measurable and integrable boundary functions; and Hf =
120
I. VIII The Dirichlet Problem for Relative Harmonic Functions
PI(B, f ). The class of PWB solutions Hf is thus [Theorem 11.14(b)] the class
D(p8_) which includes all bounded harmonic functions on B. Hence the Euclidean ball boundary is internally resolutive. The generalization of Theorem 11.14 to h-harmonic functions in Section IX.12 will make this reasoning applicable to show that the Euclidean ball boundary is universally
internally resolutive. The following example exhibits h-resolutivity and internal h-resolutivity in an extreme case. If C is a Euclidean ball boundary point and if h = K(C, ), then (9.2) implies that 1. In this case every boundary function f finite at a is h-resolutive, and Hf = f(C); the class of PWB' solutions is the class of finite constant functions. Since h is minimal (Section 11.16), a bounded h-harmonic function is necessarily a constant function; so we have proved that the Euclidean boundary is h-resolutive and internally h-resolutive for this special choice of h.
Application to Half-spaces
Denote by dd the Nth coordinate of the point of URN and define D = dd > 0}. For q in D let q* be the reflection of q in the boundary hyperplane of D. Then GD is given by logl-q*l f
- nl
ifN=2
IS - nl2-N - is - q*l2-N if N>2
(9.3)
because, as so defined, is harmonic on D with the right singularity at and has limit 0 at every boundary point of D, including oo. In view of 1(8.5) it is to be expected that (Euclidean boundary) is given by augmented possibly by a contribution from the singleton { oo }. Evaluation of this normal derivative leads to the density function in (9.4). The 'N_I integral over URN-I of this density is I ; so the natural conclusion is that dq) is given by 2dr
nNl - nl
N'N-I(dn),
=0.
(9.4)
In fact this evaluation of µD is correct because it is easily checked that if f is a finite continuous function on OD, the function PD(-J), as defined using (9.4). is harmonic on D with boundary limit function f. The function pD( J) is called the Poisson integral off, just as in the ball case. Just as in the ball case, to each boundary point q of D corresponds a minimal positive harmonic function K(q, ) on D, a constant multiple of the harmonic measure density in (9.4) for that value of q, with a special provision for q = oo. More specif-
9. h-Resolutive Boundaries
121
ically, if o is a point of D with duo = I and if K is normalized to make K(-, 0) = 1, then
K(q,
1x
I tl - Col x
if
00
117
if q = 00.
dS
It is shown that each function K(q, ) is minimal harmonic by showing that
there is a Riesz-Herglotz-type representation of an arbitrary positive harmonic function on D by means of a unique measure M. on aD:
u() =
K(C,
(9.6)
JdD
The proof follows that in the ball case and is omitted. Just as in the ball case, it is shown that the Euclidean boundary is universally resolutive and that (9.2) is true in the present context. Some of these results are easily reduced to the ball case by means of an inversion in a sphere taking D into a ball. It will be shown in Chapter XII that if D is an arbitrary Greenian subset of R', there is a universally resolutive and universally internally resolutive boundary aMD, the Martin boundary, and a function K on 3MD x D, such that K(q, -) is a minimal positive harmonic function on D when q is in a certain subset ?4'D of eD, and that to each positive harmonic function u on D corresponds a unique measure
on aMD, supported by a;'D, for which the counterparts of the RieszHerglotz-type representation (9.6) (known as the Martin representation in this general context) and of (9.2) are valid. The Martin boundary reduces to the Euclidean boundary if D is a ball or half-space, in which cases a;'D = aTMD.
The PWB Method and the Fatou Boundary Limit Theorem According to Theorem I1.15, if h is a strictly positive harmonic function on a ball B, the Dirichlet problem solution HI = PI(B, fdMM)/h has nontangential limit f(%) at M,, almost every boundary point C. This fact is at least a partial justification of the PWB method. Furthermore, if D is an arbitrary Greenian subset of RN, provided with a boundary by a metric compactification, and if f is an h-resolutive boundary function, then (Theorem 3.11.2) H f" has f as a boundary limit function along h-Brownian paths from a point of D to the boundary in the sense that almost every such path tends to some boundary subset A depending on the path (A is a singleton if the boundary is h-resolutive), f is constant on each set A, and Hf has limit f(A) along the path.
122
1 VIII The Dinchlet Problem for Relative Harmonic Functions
10. Relations between Reductions and Dirichlet Solutions Let D be a Greenian set provided with a boundary 8D by a metric compactification of D, and let B be an open subset of D with boundary 8B relative to D u D. Let h be a strictly positive harmonic function on D, and let u be an h-superharmonic function on D. Define a function f on 8B by setting J'= u on D n 8B and (if B is not relatively compact in D) f = 0 on OD n BB. Observe that a function on B in the upper or lower PWB"
class for f will be in the same class for any other choice of dD and corresponding choices of aB and off. The simplest choice for 8D in this context is the Alexandrov compactification one-point boundary. (a) If u z 0, the function f is an h-resolutive boundary function for B, with PWB" solution the restriction to B of "R°-B and u(S) ? N"9(4,u1D) _
{tB(h,U) ID)
(EB).
(10.1)
If B, and B2 are open subsets of D with B, (-_ B2, then NB,( ,u1D) ?
(DEB,).
(10.2)
In proving these assertions we assume, as we can without loss of generality,
that DD is the one-point boundary. Since this boundary of D is trivially h-resolutive, it follows [Section 8, continuation of Example (b)] that 8B is h-resolutive. If u' is a positive h-superharmonic function on D and majorizes
u on D - B, then the function u'lB is in the upper PWB" class on B for f; so "R °-B z Hf. Since 0 < Hf 5 Hf , we conclude (Theorem 8) that f is h-resolutive with Hf < 'R° B. In the other direction, if u' is now a function on B in the upper PWB" class on B for the boundary function f, then the function u' A (u1,) is also in this class and, when extended to D by u, is a positive h-superharmonic function on D majorizing u on D - B and therefore majorizing "R°-B on D. Hence Hf = Hf z on B. so there is equality on B; that is, in terms of h-harmonic measure, on B. The inequality in (10.1) and the inequality (10.2) are now both trivial. "Ro_n
"RD-B
The equality in (10.1) follows from the relation between h-harmonic measure and harmonic measure on D n 8B established in Section 8, Example (b). (b) The positivity hypothesis imposed on u in (a) was made only to allow the use of reductions. We now drop this hypothesis on u but suppose that B is an open relatively compact subset of D. Then a trivial modification
of the proof in (a) shows that the restriction to B of the infimum of the class I- of h-superharmonic functions on D majorizing con D - B is U), the PWB" solution on B for the boundary function f = ulaB. Finally, (10.2)
remains true in the present context, and in this context u1D = U on B. If B, and B2 are balls of center S and if h = 1. the inequality (10.2) reduces
123
11. Generalization of the Operator TB' and Application to GM`
to the fact that the function
r) is a decreasing function for
11. Generalization of the Operator TB and Application to GM" If u is an h-superharmonic function on an open subset D of R' and if B is an open relatively compact subset of D, we define rru as the smoothed infimum of the class of h-superharmonic functions on D, majorizing u on D - B, so that tBu is h-superharmonic on D, equal (Section 10) on B to the PWB" solution and equal to u quasi everywhere on D - B, in particular, equal to u on the interior of D - B. When B is a ball, this definition agrees on D - d B with the Section 11. 1 definition of zeu and the Section 1
definition of rBu, and therefore the definitions agree everywhere on D because two h-superharmonic functions equal 1, almost everywhere are equal everywhere. Obviously, in all cases >a,u a tBzu when Bt c B2. If u is h-subharmonic, 4Bu is defined as -TB( -u). In view of Section 111.1 [especially Observation (a)] as generalized trivially to allow arbitrary h and of the present extended definition of rB,
if D is a Greenian set, if B is an increasing sequence of open relatively compact subsets of D with union D, and if u is an h-superharmonic function on D, the limit of the decreasing sequence rB u of h-superharmonic functions
is, on each open connected component of D, either identically - 00 or h-harmonic and, if h-harmonic on D, is GMpu. That is, GMDu = iim tB,,u = limp
(11.1)
u)
if u has an h-subharmonic minorant. In particular, an h-potential u is characterized among the positive h-superharmonic functions on D by the
condition lim, p
u) _- 0.
A related result is the following, in which u is supposed positive to make reduction notation possible. Let D, h, and B. be as in the preceding paragraph, let u be a positive h-superharmonic function on D, define a bounary OD for D by a metric compactification, and write D for D u aD. Then if A c D,
GMo''R. = lim yM "R"-B +U
(11.2)
We prove this for h = I to avoid irrelevant notational complexities, and we use the alternative reduction notation because iterated reductions will be needed. Observe that D"-Bft
Db-N
DD-B,=
D OuO"
D Duo"
z D Duo"
=
QUBA-B",
and we have just proved that the limit (n - oo) on the left is GM°OuIA; so in (11.2) the left side is at least equal to the right side. In the other direction,
124
I. VIII. The Dirichlet Problem for Relative Harmonic Functions
D DUD" D°-B' 5
5
Du0"_B, Duo"-B.,
0°_1n
+ ll DUD"-B^ 0D-B1'
+ 0 DUD""B'" pD-H.
When n -+ oo, this inequality yields GM0DuD" 5 Jim
M-OD
DUD"-B"
+ GMDDUD"-,
and the last term vanishes because A n B. is relatively compact in D; so DUD"' " is a potential. Hence in (11.2) the left side is at most equal to the right side so there is equality.
A Local Property of T' If u is an h-subharmonic or h-superharmonic function and if B. is a decreasing
sequence of open relatively compact subsets of D with intersection , then In fact the proof for h = 1 and B. a sequence of balls of center [see Section II.6(f) for this result in a slightly different context] is applicable in the general case.
12. Barriers Let D be a Greenian subset of 18", coupled with a boundary OD provided by a metric compactification, let h be a strictly positive harmonic function on D, and let C be a point of aD. Usable conditions that C be h-regular are most easily formulated in terms of "h-barriers" (see Section 13). A strictly positive h-superharmonic function u on D will be called an h-barrier for D at { if limn-, U(q) = 0 and if (*) info-Bu > 0 whenever B is a neighborhood of If the condition (*) is omitted, u will be called a weak h-barrier for D at By an easy application of the h-superharmonic function minimum theorem, (*) is true if and only if Iim infn.E u(q) > 0 for in OD - {C }. As usual, h will be omitted from the notation and the nomenclature when
h-1.
Local Nature of the Existence of an h-Barrier
Let Do be an open subset of D with boundary that relative to the topology of D u aD, and let ho be the restriction of h to Do. Then if C is a common boundary point of Do and D, the restriction to Do of an h-barrier for D at is an ho-barrier for Dp at C. Conversely, if some neighborhood of C has the same trace on Do as on D, the existence of an ho barrier uo for Do at C implies the existence of an h-barrier for D at C. In fact, let B be the trace
125
12. Barriers
on Do of some open neighborhood of? so small that D n O B = Do n O B # 0,
and set a = infD0_Buo. Then a > 0, and if u is defined as a A ua on D n B and as a on D - B, the function u is an h-barrier for D at C. Thus the existence of an h-barrier for D at C is a local property of D near C, depending on h. Lemma (Euclidean Boundary, h =- 1). If there is a weak barrier at a boundary point {, there is a barrier at the point.
If N > 2 every unbounded open subset D of R' has a barrier at the point ao, namely, the restriction of the function G(0, ) to D. An unbounded open subset D of B2 has a [weak] barrier at the point ao if and only if the image of D under an inversion in a circle has a [weak] barrier at the circle center. Hence we can assume in the following proof that the boundary point C in question is finite. In view of the fact that the existence of a barrier at C is a local property of D, it is sufficient to show that there is a barrier on the trace on D of an open neighborhood of C, so that D can be supposed bounded. Suppose then that D has diameter 6 < + oo and that u is a weak barrier for D at the boundary point C. Define O(PI) = J o7 - CI and f = 01,D. The function
46 is subharmonic, and OID is in the lower PWB class on D for f. Hence Hf z ¢ on D, and it will be shown that Hf is a barrier for D at C by showing that H. has limit 0 at C. Fix r > 0, let B = B(C, r), let A be a compact subset
of D n aB, and let 0 be the indicator function of (aB - A) n D on t?B, so that PI(B, qi) is harmonic on B with limit I at every point of aB - A. If uo is in the lower PWB class on D for f, the function uo on B n D is at most
r if 8B does not meet D and (by the maximum theorem for subharmonic functions on B n D) is at most r + Su/infA u + SPI(B, 4i) if aB does meet D. Thus in both cases
Hf < r +
8u + (Pl(B, +i) inf u
(12.1)
A
on B n D. The sum on the right has limit r + RI (B, 4/) (C) at C, and this limit is at most 2r if A is sufficiently large. Since r is arbitrary, the function Hf has limit 0 at C, as was to be proved. EXAMPLE (a). If h is a strictly positive harmonic function on D with a finite
continuous strictly positive extension to b, then if u is a barrier for D at a boundary point C, the function u/h is an h-barrier for D at C.
EXAMPLE (b) (Euclidean Boundary, h = 1). If D is unbounded and N > 2,
we have already noted that the function G(0, ) is a barrier for D at the point oo. EXAMPLE (c) (Poincare) (Euclidean Boundary, h = 1). If C is a finite boundary
point of D with the property that some closed ball meets D at C but at no
126
I VIII. The Dirichlet Problem for Relative Harmonic Functions
other point the restriction to D of the function ball center, is a barrier for D at C.
C) -
with
the
We shall show in Section 15 that it is sufficient for the existence of a barrier at C if in this Poincare criterion the ball is replaced by a cone with vertex C.
is a finite boundary point of D and if there is a simple continuous arc with initial point C, in EXAMPLE (d) (N = 2, Euclidean Boundary, h = 1). If
CBZ - D except for C, then there is a barrier for D at C. In fact, if B is a ball of center C so small that the arc hits 8B, let Bo be B less the part of the arc from 1; to the first hit of aB. Then Bo is simply connected; so the function
log(- - C) has a single-valued analytic branch 0 in Bo. The restriction to B n D of the real part of 1/0 is a barrier for B n D at i ; so there is a barrier for D at C.
13. h-Barriers and Boundary Point h-Regularity Theorem. Let D be a Greenian subset of OB", coupled with a boundary 8D provided by a metric compactification. If there is an h-barrier for D at the boundary point ( and if f is an upper bounded boundary function, then
lim ;PHI(,) 5 lim Sup f(,) vf(C);
(13.1)
in particular, if f is bounded and is continuous at C, then
lim HH(q) = lim HI(,) = f(c),
r{
(13.2)
n-C
and therefore C is h-regular.
The second assertion follows from the first applied to f and -f. To prove the first assertion, let u be an h-barrier at C. Let b be any number strictly larger than the right side of (13.1), let B be a neighborhood of C so small that f S b in the neighborhood, and let b be the infimum of u outside D n B. Choose n so large that b + no exceeds the supremum of f. The function b + nu is in the upper PWB' class on D for f and has limit b at C. Hence the left side of (13.1) is at most b; so the theorem is true.
Extension. Since a change off on a set of h-harmonic measure 0 does not change HI or H f, the point , can tend to C on the right side of (13. 1) on the complement of such a boundary set, and f(C) can be omitted on the right hand side if {C} is h-harmonic null, as is true when h - I and 8D is the Euclidean boundary, unless N > 2 and C = oo. This extension of Theorem 13 reduces to Theorem 11. 1 when D is a ball, h 1, and 8D is the Euclidean boundary.
127
14. Barriers and Euclidean Boundary Point Regularity
14. Barriers and Euclidean Boundary Point Regularity Theorem. (Euclidean Boundary, h = 1). A boundary point is regular if and only if there is a barrier at the point. If there is a barrier at a boundary point, the point is regular by Theorem 13.
Conversely, suppose that the point C is a regular boundary point of the Greenian set D. If N > 2 and if S = oo, there is a barrier, exhibited in Section
12, Example (b). If N = 2 and C = ao, make C finite by an inversion in a sphere with center a finite boundary point. Thus C can be supposed finite in proving the existence of a barrier there. If D is bounded, define f = I - C1 ia, and u = I - CIID. Then u is subharmonic and is in the lower PWB class on D for f ; so Hf Z u, and Hf is a barrier at C because (regularity of C) Hf has limit 0 at C. If D may not be bounded, the
cases N > 2 and N = 2 will be treated separately. If N > 2, define f and u as the restrictions to 8D and D, respectively, of E; n-3 [n - G(C, ) A n], with f(oo) = ET n-Z if oo a D. The function f is a finite-valued continuous positive boundary function vanishing at C and only there, and the function u is a continuous positive subharmonic function in the lower PWB class for f; so Hf z u. Moreover Hf has limit 0 at C because C is regular. Hence Hf is a barrier at C. If N = 2, let B be a ball of center C so small that Dt = D U B is Greenian. The function Zr n-3 [n - GD,(C, ) A n] is a positive bounded subharmonic function on D1. Define u as the restriction to D of this function, and define f at each point ?l of OD by f(q) = lim sup;._
Then f is positive,
bounded, upper semicontinuous, and vanishes at C and only there. The function u is in the lower PWB class for f on D; so Hf z u. Moreover Hf has limit 0 at C because C is regular. Hence Hf is a barrier at C.
Application (a) : Local Property of Regularity. (Euclidean Boundary, h = 1.) Since the existence of a barrier at a boundary point is a local property
of a Greenian set D near the point, regularity at a point of the Euclidean boundary is also a local property of D near the point. Application (b) : Regularity of a Boundary Point of a Disconnected Set. (Euclidean Boundary, h - 1.) Let D be a Greenian subset of R', let C be a boundary point of D, and let D1, D2, ... be the open connected components of D with boundary point C. Then C is a regular boundary point of D if and only if C is a regular boundary point of each set Dk. In one direction, if C is a regular boundary point of D, the restriction to Dk of a barrier for D at C is a barrier for Dk. Conversely, if C is a regular boundary point of each set D. and if uk is a weak barrier for Dk at C, then the function u defined on D by setting u = 2-k(uk A 1) on Dk is a weak barrier for D at C.
Application (c) : Regularity of a Boundary Point in Terms of the Green Function. (Euclidean Boundary, h =_ 1.) A boundary point C of a Greenian set D is regular if and only if q) = O for some (equivalently every)
128
1.VIII The Dinchlet Problem for Relative Harmonic Functions
point of each open connected component of D with boundary point C. In view
of application (b) we can assume in the proof that D is connected. If the condition on GD is satisfied for a single point (, the function is a weak barrier at (; so ( is regular. If N > 2 or if N = 2 and if D is bounded, the expression for GD((, ) in terms of a Dirichlet solution (Section 3) implies
the truth of the converse, for all . The following proof of the converse is valid in all cases. If ( is regular and if u is a barrier for D at (, let ( be a point of D, and let B, be the set c}, where c is a constant chosen so large that B, is relatively compact in D. Such a choice is possible because GD((, ) is bounded in a neighborhood of OD. Choose n so large that nu > c on B,. Then (Section VII.3) GD((, ) S nu outside BB; so has limit 0 at C.
Application (d): Relative Boundaries. If D is a Greenian subset of R', provided with a boundary t3D by a metric compactification, if h is a strictly positive harmonic function on D, and if B is an open subset of D with relative
boundary 8B in D v aD, then quasi every point of t3B in D is h-regular because if u is a local barrier for B at a boundary point of B in D, then u/h is a local h-barrier for B at the point and (Section 12) can be extended to B to be an h-barrier there. Application (e) : The Kelvin Transformation and Regularity. (Euclidean Boundary, h = 1.) We use here the notation of the discussion in Section 2 of the PWB method and the Kelvin transformation. If ( is a regular finite boundary point of D, then its image under inversion in a sphere is a regular boundary point of D' because if u is a barrier for D at (, the Kelvin transform of u is a barrier for D' at O(() except possibly when N > 2, and the inversion sphere has center C. In this case, however, the image of ( is the point oo, which is a regular boundary point of every unbounded Greenian set when N > 2 (Theorem 4). If N > 2 and if oo is a (necessarily regular) boundary point of D, the image of oo under an inversion may or may not be a regular boundary point of D'. If N = 2 and if oo is a regular boundary point of D, the image of oD under an inversion is a regular boundary point of D' because the Kelvin transform of a barrier for D at co is a barrier for D' at the image of co.
15. The Geometrical Significance of Regularity (Euclidean Boundary, h = 1) If S is a boundary point of D with the property that some neighborhood of ( meets OD in a harmonic measure null set, the point ( is irregular because a PWB solution Hf is not affected by a change off on such a set. This fact together with Poincar&'s criterion [Section 12, Example (c)] for the existence
15. The Geometrical Significance of Regularity (Euclidean Boundary, h =- 1)
129
of a barrier suggests that regularity of C, equivalently, the existence of a barrier at C, amounts to the requirement that D not fill up "too much" of a neighborhood of C. Thus an isolated finite boundary point C is irregular because {C} is polar and therefore harmonic measure null, and when N = 2, the point oo (if an isolated point of OD) is irregular because D can be mapped
into a Greenian set by an inversion leaving harmonic measure invariant and taking oo into a finite boundary point. The point oo is exceptional in that when N > 2, the harmonic measure {oo}) may be strictly positive
and, in fact, is identically I when D = l'. (Recall that when N > 2, the point oo is a regular boundary point of every unbounded Greenian set.) It will be shown in Section XI.12 that a finite boundary point C is regular if and only if C is a limit point of 68" - Din the fine topology, and this criterion will be stated probabilistically in Section 2.IX. 15.
Poincare-Zaremba Regularity Criterion Poincare's regularity criterion [Section 12, Example (c)] was improved to the following: if C is a finite boundary point of D with the property that some open solid cone of revolution with vertex C does not meet D in a neighborhood of C, there is a barrier at C (so C is a regular boundary point). In view of the discussion of barriers in Section 12, it is sufficient to prove
that if A is a closed cone of revolution with vertex the origin and if D = B(0, 1) - A, then D has a barrier at the origin. Define ¢() = and f = ¢iaD. Since is subharmonic on D, 41D is in the lower PWB class on D for f; so Hf z 01D, and we show that Hf is a barrier at the origin by showing that Hf has limit 0 there. Let Do be the part of D at distance 0, the function u' + e - f is lower semicontinuous on D u aD, strictly positive on OD, and therefore also strictly positive on W. for sufficiently large n. Hence for sufficiently large n the restriction to D. of u + e is in the upper PWB" class on D. for the boundary function I,BD; so ju%(-, f) S u + e on D,,. It follows that lim sup.-. ,u' f) S f), and this inequality together with the corresponding inequality for -f yields (17.2). Application. In Theorem 17 suppose that It = I and that OD is the Euclidean boundary. Let D; be an increasing sequence of subsets of D U (D, open
relative to D u 8D, define D. = D n D., and suppose that Uo D. = D. Suppose that A 1 is a subset of D u OD, open relative to D u aD, and that Al c D for sufficiently large n. Then if A is a Borel subset of Al n 8D and
if eeD,
132
I. VIII. The Dirichlet Problem for Relative Harmonic Functions
(17.5)
A).
A) =
li m
To see this, define A2 = At n 8D, and observe that if n is so large that e Dn and A t c D,;, then A2). In view A 1) = pDn(4, A2) and At) = of the vague convergence in Theorem 17(b),
liminfpD(
n-.O
uD( ,At)=uD( ,A2), (17.6)
"
and in view of (17.3) with A = A2, inequality (17.6) implies that (17.5) is true when A = A2. Now by the monotoneity in (17.3), lim
A) _< ID(b, A),
A2 - A) c}, for c > 0. Then DD is an open subset of D, and we now prove that if DD is not empty, then (Euclidean boundaries)
Define v = GDv. We can assume that u S c + 1, that is, v 5 (c + 1)h, because DD is unaltered if we replace u by u A (c + 1). The measure v vanishes on polar sets because v is finite valued; so (Theorem 18) GMD V =
(18.14)
1.VIll. The Dirichlet Problem for Relative Harmonic Functions
136
Now on the one hand (GMD v)/h = hGMDDv on D, and on the other hand vl p)h combined with (18.14) (Section 10(a)) the evaluation µ4,(, ul p) = yields "GMDDu =
(18.15)
ul p).
The left side of (18.15) is a majorant of the constant function c and u 5 c on D n OD, by lower semicontinuity of h-potentials, so (18.15) implies that
c5 that
D n 8D,) and we conclude that there is equality here and therefore dD n 0, as asserted.
19. Modification of Section 18 for D = R2 In Section 18 the set B is Greenian and so cannot be chosen to be RN unless
N > 2. In this section N = 2, the set D is a Greenian subset of R2, and the work in Section 18 is adapted to the choice B = R2. As in Section 18 the topology defining boundaries is the Euclidean topology. We adopt the oo) = - oo. The counterpart of the Section 18 set convention that a B r OD is the empty set if D is bounded and the singleton (00) if D is unbounded. Recall from Section 5, Example (a), that when D is unbounded, the singleton {oo} is a µp null boundary set. According to the following theorem, an awkward new term appears in the counterpart of (18.2). Theorem. (a) For each n in R2 the function G(n, -) is a resolutive boundary function; that is, this function is in L'(0). (b)
If e D, then GD
n) =
n) - pD(S, G(n, ')) + 0D(')
(n a 022),
(19.1)
where ¢D is a positive harmonic function on D, defined in (19.10), and
(bl) OD has limit 0 at every finite regular boundary point of D. (b2) Op is bounded on bounded sets. (b3) (pp = 0 if D is bounded. (b4) if D is unbounded and connected, OD - 0 if and only if ae is a regular boundary point of D. (c) If v = Gv is a superharmonic potential on R2 and if v' is the projection of v on the set of finite irregular boundary points of D, then on D, GMDv = PD(-1 V) + Go v' - v(R2)OD.
(19.2)
If D is bounded, (19.2) is true with Op 0 whenever v is a superharmonic function on D and v is the associated measure.
137
19. Modification of Section 18 for D = R2
Observation (1). If q e D, then (19.1) becomes [(c, q) E D x D].
GD(4, q) = G(4, q) - YD(', G(q, )) +
(19.3)
According to (b3) and (b4), the following two important implications of (19.3) are true if and only if OD = 0, that is, if and only if D is bounded or is unbounded with regular boundary point oo : q) =
q) -
q) E D x D],
G(q, ))
q) e D x D].
G(q, )) = pD(q,
(19.4)
(19.5)
According to Section 18, Observation (1), both these relations are true when N > 2 with no restriction on the nonempty open subset D of R". Equation (19.4) is a natural approach to finding the Green function GD, and Theorem
19 exhibits the conditions under which it is valid, that is, under which G(q, )) when N = 2. If there is no restriction on D, the difference between left and right sides of (19.5) is 4D(S) - OD(q)
Observation (2). If C is a finite regular boundary point of D or is an inner point of R2 - D, equation (19.1) yields C) =
(19.6)
G(g, )) - 4 D(4).
Observation (3). According to Theorem 19, the integral
G(q, )) = G
t) e D x R2]
(19.7)
is well defined and finite; so for fixed in D the potential is a finite-valued superharmonic function on R2. [There is a slight abuse of language here because when D is unbounded, is not a measure of subsets of R2, but we have already noted that oo) = 0 in the present context.] In view of (19.7) Theorem 19(b) implies that the Riesz measure on R2 - {} associated with the superharmonic function on that set is
Proof of (a). For q e R2 - D and, if D is bounded, for q e 6D. If we define f = G(,, )iaD, the function f is resolutive under the stated restrictions on q and D because Euclidean boundaries are resolutive and
- co < GM
sup f, < + oo
if q e R2 - aD,
G(q, ),D < + co if q e aD, D bounded. (19.8)
138
1 VIII. The Dinchlet Problem for Relative Harmonic Functions
The function q -
G(q, )) =
f) is the potential for the kernel G
of the measure pn(C, ) and is superharmonic because it is finite on R2 - 8D. Proof of (19.1) when D is bounded. If D is bounded and if B = B(0, 8) contains D, then GD (C, ) = GB(C, ) - PD(C, GB(l,'))
(19.9)
according to Theorem 18. Furthermore according to the formula for GB in 11(1.1), GB(C, q) = G(C, q) + ha(C, q),
where ha is a symmetric function on B x B and ha(C, ) has a harmonic extension to a neighborhood of B. If this evaluation of GB is substituted in (19.9), we obtain (19.1) with OD = 0.
Proof of (a) and (19.1) when D is unbounded. If D is unbounded, let B; be an increasing sequence of relatively compact open subsets of R2 with union 982 and define B. = B. n D. If we write (19.1) for B,,, let n ao, and take into account the fact that PD(C,ft) > - oo, we find (19.1) first for quasi every q and then for all q, with OD(S) = - lim
f
G(q,
dC);
(19.10)
[B = D n B(0, n)].
(19.11)
for example, (19.10) can be particularized to AD(C) = li m [p8.(C, D n OB.) log n]
Thus (a) and (19.1) are now completely proved.
Proof of (bl)-(b3). If C is a finite regular boundary point of D, fix q in D G(q, )) and observe that since ft is upper bounded, the PWB solution has superior limit 5 G(q, C) at C (Theorem 13). Moreover GD(q, ) has limit 0 at C; so (19.1) implies that
0 z G(C, q) - lim sup ID(, G(q, )) + lim sup WD(S), r-c
a-c
and so (bl) is true. The function OD is bounded on bounded sets, that is, (b2) is true, because in (19.1) for fixed q in D the function G(q, ) is bounded q) is bounded above on OD and bounded below on bounded sets, and outside each neighborhood of q. Finally we have already proved (19.1) with OB = 0 when D is bounded; that is, (b3) is true.
20. Interpretation of 0D as a Green Function with Pole oo (N = 2)
139
Proof of (b4). Observe first that with no hypotheses on D, 1im sup D9il-ao
q) -
G(q, ))] = li%!up f,,) log w
71
I - ql
! D(' ,''S) S 0 (19.12)
in view of Fatou's lemma, because the integrand has limit 0 when q - o0 and is at most log 1 + C
I{ -I 5 log(1 + I -I) if I
I - nl l
- ql > 1.
(19.13)
Here the right side of (19.13) defines a function of in L'(t0), according to Theorem 19(a). If ¢D - 0, inequality (19.12) combined with (19.1) shows that has limit 0 at oo ; so (Section 14) oo is a regular boundary point of D. To prove the converse, apply in (19.1) the evaluation of limd.. L(u, 0, S) with u a potential (kernel G) on 682 to find (19.14)
lim L(GD (, ), 0, b) e-M
In particular, if oo is a regular boundary point of D, the function Go has limit 0 at oo; so the left side of (19.14) is 0, and OD - 0, as was to be proved. Proof of (c). See the proof of Theorem 18(c).
20. Interpretation of OD as a Green Function with Pole oo (N = 2) If D in Section 19 is a deleted neighborhood of the point oo, this point is an irregular boundary point of D, and the limit equation (19.12) simplifies to lim
PI) -
qm
G(q, ))] = 0.
(20.1)
Equation (19.1) now yields lim
'-m
(20.2)
q) = 0D(0-
It is natural to write the limit on the left as
oo) and to think of
oo)
as the Green function of D with pole ao. See Section XIII.18 for further remarks on this Green function.
140
I. VIII. The Dirichlet Problem for Relative Harmonic Functions
21. Variant of the Operator TB If u is a superharmonic function on an open subset D of RN (N z 2) and if B is an open relatively compact subset of D, we have defined TBu in Section 11
as the smoothed infimum of the class of superharmonic functions on D majorizing u on D - B. Under this definition TBU = u) on B and TBu = u quasi everywhere on D - B. Define TBu as the superharmonic function on D equal quasi everywhere on D - B to u and equal on B to GMBu. The construction of GMBu in Section III.1 in terms of a decreasing sequence of superharmonic functions shows that TBu exists. Since TBu 5 u on B, it follows that TBu S rBu. According to Theorem 18(c) (in which the roles of B and D are reversed), TBU = rBu if and only if the set of irregular Euclidean boundary
points of B is a null set for the Riesz measure associated with u; so there is equality if 8B is regular. In the general case both TBu and TBu decrease when B increases. Moreover, if Bt and B2 are open relatively compact subsets of D, with Bt c B2, then TB,u Z TB,u Z TB2u -- TB2u.
(21.1)
For many purposes we can use tB and tB interchangeably. For example, if B varies through the open relatively compact subsets of D, then (21.1) shows that r8u and TBu as B varies have a common infimum; the infimum is GMDU if this harmonic minorant exists.
Chapter IX
Lattices and Related Classes of Functions
1. Introduction In this chapter certain function classes that arise naturally in potential theory will be discussed. These classes, the corresponding identically named
classes in parabolic potential theory (Section XVIII.19) and in stochastic process theory (Chapter V of Part 2), are discussed together in Chapter I of Part 3. Throughout this chapter D is a Greenian subset of R', N z 2, and h is
a strictly positive harmonic function on D. Both D and h are held fast throughout the chapter.
2. LMDu for an h-Subharmonic Function u Suppose that D is connected and that u is an h-subharmonic function on D. According to Section VIII.l 1 (where the discussion is in the dual context,
in which u is h-superharmonic), if A and B are open relatively compact subsets of D with A c B, then u5 TAU 5 rq'u, and if B is an increasing sequence of open relatively compact subsets of D with union D, then tB,u is an increasing sequence of h-subharmonic functions, and either LMDu exists and is given by tu, an h-harmonic function, or this limit is identically
+oo. An equivalent formulation is: the set of functions teu for B ranging through the class of open relatively compact subsets of D is directed upward and either LMDu exists and is given by LMDu = sup {
u) : B open relatively compact subset of D},
(2.1)
or else the supremum in (2.1) is identically +oo. [We adopt the convention here and in similar contexts below that the domain of µ8(-, u) is B and that the supremum of a set of functions at a point is the supremum at the point of the values of those functions defined there.] Observe that the supremum (directed limit) in question is unchanged if the sets B are all supposed to contain a specified compact subset of D; that is, the analysis relates to the properties of u near (any choice of) the boundary of D.
142
1. IX. Lattices and Related Classes of Functions
If u is positive, the existence of LMDu is equivalent to the L' boundedness of the class
{ [ulaB, µe(, )] : B open relatively compact subset of D}
(2.2)
of functions coupled with the indicated measures, for some, equivalently each, point of D. Alternatively the set B can range through a nested sequence B, with union D, as above. In particular, suppose that u is positive and that the class (2.2) is uniformly integrable for some point in D, that is, that there is a uniform integrability test function fi for which c(u)): B open relatively compact subset of D} < + oo.
sup
(2.3)
Then the class (2.2) is L' bounded; so LMDu exists, and since ((u) is a positive h-subharmonic function, the supremum in (2.3) is Thus in this case LMDF(u) exists, and the supremum in (2.3) is finite for all ; that is, the class (2.2) is uniformly integrable for each point in D. Furthermore the h-harmonic function LMDu satisfies the same uniform integrability condition as u, with the same test function 0; (2.3) is true with u replaced by LMDu. In fact we now show, under the hypothesis that (2.3) is true as written, that sup
b(LMDu)): B open relatively compact subset of D} (2.4)
= LMD(O[LMDu]) = LMDt(u). To see this, let B and B' be open relatively compact subsets of D. Then as B' varies,
r
[LMDu]) = µe
1l
-0 [suP µa4(', u)J \\
=
(suPP.(.G(u)))
B'
(2.5)
LMD(D(u)) = LMDt(u)O < +oo.
Take the supremum as B varies to find that the first two terms in (2.4) are majorized by the third. The reverse inequality is trivial.
3. The Class D(4D_ ) (See the corresponding stochastic process class in Section 2.11.11. This class is the linear class of real-valued Borel measurable functions u on D for which if is in D and if B is an increasing sequence of open relatively compact subsets of D with union D, then the sequence (3.1)
143
3. The Class D (t4-)
of coupled functions and measures is uniformly integrable. It follows from the definition of uniform integrability that a Borel measurable function u
on D is in D(µ4_) if and only if, for each point t in B, pB(, Jul) < + oo whenever B is an open relatively compact subset of D and there is a compact
subset A = A4 of D for which the family (2.2) of coupled functions and measures, under the added restriction that A c B, is uniformly integrable. This condition is satisfied if and only if there is a uniform integrability test function D = '{ such that sup { lt"B(1, (Dfl u1)) : A c B, B open relatively compact subset of D} < + oo. (3.2)
The notation D- is to remind the reader that not h-harmonic measure on some boundary of D but h-harmonic measures on the Euclidean boundaries of open relatively compact subsets of D are involved. When D and At are specified and there is no danger of confusion, we shall sometimes write D a function u is in D(pD_) if instead of D(µo_). Since and only if uh is in D(µD_). (See Section 11. 14 for the D class of harmonic functions on a ball.)
When the function u on D is h-subharmonic the situation is simpler is monotone increasing. In particular, because then the map if u is positive and h-subharmonic, the supremum in (3.2) is independent of the choice of A, and (Section 2) if (3.2) is true, the left side of (3.2) is It is thus natural that the case of most interest in potential theory is that in which Jul is h-subharmonic, in particular, if u is h-harmonic.
Theorem. If u is a Borel measurable real-valued function on the connected Greenian set D and if I uI is an h-subharmonic function on D, the following conditions are equivalent :
(c)
ueD(µo_). The family (2.2) of coupled functions and measures is uniformly integrable for every point in D. Condition (b) is satisfied for a single point .
(d)
There is a uniform integrability test function 0 such that the h-
(e)
subharmonic function ((jui) has an h-harmonic majorant. (If u is h-harmonic) u = u1 - u2, where ut is a positive h-harmonic function in D(µo_).
(a)
(b)
Moreover, if u satisfies (a)-(c) and if (V satisfies (d), then LM'Dju' e D(µD_), LMDN(LMoJul) = LMo D(Jul),
(3.3)
and each function u, in (e) can be chosen so that the h-subharmonic function '(u;) has an h-harmonic majorant.
144
1.1X. Lattices and Related Classes of Functions
If Jul here is identified with u in Section 2, this theorem follows at once from the discussion in Section 2, except for the assertion involving (e). To prove that an h-harmonic function u in D(µ4_) has the representation (e), observe that since Jul is h-subharmonic and LM4IuJa D(µ4_), it follows that
u = LM4Iul - (LM4Jul - u) is the desired representation; with this choice of u1, equality (3.3 shows that (D(u;) has LM"(D(Iu1) as an h-harmonic majorant. Conversely, if u = ut - u2 with u; positive h-harmonic and in D(µ4_ ), then I uI 5 ut + u2 ; so u e D(µ4_ ). It will be shown in Section 3.1.9 that an h-harmonic function on a Greenian subset D of tll' is in D(µ4_) if and only if the function is quasi bounded (a property defined in Section 9). More detailed results on the class of harmonic functions in D(µo_) for D a ball were obtained in Section 11.14, and these results will be extended to h-harmonic functions on a ball in Section 12, to h-harmonic functions on a Greenian set in Section XII.9.
4. The Class LP(pD_) (p >_ I ) This class is the linear class of extended real-valued Borel measurable functions u on D for which if is in D and if B is an increasing sequence of open relatively compact subsets of D with union D, then sup._,, JulP) < + eo.
It follows forp > 1 that LP(µ4_) c D(µD_), because the function s'-4s' is a uniform integrability test function. A Borel measurable function u on D is in LP(µ4_) if and only if, for each point 1; in D and each open relatively compact subset B of D with in B, µB( , uI P) < + co and there is a compact subset A = A, of D for which
sup {µB(1;, Jul"): A c B, Ban open relatively compact subset of D} < + eo (4.1)
When D, h, and p are specified and there is no danger of confusion, we shall sometimes write LP instead of LP(p4_ ). A function u is in LP(µ4_) if and only if uh is in LP(po_). As in Section 3, the case of most interest in potential theory is that in which Jul is h-subharmonic, in particular, if u is h-harmonic. The following theorem is in part a specialization of Theorem 3.
Theorem. If u is a Bore! measurable real-valued function on the connected Greenian set D and if Jul is h-subharmonic, the following conditions are equivalent :
(a) ueL1'(µ4_). (b) The family (2.2) of functions coupled with the indicated measures is LP bounded for every point l: in D. (c) Condition (b) is satisfied for a single point .
5. The Lattices (S2, 5) and (S', 5)
(d) (e)
145
The function IuIP has an h-harmonic majorant.
(If u is h-harmonic) u = ul - u2, with u; positive, h-harmonic, and in LP(1UD-)
If u satisfies (a)-(d), then LMolul e L'(4_), and in fact LMD(LMDIuI)P =
LMDIuIP.
(4.2)
The proof of this theorem is left to the reader because the theorem follows easily from the discussion in Section 2. Observe that part (e) of the present theorem is slightly stronger than Theorem 3(e). In fact, in Theorem 3(e) it is not asserted that if u = u! - u2 and if d>(u1) and'(u2) have h-harmonic majorants, then c(Iul) has an h harmonic majorant, although the counter-
part of this assertion is contained in Theorem 4(e) with ((s) = sP. However, for 0(s) = sP and p z 1 this assertion is true because then (ut + u2)P 5 2P-'(ui + u2P). In Theorem 4, as in Theorem 3, the condition (b) involves a set B in (2.2) increasing to D, but as in Theorem 3, it is sufficient if the set B runs through
a nested sequence with union D as described in Section 2. If D is a ball, it is natural to choose B to increase through balls concentric with B so that if D = B(0, S), the subharmonic function Jul on D is in LP(z _) if and only if sup,L(I u I P, 0, r) = 1im,t,,L(l u I P, 0, r) < + oo. The class of harmonic functions was discussed in Section 11. 14.
on B(0, S) in the class L'
EXAMPLE. Let u be a harmonic function on the Greenian set D. Then the func-
tion u2 is subharmonic with associated Riesz measure d A = (Au2/trN) dIN, so that A(D) is a multiple of the Dirichlet integral of u,
).(D)=2 f Igradul2d/Nlx,,S +oo. D
We now prove that ueL2(1LD_) if and only if GD). is superharmonic. In the one direction if u e L2(JD_), that is, if LMDU2 = v exists, then GMD(v - u2) =
0; so v - u2 is a potential and necessarily v - u2 = GD2 by direct calculation of Due. Conversely, if GDA is superharmonic, the function u2 + GD). is a majorant of u2 and is harmonic because i(u2 + GD).) = 0; so uEL2(AD_). In particular, if the Dirichlet integral of u is finite, that is, if ).(D) < + oo, then GD). is superharmonic; so ueL2(pD_).
5. The Lattices (St, 1. We now show that ue S. n LP(µ4_) if and only if u = PI(B, ¢dM,,)/h for some function 0 in L°(OB, Mb). Observe first that if u has this form, then lulP
_ - lim en = 0.
(4.2)
n-W
n-ao
Apply Section VI.3(n) again to obtain, for f in t with approximating sequence f = u, - v,, L.(f)I = lim IL.(f)I = urn IIunDA - kVjA I s lim supIII = SUP n-m n-m n-m D D
If 1;
(4.3)
that is, the linear functional LS has bound 1. Thus (Riesz representation on D with doA(r;, D) 5 1 for which theorem) there is a measure L4(f) = ODA(F, f) whenever f is continuous on D and has compact support. f) is Borel measurable, SDA is a kernel. Since this equation shows that Now suppose that u = GDp is the potential of a measure p with compact support S, and let D. be an increasing sequence of open relatively compact subsets of D with union D, for which S c Do and B. c Dn+t . According to Theorem IV.10, there is an increasing sequence u, of continuous potentials with limit u for which u = u on D - Do, and there is an increasing sequence un of continuous potentials with limit tDnu for which u k = TD.u = u on D - Dn+t. Then u. - u,. is in r, so A
A
(4.4)
Moreover u is a potential; so (from Section V111. 11) limn-m TDnu = 0, and therefore lira sup ou,,,,8 A 5 Jim sup U. 5 lim TDnu = 0. n-W
n-M
n-m
(4.5)
Hence (4.4) yields 8uj" = 6DA(',u)
(4.6)
160
1.X. The Sweeping operation
In particular, when u =
we find with the help of sweeping symmetry that 6DA satisfies the defining equation for bD ; so SD = SD" and is a kernel.
5. Swept Measures and Functions Theorem. Let D be a Greenian subset of I8", and let A be a subset of D. Let u = GDµ be a superharmonic potential, and let v be a positive superharmonic function on D with associated Riesz measure v. Then Quo"=GDOruIA=GDp=5D(',u),
(5,1')
0v0" = .5D(-,v),
JDv4d D
(5.1)
(5.3)
= fD v dQNQ".
In particular, if v is a superharmonic potential, (5.3) can be continued: f QvQ"dµ D
=
f VA'UQA D
(5.3')
= ID
ID
ID
I n the course of proving Lemma 4 we proved (4.6); that is, the first and fourth terms of (5. 1) are equal if µ has compact support. Since every positive superharmonic function is the limit of an increasing sequence of potentials
of measures with compact supports (Section IV. 10), the first and fourth terms of (5.1) are equal, and (5.1') is true. The first term is (5.1) is equal to the second by definition of Qµ0". The third term is equal to the fourth because
f
6D N' u) = J D
GDOI, C)µ(dd,
D
and the iterated integral becomes GDµ on reversal of the order of integration. Equation (5.2) follows immediately from (5.1); (5.3) and (5.3') follow from (5.1), (5.1'), (5.2), and the fact that J Qv0"lA = 0vO" Generalization of Q p Q"
It is convenient to define Qµ0" by (5.2) when µ is a measure on D even when
GDµ is not superharmonic. Under this extended definition, (5.3) remains true.
161
6. Some Properties of bn
Application to the Representation of a Reduction. Let D be a Greenian sub-
set of R', and let A be a subset of D. Then if v is a positive superharmonic function on D and if vA is the Riesz measure associated with DvOA (reduction relative to D), OvDA=GDVA+h,
vA=DvADA,
h=GMDOvIA=ihOA.
(5.4)
The first equation in (5.4), with the stated identification of h, is simply the Riesz decomposition of OvOA. Since the smoothed reduction operation is idempotent, (5.4) leads to (5.5)
GDOVAUA + OhOA = G0A. + h',
OVDA =
where A is a measure on D and h' = GMAh JA :!s; Oh OA :!g h.
A comparison between (5.4) and (5.5) shows that h' = h and A = O VADA = VA,
so that the second and third equation in (5.4) are true. ExAMPUS. If A is relatively compact in D, then [VIA is a potential; so h = 0. If A is sufficiently large, say A = D, then D V O A = v, and (5.4) becomes the Riesz decomposition of v.
6. Some Properties of SD (a)
Subadditivity of 6L. This set function is subadditive in the sense that bD B 5 bD + Sn
(6.1)
for all subsets A and B of D. In fact [Section VI.3(i)] 5DAvB(,, v)
+5
Dv0A
A OvD ) = c5
v) +
v)
(6.2)
whenever v is positive and superharmonic on D; so if f is the difference between two finite-valued positive superharmonic functions on D, with f -e 0, it follows that
s'
s
(6.3)
and this inequality implies (6.1) in view of the approximation application in Section VII.9. (b) For each point of D and each subset A of D the measure SD(, ) vanishes on polar sets not containing . In fact, if B is a polar subset of D
I. X. The Sweeping Operation
162
not containing , there is (by Theorem V.2) a positive superharmonic function on D, finite at but identically + o0 on B, and the inequality + 00 > v( ) 2t Dv[ 4( ) = 6D(S, v)
B) = 0. implies that (c) A zero-one law. Let be a point of D, let A be a subset of D, let B be a neighborhood of , and let v be a positive superharmonic function on DvDA^B() D. Then either the equalities 6D^1(, I and = v() are true
for all B and all v or SD^B(, {}) = 0 for all B, in which case whenever Dv0A11 tends to 0 on D when B shrinks to . The oo, the function 1 will be given a topological interpretation in property that Section XI.3, where incidentally it will also be shown that this property is independent of the choice of D containing . Thus this property is a local property of A near . To prove (c), we first observe that, using the notation , decreases when B increases because when B, and B2 are neighborhoods of with B, c B2 subadditivity of bn implies
of (c), bD^B(
bD2( ,
{
})
65"(S, M) + 6 D
and the second term on the right is 0 because the measure SD= Bo(, ) is supported by the closure of B2 - B, . Next observe that since J V O A B = DvOA
QA^B
it follows that
D^B(, {})
= JD' {})bB( dq)(6.4)
and since the measure 6'(ry, ) vanishes on polar sets not containing q, equation (6.4) reduces to bD^B(, {
}) = D(
})bD '8(
,
{l}).
(6.5)
When B = D, we conclude that must be either 0 or 1. In the first SD^B(, {}) = 0 for all B. On the other hand, if case (6.5) implies that 8D(, { }) = 1, then monotoneity of as B varies implies that 8D^B(, { }) = I for all B. 1, then IV I A-8g) = SD-B( z , v) = vO. If SD( , { }) = 0, If let B be a sequence of balls of center , relatively compact in D, with radii tending monotonely to 0. The sequence DvOA^B is a decreasing sequence of
potentials on D with limit a positive function vo, harmonic on D The restriction of vo to D has a positive superharmonic extension v, to D, necessarily of the form const. and the constant is 0 if as we suppose from now on v 1() < Hence vo = 0 on
D-
so
163
7. Poles of a Positive Harmonic Function hmOvlA'
'(7)=
The last term on the right is 0 by hypothesis when ry = and vanishes at all other points ry because according to (b) above, the measure c5(,) vanishes on polar sets not containing q.
7. Poles of a Positive Harmonic Function Let D be a connected Greenian subset of R", coupled with a boundary provided by a metric compactification, and let A be a boundary subset. If h is a positive harmonic function on D, if Ra = ch (as is true, for example, if h is minimal), and if h * 0, then c is either 0 or 1 because the smoothed reduction operation is idempotent [Section VI.3(h)]. The boundary point C is said to be a pole of h if Rti = h. The set of poles of h is obviously closed. If C is a pole of h, then C is also a pole of every positive harmonic minorant of h. In fact, if h, is such a minorant, the equality OhiO"'+Oh
-h,O1°=h'°=h
implies that the reduction operations performed on the left have not decreased ht or h - h,, so h, = Oh, Out.
Poles of a Minimal Harmonic Function If h is minimal, if C is a pole of It, and if A is a boundary subset containing C, then
hzRh - RIo=h; so R," = h. On the other hand, if C is not a pole of h, there is a positive super-
harmonic function v on D majorizing h near C and strictly less than It at some point of D. It follows that RtiA 5 v < h at some point of D; so RA 0 if A is a sufficiently small boundary neighborhood of C. Furthermore, if B is a boundary subset containing no poles of It, then Rh =- 0 because B can be covered by a countable collection A. of boundary neighborhoods of nonpoles, with R,"i =- 0; so R':!9 EJR,'J =_ 0. This result implies that there is at least one pole of a minimal harmonic function h because Rh° = h. It is left to the reader to check that the boundary is h-resolutive if and only if h has only one pole and that if C is this pole, the singleton JC) hash-harmonic
measure 1. If It has more than one pole, the indicator function of the set of poles is h-resolutive, and the set of poles has h-harmonic measure 1.
164
1. X. The Sweeping Operation
8. Relative Harmonic Measure on a Polar Set Let D be a connected Greenian subset of I8", let A be a compact polar subset of D, and let h = GDvh be the potential of a not identically zero measure
supported by A. Define Do = D - A and aDo as the boundary of Do relative
to the one-point compactification of D. Since A is nowhere dense, 8Do consists of A and the Alexandrov boundary point at of D. Let ho be the restriction of h to Do. Recall (from Section VII. 1) that GDo is the restriction of GD to Do x Do. We now prove that 8Do is an h-resolutive boundary for {a}) 0, and that for n in A and in Do Do, that p h(
).
(8.1)
Reductions below not otherwise described are relative to Do. If F is an open
relatively compact subset of D containing A, the set Do - F is a deleted neighborhood of a relative to Do u aDo, a minorant of ho and is harmonic and bounded on F, and QhoI D0-F/ho is in the upper PWB"o class QhoQDo-F is
on Do for the boundary function 1M. Moreover (by Theorem V.5) the function an extension to D, harmonic on F and majorized by h. When F increases, tending to D, this extension decreases, tending to a harmonic minorant of the potential h, that is, tending to the zero function. Thus {a} is an ho-harmonic measure null set. Next we prove that the Qh0QDo-F has
evaluation J Gd , n)v"(dn) A
implies that if C is a compact subset of A, then Qhr =
JG(.tl)v(dll).
(8 .2)
If Bo is the trace on Do of a neighborhood B of C, bno(
QhoOBO( ) =
,
(
IG.. ,q) QBO (S)vh( dn)
=
8.3)
fA
If smoothed reductions relative to D are denoted by Q QD, an easy argument
involving the superharmonic function extension theorem (Theorem V.5) shows that for , in D
165
8. Relative Harmonic Measure on a Polar Set D
on B. Then for
in B0 (8.4)
DhoOB0( ) = J
Now [by Theorem VII.3(b)] when PI is in C, the integrand is equal to
n),
and on the other hand [Section VL3(m)], when q is in A-C and B shrinks which vanishes because C is to C, the integrand tends to polar. Thus (8.2) is true, and since Ih0Dd/ho is the upper PWBkQ solution on
Do for the boundary function lc, we have proved that this upper solution is an additive function of C. Since we can ignore the ho-harmonic measure null singleton {a}, it follows from Theorem VIII.9 that Do is ho resolutive. Finally (8.1) is now equivalent to (8.2).
Chapter XI
The Fine Topology
1. Definitions and Basic Properties The topology of a topological space is the class of open subsets of the space.
If T, and T2 are topologies on a space, T, is said to be finer than T2 (and then T2 is said to be coarser than T,) if T2 c Ti. For any family of extended real-valued functions on a space there is a coarsest topology making every
member of the family continuous, namely, the intersection of all the topologies doing this. The fine topology of classical potential theory is defined as the coarsest topology on lf8" making continuous every superharmonic function on R'. It is easy to verify that the fine and Euclidean topologies coincide when N = I (see Chapter XIV for classical potential theory on t8), and we suppose from now on in this chapter that N > 1. Concepts relative to the fine topology will be distinguished by an "f," for example, f lim sup, afA. From now on any otherwise unqualified topological
concept will refer to the Euclidean topology. Since the fine topology is defined intrinsically in terms of superharmonic functions, it is not surprising that this topology plays a fundamental role in classical potential theory.
If u is a superharmonic function, the function u A c is also, for every constant c, and it follows that the fine topology is the coarsest topology making continuous every upper-bounded superharmonic function on 68". is fine continuous and since c) is a ball with center and an arbitrary radius depending on c, balls are fine open, and therefore Since
the fine topology is at least as fine as the Euclidean-topology. Since there are
discontinuous superharmonic functions, the fine topology is strictly finer than the Euclidean topology. Let D be a nonempty open subset of IZ". Then the restriction to D of the fine topology is the coarsest topology making continuous every superharmonic function with domain D. To prove this, it is sufficient to prove that if u is superharmonic on D, then u is fine continuous, and it is even sufficient
to prove that u is fine continuous on every open relatively compact subset Do of D. Now if µ is the measure associated with the restriction to Do of u, the potential Gp is superharmonic on !fF", differs from u on Do by a continuous (harmonic) function, and is fine continuous on R'. Hence u is fine continuous on Do, as was to be proved.
167
1. Definitions and Basic Properties
It will sometimes be useful to extend the fine topology to R' v { ao } by defining a set to be a deleted fine neighborhood of co if after inversion in a
sphere the image of the set is a deleted fine neighborhood of the sphere center (see Section 5). This criterion will be seen to be unaffected by the choice of inversion sphere, and in fact it will be seen that under an inversion the fine topology is invariant. The fine derived set of a set A, that is, the (fine-closed) set of fine limit
points of A, will be denoted by Af. It will be shown (in Theorem 6) that Af = (Af)f ; that is, Af is fine perfect. It is immediate that one base for the class of fine neighborhoods of a finite point is the class of sets of the form n {neB: uj(q) < cj}, t
where B is a ball containing , u; is an upper-bounded superharmonic function on B, vanishing at t, and each constant cj is strictly positive. The neighborhood (1.1) is fine open. It follows that another base for the class of fine neighborhoods of is the somewhat more convenient class of sets of the form
n
u;(n) < c},
(1.2)
t
where B is a ball containing , u; is a bounded superharmonic function on a neighborhood of B, vanishing at , and c is a strictly positive constant. The neighborhood (1.2) is fine closed and is Euclidean topology compact. It follows that a point is a fine limit point of a set A if and only if is a fine limit point of every open superset of A. A point of R' is a fine limit point of a set A if and only if is a Euclidean limit point of A and if each superharmonic function u defined on an open neighborhood D of has as a cluster value at along A. For example, according to 11(6.1), the complement of an IN null set is fine everywhere dense.
A subset A of RN is said to be thin at a finite point c if is not a fine limit point of A, that is, if is not in Af. According to the above remarks on fine neighborhoods, if A is thin at , there is an open superset of A which is also thin at . The corresponding remarks for = co are left to the reader. The Baire Property of the Fine Topology The intersection of a sequence of fine-open fine dense sets is fine dense. To prove this assertion, observe that the fine topology has a basis of Euclidean compact sets so that if B, is a sequence of fine-open fine dense sets and if B
168
1. Xl. The Fine Topology
is a nonempty fine-open set there is a sequence B; of compact sets with nonempty fine interiors such that Bo c Bo n B and that is a subset of the fine interior of B;, n B. for all n. Then W
0#nB, cnB,, B 0
0
so no B. is fine dense.
2. A Thinness Criterion Theorem. If a set A has finite limit point , then A is thin at S if and only if there is a superharmonic function u defined on an open neighborhood of 4 such that
liminfu(,) > A is thin at and if D is a Greenian set containing , there is a potential u = GDv satisfying (2.1) for which v has compact support and the left side of
(2.1)is+oo. If is a fine limit point of A and if u is superharmonic on an open neighborhood of , u has fine limit at and is lower semicontinuous; so (2.1) is impossible. Conversely, if is a limit point of A and if A is thin at , some fine neighborhood (1.2) of does not meet A except possibly at The functions u = L' uj and minjs uj are superharmonic on B with minjsn 0, and
u>c+(n- l)minuj jsn on
B; so
lim inf u(,) z c > A3rS
0.
To prove the second assertion of the theorem, let u satisfy (2.1), and suppose first that b is bounded and contained in the domain of u. Then u is bounded
below and can be made positive on D by addition of a suitable constant and radius r < 8Dl and let u, be the projection on B, of the measure associated with u. Then u differs from GDp, by a function harmonic on 5 B,; so if b (5 + co) is the strictly positive difference between the left and without affecting (2.1). Let B, be the ball of center
right sides of (2.1),
3. Conditions That e A'
169
lim infGDµ,(q) = Aa4-4
S z S.
Choose r = r" so small that
2-". The superharmonic potential GD(100 µ") has value 5 2 at and has limit + oo at along A as desired, and the associated measure has compact support in D. If b is not bounded and is in the domain of u, choose a ball B containing , with Bin the domain of u. Then the potential Gev of some measure v with compact support in B is finite at and has limit + oo at along A. The potential GDv has the same property because GDv differs from GBv by a function harmonic on B. Observation. If N > 2, the set D can be chosen to be l8" in the theorem. If N = 2, the proof yields a superharmonic potential u = Gv on R" for which the left side of (2.1) is + co. Application to Polar Sets and Reductions A polar set Ao has no fine limit points because according to Theorem V.2, if Ao is polar and if e Al, there is a superharmonic function defined on an open neighborhood of , finite at , and identically + oo on the part of A 0
in a deleted neighborhood of ; so (2.1) is satisfied, contradicting the hypothesis that E Ao. (In the fine topology of R" u { oo } as defined in Section 5 a polar set cannot have oo as a fine limit point; this assertion is reduced by an inversion in a sphere to the one just treated.) The converse result that Al = 0 implies that Ao is polar will be proved in Section 6. The fact that a polar set has no fine limit point implies that if A is an arbitrary subset of a Greenian set D, then (reduction relative to D) OVOA = v on Af An D.
In fact this equality is true quasi everywhere on A, that is, everywhere on A - Ao for some polar set AO; so the fine continuity of superharmonic functions implies that u = OvQA on (A - Al) = Af
3. Conditions That t e Af Let D be a Greenian subset of R", let A be a subset of D, let be a point of D, and let v be a positive superharmonic function on D. All reductions will be relative to D. We shall write lim81, to mean the limit as B, a neighbor-
hood of , shrinks to , that is, as its diameter tends to 0. Define vo = lim8J, Ov0A^1, that is, vo is the infimum of the indicated class of smoothed reductions. The positive function vo is harmonic on D so (by Theorem V.5) the restriction of vo to D has a superharmonic extension to D,
and (by the Riesz decomposition) this extension must have the form cG R,') + h, where c - 0 and h is a positive harmonic function on D. Since this superharmonic function is majorized by a potential BvrB for B relatively compact in D, the function h must vanish identically. The inequality vo 5 u
170
I.M. The Fine Topology
oo and more generally c = 0 if OvO "^B(,) < +00 implies that c = 0 if for some B. According to the following theorem, if v is the Riesz measure associated with v, then either c = v({ }) or c = 0.
Theorem. If e Af, then for every neighborhood B of
(a) (b) (c)
.
1.
Gii(, ) =
If 0 Af, then (a') limBl4 Dv0"nB = 0 on D (b') 6A(4, J })=0. (c') Gp( , ) # GD(4, ) on of D containing
and the limit is 0 at
if
+00.
0}, the open connected component
Observation (1). The fact that when B is a neighborhood of , the relations e Af and e (A n B)f are equivalent implies that properties (c) and (c) are
true under the stated conditions if A is replaced by A n B, and B need not appear in (a). Property (a) was phrased using B to contrast with (a'), in which B is essential.
Observation (2). We have already noted in Section X.6 the fact that the only possible values of are 0 and 1. We now see that the value is 0 if and only if A is thin at .
Proof. (a)-(c) Suppose that e Af. Then e (A n B)f, and as pointed out in Section 2, it follows that v(); so (a) is true, and in particular, GD(,) D; that is (sweeping symmetry), (c) is true. Finally, (c) ca (b) by definition of bD.
(a')-(c') Suppose that c e D - Af We can suppose that is a (Euclidean topology) limit point of A, adjoining a countable set with limit point to A if necessary to achieve this. The adjunction does not change smoothed reductions on A and does not change 61. Then (by Theorem 2) there is a positive superharmonic function u on D, finite at but with limit + 00 at along A. If t > 0 and if B is so small that u > 1/E on A n (B - {%}), it follows that SDnB(S, { } ) S (7"D-B(S, D) = O I O
Hence
D
(S) 5
cDuDAnB(
) S EU(S)
(3.1)
0, and it follows from Section X.6(c) that
0, as asserted in Theorem 3(b'). Theorem 3(a') now also follows from Section X.6(c), and Theorem 3(c') follows from the definition of So.
0
171
4. An Internal Limit Theorem
Application. Let A be a Borel subset of O", and let A is thin. Then if A, = A n t3B(t, r),
be a point at which
lim l"-t (Ar) = 0. r-.0 rN-t
In fact, in the first place according to Theorem 3 with reductions relative to
D= lim O I DAr(S) < ltm I oA,8(0.2r)( ) = 0. r-.o r+0
In the second place the rth smoothed reduction on the left obviously majorizes the smoothed reduction at relative to r) of the function 1 onto the boundary set A,; the value of the latter smoothed reduction is IN_t(A,)/ aNr'-t according to 111(4.3). This application of Theorem 3 implies that a solid cone of revolution in R" is not thin at its vertex. According to the criterion of Theorem 12 for Dirichlet problem regularity of a boundary point in terms of the fine topology, the nonthinness of a solid cone of revolution at its vertex is equivalent
to the Poincare-Zaremba criterion for Dirichlet problem regularity of a boundary point (Section VIII.15).
Strengthening of (a'). The following slight strengthening of (a') will be needed below: if e Ar and if v() < + oo, then lliim 0
0.
(a")
To see this, observe that this limit is limBl, 0 vrr), that according to the remarks at the beginning of this section, lim84,8vOB = 0 on D and 0. that according to (b'),
4. An Internal Limit Theorem If D is an open subset of R", if v is a positive superharmonic function on D, and if h is a strictly positive superharmonic function on D, then u = v/h is defined in the obvious way on the set of points at which at least one of the functions v, h is finite valued. As so defined, the function u is fine continuous.
The following theorem provides information on the character of u near a point of the set of common infinities of v and h. Let v and v,, be, respectively,
the Riesz measures associated with v and h. Recall our convention that dv ldvh is the Radon-Nikodym derivative of the absolutely continuous component of v with respect to vh. The singular component of v with respect to vh will be denoted by v.
I. XI. The Fine Topology
1 72
Theorem. (a)
The function u has a fine limit u*(C) at quasi every and vp + v,,
almost every point C of D. Moreover u* < + oo quasi everywhere and v,, almost everywhere on D, and u* = + oo vv almost everywhere on {C: v(C) _ +oo}. (b) At v,, almost every point C of the set {C: h(C) = + oo}, equivalently, at v,, almost every point C of an arbitrary polar subset of D, u*(C) = (c)
(C)
(4.1)
In particular, if C E D, f lim
v(q)r = vo({C }) = inf
D- c
7-c GD(n, C)
(d)
dvn
v y
(4.2)
S)
Let C be a point of D, and let F be a subset of D with (Euclidean
topology) limit point C. If F is thin at C, there is a superharmonic potential v on D for which v(n) _ +ao. 4GD(7,C) -
F191
(4.3)
Conversely, if there is a positive superharmonic function v on D - {C} satisfying (4.3), then F is thin at C.
Observation. In the first term of (4.2) and in (4.3) GD can be replaced by G without altering the validity of these equations because 1im,,..t GD(n, C)/ G(n,C) = 1. Since the theorem is local, we can suppose in the proof that D is connected.
Proof of (c). The special result (4.2), in which h = C) and so v,, is the unit measure supported by {C}, is so much easier to prove than the more general case (4.1) that we prove (4.2) separately, as follows. We have already seen (Section VII.IO) that the last two terms in (4.2) are equal. In proving (4.2) it can be supposed, replacing v by v C) if necessary, that these two terms vanish; under this hypothesis, unless (4.2) is true, there is a strictly positive number b such that the set B = {n: v(n) > bGD(n, C)} i3 not thin at C. Apply Theorem 3(c) to find that vZ
1v08
z
contrary to the hypothesis that
C) has infimum 0. Hence (4.2) is true. 13
173
4. An Internal Limit Theorem
Proof of (d). If v is a positive superharmonic function on D - {C} and if (4.3) is true, then since v has a positive superharmonic extension to D (Theorem V.5), the set Fmust be thin at C, in view of (4.2). Thus the converse
half of (d) is true. In the other direction, if F is thin at C, choose n(96 C) in D. Then [by Theorem 3(a')] there is a decreasing sequence B. of neighborhoods of C. shrinking to C, with G
C) < 2-".
There is therefore a positive superharmonic function v on D, majorizing 0) if C) on F n B. with v (q) < 2-". After replacing v by V. A necessary, we can suppose that v is a potential on D. The potential v = Eo v on Fn B. and therefore satisfies (4.3). is at least (n + 1) Proof of (a). If 0 5 a < b, define
S
A =
B = {
:
Z bh(p)}.
(4.4)
According to Section V1.3(o), V A bh hAB + hABAB + ... G
b-a
(4.5)
Now hA = U' VA Z UGDVkUA -
GD(V,.bA
and bp( , { }) = 1 when e Af n D; so if VhA is the projection of vh on Af n D, it follows that hA GDVhA. Similarly hAB Z OGDVhAOB - GD(VhAaD)+
so if vhc is the projection of vh on C = Af n Bf n D, that is, if vhc is the projection of vhA on Bf n D, it follows that hAB Z GDVhc. Similarly all the other summands in (4.5) majorize GDvhc, and we conclude that vhc - 0. Since every
point of D at which u* does not exist is in C for some rational pair a and b, we conclude that u* exists vh almost everywhere. Apply this result to h/v to show that u* exists v almost everywhere. As noted at the beginning of this section, the function u is fine continuous at every point at which it is defined ; that is, u* exists and is equal to u quasi everywhere on D. Moreover u* < + 00 quasi everywhere on D because u* = u = + oo can be true only on the polar
infinity set of v. The rest of the last sentence of (a) will be proved in the course of proving (b).
174
1.X1. The Fine Topology
Proof of (b). We show first that if F is a Borel polar subset of D and if v,(F) = 0, then u* = 0 v,, almost everywhere on F. It is sufficient to prove this for F compact. Under this hypothesis on F, we can assume that v and h are potentials because if v and h are replaced by their reductions on some open neighborhood of F, relatively compact in D, then v, h, v,,, and va are unchanged on this neighborhood, and v and h become potentials. Define B by (4.4). Then (reductions relative to D) v > 6vD8 z
bDhOB = bGD(VhSD)
Since I when e Bf, we can continue this inequality to find that if vM is the projection of vh, on BJ n F, then v ,-a bGDVV,. Now GD_F is the
restriction to (D - F) x (D - F) of GD, and the restriction to D - F of bGDvv is harmonic and is majorized by a potential, the restriction to D - F of v. Hence GDv; = 0 on D - F; so G0v = 0 on D, and therefore v,,(Bf n F) = 0; that is, f lim sup,,.., u(q) 5 b for vh, almost every point of F for all b>0. Hence u* = 0 v,, almost everywhere on F, as asserted. If this result is applied to h/v with F the trace of a v,, null support for vv on the set of infinities of v, we find that u* = + oo 4 almost everywhere on F, as asserted in (a). There remains the proof of (4.1). Let ¢ be the Radon-Nikodym derivative in (4.1). It is sufficient to show that if Fis a vv null Borel polar subset of D, then u* = 0 v,, almost everywhere on F. By what we have just proved, if vw is the projection of v,, on F, f lim h(q) - GD Vti (q) = 0 ,-.c
NO
at vw almost every point C of F; so in discussing u* on F we can assume that F supports vh and that h = GDvh. Define F. = v,
4,(i) 5 a},
(S,
5 ah; so f lim,, va(ry)/h(q) 5 a at v1, almost every point C of F.
Moreover, by what we have proved above, f lim,,..., [v(q) - va(ry)]/h(q) = 0 at v,, almost every point i, of F. n F. Hence u* 5 a v,, almost everywhere on
F that is, u* 5 a vv almost everywhere on F where 0 5 a, and therefore u* 5 ¢ v,, almost everywhere on F. A similar argument gives the reverse inequality; so u* = 0 vw almost everywhere on F, as was to be proved. Incidentally it now follows that u* < + oo v,, almost everywhere on D, as asserted in (a), because u* = u < + oo quasi everywhere on D, and we have which is finite vw just proved that on the remaining polar set, u* = almost everywhere.
175
S. Extension of the Fine Topology to RN u { co }
5. Extension of the Fine Topology to IAN v { oo } Theorem. Let A be a subset of 081, and let ¢ be an inversion of 081 in a sphere.
(a) (b)
Either (al) ¢(A) is thin at 4,(co) for every inversion 0, or (a2) 4(A) is thin at 4,(oo) for no inversion tp. An unbounded set A is under case (al) if and only if there is a positive superharmonic function u on some deleted neighborhood of the point oo with the property that lim
u(q)
_ =- 4 -0 0
A 37-m log 1111
lim u(ri) = + oo
A3q-m
(N= 2) , (N > 2).
(5 . 1)
(5.2)
Moreover, under case (a 1) with N > 2, there is a positive superharmonicfunction on 981 satisfying (5.2). (c)
If C and d(C) are finite, 4,(A) is thin at 4,(C) if and only if A is thin at
Proof of (a) and (b). (For N > 2). To avoid trivialities we assume that A is unbounded. Let 0 be an inversion in aB(C, b). Then 4,(A) has limit point C, and we prove first that if ¢(A) is thin at C, there is a positive superharmonic function u on 08' satisfying (5.2). In fact by Theorem 4(b) there is a positive superharmonic function v on D = R N such that (4.1) is true, and therefore the function bz1
G(C, 0)
has limit + oo at the point oo along A. The function on the right is a positive superharmonic function on 08' - {C} and in fact is a multiple of the Kelvin transform of v. This function has a positive superharmonic extension u to 081 (Theorem V.5), and u is the desired function. Conversely, if there is a positive superharmonic function u on some open deleted neighborhood of the point ao satisfying (5.2) and if 0 is an inversion in aB(C, b), we prove that 4,(A) is thin at C. The Kelvin transform v of u is a multiple of u(4,)G(C, ), defined positive and superharmonic on a deleted open neighborhood B of C, has limit +oo at along ¢(A); so [by Theorem 4(b)] the set and A is thin at C, as was to be proved. Thus (a) and (b) of the theorem are true
forN>2. o
Proof of (c). (For N z 2). Under the hypothesis of (c), the set A is thin at C if and only if (Theorem 2) there is a superharmonic function u defined on
176
I . XI. The Fine Topology
a neighborhood of C such that (2.1) is true. The Kelvin transform of u under an inversion 0 is a positive superharmonic function defined on a neighborhood of O(C) and satisfies the version of condition (2.1) at 4(C) if and only if u satisfies (2.1). Hence (c) is true.
The proof of Theorem 5(a) and (b) for N = 2 follows the proof for N > 2 and is left to the reader. Extension of the Fine Topology to R" u { oo }. We make the definition that a subset A, of R" u {ao} does not have fine limit point co, and we describe A, as thin at oo if A = A, n R" is subsumed under case (al) of Theorem 5.
The fine topology is thereby extended to R" u {oo}, and according to Theorem 5, an inversion is a fine topology homeomorphism of I2" u {ao} onto itself. The Limit Relation (4.1) at the Point co. Let v be a positive superharmonic
function defined on a deleted neighborhood of a finite point C. Then (by Theorem V.5) if v(C) is defined as lim inf,,..{ v(q), the extended function v is positive and superharmonic on a neighborhood of C, and v satisfies the fine limit relation (4.1). Trivially, v also satisfies the fine continuity relation f lim v(q) = lim inf v(q). o-C
11-C
If u is a positive superharmonic function defined on a deleted neighborhood of the point oo, then inversion of R" in a sphere yields a context to which (4.1) and (5.3) can be applied to yield the following. If u is a positive superharmonic function on R" - B(0, 6), then f"_'O lim u(g) = lim inf u(q) (< + oo) if N = 2, I,- ID
(5.4)
if N> 2,
flimu(q)IgIN-2 = liminfu(,i)I,I" 2 u(g) f,-M limlog61171
- inf
u(q)
lel>alogBl'1I
if N = 2,
flim u(,) = inf u(q)[l - (6117 l)2-"] 'rm hI>a
(5.5) 1
if N > 2.
In fact, if R" is inverted in B(0, 1), the Kelvin transform of u has domain of definition B(0, 1/6) - {0} and is given there by q i-, v(g) = u(glgl-2)I,I2-1.
The relation (5.3) with t = 0 yields (5.4), and (4.1) with C = 0 and D = B(0, 1/6) yields (5.5). The Green function GD is evaluated in Section I1.1. If the domain of definition of u is an arbitrary Greenian set that is a deleted neighborhood of the point oo, Example (c) of Section VIII.8 can be used to evalute the right-hand side of (5.5).
7. Application to the Fundamental Convergence Theorem and to Reductions
177
6. The Fine Topology Derived Set of a Subset of R" Theorem. A polar subset of R' has no fine limit point. Conversely, a subset of R" with no fine limit point is polar. If A is an arbitrary subset of R", the set Af is a fine perfect Gs set including quasi every point of A.
The converse is true whether or not a point ao is adjoined to R". It was pointed out in Section 2 that a polar set has no fine limit point. Since Theorem 6 is a local theorem, it can be assumed in proving the last assertion that A is a subset of some Greenian set D. (We can take D = fib" if N > 2.) Reductions below are relative to D. Let B, be an enumeration of the balls with closures in D and with rational centers and rational radii. According to Theorem 3, the Euclidean Fe set
U j=0
is D - Af. Since each set in this union meets A in a polar set, the set D - Af
is an F. set meeting A in a polar set. In particular, A is polar if Af = 0. The set Af is trivially fine closed, and is fine perfect because if were a fine
isolated point of Af, that is, if some deleted fine neighborhood B of contained no point of Af, then the set B n A would contain none of its fine limit points and thus would be polar, and could not be in Af, contrary to hypothesis.
The Fine Boundary
If A is fine perfect in RN, that is, if A = Af n R", its fine boundary 8fA = Af n (R" - A)f is a Euclidean Ga set, and therefore its fine interior A - 8fA is a Borel set, the difference between two Ga sets. If A is analytic, its fine closure in R" is analytic, and its fine interior A - (R" - A)r is analytic; so its fine boundary is universally measurable. In particular, if A is a Borel set, its fine interior and fine boundary are also Borel sets according to this argument.
7. Application to the Fundamental Convergence Theorem and to Reductions In the Fundamental Convergence Theorem the lower envelope u of the specified family of superharmonic functions is equal quasi everywhere to its lower semicontinuous superharmonic smoothing u, and lim inf, -, u(q). Hence u has a fine limit at every point,
u() = lim of u(q) = flim u(q). n
178
1. X1. The Fine Topology
If D is a Greenian subset of 08", if A is a subset of D, and if v is a positive superharmonic function on D, OVBA =
UV0AfnD.
(7.1)
In fact the first two terms are equal because, on the one hand, A C A U (Af n D) and, on the other hand, a superharmonic function on D majorizing v on A necessarily majorizes v on Af n D. The first term is equal to the third because A differs from A n Af by a polar set. The fourth term is equal to the others because A n Af c Af c A u Af.
8. Fine Topology Limits and Euclidean Topology Limits A function defined on a deleted Euclidean topology neighborhood of a point and with a fine topology limit at the point need not have a Euclidean topology limit there. For example, the indicator function of a countable dense subset of R' has fine limit 0 at every point but has no Euclidean topology limit at any point. Nevertheless the following lemma makes trivial the proof of Theorem 9, which establishes a surprisingly close relation between the two kinds of limit. Lemma. Let
be a point of 98N'v {oo}, and let A. be a decreasing sequence of
subsets of I8" not containing . Suppose that each set Ak is a deleted fine neighborhood of c [has fine limit point ]. Then there is a set A with the property that for each k the part of A in a sufficiently small neighborhood of 1; is in Ak and that A is a deleted fine neighborhood of [has fine limit point f].
It can be assumed that is finite. Define B(S) = S). Suppose that each set Ak is a deleted fine neighborhood of , and define Fk = B(1) (Ak u The set Fk is thin at , and to show one part of the lemma, it is sufficient to show that there is a set F, thin at , with the property that for each k the part of Fk in a sufficiently small neighborhood of is a subset of F. If Lot all but a finite number of values of k the point is not a limit point of Fk, we can take Fas the union of those sets Fk for which is a limit point of Fk. Otherwise, it is no restriction to assume that is a limit point of Fk for all k, and we can apply Theorem 2 to find a positive superharmonic function uk on B(1) with uk() < 2-k and with limit +oo at along Fk. The function u = Io uk is positive and superharmonic on B(1) with 2 and , with limit + oo at along each set Fk. Choose rk so that I = ro > rl > limk-. rk = 0, and so that u >_ k on Fk n B(rk). If F= Um (Fk n B(rk)), the o function u has limit + oo at along F; so F is thin at , and Fk n B(rk) c F as desired. To prove the second assertion of the lemma, observe that if each set Ak has as fine limit point, then
9. Fine Topology Limits and Euclidean Topology Limits (Continued)
1=
179
1 IAk.e(,) -a(.n({) r- o
for s > 0, according to Theorem 3 and Section VI.3(e). Hence it is possible
to choose r0 = I > r, > ... successively in such a way that
r=0
and that the smoothed reduction on the right with s = rk and r = rk+t is at least #. Define w
A= 0
so that Ak r (B(rk) - B(rk+t)) c A. Then U 1 oA^&'k)() Z #. If follows (by Theorem 3) that A is not thin at , and since A n B(rk) c Ak, the proof is complete.
9. Fine Topology Limits and Euclidean Topology Limits (Continued) If u is a function from a deleted Euclidean topology neighborhood of a point of R' into a topological space S, the fine cluster set of u at is defined as n u(B)-, where B ranges through the class of deleted fine neighborhoods of . In particular, if S is metric a is a fine cluster value of u at if and only if f lim infq_, dist (u(rl), a) = 0.
Theorem. A function u from a deleted Euclidean neighborhood of into a metric space has fine limit [fire cluster value] a at if and only if u has limit a at along a subset of R' that is a deleted fine neighborhood of [that is not
thin at ]. These conditions are trivially sufficient. Conversely, if u has fine limit a at , that is, if for n > 0 the set of points
A. _ q: dist (u(q), a) < n} is a deleted fine neighborhood of , the set A of Lemma 8 is a set along which u has limit at at . The cluster value assertion is treated similarly.
If the domain of u merely has as a fine limit point, the cluster value definition and Theorem 9 have obvious rephrasings. Continuity of a Superharmonic Function
If v is a function defined and superharmonic on an open neighborhood of a point and if v has associated measure v, the function v is lower semi-
180
1. XI. The Fine Topology
continuous by definition. In addition the following properties have now been proved or are now trivial. (a) The function v is fine continuous at , and therefore (by Theorem 9) there is a fine neighborhood A of such that VIA has (Euclidean topology) limit at . (b) If v(c) = + oo, then by lower semicontinuity v is continuous at , but v cannot become infinite too fast. More precisely (Theorem 4), there is a has a (Euclidean fine neighborhood A of with the property that
topology) finite limit v({c}) at . If D is a Greenian set containing , G here can be replaced by GD.
10. Identification of Af in Terms of a Special Function u" Lemma. If D is a Greenian subset of U8^', there is abounded continuous potential
u' = GDu with the property that, for every subset A of D,
(reduction relative to D).
Let B. be a sequence of balls with closures in D, forming a basis for the topology of D, and define W
uE2-n01Is
0
The summands are continuous potentials because 010k" =
on
D - B. according to Section VIII.10, and the points of 8B are regular boundary points of D - B,,. The function u' is therefore a bounded continuous potential on D and if A c D
0
If E Af n D, the left side of (10.2) is trivially u'(S) (Section 3). If E D - Af and if e B,,, then 01 Q8.Q A( ) is arbitrarily small for small B. [Section 3(a")]. Hence the right side of (10.2) is strictly less than as was to be proved.
11. Quasi-Lindelof Property An arbitrary union of open subsets of a second countable Hausdorff space is equal to some countable subunion (Lindelof property). The fine topology has a slightly weaker property (quasi-Lindelof property) as follows.
181
12. Regularity in Terms of the Fine Topology
Theorem. An arbitrary union of fine-open subsets of R" differs by a polar set from some countable subunion.
We shall prove the corresponding complementary assertion : an arbitrary intersection n7E,A7 = A of fine-closed subsets of R" differs by a polar set from some countable subintersection. It is sufficient to prove this assertion for subsets of some Greenian set D. Reductions below are relative to D. Let u' be a superharmonic potential on D with the properties described in
Lemma 10. According to the Fundamental Convergence Theorem, the family A. has a countable subfamily {A7, ate J) such that
rinfOu'i"al = (infOu'OA `aEf
261
+
+
To prove the theorem, we define B = n7E./ A. and prove that Bf = Af. Observe that
5
Qu"OB 7 8.1
+
7E1
5 U'
(11.1)
+
for a e 1, and we conclude that Bf c Aa c A. for a e 1. Hence Bf c A, and therefore (Bf)f = Bf c Af. Since A c B, it follows that Bf = Af ; so B - A is polar, as was to be proved. Application to the Approximation of a Fine-Open Set by (Euclidean Topology) Compact Subsets
If A is a fine-open subset of R" to each point of A corresponds (Sectionl), a Euclidean topology compact subset of A which is a fine neighborhood of the point. The union of these compact sets covers A, and therefore (quasiLindelof property) some countable subunion covers A up to a polar set.
12. Regularity in Terms of the Fine Topology Theorem. A finite Euclidean boundary point C of a Greenian subset D of R"
(or the point oo if N = 2 and D is unbounded) is regular if and only if
e(R"-D)f.
Recall that if N > 2, the point oo is a regular boundary point of every unbounded open set. Since the case of an infinite boundary point when N = 2 can be reduced to that of a finite boundary point by an inversion, only finite boundary points will be considered in the following proof. Suppose then that t; is a finite Euclidean boundary point of D, and let B be a ball of center C, so small if N = 2 that B u D is still Greenian. According to Section VIII. 10, the restriction to D of Q 1 Q B-D (reduction relative to B v D)
182
I.M. The Fine Topology
B n aD). If C is a regular boundary point of D, it follows that
is
this smoothed reduction has limit I at C on approach along D. Moreover 91 B-D = I quasi everywhere on B - D; so [111-D has limit I on approach to C excluding a polar set. Hence (fine continuity of the smoothed reduction) 8-D(C) = 1. This smoothed reduction is majorized by the smoothed reducQ1
tion on B - D relative to a Greenian superset D, of B u D; so the latter smoothed reduction is also I at C. Since this is true for arbitrarily small B, it follows (from Theorem 3) that C e (R" - D)f . Conversely, if ( e (R" - D)f, then 0 I o8-D(C) = l (reduction relative to B u D) by fine continuity; so the lower semicontinuity of superharmonic functions implies that this smoothed B n OD) has limit I at C for every reduction is continuous at C. Hence B, and this condition implies regularity of C (Section VIII.8).
Observation (a). Regularity of a finite Euclidean boundary point C requires only that C be in (R" - D)f but quasi every finite point of OD is not only regular and therefore in (R' - D)f but is even in (OD)f (Section 6).
Observation (b). Theorem 12 shows that every condition that a point be a fine limit point of a set is a condition for Euclidean boundary point regularity. For example (Theorem 3), a finite Euclidean boundary point C of D is regular if and only if whenever D' is a Greenian superset of D containing C. it follows that Ga .°({, ) = GD.(C, ). In fact, since regularity is a local property, D' need not be a superset of D (but of course must contain 0).
It is natural as a counterpart of Theorem 12 to investigate the set of points of aD in D! The basic result is the following.
13. The Euclidean Boundary Set of Thinness of a Greenian Set Theorem. A Greenian subset D of RN` is thin at p° almost no finite Euclidean boundary point of D.
If N>2define D,=R";if N=2defineD,=DuB,where Bis aball with center a finite boundary point of D and is so small that D U B is Greenian. It is sufficient to prove that D is thin at pD almost no Euclidean boundary point of D in D1. Let ul be the positive superharmonic function described in Lemma 10 but associated with D, rather than D, and define u = Qu'Q° (reduction relative to D,). Also define u = u = 0 on OD n OD,. To prove the theorem, it is sufficient to prove that u = u at PD almost every point of D, n OD, equivalently, since u 5 u , to prove that u) = U 0) on D. Now (by Theorem V.11) the Riesz measures associated with u and
u vanish on polar sets because u and u are bounded; so [by Theorem VIII.18(c)]
u) = GMDU and PD(', u0) = GMDUL' on D. These harmonic
minorants are equal because u = u+ on D; so the harmonic averages are also equal, as was to be proved.
183
15. Characterization of 0µQ A
14. The Support of a Swept Measure Theorem. Let D be a Greenian subset of R N, let A be a subset of D, and let be a point of D. (a)
If l; eAJ then bD(,
(b)
If µ is a measure on D, then the measure 8µQ" [in particular, the measure l (, )] is supported by Af n D. If A is a Borel set, if A is fine dense in itself, and if µ is a measure
(c)
supported by A, then y = Dµ0" (d)
If v is a positive superharmonic function on D, the Riesz measure associated with Dv0" is supported by Af n D.
Observation. We shall sharpen (b) in Section 18 by proving that a supported by the fine boundary of A relative to D whenever e D - Af.
is
Proof. (a) If a Af then bD(, { }) = I according to Theorem 3(b). (b) If u is a positive superharmonic function on D, then Du 'D" = J Du'D"0" =
(14.1)
so if u' has the properties described in Lemma 10, the set
{u'>Du'D"}=D-Af is aD(, ) null for every ; that is, SD( , ) is supported by AfnD. The evaluation of Dµ0" in X(5.2) shows that this swept measure is supported by
AfnD. (c)
The evaluation of 0µD" in X(5.2) yields (b).
(d)
If v" is the Riesz measure associated with OvO", then VA = OVA DA
according to Section X.5; so (d) follows from (b).
o
15. Characterization of D be a Greenian subset of RN, let A be a subset of D, and let µ be a measure on D with superharmonic potential GDµ. Then the swept
measure Oµ0A is characterized uniquely by the following properties:
(a) (b)
DµO" is supported by AfnD. GoDpO" = GDµ on Af n D.
We have already proved that Oµ0" has these two properties. Conversely, if a measure v on D has these properties, then v = OvO" by Theorem 14; so since is supported by Af n D,
184
l.Xl. The Fine Topology
/ 5D(.,GDV) = SD(.,GD,) = GDOPOA,
GDV =
and it follows that v = OµOA
Observation. This theorem extends and makes more precise the characterization of OµOA for A closed in D made in Section X.1. In comparing the two characterizations when A is closed in D, observe that in Theorem 15(b) the equation holds everywhere on Af n D if it is known to hold quasi everywhere on this set because Af is fine perfect.
16. A Special Reduction The following lemma is the first step in a delicate analysis of the support of a swept measure. Lemma. Let D be a Greenian subset of R', let h be a strictly positive harmonic function on D, and let u = GDp/h be an h-superharmonic h potential on D. Then for even' positive constant c (< + oo) hpunlusc) =
Observe that if v =
GDllv,lllla
U A C.
(16.1)
rephrasing of (16.1) is
P)vsch)
=
V A (ch),
(16.1')
valid for every superharmonic potential v. To avoid trivialities, it will be assumed in the proof of (16.1) that 0 < c < +oo. It is sufficient to prove equality of the first and third terms in (16.1) because this equality with c(1 - 1 /n) instead of c yields equality of the second and third terms when n + oo. Now hlUplusc) = h]U A Cp)uc) S U A C.
(16.2)
Furthermore there is equality in (16.2) quasi everywhere on the closed in D set {u S c}, which is a support of the measures associated with the finitevalued h-potentials in (16.2). Hence (by the domination principle) there is equality in (16.2); so the lemma is true.
17. The Fine Interior of a Set of Constancy of a Superharmonic Function Corollary. Let D be an open subset of R', let v be a superharmonic function on D with associated Riesz measure v, and let c be a constant. Then the fuze interior of the set {v = c} is v null.
18. The Support of a Swept Measure (Continuation of Section 14)
185
This corollary is a strengthening of the result that the Euclidean interior of the set {v = c} is v null because v is harmonic on this interior, and the corollary suggests that Euclidean open sets and fine-open sets play similar roles relative to superharmonic functions. This idea will be discussed further in Section 19.
Since the corollary is local, it is no restriction in its proof to assume that D is a ball on which v is lower bounded. Without loss of generality we can then assume that v is a potential, v = GDv, because we can first add a constant to v to make v positive and then replace v by its smoothed reduction on a strictly smaller ball than D, concentric with D. The resulting function v is a potential on the ball D and on the smaller ball differs by a constant from the original function. To prove the corollary in this context, let A be the fine interior of the set {v = c}. Lemma 16 applied to the function v A C yields Dv A cOt""1 = v A C. Since (by Theorem 14) the Riesz measure vv associated with this smoothed reduction is supported by D n {v < cy, which
contains no point of A, it follows that vv(A) = 0. Next apply Lemma 16 to derive V A C,
QvjtaS`f = VC,
(17.1)
so that
0 = vv(A) = J
A)
(17.2)
D
Now A c {v < c}f, and therefore [by Theorem 3(b)] the integrand in (17.2) is 1 when e A ; so "(A) = 0, as was to be proved. Pseudogeneralization of the Corollary to h-Harmonic Functions
If h is a strictly positive harmonic function on D, the function u = v/h is h-superharmonic, and the fine interior of the set {u = c} = {v - ch = 0} is v null according to the corollary because the superharmonic function v - ch has the same associated measure as v. Thus the corollary is not effectively more general when stated for h-superharmonic functions.
18. The Support of a Swept Measure (Continuation of Section 14) If A is a subset of RN, we denote the set of fine limit points of A, the fine interior of A, the fine boundary of A, and the Euclidean boundary of A by Af, Af', afA, and 8A, respectively, we leave to the reader the proof that afAf c 8f (A . Af) c VA. Recall from Section 6that Af, Af', and afAf are Borel sets.
1.X1. The Fine Topology
186
is supported by Af n D According to Theorem 14, the measure in D, and if v is a positive superharmonic function on D, the Riesz measure associated with the smoothed reduction DvCA = 6D'(', V) is also supported by Af n D. The following theorem sharpens the more elementary result (given in Section X.1) that is supported by D n aA when e D - A by putting the latter result into the context of the fine topology and thereby also sharpens Theorem 14. for all
Theorem. Let D be a Greenian subset of R v, and let A be a subset of D.
1. If e D - AJ the measure
(a)
If e D n Af, then
(b)
is supported by D n afAf. If v is a positive superharmonic function on D, the Riesz measure vA
associated with [VIA is supported by D n Af. In particular, if v is harmonic on a Euclidean open superset of A, the measure vA is supported by D n afAf.
Proof of (a). The first assertion of (a) is Theorem 3(b). To prove the second assertion, let be a point of D - Af, and observe that the restriction v to Dis superharmonic with associated of the function Riesz measure the restriction of to the class of measurable subThe set {v = 0) is a Borel superset of the set Af n (D sets of D -
and it follows from Corollary 17 that the fine interior of this superset is 5 null. Since it is already known (Theorem 14) that SD(,) is supported by D n Af, it follows that this measure is supported by D n afAf, as was to be proved.
Proof of (b). The first assertion of (b) is included in Theorem 14(c). To prove the second assertion, observe that if v is harmonic on the open superset B of A, then the set { pvQA = v} n B is the set on which the function IvIA - v, superharmonic on B, vanishes. The fine interior of this set includes Af' and is null for the Riesz measure associated with CvDA - v on B. Hence vA is supported by D n afAf.
Extension of Corollary 17. If u, and u, are superharmonic functions on an open subset of R', if p, is the Riesz measure associated with u,, and if A = {u, = u2 }, then p, = µ, on the Borel subsets of the fine interior of A. Since this extension of Corollary 17 (like the corollary) is local, it is no restriction in the proof to assume that u, and u2 are defined on a ball D on which u, and u2 are lower bounded and even (after addition of a sufficiently large constant) positive. Thus we shall suppose that u, is a positive super)ujDD-'(x) = harmonic function on a ball D. Now for quasi every point of D - A and therefore (fine continuity f superharmonic functions) also for in D and a fine boundary point of D - A, that is, o\
187
20. A Generalized Reduction
a fine boundary point of A. For
a fine interior point of A the measure
aD-"(t, ) is supported by the trace in D of the fine boundary of A, on which u1 = U2; so Du10D-" + u2 = u1 + lu2OD-" Since (by Theorem 18) the measure associated with a reduction on D - A vanishes on Borel subsets of the fine interior of A, we conclude that pl = p2 on these subsets.
19. Superharmonic Functions on Fine-Open Sets Let D be a Greenian subset of R' and let B be an open relatively compact subset of D. If e B, Theorem 13 implies that the measure is supported by MB. If v is a superharmonic function on D then v() ;->
bD-B(,V),
(19.1)
and there is equality if v is harmonic. Suppose now that B more generally is a fine-open relatively compact subset of D and that e B. Then is no longer defined, but (by Theorem X.5) if v is a positive superharmonic function on D, -f
SD-1(S' V).
(19.2)
It follows easily that v() z
6DD-8( ,v)
(19.3)
whenever v is a superharmonic function on D. In particular, there is equality in (19.3) when v is harmonic because (19.3) in then true for both v and - v. is supported by D r M(D - B) = MB, Observe that the measure according to Theorem 18. Thus the superharmonic function inequality and the harmonic function equality can be generalized to be valid on fine-open sets, and this suggests that much of classical potential theory can be extended by replacing Euclidean open sets by fine-open sets. The next step is to develop a fine potential theory based on generalizations of superharmonic and harmonic functions to fine superharmonic and harmonic functions, defined on fine-open sets. We omit further discussion in this direction.
20. A Generalized Reduction If D is a Greenian subset of R", if A is a subset of D, and if v is a positive superharmonic function on D, we have defined the reduction Rv relative to D as the infimum of the class of positive superharmonic functions on D
188
I. XI. The Fine Topology
majorizing v on A, equivalently, as the infimum of the class of superharmonic
functions on D majorizing v1A on D. In this section we replace VIA by a function on D that is arbitrary except for boundedness and smoothness conditions. Let D be a Greenian subset R', let f be a locally lower bounded function from D into A majorized by some superharmonic function on D, and let Rf be the smoothed infimum of the class of superharmonic functions on D majorizing f. According to the Fundamental Convergence Theorem, Rf is superharmonic and Rf z f quasi everywhere on D; so Rf is the infmum of the class of superharmonic functions on D majorizingf quasi everywhere on D. Let u f be the Riesz measure associated with R.. Recall that when C is a subset of R', the notation Cf refers to the set of fine limit points of C. Theorem. In the above context, (a)
For c > 0 the measure p f is supported by the set
{Rf= +oo}u n {f5Rf5f+e< +oo}f. L>o
(b)
If f is fine upper semicontinuous, the measure M f is supported by the
set {Rf=f}f. This theorem generalizes Theorem 18(b). In fact, suppose that A is a subset of D, nonpolar in each open connected component of D to avoid trivialities, and suppose that v is a strictly positive superharmonic function on D. Define f = 1,,v; so Rf = Rr , and suppose first that A is fine closed in D. Then f is fine upper semicontinuous, and according to the present theorem,
pf is supported by the set {Rf =J}1 and so is supported by {Rf =1) because this set is fine closed in D and so is supported by Af because Rf >f = 0 on D - A. If A is not fine closed in D, we can replace A here by the set Af n D, fine closed in D. because by fine continuity of superharmonic functions any superharmonic function majorizing v on A majorizes v on Af n D; so Rf and R" are not changed by this set change. The result for fine-closed sets shows that µ f is supported by Af n D as stated in Theorem 18(b). Proof (a)
To prove Theorem 20, suppose first that f z 0, define
Ata={f 0, and observe that the function
1+eaRf+l+rbRf
A
189
20. A Generalized Reduction
is superharmonic on D, is majorized by Rf, is equal to Rf quasi everywhere
on Ath, and majorizes f quasi everywhere on D - A,, because the first summand does. Hence this superharmonic function coincides with Rf; so Rf =Q Rf D A . According to Theorem 18, it follows that RJ is supported by A for all strictly positive a and S. Now
up to a polar set; so
1RfZ- I
f-{RJZ},
and when b - 0, the set on the right decreases to { Rf = + oo }. Thus Theorem 20(a) has now been proved for f z 0.
To reduce the general case to the case of positive f, observe that if B is a compact subset of D and if Do is an open neighborhood of B, relatively compact in D, then the restriction of Rf to Do is the smoothed infimum of the class of superharmonic functions majorizing R f on Do - B and majorizing f on B. Without loss of generality in the discussion of of we can suppose, adding a constant to f if necessary, that the restriction to Do off is positive. It now follows from what we have already proved that for every B the projection of u f on B is supported by Af,6 n Do for all strictly positive E and S; so µf is supported by A' for all strictly positive a and 6, and the rest of (a) follows as in the positive case. (b) If f is fine upper semicontinuous, the set
{Rf Sf+e < +oo}v{fZ E}
differs from A. by a polar set and is fine closed in D; so
Au=AucAu, and part (a) of the theorem implies, when c - 0, that the measure µf is supported by the union of {Rf =A and the polar set {R 0, fPD(S, {S}) !5 fUD(S, ev') -5 lim
inf S-{
ev') 5 ev(O.
193
22. The Limit Harmonic Measure JµD
Here we have used the fact that v" = v' off the harmonic measure null set {C} and that v is in the upper PWB class on D for the boundary function ev'. Inequality (22.6) implies that f1D(C, {C}) = 0, as asserted. From now on when u is a function in D, and Be f, the averages pa(C, u) and fµ8(0, u) will be used under the convention that in these averages u(PI) is to be taken as lim infs j4.,u(C) when q e aB n aD. If u is superharmonic,
this convention means that the function to be integrated on aB is lower semicontinuous. (c) If u is a lower-bounded superharmonic function on D and if B, and
B2 are in f with B, c B2, then f u(C) ? fAB' (C, u) Z fµs,(C, u),
(22.7)
and fu(C) = sup {µ8(C, u): Be f}.
Recall the convention just made for u on aB, n aD, and observe that (from Section VIII.8) u z µs u) on B, . (Here are below the convention on the values of u assigned at boundary points of the sets involved may not be the same as those in referenced inequalities, but the differences are in favorable directions.) On approach to C along a suitable deleted fine neighborhood of C this inequality yields the first inequality in (22.7). Similarly u) on D; so (22.5) with f replaced by u yields the inequality rILD(C, u) uZ
5 fµ8(0, u) and thereby yields the second inequality in (22.7) if the pair B,
D is replaced by the pair B1, B2. Finally, if B= {CeD: u(C) > a} with a < fu(C), then fµ8(C, u) z a; so the last assertion of (c) is true. It is trivial that the supremum in this assertion is not decreased by the additional condition on B that aB n aD = {C}. is lower semicontinuous on the space of posi(d) The function tive superharmonic functions on D in the topology of pointwise convergence. In fact, if u, is a convergent sequence in this space, with limit u, and if a < fu(C), choose B in r in such a way that aB n OD = {C} and that fue(C, u) > a. Then apply Fatou's lemma to obtain lim inffun(C) z lim inf fu8(C, un) Z fus(C, u) > a. ft-.0
n-m
This inequality implies the stated lower semicontinuity. (e) By definition of regularity, if q is a regular boundary point of D, then limC . µD(C, ) in the sense of vague convergence is the unit measure supported by {q}, whereas if q is an irregular boundary point of D, then ) = fuD(q, ). That is, in terms of Dirichlet solutions if f is a f lim finite-valued continuous function on OD, then limt.,n Hf(C) = f(q) if q is a regular boundary point, but f lim4-,, Hf(C) = 'PAID if q is an irregular boundary point. (f) If C as above is an irregular boundary point of D and if q e D, then according to Theorem 21, the function has a fine limit at C; we
denote this fine limit by fGD(C, q). In view of the extension Go of GD defined in Section VII.4, not only does this fine limit exist, but
1 X1. The Fine Topology
194
IGD(C, 1) = lira sup GD(b, p), S»;
(22.8)
and VIII(18.2) implies that IGD(C, ) is harmonic on D, as is also easily proved
directly. When D is connected, the superior limit in (22.8) must be strictly positive; that is, fGD(C, ) must be strictly positive, in view of the criterion [Section VIII.14, Application (c)] in terms of GD that a boundary point of D be irregular.
23. Extension of the Domination Principle Recall (from Section 4) that if h and v are strictly positive superharmonic functions on a Greenian subset D of R' with respective associated measures vi, and v, if u = v/h wherever either h or v is finite valued, and if u*(C) is defined as the fine limit of u at C wherever this fine limit exists, then u* is defined quasi everywhere and vk + v almost everywhere on D. The following extension of the domination principle (Theorem V.10), unlike Theorem V.10, does not suppose that polar sets are v,, null. 't'heorem. Let h = GDvi, be a potential on a Greenian subset D of R", and let v be a positive superharmonic function on D. Then h 5 v if (fine limit function notation) (v/h)* z I vA almost everywhere. To prove this theorem, which obviously includes Theorem V. 10, let B be a Borel support of vi, on which (v/h)* is defined and z 1. If 0 < c < 1, the set A = {v z ch} is a deleted fine neighborhood of every point of B; so B c Af, and therefore bD(C, {C}) = I when C e B. Hence v z 11v0" z cDhD" = cGD(vi,bn) = ch;
so vzh.
Chapter XII
The Martin Boundary
1. Motivation Let D be an open subset of R'. If D is a ball, its Euclidean boundary is so well adapted to it from a potential theoretic point of view that the following statements are true. (a) The class of minimal harmonic functions on D is in a one-to-one correspondence with the Euclidean boundary points, q a K(ij, ) (see Section 11.16), and every positive harmonic function u on D is representable as an integral J8D K(ry, )M (dq) with M. a uniquely defined measure on the ball boundary. The lattices of positive harmonic functions on D and of measures on aD are thereby in a one-to-one order-preserving correspondence, and the vector lattices of differences between positive harmonic functions on the ball and signed measures on the ball boundary are linearly and order isomorphic. (b) The Euclidean boundary of the ball is universally resolutive and universally internally resolutive (Section IX.12). (c) Every boundary point is regular, so that has limit 0 at every boundary point. Moreover (up to a multiplicative constant) the normal derivative of at the boundary is the density relative to /N_, of harmonic measure. (d) The Fatou theorem and its generalizations (Section 11.15) assign to each positive h-harmonic function, and therefore to each h-harmonic function in L'(µo_) on the ball, a boundary function. In particular, the Fatou boundary function of a PWB" solution Hf is equal M almost everywhere to f The Martin boundary of a Greenian subset D of R" is defined in such a way that it has some of these properties. Observe, however, that other compactifications may be better adapted to other properties. For example, the Kuramochi boundary is specially well adapted to the second boundary value problem. The only restriction to be imposed on D is that it be Greenian and connected. It will be necessary to generalize two concepts playing a central
role in the ball case: special approach to the boundary (radial or non-
196
1.Xtl. The Martin Boundary
tangential) and the normal derivative of the Green function at the boundary. The fine topology will be extended to the Martin boundary, and approach
to a Martin boundary point in this topology will replace both radial and nontangential approach to the boundary in the ball case. If D = B(0, r) and if I e D, then const. (r - j') when j>ij --i r so that if v is a measure on D with compact support, const. (r - Iryf) near the boundary. This fact suggests that the normal derivative of GD(i;, ) at the boundary point C should be replaced by h) lim rc GDv(ry)
when c is a Martin boundary point, if this limit exists, for some convenient measure v. It was Martin's idea to define a boundary in such a way that this
limit does exist and in fact to use the existence of this limit to define a boundary.
Topological Conventions
Recall that throughout this book in dealing with R" and its subsets, if no topology is specified, the topology of RN is to be understood as the usual topology, sometimes identified as the Euclidean topology, compactified by a point at infinity when boundaries of unbounded sets are in question.
2. The Martin Functions Let D be a connected Greenian subset of R', let v be a not identically 0 measure on D with compact support a, c D, and define K,(17,
) = GD(n, ) GDV(q)
for , and
in D except that is left undefined when co. The function K, will be called the Martin function based on v, and K will sometimes be written for K, when the choice of v is irrelevant. If v is the probability measure supported by a singleton so that K,(rl, GD(rl, )/GD(rl, a), the Martin function will be said to be based on o. For h a point of finiteness of Gov the function K,(r1, ) is a strictly positive
harmonic function on D - {, }, and for in D the function K is continuous on D - a,. In view of the properties of GD listed in Section VIL3, if A is a compact subset of D and if A, is a neighborhood of A u or, the functions K, and 1/K,, are bounded on (D - A,) x A. Moreover
197
3. The Martin Space
f K,(n, Ov(dd) = 1
(2.2)
,
if GDv(n) < + OC.
3. The Martin Space Theorem. Let D be a connected Greenian subset of RN. There is a unique up to homeomorphisms metrizable compactifcation DM of D with the following properties. (a) Each Martin function K,, on (D x D. (also denoted by to (DM (b) K,,(gl, ) = K,(n2, ) if and only if ni = n2
a continuous extension
The boundary aMD = DM - D obtained in this way will be called the Martin boundary, and any metric for DM compatible with its topology will be called a Martin metric. The function K. is now defined on its original domain augmented by 0MD x D. Observation. According to the theorem, if n, is a sequence of points of D with limit n in aMD and if K,, is a Martin function, then
K,,(n,,, ) =
K,(n, ) is a strictly positive harmonic function satisfying (2.2) because in any open relatively compact subset of D the above limit is the limit of a uniformly bounded sequence of positive harmonic functions satisfying (2.2). Moreover, because of (b) and the convergence remarks in Section 2, a sequence n, in D with no limit point in D converges to a boundary point if and only if lim,,..,, K,,(n,,, ) exists; the limit function is then K,(n, ) with n,,. Thus the existence assertion of the theorem implies that n= each point n of DM - a,, is associated with a function K,(n, ), harmonic on D if n E 3MD, harmonic except for an infinity at n otherwise, and the class of associated functions under the topology of pointwise convergence is homeomorphic with DM - a,. This homeomorphism implies the uniqueness of a compactification satisfying (a) and (b) above. Roughly, the Martin space DM is the compactified space of the set of Green functions {GD(n, ), n e D} normalized to be conditionally compact under pointwise convergence.
Observe that if K, and K are Martin functions, then Kti(n, ) J
[17 c- D - (a, u o,.)].
(3.1)
C)u(dZ) oN
Hence for n in D - (a u a,,) the function KM(n, ) associated with the point n and the Martin function K,, is a constant multiple of the function K,.(n, )
198
1.X11. The Martin Boundary
associated with q and K.. The choice of reference measure for a Martin function is a matter of convenience, although the most common choice is one with a, a singleton, that is, with K, based on a reference point.
Proof of theorem. Let K. be a Martin function, and suppose that v has been chosen to make Gov finite valued and continuous. Let f be a strictly positive continuous function on D, IN integrable over D, and define d,(C, q) =
[I A IK,(C, C) - K,(q, C)I)f(C)IN(dC).
(3.2)
fo
T he function d, is a metric for D, endowing D with the Euclidean topology: so if q, is a Cauchy sequence ford, and if some subsequence q,, has Euclidean
topology limit q in D, then lim, ,, d,(q, qa.) = 0. It follows that q, is convergent to q in the d, metric and therefore also in the Euclidean metric. If q, is a sequence of points in D and if no subsequence of q, has a Euclidean topology limit in D, then according to Section 2, if B is an open relatively compact subset of D, the sequence n i-. K,(,, ) on B is a uniformly bounded sequence of harmonic functions if finitely many values of n are omitted. It follows from the convergence properties of harmonic function sequences (Section 11.3) that the sequence n i-4 K,(q,,, -) has a subsequence j i -+ K,(q, , -)
converging locally uniformly on D to a harmonic function h. This subsequence is a Cauchy sequence for d,. In particular, if q, is a Cauchy sequence ford, and if the pair (C, q) in (3.2) is replaced by (q_, q,J ), we find when j - oo
that the integral
[ I n IK,(q,.,C) - h(OI1fC)lN(dC) fo tends to 0 when m -+ oo. We conclude that every convergent subsequence of the sequence n - K,(q,,, ) has limit h, so that lim, K,(q,,, ) = h locally uniformly on D. o This discussion shows that if D is completed in the d, metric, the resulting space DM is a compact metric space and that each point C adjoined to D in the completion can be identified with a positive harmonic function, which we denote by K,(C, ), identified by the fact that q -+ C in the DM metric if and only if K,(q, ) -+ K,(C, -) locally uniformly on D. Moreover, integrating to the limit shows that (2.2) is satisfied when q = C. The conditions (a) and (b) of Theorem 3 are satisfied by K, and therefore also satisfied by an arbitrary Martin function K,, in view of (3.1). EXAMPLE (Balls and Half-Spaces). Let D = B(0, S), and let v be a unit mea-
sure supported by the origin. Then K,(q, 0) = I for q 0 0 and
4 Preliminary Representations of Positive Harmonic Functions and Their Reductions
S-Inl G.(n, 0)
199
ifN=2 (InI-'b),
N-2
SN_, (S - Inl) if N > 2
so that when n tends to { on the Euclidean boundary, the value K,(n, ) 2)6'-N if N > 2] of the inner normal tends to a multiple [S-t if N = 2, (N at C, and the Martin function K becomes the Poisson derivative of kernel density denoted by K in Section 11. 1. Thus the Martin boundary of a ball is the Euclidean boundary. Similarly, if D is a half-space, the Martin function based on a suitably chosen reference point becomes the Poisson kernel density denoted by K in Section VIII.9, and again the Martin boundary is the Euclidean boundary. Alternatively the Martin boundary of a halfspace can be derived from that of a ball since a ball can be mapped onto a half-space by an inversion. The same reasoning, based on the Riemann mapping theorem, shows that the Martin boundary of any plane Jordan
domain is its Euclidean boundary and that more generally the Martin boundary of an arbitrary simply connected plane Greenian open set other than 682 is the Caratheodory prime end boundary.
4. Preliminary Representations of Positive Harmonic Functions and Their Reductions Lemma. Let D be a Greenian subset of 68" with Martin function K,. If u is a positive harmonic function on D and if A is a subset of D with (8A) n ar = 0, on DM, supported by the boundary of A relative to DM, there is a measure for which (reduction relative to D) KV({,
DUD" =
Z. t(DM) =
J DM
In particular, there is a measure A. (=
u=f
K,,(C,)
(d{),
lu8" dv.
(4.1)
fD
supported by aMD for which
.(aMD) =
"MD
udv.
(4.2)
JD
A measure A. on 3MD satisfying (4.2) must also satisfy
OKr(t,')O'(4)
ouOF =
J aMD
for every Bore! subset F of 61D.
(4.3)
200
l.Xlt. The Martin Boundary
Observation. In general the measure J.,, is not uniquely determined by u in (4.2), but it will be seen in Section 9 that A. can be chosen to be supported
by the set of minimal Martin boundary points (to be defined in Section 5) and that if so chosen, A. is uniquely determined. If A is relatively compact in D, the function Quo" is a potential, say GDA', A' is supported by OA, and (4.1) is true with A(dC) = GDv(C);i (d1). If A is not relatively compact in D, let B. be an increasing sequence of open subsets
of D, relatively compact in D and with union D, so large that o c B0. According to what has just been proved, the potential Ju6B"^" can be represented in the form
Me""" =
K,. ((,)A(d(),
(4.4)
JOM
where A. is a measure supported by a(B" n A) and (4.5) JDM
The sequence A is a bounded sequence of measures on the compact space DM. If A. is the limit of a vaguely convergent subsequence, it is trivial that
i is supported by the boundary of A relative to DM and that (4.4) yields (4.1).
We shall only need (4.3) when Fis compact, but this equation has general interest and so we prove it as stated. In view of the properties of the reduction
operation the class of sets F satisfying (4.3) includes limits of monotone sequences of its members; so it is sufficient to prove (4.3) for compact boundary subsets F. If F is such a set, let F; be a decreasing sequence of compact neighborhoods of Fin DM with intersection F, and set F. = D n F,. Then [from Section 111.5(a)] lim, urr" = our. Now bn"(', u) = J
KY((,
(4.6)
and (4.6) yields (4.3) when n -+ co.
5. Minimal Harmonic Functions and Their Poles Let D be a Greenian subset of IB" with Martin function K (= K,,). A Martin
boundary point (is called minimal if its associated function K({, ) is a minimal harmonic function for one and therefore for every choice of the Martin function K. The set of minimal boundary points is called the minimal Martin boundary and is denoted by 8;"D. In the following theorem K is an
arbitrary Martin function for D. Recall from Section X.7 that a Martin boundary point C is said to be a pole of a positive harmonic function u if
6. Extension of Lemma 4 OuO141
201
= u and that then C is also a pole of every positive harmonic minorant
of U.
7lteorem. (a) Every ( 0) minimal harmonic function on D has a unique pole on aMD. If a Marlin boundary point C is the pole of a positive (0 0) harmonic function u, then C is the only pole of u, u = const. K(C, ), and C is a minimal boundary point. In particular, if n is a minimal Martin boundary point, the function K(q, ) has pole p. (b) If C is a minimal Martin boundary point and if A is a set of minimal Martin boundary points, then O K(C, ) 0A is either K(C, ) or 0 according as CeAorCOA.
Proof of (a). According to Section X.7, whatever the boundary assigned to D by a metric compactification, a minimal harmonic function on D has at least one pole. Now let u be a strictly positive harmonic function on D with pole C. We can assume in proving (a) that K = K and that u has been normalized so that Jo u dv = 1. Apply Lemma 4 with A the trace on D of a neighborhood of C to derive
u = HA = J
K(7,')2UA(d4),
AUA(DM) = 1.
(5.1)
DM
The measure 2gA is supported by the boundary of A relative to DM ; so tends (vague convergence) to the when A shrinks to C, the measure probability measure supported by the singleton {C}, and therefore (5.1) becomes u = K(C,'). Thus C is uniquely determined by u. Moreover, as already recalled at the beginning of this section, if v is a positive harmonic minorant of u, then v also has pole C. Hence v = const. K(C,'), and therefore
u is a minimal function; that is, C is a minimal boundary point. Finally, if PI is a minimal Martin boundary point, then according to what we have just proved, K(q, ) has a pole C and K(rl, ) = const. K(C, ). The constant
must be l;soq=C. o Proof of (b). If C is a minimal Martin boundary point, the function K(C, ) has pole C according to (a) ; so surely 8 K(C, ) IA = K(C, -)if Cc-A. Furthermore
the set B = 8MD - {C} is open in aMD and contains no pole of K(C, ); so (from Section X.7) Q K({, ) UB = 0, and therefore O K(C, ) QA - 0 if A (-- B.
o
6. Extension of Lemma 4 Lemma. (Context of Lemma 4). If F is a compact subset of aMD, there is a measure A. supported by F such that BUOF =
JK(c.)A(dc) = IF
(6.1)
202
LXII. The Martin Boundary
Let F; be a decreasing sequence of compact neighborhoods of F in DM with intersection F, and set F. = D n F,;. Then [Section III.5(a)] lim, OuOF" = and according to Lemma 4, there is a measure Y. supported by the boundary of F. relative to DM such that µ"(D" ') 0 and R" = K(4, ), in which case c = 0 and R" +K((, ) +K({. )
c=Iandv=0.
Step 2. Proof of (c2). Without loss of generality we can assume that B is so small that the compact support of the measure on which K is based does not meet B. According to Lemma 4, there is a measure A on DM, supported by a(D - B) (boundary relative to DM), such that RD-B =
fDM
J
K(n, )A(dp).
If there were equality in (c2), the integral would define a harmonic function on D; so the measure A would be supported by OmD. However, according to Section 9, such an integral representation of a minimal harmonic function K(C, ) is possible only if A is supported by {C}, contrary to the definition of B. Hence there cannot be equality in (c2). Step 3. Proof that (a 1) ca (a2). The implication (a2) (a 1) is trivial. To
prove the reverse implication observe that if (al) is true and (a2) is false, then R"^B is a potential for sufficiently small B by Step 1, and R'-' is a potential according to Step 2, because this smoothed reduction is a positive superharmonic function majorized by the potential R'-,'. Hence by set subadditivity of reductions is a potential, contrary to hypothesis. Step 4. Proof of (c 1). Assertion (c 1) is trivially true when B = D and therefore true for arbitrary B by (a), which we have just proved. Alternatively (cl) is true because [by Theorem 5(a)] C is a pole of K(C, ).
Step S. Proof that (bl) (b2). The function Rg(GB) is harmonic and positive on D - B. Let B shrink to C, say along a sequence of balls of center C and radii tending to 0 (in terms of some Martin space metric). Then the limit of the corresponding sequence of reductions in (b2) is the indicated infimum and is a positive harmonic function, majorized by RK"(S ). Since this smoothed reduction is a potential by Step 1, the harmonic function vanishes identically, as was to be proved. Step 6. Proof that (b2) . (b3) . (b1). These implications follow trivially from Step 1 and the equivalence of (al) and (a2).
210
1 X[[. The Martin Boundary
The proof of the theorem is now complete, and we turn to the definition of the minimal-fine topology of DM.
12. The Minimal-Fine Topology Let D be a connected Greenian subset of R", let K be a Martin function for D, and let C be a minimal Martin boundary point. The class MT(O of subsets of D minimal thin at C has the following properties. Every subset of a set in MT(h) is itself in MT(C) (trivial). (P2) A finite union of sets in MT(C) is in MT(C), because the set function A )-4 is subadditive. (P3) If A E MT(C), then the union A' of A and its set of fine limit points in D, as defined in Section XT. 1, is in MT(h), because according to Section XI.7, (P1)
RA
(P4)
= RA' +K({..).
Every set in MT(C) has an open superset in MT(i;), because according to Section 111.5(e),
RK(i.) =inf{RK((.):Bopen, AcBcD). The Minimal-Fine Topology We define the minimal-fine topology of DM by the following conventions:
A point C of D is a minimal-fine limit point of a set A if A r D is not thin at C.
A point C of a'('fD is a minimal-fine limit point of a set A if A n D is not minimal thin at C.
Each nonminimal Martin boundary point is a minimal-fine isolated point of DM.
The minimal-fine topology of DM has as relative topology on D the fine topology already defined on D in Section XI. I. According to Theorem
1 l (c), if B is a Martin topology neighborhood of the minimal Martin boundary point C, then B n D is a deleted minimal-fine neighborhood of C, and D - B does not have C as a minimal-fine limit point. Thus the minimalfine topology of DM is a (Hausdorff) topology finer than the Martin topology of DM.
For some choices of D, for example, when D is a ball (Section 3). the Martin space DM can be identified with the Euclidean closure of D. Observe
that for such a choice of D if C is a boundary point and if A is a subset of
12. The Minimal-Fine Topology
211
D, then thinness of A at C in the fine topology of R' need not be equivalent
to minimal thinness of A at C. For example, if D = B(0, S), then every boundary point is a minimal Martin boundary point, and if B is a ball internally tangent to BD at C, then D - B is minimal thin at C but is not thin at C. In fact, in this case K(C, ) is a constant multiple of the function
62 - 112 1{
01
(Section 11. 16), and D - B is the locus of the inequality K(C, -) 5 c for some strictly positive constant c. Since we shall prove [equation (12.3)] that the minimal-fine limit of K(C, ) at C is + oo, the set D - B is minimal thin at C. The set D - B is not thin at C because it contains the trace on a neighborhood of C of an open cone with vertex C (see Section XI.3). We shall use the notation "mf lim" for minimal-fine limits. EXAMPLE (a). Let D be a Greenian subset of RN, let C be a point of D, and
define Do = D - {C}. Then (from Section VII.1) GD° is the restriction of GD to Do x Do, and (from Section VII.10) the restriction of GD(C, ) to Do is a minimal harmonic function on Do. We conclude that the point C can be identified with a minimal Martin boundary point of Do, the pole of the restriction of GD(C, ) to Do. The Martin topology of Do coincides on D with the Euclidean and Martin topologies of D. Thus the Martin space Do can be identified with D". Finally, the minimal-thin topology of Do on a Martin neighborhood of C is identical with the fine topology of DM and of R" on that set. Hence minimal-fine limit concepts on Do at C coincide
with fine limit concepts on D at C. More generally, a trivial refinement of this reasoning shows that if A is a closed relative to D polar subset of D
and if Do = D - A, the Martin space Do can be identified with D" by identifying each point C of A with a point of BMDo, the pole of the restriction
to Do of GD(C, ); minimal-fine limits on Do at C coincide with fine limits on D at C. Finally, suppose that v is a positive superharmonic function on D whose Riesz measure v is supported by A. The function v is harmonic on Do and thus has a Martin representation there in terms of a measure Mo° on O "Do = A v arD. Choose a Martin function K° for Do based on a point o in Do. Then it is clear that for C in A, K°(C,C)Mo°(dC) =
GD(C,Q
GD(4o, 0
Mo°(dC) = GD(4,Ov(dO,
so that Mo°(dC) = GD(40,C)v(dC) for C in A. It is important that Mo,, and v are mutually absolutely continuous on A. Minimal--Fine Limits at an Isolated Boundary Point. Example (a) implies that to each theorem on minimal-fine limits at a minimal Martin boundary
212
1.XII. The Martin Boundary
point of a Greenian set D corresponds a theorem on fine limits at a point of D. For example, the fact that (Section XI.1) a set thin at a point of D has an open superset thin at the point corresponds to the fact (P4) that a set minimal thin at a minimal Martin boundary point of D has an open superset minimal thin at the point. The fact that [Theorem XI.4(a)] if v is a positive superharmonic function on D and if C is in D, then v/GD(C, ) has fine limit the value infD_it, v/GD(C, -) at C corresponds to the fact that if Z is a minimal Martin boundary point of D and if K is a Martin function for D, then v/K(i;, -) has fine limit the value infD v/K(C, -) at C. The latter result is proved in Section 13, and a dual result is proved in Section 14. EXAMPLE (b). Denote by dt the Nth coordinate of the point
of R', and
define D = { : dc > 0}. Then (from Section 3) DM is the closure of D in the one-point compactification of RN ; so the Martin boundary is the Euclidean boundary. If u is a positive superharmonic function on D, if c > 0, if u, = u(/c), if A is a subset of D, and if cA has the obvious meaning,
In particular, if u() = L-Ndd, that is, if u is a minimal harmonic function on D with pole the origin, then uc = cN-'u and
then RA
RYA(D = C' _NRY C
If A is relatively compact in D and not polar, then Rr is a nonzero potential, and the evaluation of Go in Section VIII.9 shows that RA(cC) c 0, with ¢ a strictly positive finite-valued function. It follows that if c is an arbitrary sequence of strictly positive numbers with limit 0 and if B is the intersection of Uo (c"A) = A' with a Euclidean neighborhood of the origin, then RB
z RM"A(C) >
2
for sufficiently large n. Hence A' is not minimal thin at the origin.
A trivial example of the application of Example (b) shows that if N = 2, no initial segment of a ray from the origin into D is minimal thin at the origin.
An analogous argument shows that if C' is a boundary point of a disk D', then no initial segment of a ray from C' into D' is minimal thin at C'. In view of Example (b) if a function from the upper half-space of R' into a Hausdorff space has both a minimal-fine and a nontangential limit at a boundary point, the two limits must be the same. An analogous argument leads to the same conclusion if D is a ball.
213
14. Second Martin Boundary Counterpart of Theorem X1.4(c)
13. First Martin Boundary Counterpart of Theorem XI.4(c) and (d) Theorem. Let D be a connected Greenian subset of R', let K be a Martin function for D, and let C be a minimal Martin boundary point of D. (a)
If v is a positive superharmonic function on D, then v(7)
K((,7)
= 1Df
v
K((,')
= M-(")
)
(b) Let A be a subset of D with Martin topology limit point C. If A is minimal thin at C, there is a positive superharmonic function v on D for which lim v(7) A.o-K K(C,7)
- +cc.
(13.2)
Conversely, if there is a positive superharmonic function v on D satisfying (13.2), then A is minimal thin at C.
We shall see (in Section 19) that Theorem 13(a) is a special case of the Fatou boundary limit theorem for a Martin space. To prove Theorem 13, translate the proof of Theorem XI.4(c), (d) into the present context, replacing GD(C, ) in that proof by K(C, ) and "thin" by "minimal thin." See Example (a) in Section 12 for a discussion of the relation between theorems on limits at a minimal Martin boundary point and limits at a point of D. Application. If v is a potential in (a), we find that the minimal-fine limit is 0. If v =_ I in (a), we find that mf lim K(C, PI) = sup K(µ, ). q-c
(13.3)
D
14. Second Martin Boundary Counterpart of Theorem XI.4(c) Theorem. If D is a connected Greenian subset of RN, if v is a strictly positive superharmonic function on D, if C e BD, and if e D, then
0 < mf lim
v(7) = lim inf n-c 7)
v(7) S + oo. 7)
(14.1)
Observe that (14.1) is trivial if the indicated inferior limit is + oo. We can therefore ignore this case and prove the equality in (14.1) by showing that
214
1.X11. The Martin Boundary
whenever c is a finite number strictly larger than the inferior limit in (14.1), it follows that the set A = {n: v(n) z cGD(4, n)) is minimal thin at C. Let K be a Martin function for D based on the point . By definition of the smoothed reduction on A the inequality v z cOGD(4, .) DA is valid on D; so in view of sweeping symmetry, if n e D v(n) n)
GD((, h
(14.2)
n)'
and therefore since OK(n, ) r() = bo ((, K(n, -)), Fatou's lemma is applicable when n
C in (14.2) and yields
cOK(S, )OA( ) = cbn"( , K(C, ))
5 lim of Gn(i)n) < c.
(14.3)
I-C
I = K(i(, ); so (from Section 11) the set A is minimal thin at (, as was to be shown in proving the equality in (14.1). To show that the minimal-fine limit in (14.1) is strictly positive, it can be assumed that v is a strictly positive potential GDV, after replacing v if necessary by its reduction on a ball relatively compact in D. Under this hypothesis, Hence OK(C,
lim of GD(S)) q-c
fD K((, (')v(d(') > 0,
as was to be proved. Special case: v =_ 1. If GD((, ) has minimal-fine limit 0 at i;, as we shall
prove (Section 18) is true at µD almost every minimal Martin boundary point (, it follows from Theorem 14 that has limit 0 at Con approach to C in the Martin topology. [Incidentally, this application of Theorem 14 exhibits the fact that the minimal-fine limit + oo to the function at cannot be excluded in (14.1).] This vanishing of the Green function
the Martin boundary (to be extended by relativization in Section 18) is one indication that the Martin boundary is well adapted to classical potential theory.
Relation between Theorems 13, 14, and XI.4
The fact that the Laplacian is a self-adjoint differential operator leads to the symmetry of the Green function GD, absent in the potential theory generated by a non-self-adjoint differential operator. In such a theory the counterpart of Theorem XI.4 splits into two theorems. See Section XVIII.14
for the versions of Theorem XI.4 in the potential theory corresponding to the heat equation and its adjoint. The self-dual character of classical
15. Limits at a Minimal Martin Boundary Point
215
potential theory is lost at the Martin boundary of a Greenian domain, however, and in fact Theorems 13 and 14 are dual to each other. This is suggested by the fact that the proof of Theorem 14, unlike the proof of Theorem 13, uses the sweeping symmetry of classical potential theory. In the probabilistic versions of these theorems (Theorem 3.111.5) Theorem 13 states that the function v/K(C, ) has the indicated limit at C along almost
every Brownian path conditioned to go from a point of D to C, whereas Theorem 14 states that v/GD(1;, ) has the indicated limit at C along almost every Brownian path conditioned to go from C to a point of D. If the two points of D here are taken to be the same, these conditional Brownian paths to C can be identified with those from C; so the same limit concept is involved, corresponding to minimal-fine limits at C. In particular, in classical potential
theory if C is a point of a Greenian set D, then C can be identified with a minimal Martin boundary point of D - {C} [Section 12, Example (a)], the corresponding minimal harmonic function is a multiple of GD(C, ), and Theorems 13 and 14 coalesce to Theorem XI.4(c).
15. Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal Martin Boundary Point Let D be a connected Greenian subset of I8", let K be a Martin function for D, and let C be a minimal Martin boundary point. The following lemma is the analog of Lemma XI.8 in the present context. Lemma. Let A be a decreasing sequence of subsets of D, and suppose that each set Ak is a deleted minimal-fine neighborhood of C [has minimal-fine limit point C]. Then there is a subset A of D with the property that for each k the part of A in a sufficiently small Martin topology neighborhood of C is in Ak and that A is a deleted minimal-fine neighborhood of C [has minimal-fine limit point C].
The proof is similar to that of Lemma XI.8; so only the unbracketed assertion will be proved. Suppose that each set Ak is a deleted minimal-fine neighborhood of C, and define Fk = D - Ak. The set Fk is minimal thin at C, and it is sufficient to show that there is a set F, minimal thin at C, with the property that for each k the part of FA, in a sufficiently small neighborhood of C is a subset of F. If for all but a finite number of values of k the point C
is not a limit point of Fk, we can take F as the union of those sets Fk for which C is not a limit point of Fk. Otherwise, it is no restriction to assume that
C is a limit point of F. for all k. Let B. be the intersection with D of a ball (in terms of some metric on DM) of center C and radius r. Let Co be a point of D, and applying Theorem 13, let uk = vk/K(C, ) for keZL+ be a positive K(C, )-superharmonic function on D with u(C0) < 2-k and with limit + 00 at C on approach along Fk. The function u = Eo uk is positive and K(C, )-
216
LXII The Martin Boundary
superharmonic on D with limit + oo at C on approach along each set Fk. Choose rk so that limk-m rk = 0 and that u Z k on Fk n B,k. If
F= U(FknB,k), 0
the function u has limit +oo at C along F; so [by Theorem 13(b)] F is minimal thin at C, and Fk n B,k c F, as desired.
16. Minimal-Fine Topology Limits and Martin Topology Limits at a Minimal Martin Boundary Point (Continued) Theorem. Let D be a connected Greenian subset of RN, and let C be a minimal
Martin boundary point of D. A function u from the trace on D of a Martin topology neighborhood of C into a metric space has minimal-fine limit [minimal-
fine cluster value] a at C if and only if u has limit a at C on approach along some subset of D that is a deleted minimal-fine neighborhood of C [is not minimal thin at C]. The proof is that of Theorem XI.9 with trivial changes corresponding to
the change of context. Observe that if the range space of u is E8, then mf lim sup,_, is a minimal-fine topology cluster value of u at C, and therefore u has this value as a limit on approach to C along some subset of D
that is not minimal thin at C; that is, this subset has C as a minimal-fine topology limit point.
17. Minimal-Fine Martin Boundary Limit Functions Let D be a connected Greenian subset of D. If A is a subset of D, denote by A'"f the set of minimal-fine limit points of A in DM. Recall (Section 12) that Dmfn aMD = D. Lemma. (a)
If A is a subset of D, the set A'"f is a Martin topology Ga subset
of DM.
(b) If u is a function from D into 08, the function lim sups-t u(l; ) on the minimal Martin boundary arD is Borel measurable (Martin topology of DM). (c) If u is a Junction from D into a compact metric space, the set of
Martin boundary points at which u has a minimal-fine limit is a Borel set, and the limit function on this set is Borel measurable (Martin topology of DM).
217
17. Minimal-Fine Martin Boundary Limit Functions
Proof of (a). According to Theorem XI.6, the set Amf n D is a Euclidean Gs set, and this set is therefore a G8 set in the Martin topology. To prove that A'f n aMD is a Ga set, let K be a Martin function for D, and let o be a point of D. If B is a subset of D, let B' be the class of Martin boundary points C 5 K(l, X0)/2. Since the smoothed resatisfying the inequality OK(C, ) K(C, )), Fatou's lemma implies that the set B" is duction on the left is
compact. Let B. be the sequence of traces on D of the sets of a countable topological base for DM. If A is a subset of D, the set Uo (A n Be)' is an F. set, the set of Martin boundary points that are not minimal-fine limit points of A.
Proof of (b). Assertion (b) follows from (a) because
l
s c}= n {eD:
{Ea'D:
c+ n}mf n0 MD. (17.1)
C-C
Proof of (c). According to (b), the boundary set on which u has a minimalfine limit is a Borel set if u is extended real valued, because the set in question is the set on which the minimal-fine superior and inferior limits of u are equal. Moreover (b) implies that the limit function on this set is Borel measurable. If the range space S of u is compact metric and if 0. is a sequence of functions
dense in C(S) in the metric of uniform convergence, then the map {O (t)/sups n e V+} is a one-to-one bicontinuous map of S onto a compact subset of the compact metric space [0,11 z; so an application of (a) and (b) to each function 0.(u) yields (c).
EXAMPLE. If h is a strictly positive harmonic function on the connected Greenian subset D of 08" and if A is a subset of D, then A)
GMD(''RI) = GMh(R
= ND(-. Amf n aMD).
(17.2)
Observe that the first equality is trivial and that the second equality asserts that the function h1D(-, Amf n aMD) is the harmonic component of RA (Riesz h decomposition). To prove this assertion, apply the kernel operator 6D to the
Martin representation of h to find +h
LD +RtC.'1
K(g, -) Mh(dd)
h
Amf 0 M D
+ aMD-Amt
(17.3)
R'
I
Mh(dC).
218
I. X11. The Martin Boundary
Af n aMD) according to Section 10. According to Theorem 11, the function RA is a potential whenever C is not minimal or is not in Am!, so the last integral is a potential. Thus (17.3) displays the Riesz decomposition of R' and thereby yields the second equality in (17.2). The first integral after the second equality sign is equal to
Observe that according to this example, the function Oh Q" is a potential if and only if A'f n aMD is y 4 null.
Application. If D is a connected Greenian subset of R, if h is a strictly positive harmonic function on D, if u is a positive h-superharmonic function on D, and if
B2 _ {(earD: mflimsupu(ry) > c},
r[
then u > cµ4(-, B2). We can suppose in the proof that c > 0. If A, a D: a}, then according to the preceding example and the relation (Section VIII.2) between reductions and h-harmonic measure, if 0 < a < c, u>"RA8>a"R A0
(17.4)
Let a tend to c to obtain the desired inequality.
18. The Fine Boundary Function of a Potential Theorem. If D is a connected Greenian subset of I3" a superharmonic hpotential, u = G0p/h has minimal-fine limit 0 at µ4 almost every (equivalently Mh almost every) point of 3MD.
Define AL = {Cea"D: mflimsup,-t > s}. Then according to Section 17 (Application), u >_ sµ4(-, A), and this is impossible unless A, is 1 2. In view of
the Riesz decomposition theorem it follows that for h = I and D a ball
222
I.XII. The Martin Boundary
Theorem 19 is true for radial as well as minimal-fine approach to the boundary. The following example shows that this assertion is false for general h.
EXAMPLE (b). Let h be a strictly positive minimal harmonic function on a ball D, corresponding to a boundary point . Then (from Section VIII.9) uD
is supported by the singleton {C}, and Theorem 19 states that a positive h-superharmonic function u = v/h has a finite minimal-fine limit at C. This limit is info u according to Theorem 13 and is 0 (by Theorem 19) if v = GDv is a potential. Since [Section 12, Example (b)] the radius to C is not minimal
thin at C, the function u has radial limit inferior 0 at C. However, if v is supported by a countable dense subset of this radius, the function u has radial limit superior + oo at C.
Thus the fine topology is better adapted to Fatou-type boundary limit theorems for a ball than nontangential and radial approach topologies. The difference is that the latter approach topologies are adapted to the ball rather
than to the properties of the functions involved. To show the relation between Theorem 19 for a ball and the corresponding results for nontangen-
tial and radial approach to a ball boundary, it will be shown in the next sections that a positive h-harmonic function has a nontangential limit fi at any ball boundary point at which there is a minimal-fine limit Ii and that a superharmonic potential has radial limit 0 at 1,4_1 almost every boundary point at which there is minimal-fine limit 0. Thus Theorem 19 for a ball implies the truth of the corresponding partial results described above for nontangential and radial boundary approach. The converse is false. More exactly, it will be convenient to prove the stated implications for a half-space rather than a ball. Since half-spaces and balls can be mapped onto each other by inversions, the assertions on nontangential convergence to boundaries correspond exactly in the two contexts. The assertion on radial convergence to a ball boundary corresponds very nearly to the corresponding assertion on convergence to a half-space boundary along normal lines.
21. Nontangential and Minimal-Fine Limits at a Half-space Boundary Let dd be the last coordinate of the point of R', and let D be the half-space {d, > 0). Theorem. If u = v/h is a strictly positive h-harmonic function on the half-space D and if u has minimal-fine limit fi at the boundary point i;, then u has nontangential limit ft at C.
The converse theorem can be shown to be false. Theorem 21 implies Theorem I1.15, that u has nontangential limit dM,,/dM, at Mh almost every
223
22. Normal Boundary Limits for a Half-space
boundary point, in view of the minimal-fine topology Fatou theorem (Theorem 19) and the correspondence between ball and half-space given by an inversion. We can assume in the following proof that the specified boundary point { of D: {dd > 0) is the origin. Let . be a sequence in D with nontangential limit the origin, and suppose that fi'. To prove Theorem 21, it will be shown that fi' = P. Assume from now on that 0 < fi < + 00. The modifications to be made in the argument if fi is 0 or + oo will be obvious. It can be assumed that the ray from the origin through tends to a limit ray L as n ao. Let ;, be the point of L at minimum distance from c,,, let A be a ball, closure in D, center o, let Ao be a smaller concentric ball, and
define c = The ball c,A has center ;,, and ec.A;, for sufficiently large n. Finally (by the Harnack inequality), there is a strictly positive constant y depending only on the ratio of the radii of A and Ao such that 1 /y 5 u(q)/u() 5 y for both and >l in Since Uo (c Ao) is not minimalthin at the origin [Section 12, Example (b)], there is a sequence 17. along which u has limit fi, with q E Hence 1/7 5 fl/li' S y. Since the Harnack constant y can be made arbitrarily near I by choosing Ao sufficiently small, it follows that fi = fi', as was to be proved.
22. Normal Boundary Limits for a Half-space Define the half-space D as in Section 21. A boundary point C will be called a normal limit point of a subset A of D if C is a limit point of the part of A
on the normal to 8D through C. A function on D will be said to have a normal limit at t if it has a limit at C along the normal. Lemma. If A is an (open) subset of the half-space D, IN-, almost every normal boundary limit point of A is a minimal-fine limit point of A.
Warning: "Open" was enclosed in parentheses in this statement because (Section 23) the lemma is true for an arbitrary subset A of D. It is sufficient to prove the lemma for a bounded open set A. A careful application of the Vitali covering theorem shows that there is a subset Ak of A with the following properties: Ak is a countable union of closed (N - I)dimensional intervals, each on a hyperplane dd = const < 1/k), each
with edges parallel to the coordinate axes; these intervals have disjoint projections on 8D; the projection on 6D of Ak covers IN_, almost every point of the projection of A n do < l/k}. Define the potential vk by ,)lN-t(dn) vk(0 = fAk G dd
(22.1)
The remainder of the proof assumes that N > 2. The argument is similar when N = 2. We first prove that there is a constant c, depending only on N
224
I.XII. The Martin Boundary
such that uk 5 c, for all k and choices of Ak. To see this, let be a point of D, and define
D=
n)
hi(s) = sup
s),
d,,
qe DrOD,
The evaluation (see Section VIII.9) s-(N-2)
- (s2 +
d
d.
shows that the left side decreases when d,, increases, from which it follows that
2(N - 2)ds-" cur(s) = s-iN-2) - (2ds - s)-(x-2)
d4 -s so that
if
s > d4,
if
s 5 d{,
(22.2)
is monotone decreasing. Thus
rk()
0 the open set Ae = {u > e} has IN_, almost no normal boundary limit point, and in view of Lemma 22 this follows from the fact
(Theorem 18) that u has minimal-fine limit 0 at µD (equivalently, IN-t) almost every boundary point, so that IN_, almost no boundary point is a minimal-fine limit point of A, Observation. We have now proved that a positive superharmonic function on a half-space of 1' has a limit at IN_, almost every boundary point on both minimal-fine and normal approach to the boundary point and that the boundary limit functions in the two approaches are equal up to IN_, null sets. In fact (Riesz decomposition) it was sufficient to prove the theorem separately for u harmonic and u a potential. The harmonic case was covered in Section 19, in which case nontangential boundary approach was admissible; the potential case, in which the limit vanishes 1N_, almost everywhere on OD, is covered in (a) of the present theorem.
Proof of (b). Let An [A'] be the set of normal [minimal-fine] boundary limit points of A. The function 018" is a positive superharmonic function on D, equal to I quasi everywhere on A. Let Ao be the polar subset of A on which 818" < 1, and let vo be a superharmonic potential on D, identically
+ oo on Ao. According to (a), the function vo has normal limit 0 at IN_, almost every point of BD; so 1N_, almost no point of aD is a normal limit point of Ao, and we shall therefore assume from now on that Ao is empty. In view of the Riesz decomposition theorem and (17.2), 818A = v + PD(-, A'),
_
(23.1)
where v is a potential on D. The smoothed reduction on the left side of (23. 1)
has normal boundary cluster value I at every point of A" and therefore by (a) and the above observation has normal and minimal-fine limit 1 at 'Ni every point of A". The function on the right side of (23. 1) has minimal-
fine boundary limit function 1A. up to an IN_, null set; so A" a A' up to an 1N_, null set, as was to be proved.
Chapter XIII
Classical Energy and Capacity
1. Physical Context Consider a distribution of positive and negative electric charges on IV and the electrostatic potential induced by this charge. By definition of a conductor, if A is a connected conducting body in R', the charge on A distributes itself
in such a way that the net effect is that of an all-positive or all-negative charge. and the distribution on A is in equilibrium in the sense that the restriction to A of the potential of the charge distribution in I8' is a constant function. Let D be an open subset of R with a conducting smooth boundary, and suppose that the boundary is grounded. The significance of grounding is that
if a positive charge p is imposed on D, an induced negative charge - p* appears on W. and the potential G(p - p*) is identically 0 on OD. Thus, if p is a unit positive charge at in D, the restriction to D of the potential and p* is identified G(p - p*) is identified with the Green function with the sweeping of p onto aD (relative to R'); that is, p* is identified with p the measure p* is the harmonic measure identified with the sweeping of p onto aD and that G(p - p*) = GDp on D. In view of this physical context the existence of a mathematical version of the Green function of a reasonable set D was obvious long before there was a rigorous existence proof, and the sweeping of a measure was a natural concept to formalize. Now suppose that a connected conducting body is introduced into D and given a positive charge p. This charge necessarily distributes itself in such a way that G(p - p*) = GDp is constant on A. Such a charge p, that is, such a measure, is called an equilibrium charge (or measure, or distribution), and the corresponding potential GDp is called an equilibrium potential for A. Two
equilibrium potentials for A are proportional, as are their potentials, and if the potential on A of an equilibrium measure has the constant value I the equilibrium measure [potential] is called the capacitary measure [potential]. In view of this physical context it was clear to Gauss that there must be a capacitary distribution in a suitable mathematical context for any reasonable pair A and D. The pair (A. OD) is a condenser in the physical context, and the capacity of this condenser is defined as
227
2. Measures and Their Energies
Total charge of an equilibrium distribution on A Value on A of the corresponding potential = total charge of the capacitary distribution on A.
The mathematical model of this physical context has already been discussed, at least in part. The set D is supposed Greenian. The Green function has been shown to have limit 0 GD has already been defined, and at every regular boundary point of D. If p is a measure on D, the function GDµ is the mathematical version of the electrostatic potential generated by a distribution p of electric charge. More generally p will sometimes be allowed to be a suitably restricted signed measure or charge in the sense of Appendix IV.7. The energy of a charge and the mutual energy of a pair of charges will be defined, following mathematical tradition, as twice the values assigned
by physicists. Equilibrium distributions will be derived, and the capacity of a subset A of D will be defined by (1.1) when A is analytic.
2. Measures and Their Energies If D is a Greenian subset of RN, the mutual energy [p, v] of a pair of measures on D is defined by
[µ, v]
=f
GDµdv
=
J
fG(. 1)µ(d)v(drl),
(2.1)
and the energy lip 11' of a measure p on D is defined as [µ, µ]. The form is symmetric,
[µ, v] = [v, µ],
(2.2)
and this symmetry is sometimes dignified by the name reciprocity law. The symmetry of the Green function GD is a special case.
The Space 8' The space of measures on D of finite energy will be denoted by 8'. It is trivial that a positive constant multiple of an element of 8' is in 8', but it is a much deeper fact that (Theorem 7) the sum of two elements in 8' is in 8', equivalently, that the mutual energy of two measures in 8' is finite. EXAMPLE (a). If p is a measure on D and if A is a polar subset of D that is not p null, then lip 11 = +oc because (Theorem V.11) GDµ = +oo p almost everywhere on A. Thus a measure of finite energy vanishes on polar sets. Conversely, if A is an analytic subset of a Greenian set D and is null for every measure on D
228
1. XIII. Classical Energy and Capacity
of finite energy supported by A, then A is polar in view of Corollary V.9 and the fact (Theorem VI.2) that an analytic nonpolar set has a compact nonpolar subset. It will be shown in Section 3.1.9 that a superharmonic potential GDµ is quasi bounded if and only if u vanishes on polar sets. It was noted in Section V.10 that the domination principle as applied to potentials of measures p vanishing on polar sets has a simple form: if GDµ is majorized u almost everywhere on D by a positive superharmonic function v, then GDu < v on D. EXAMPLE (b). Let
be a point of D, choose a > 0, and define B =
a), A = D n aB (Euclidean boundary), and u = SD(I, ). Since has limit 0 at quasi every point of aD, aB - A is polar and thus is a null set; that is, 1. (This fact is a special case of the application in Section VIII.18.) The measure y is supported by A, and a on A; so
A) = a.
lip 02 = aba(:, A) =
In particular, if D =
B=
(2.4)
fl) with fi < b, then
if N = 2, {G(S,') > R2-N - 62-N}
if N > 2;
so the energy of the uniform distribution on aB of total value I is log (S/f) if N = 2 and is R2-1 - 62-N if N > 2. The latter energy decreases to the energy of u relative to RN, when b - oo. Y{r2-N,
EXAMPLE (c). The general case of the preceding example is the following. Let D be a Greenian subset of RN, let B be an open subset of D, let be a
point of B, and define A = D n aB (Euclidean boundary) and u = Sp(, ). Then IIull2
= 6A (S, GD(c,')) = GD(S.
1D)
NB(S,
(2.6)
3. Charges and Their Energies A charge (Radon measure) ,u on the open subset D of R' is (see Appendix IV.3) a pair (ut,u2) of measures on D under the identification (u1,u2) _ (vt, v2) when pt + v2 = u2 + v1 and the operations (PI, 142) + (v1, v2) = (101 + v1, uz + v2), (cut, cue)
c(ut, uz) _ _
ifc-0,
(-cue, -cut) ifc < 0.
(3.1)
4. Inequalities between Potentials, and the Corresponding Energy Inequalities
229
A charge has a minimal representation whose component measures have disjoint supports. A charge is called positive if its minimal representing pair has the form (p,, 0) and is then identified with the measure p, .
Let N be the space of pairs (f,,f2) of positive extended real-valued functions under the equivalence relation (f, ,f2) = (g,, g2) when f, + g2 = f2 + g, and the operations (3.1) interpreted in the present context. The potential GDµ of a charge p is defined as the point (GDµ,, GDP2) of N and is informally identified with the function GDp, - GDp2 where this difference is defined. The only charges of interest are those charges p the components of whose minimal representing pair (p, , p2) have superharmonic potentials, in which case GDp, - GDp2 is defined quasi everywhere on D. The potential GDp of a positive charge (p, , 0) is identified with the function GDp,. The space of charges that have representations with both components in d'' will be denoted by d, and for elements of if only representations with components in 8' will be used. If p: (p,, 142) and v: (v,, v2) are charges in 8, their mutual energy is defined following the definition in the positive case,
[p, v] = [p., vt] + [p2, v2] - [PI, v2] - [p2, vt], and the energy of u is defined as [p, p]. To justify this definition, it must be
proved that the mutual energy of a pair of elements in 8' is finite (see Theorem 7), and until this fact is proved, the definition (3.2) will be used only when the finiteness of the summands is obvious from the context. The form is symmetric (reciprocity law). It will be proved that the energy of a charge in 8 is positive, and the energy will then be written This notation will be used before positivity is proved whenever positivity is obvious from the context.
4. Inequalities between Potentials, and the Corresponding Energy Inequalities Lemma. (a) The inequalities GDµ' 5 GDV and GDP" S GDv" for potentials of measures imply that [y', p"] 5 [v', v"]. (b) The inequality GDµ 5 GDv for potentials of measures implies that ((p((2 5 ((v((2 and that y = v if these energies are finite and equal. (c) Let GDp, and GDv, be increasing [decreasing] sequences of potentials of measures, with respective [smoothed] limits GDp and GDv. Then lim,,..m [Ati., v.] = [p, v]
in the increasing case and, if [pa, vo] is finite, also in the decreasing case.
230
I .X111. Classical Energy and Capacity
Proof of (a). Apply the symmetry of GD to obtain
[i'f'] =
GDµ' dµ" < fD
f
GDv' dp" = JGoKdv'
D
D
(4.1)
5 f GDv" dv' _ [v', v "]. D
Proof of (b). If p' = p" = p and v' = v" = v in (a), then l1µ11 s 11v11, and with
this specialization if the energies, that is, the integrals in (4.1), are finite, equality in (4.1) implies that GDµ = GDv both p almost everywhere and v almost everywhere, so that GDµ = GDv by the domination principle for potentials of measures of finite energy (Section 2), and therefore µ = v. Proof of (c). In the increasing case in view of (a) the sequence [µ,, v.] is an increasing sequence with limit at most [µ, v]. The limit is [p, v] because for every m,
lim [µ
n+m
Z lim inf f GDµ dv = lim inf n-aD
=
f
n-.0J, D
o
GDv dµ,,,
GD, dµm = J GDPm dv,
D
D
and the last integral can be made arbitrarily near [p, v] by choosing m large. The decreasing case is treated similarly, with the help of the fact that polar sets are A null if A is a measure of finite energy.
5. The Function D (a)
G0µ
Let D. be an increasing sequence of open subsets of W4 with union a
Greenian set D. Since GDS 5 GDM+t on D. x D., it follows that if µ and v are
measures on D and if µ is the projection of u on D, then lim
f
e-m JD
G
dv;
D
that is, mutual energy relative to D. increases to mutual energy relative to D. Observe that we have not supposed that the integrals in (5.1) are finite valued but that once we have proved (Section 7) that when the measures µ and v are in tf + for D, their mutual energy for D is finite valued, it will follow trivially that (5.1) is valid for p and vin 4f for D. although the sequence GD,p,
need then not be monotone. (b) Let D2 be a Greenian subset of 08', let Dt be a relatively compact
231
6. Classical Evaluation of Energy; Hilbert Space Methods
open subset of D2, and let p be a measure on D2 with compact support in D, . Energies, reductions, etc., relative to D, will be distinguished by the subscript i: 8.0;, ffi, .... According to Section VII.5, GD,µ = GD p + GD,µt
(5.2)
on D, , where p, = 0µ0°=_°j is supported by OD, . Equation (5.2) is obviously
also valid for a charge p supported by a compact subset of D, , with µ, a charge supported by OD,. Since GD,µ, is bounded on the compact support of p,, equation (5.2) implies that p e 62 if and only if µ E 4, . (We are abusing
language slightly here in considering p both as a measure on D, and as a measure on D2.) If we assume that all the following mutual energies are well defined, we can combine (5.2) with X(5.3') as applied to the measure components of the charges to find that if p and v are in S2 with compact supports
in D,, then [p,v] I = [p - µt+V - VIJ2,
(5.3)
where v, is defined in terms of v in the way µ, was defined in terms of µ. Thus, if all the mutual energies involved are well defined (and according to Theorem 7, this is true if µ and v are in 82), the map µ'--. µ - µ, is a linear mutual-energy-preserving map from a subset of 8, into d'2.
6. Classical Evaluation of Energy; Hilbert Space Methods If D is an open subset of R" and if u and v are functions from D to the extended reals, define
2(u, v) _ f (grad u, grad v>&",
em(u) = 2(u, u)
(6.1)
D
whenever the integrals are meaningful. The value 2(u) is the Dirichlet integral of u. If u = GDµ and v = GDv are the potentials of charges in if, on the Greenian set D, and are in class Cj2'(D), then nH dv = -Av d1N (Section 1.7), so that
[µ,v] _ -n uAvd1N.
(6.2)
D
If D is bounded and smooth enough for the application of Green's identity and associated theorems and if the potentials u and v when defined as 0 on 3D are in class Ct21(D), (6.2) yields
[ p, v] = 2(n'
u)
[ µ 1 1}, an open set which covers quasi every point of A, let u be a superharmonic potential on D, identically + oo on the polar set A - B, , and define B = B, u {u > 11. Then B is an open superset of A and is in rp because 116' is majorized by the potential 2 111 0 A+ U.
For A in f, the measure .A is supported by D n aA (Euclidean boundary) and even (Corollary XI.18) by D n afA1. Recall (Section XI.6) that for an arbitrary subset A of R' the sets Af and afAf are Borel sets. Since D 1 Q A = 1
on D n afAf we conclude that O1QAd1A=2A(D)
IIAAII2=
(Aerr).
(10.1)
Ond fAf
If A is a subset of D not in fr,, we do not define AA, but it is convenient to define I12A II2 = + oc to obtain a set function II2.112 defined on every subset of D.
Theorem. (a) The set function II2.112 is a countably strongly subadditive Choquet capacity on D relative to the paving of compact sets, and for A in r'p inf {II28112 : A c B, B open} = inf {2B(D): A c B, B open}. (b)
(10.2)
If A is II2.112 capacitable, then
II2AII 2 = sup {2(F): F c A, Fcompact, A supported by F, GDA < I on D}. (10.3) (c)
A subset A of D is polar if and only if II AA II2 = 0.
Observation. Assertions (a) and (c) together are equivalent to the assertion that the restriction of II2.112 to the class of compact subsets of D is a topological precapacity and that II2.112 is the Choquet capacity generated (Appendix 11.8) by this topological precapacity.
Proof of (a). The set function II2.112 is obviously an increasing set function, and if A is an increasing sequence of sets with union A (c D), then lim,,..," II2AII2 because this limit equation is trivial if the limit is + oo and the limit equation follows from Theorem 9 otherwise. Next let B be an open relatively compact subset of D, and let A be a compact subset of B. Then AA is supported by A, GDAA = 1 quasi everywhere on A, GD2B = 1
239
10. The Classical Capacity Function
on Af and [Section VI.3(b)] GDAA = R' on D - A and so surely on the compact support of 2B. Since polar sets are 2A null, it follows that GDAA d2A =
112A 112 =
J
GDAA d i8 =
GRAB d2A =
Ji)
fD
f Ri dAB.
(10.4)
JD
Since (Theorem VI.5) the set function R, (c) is a topological precapacity on the class of compact subsets of D, it follows that if A. is a monotone sequence
of compact subsets of B with limit the compact subset A of B, then limw_. IIAA,I12 = 1124112. The proof that 112,112 is a Choquet capacity relative
to the paving of compact sets is now complete. To prove (10.2), observe first that the second equality follows from (10.1) and that the first equality is trivial when II2A112 = +00. If 1124112 < +oo, the set A must be in fo, and we have already noted that then Io contains open supersets of A. Furthermore, according to Section VI.3(m),
0104=[inf{p10B:A cB,Bopen}]+,
(10.5)
and therefore according to the Fundamental Convergence Theorem, there is a decreasing sequence B. of open supersets of A such that DIIA=ClimDI p"] w
ID
(10.6) +
The first equality in (10.2) now follows from Lemma 4(c). What we have proved implies that the restriction of the capacity 112,112 to the class of compact subsets of D is a topological precapacity generating 112,112 ; so this capacity is countably strongly subadditive. Proof of (b). If F is a compact subset of D, if A is a measure supported by F, and if GDA 5 1 on D, then we can apply (10.1) and Lemma 4 to obtain A(F) 5 SF GDAFd2 S IIAF112 = 2F(F).
(10.7)
Moreover, if A is 1P2.112 capacitable, then by definition of capacitability 1124112 = sup{112F112: Fc A,Fcompact}
= sup {2F(F): F c A, Fcompact}.
(10.8)
The combination of (10.7) and (10.8) yields (b).
Proof of (c). It is sufficient to prove (c) for relatively compact subsets A of an arbitrary relatively compact open subset B of D, and we apply (10.4).
240
1. XIII. Classical Energy and Capacity
If A is polar, then RtA = 0 quasi everywhere on D (Theorem V.4); so (10.4) implies that IIAA112 = 0. Conversely, if IIA4112 = 0, then equality of the first and third terms in (10.4) implies that AA(D) = 0; so !2' vanishes identically, and therefore (by Theorem V.4) A is polar. Unique Characterization of AA
If A E r,, then a measure A on D is AA if and only if A is supported by Af and GDi : 1 with equality quasi everywhere on A. In fact, in view of Theorem XI.14 the measure AA satisfies these conditions. Conversely, if A satisfies these conditions and if A' is the projection of A on a compact subset of Af then (domination principle) GDA' 5 GDAA; so GDA S GDAA. Interchanging A and AA yields the reverse inequality; so A = AA. Observe that this argument
also shows that if A is closed relative to D in the Euclidean topology and if A is in r,, then A = AA if and only if A is supported by A and GDA S I with equality quasi everywhere on A.
11. Inner and Outer Capacities (Notation of Section 10) Let A be a subset of D, and define C*(A) = sup{11AFII2: Fc A,Fcompact},
C*(A) = 11A4112.
(11.1)
Recall that with this definition C *(A) < + oo implies that A E r,; so Rl is a potential, GDAA, that AA is supported by afAf and that C*(A) _ JJAA11' = AA(DnafAf).
On the other hand, there are sets A in TD with AA(D) = C*(A) = +oo. The values C*(A) and C*(A) are called, respectively, the inner and outer
capacities of A (relative to D). Thus C*(A) is the infimum of the inner capacities of the open supersets of A. The set A is 11 A. 112 capacitable, that is,
C* capacitable, if and only if C*(A) = C*(A), and then C(A), called the capacity of A (relative to D), is defined as the common value of the inner and outer capacities of A. If A is a subset of D with finite outer capacity and if e > 0, there are an open superset AE of A and a compact subset A, of A such that C* (A) < C(AE) + e,
C(AE) < C*(A) + e,
and there exist a G6 superset A" of A and an F. subset A' of A such that
C(A') = C*(A) 5 C*(A) = C(A").
241
12. Extremal Property Characterizations of Equilibrium Potentials
12. Extremal Property Characterizations of Equilibrium Potentials (Notation of Section 10) The canonical equilibrium measure A,*. We have used the capacitary measure AA as a canonical equilibrium measure. A second choice, which is sometimes convenient when A is known to be capacitable with 0 < C(A) < + oo, is .i;, defined by A,* = 2"/C(A). Then Z,*i (D) = 1,
4 I I AA
II Z = C(A) ,
I
GDAA*
with equality quasi everywhere on A.
A Special Class of Subsets of D
In discussing capacitary measures AA relative to D it will be convenient to restrict A somewhat. Observe (Theorem XI.6) that for an arbitrary subset A of D the set A - Af is polar and so has capacity 0 and is null for every measure that has finite energy or (Theorem V.11) whose potential is finite
valued. Furthermore (Theorem XI.6) the set Af is a Borel set and so is capacitable. Thus, if we restriqt A to be in rp so that AA is defined and further restrict A to be fine closed in D, that is, D n Af c A, then the representation
A = (A - Af) u (D n Af) shows that A is capacitable and is measurable for every measure A on A of finite energy or with a finite-valued potential. This restriction on A does not restrict the class of capacitary measures because (Section XI. 7) C 10A = I 10D A-'' Furthermore the capacitary measure A" under
this restriction on A is supported by A and in fact (by Theorem XI. 18) is even supported by D n dfA! and C(A) = AA(A) according to (10.1). In particular, C(A) < + oo if A is also relatively compact (Euclidean topology) in D. We shall use the fact that if A is a measure on D, vanishing on polar sets and supported by the fine closed in D set A, then [A, )A] =
= A(A).
(12.1)
In (a)-(c) below we characterize AA and A,; by maximal and minimal properties of measures A on D satisfying the stated side conditions under the hypothesis A e r,, (a)
D n Af c A,
0< C(A) 2.
244
I. Xl 11. Classical Energy and Capacity
14. The Gauss Minimum Problems and Their Relation to Reductions Let D be a connected Greenian subset of D. Energies and reductions below are relative to D, and charges are on D. Let A be a nonpolar Borel subset of D, and let f be a Borel measurable function from A into Ul;, not vanishing quasi everywhere, with If I majorized quasi everywhere on A by the restriction to A of the potential GD,1 of some measure A of finite energy supported
by A. Observe that if A is fine closed in D, the words "supported by A" are no restriction on f because if A is a measure on D of finite energy and if GD)l quasi If 1:5 GDa. quasi everywhere on A, then If 15 OGD).DA = everywhere on A, the measure II).0A is supported by A (Theorem XI.14), and IIAII < +oo (Lemma 4). Let I'A be the linear class of charges v supported by A, of finite energy, with 1A I f I dlvl < + oo. This class contains A. We consider two modifications of a problem studied in 1840 by Gauss (see-Historical Notes to this chapter)
under hypotheses suited to his era.
(GI)
Minimize f GDvdv=IIvII2forVEr'Awith j'fav=
1.
A
( G2)
Minimize [(GDv - 2f)dv = IIvIIZ - 2 fA fdv for ver'A.
L et r,; be the class of positive charges, that is, of measures, in rA, and let
problems (G1+) and (G2) be, respectively, (G1) and (G2) with f'A replaced by 17A. We shall treat problems (G1+) and (G2+) only when f z 0. We shall write that a charge solves one of the above problems if the charge minimizes the relevant integral under the specified side conditions. (a) A charge p solves problem (G2) if and only if
[µ, v] =
ffdv
(14.1)
for every v in r'A, equivalently, if and only if GDµ = f quasi everywhere on A. If so,
f (GDµ-2f)dµ=-1Iµ112=- Jfdµ 0 and µllµll-2 solves problem (G2), equivalently, if and only if llpll > 0 and G0µ = llpll2f quasi everywhere on A. There can be only one such charge. In fact, if p solves problem (G1), then the side condition implies that Ilpll > 0, and the minimizing property of p implies that for v in 17A,
J(Gnv
- 2f)& z (JA1)2hph2 - 2 fdv A
_
_111111-2
- Ilpll-t)2
+ (111411.14
with equality when v = pllpll-2 Thus the latter charge solves problem (G2), and the rest of (b) is now trivial. (a+)
If f z 0, a measure u solves problem (G2+) if and only if
[p, v] z f fdv,
Ilpll2 =
ffd14
(14.1+)
A
for every v in r,, equivalently, if and only if G0µ ;2: f quasi everywhere on A with equality µ almost everywhere on A. If so, (14.2) is true, and for v in r,; the identity
j(Gov_2f)dv= -Ilpll2+llv-PII2+2(IM, v]-
f fdv)(14.3+) A
shows that p is the only measure solving (G2+). Furthermore µ/jAfdµ solves problem (G1+). In fact, if p solves problem (G2') and if ce R , the integral
246
1. XIII. Classical Energy and Capacity
in (G2') with v replaced by cµ has its minimum value when c = 1, and the integral in (G2') with v replaced by µ + cv with c in R+ and v in ri has its minimum value for fixed v when c = 0. These two conditions are satisfied if and only if (14.1') is true. Conversely (14.1') implies that y solves problem
(G2+) and is thereby uniquely determined because under (14.1') in the identity (14.3') the right-hand side z -11µ1I2 with equality only when v = p.
Equation (14.1') with v replaced by its projection on an arbitrary Borel subset of A implies that G6µ 2:f v almost everywhere on A; so GDµ ;->f quasi everywhere on A. There is equality p almost everywhere on A in view of the equality in (14.1' ). Conversely, these two conditions on y imply that (14.1 +) is true. Finally, if p solves (G2' ), then µ/ f A f dµ solves (G 1') by the same proof as that of the corresponding implication in (a). (b') /f f z 0, a measure p solves problem (G 1') if and only if l1µ II > 0 and µ11µI1-2 solves problem (G2'), equivalently, if and only if l1µ11 > 0 and GDP Z
IIpII2fquasi everywhere on A with equality p almost everywhere on A. There can be only one such measure. The proof is left to the reader. (c) Suppose that f z 0, and define f as 0 on D - A. Then if µ solves problem (G2'), GDP = R . (See Section XI.20 for the reduction involved
here.) Conversely, if A is fine closed in D, then R is the potential of a measure p solving problem (G2'). In fact, if u solves problem (G2'), then GDµ = f < + oo p almost everywhere on A; so if v is a positive superharmonic function
on D that majorizes f quasi everywhere on D, the domination principle states that GDµ 5 v. Since one choice of v is G6µ, it follows that GDµ = Rf.
Conversely, R f quasi everywhere on A, and R < GDA; so R is the potential of a measure p of finite energy (Lemma 4). To prove that p solves
problem (G2+), we need only verify that A supports p, and according to Section X1.20 (special positivity case), µ is supported by cdfAf - D, a subset of A since A c Af by hypothesis. (d) Classical balayage. Suppose that A is a Borel subset of D fine closed in D and that f is equal quasi everywhere on A to the restriction to A of the potential GDµ' of some measure µ' of finite energy not necessarily supported by A. We have seen at the beginning of this section that then f satisfies the
conditions imposed throughout. In the present context, problem (Gl) becomes the problem of minimizing 11v112 for all charges v of finite energy supported by A with [p'. v] = 1, and problem (G2) becomes the problem of minimizing 11 v112 - 2 [p', v] for all charges v of finite energy supported by A. The measure 8 p, [A solves (G2) because
[G6µ'" =f quasi everywhere on A and (by Theorem XI.14) the measure iµ'0" is supported by A. The linearity properties of (G2) solutions noted in (a) imply that if f is
15. Dependence of C* on D
247
equal quasi everywhere on the fine closed in D set A to the restriction to A
of the potential of a charge u' of finite energy, then problem (G2) has a solution: in fact, if p' = p'+ - µ'- (Hahn decomposition), then µ'+8A p'- @A solves (G2).
In particular, if A is a fine-closed Borel relatively compact subset of D and if f - I on A, f is the restriction to A of the potential of a measure of finite energy; for example, f is the restriction to A of the reduction of the function 1, reduced onto an open superset of A relatively compact in D. In this special case problem (G 1) is the problem of minimizing JJvll2 for charges
v of finite energy supported by A with v(A) = 1, and problem (G2) is the problem of minimizing 11v112 - 2v(A) for all charges of finite energy supported by A. The capacitary measure of A solves problem (G2) in this context. The Frostman approach (see Historical Notes). If A is compact and if f is positive (< +oo) and continuous, it is easy to see that problems (G1+) and (G2+) have solutions. In fact, if p. is a minimizing sequence of measures for (G 1+), then p.(A) is a bounded sequence, and any measure that is the vague limit of a subsequence of p, solves (G I+).
15. Dependence of C* on D If D, and D2 are Greenian subsets of R' with D, c D2, we have noted in Section VII. I that GD, 5 GD2. If outer capacity relative to D, is denoted by
C7, it will now be shown that C,* >- CZ on subsets of D, and that to a compact subset B of D, corresponds a constant a = a(B) such that C,* 5 aC2 on subsets of B. In view of the relations between inner and outer capacities, the corresponding inequalities are true for inner capacities, and it is sufficient
to prove these inequalities on compact sets, for which C,* reduces to the capacity function C. If A is a compact subset of D, with capacitary measure 2;A relative to D;, then I >: GD2A2A > GD,A2A on D,; so by (13.1)
C(A) = sup {A(A): A supported by A, GD,A 5 1) z A2A(A) = C2(A), (15.1)
as asserted. To prove the second assertion, observe that if B is a compact subset of D, and a = sup8 K 8 GD,/GD, (the ratio is defined as 1 on the diagonal) and if A is a compact subset of B, then GD,22A 5 "GL), )2A on D, ; so GD, (rA2A)
z I quasi everywhere on A. Therefore by (13.1) C, (A) = inf (A(A): A supported by A, GD, 2 z I quasi everywhere on A) 5 0(i.2A(A) = aC2(A),
as was to be proved.
(15.2)
248
I. X111. Classical Energy and Capacity
16. Energy Relative to R2 In this section we consider potentials Gµ when N = 2. The Green function G is bounded below on compact subsets of 082 x 082, and we have seen in Section IV.I that Gp is superharmonic on 082 whenever µ is a measure on 082 with compact support. Hence, if p and v are measures on 082 with compact support, the integrals I R2 Gp dv and 11112 Gv dµ are meaningful and equal.
Their value, the mutual energy of the pair of measures, will be denoted by [µ, v]. Let 9 in the present context be the class of charges µ: (µt,µ2) with the
following properties (in which we suppose that pt and lee are minimal components) : E1.
µt and µ2 have compact support. Gp; dui < + cc, i = 1, 2.
E2. R2
E3.
µ(O82) = 0.
Then the following two assertions are true. (a) If µ and v are measures in B', then [p, v] < + oo.
This assertion follows from the fact that if D is a ball containing the compact supports of p and v the potentials Gµ [Gv] and Goµ [Gov] differ on D by harmonic functions; so (a) can be reduced to the corresponding assertion for potentials relative to D, covered by Theorem 7. In view of (a), if p and v are charges in 8', their mutual energy [µ, v] defined formally by (3.2) in terms of their minimal components is meaningful
and finite valued. The evaluation (3.2) is independent of the choice of components of the charges as long as all the mutual energy integrals are finite valued. The class 9 is now seen to be a linear class. We have not yet used the condition E3 in this discussion, but this condition is essential in the following. In view of the evaluation (6.3) of mutual energy in terms of the Dirichlet integral when the charges involved are in B and have E'21(R2) potentials, we can expect charges in tf to have positive energy, and in fact we now prove the following statement. (b) Theorem 7 is true for D = R2 and charges in 8. To prove that [p, µ] Z 0 for µ in 9°, let A be a compact support for p, of
diameter a, and let B be a ball containing A, with center in A, of radius ft > a. Define u( > h) = GM B
pB(q, G(c, ))
(16.1)
Band
- log (fl + a) < u 5 - log (fl - a).
(16.2)
249
17. The Wiener Thinness Criterion
In view of the positivity of energy relative to B (Theorem 7), if p is a charge in d (for R2) with minimal representation (p ,' p2)'
JR2
Z2pi(A)2log.
AA
Since the last term has limit 0 when i4 - oo, it follows that the energy [p,p] of p relative to R2 is positive. Schwarz's inequality (7.1) in the present context follows easily from positivity of energy [see (7.3)]. The rest of the proof of Theorem 7 in the present context follows that of Theorem 7 with one modification. To prove that [p, v] = 0 whenever v is in 8 implies that u is the zero charge, we cannot choose v as a uniform distribution on a sphere since this distribution is not in .9. Instead choose S, > S > 0, and for in R2 let v be the charge supported b) v by S,), equal on the larger sphere to the uniform distribution of a unit mass and equal on the smaller sphere to the uniform distribution of a negative unit mass. Define u = GDp. The equality [p, v] = 0 implies that L(u, , d) = L(u, , b,), and the condition E3 implies that lima,-. L(u, , S,) = 0. It follows that L(u, , b) = 0 and therefore (S - 0) that u - 0. Hence p is the zero charge, as was to be proved.
17. The Wiener Thinness Criterion Let D be a Greenian subset of R"(N z 2), let A be a subset of D, and let be a point of D. Let C* be the outer capacity !Section 11) defined on subsets of D, relative to D. Let a e ] 1, + oo [, n e 7+, and define
B"={n:
A"=Ar, B",
A"=UA,,.
(17.1)
"
Let k be an integer so large that Uk BB c D and, if N = 2, so large that this union has diameter less than 1. In the following k is fixed, and it is to be understood that A. and A" are considered only for n >_ k. Nothing in the
following theorem or its proof would have to be changed if one of the inequalities in (17.1) is changed to be a strict inequality. Observe that there is a constant c so large that
0 2, the stronger relation GD < G is valid on D x D.) In view of this inequality for B = Ak and (17.2). GD(s, q) < cc"cGD(S, q)
for q e A' - A2n, C E A2n, n > k.
Hence for n > k, GDu2" 5 const GDµ4n(S) 5 const u(i)
18, The Robin Constant and Equilibrium Measures Relative to R2 (N = 2)
251
on AZn, and the constant does not depend on n. It follows that GDµ2n Z 1 quasi everywhere on AZn for sufficiently large n; so Ri 2n 5 GDN2n quasi everywhere on A2i for sufficiently large n, and therefore this reduction inequality holds everywhere on D (domination principle) for sufficiently large n. Hence, if m is sufficiently large,
Ri "am
"am
S Y_ GDp2nS)
S GDP() < +w.
(17.6)
Ram
Thus the sum in Theorem 17(b) over the terms with even n converges, as was to be proved. Observation. In view of (17.2) the theorem is true, and the proof requires only trivial modification, if G in (17.1) is replaced by GD.
18. The Robin Constant and Equilibrium Measures Relative to R2 (N = 2) Since we shall not use the results of this section, detailed proofs will be omitted. The material is presented as an interesting and important application not readily available elsewhere.
Superharmonic and Harmonic Functions on Neighborhoods of oo A function defined on an open neighborhood of the point oo of R2 (that is, on an open set including this point) is said to be superharmonic [harmonic] there if the function is superharmonic [harmonic] on the deleted neighborhood and if the Kelvin transform of the function under an inversion is superharmonic [harmonic] on an open neighborhood of the image of oo. We shall use obvious consequences of theorems on superharmonic and harmonic functions defined on open subsets of R2 even when oo is allowed in the domains of the functions. For example, if u is a positive superharmonic
function defined on an open deleted neighborhood of oo, then u has a superharmonic extension to the full neighborhood.
The Green Function of an Open Neighborhood of oo and the Robin Constant of Its Complement
Let A be a compact nonpolar subset of R2, and define D = R2 - A. The Green function Go has a continuous extension to (D i {x}) x (D u { 00)), and we shall denote by GD(oo, ) the restriction to D of this extension with first argument fixed at oo. The function GD(oo, ) is called the Green function of D with pole oo. This function is positive and harmonic on D, is bounded
252
1. X111. Classical Energy and Capacity
outside each neighborhood of oo, and has limit 0 at quasi every finite (Euclidean) boundary point of D, and the difference has a finite limit r(A) at co. [If this difference is defined as r(A) at oo, the difference becomes harmonic on a neighborhood of oo.] The value r(A) is called the Robin constant of A. Obviously r(A) is invariant under rotation and translation of R2. The value a-"" is called the logarithmic capacity of A. Although the logarithmic capacity has the advantage of positivity, we shall see that the key set function in this context is - r(-). EXAMPLE. If A = B(O, S), then GD(oo, ) = log (J 1/b), r(A) = -logo, and the logarithmic capacity of A is therefore 0. If o a A, then [all functions in (18.1) are defined on D]
GD(oo, ) = inf {u > 0: u = log
= inf {u z 0: u = log
1 + h, h superharmonic} I + h, h harmonic}
= inf {u z 0: u superharmonic,
(18.1)
u Z log I I on a deleted neighborhood of co}.
If D is not connected the function GD(oo, ) vanishes identically on every open connected component of D except the component D. containing a deleted neighborhood of oo.
Canonical Equilibrium Potential of a Compact Set The function GD(oo, ) can be extended to a positive subharmonic function G, (oo, ) on R2, vanishing quasi everywhere on A. The measure A, on R2 associated with the superharmonic function is supported by A, We can write in fact by the Euclidean boundary of Dc,, and even by GD (co, ) = -CAA + h, where his a harmonic function on R2. In view of the form of GD near oo, it follows that AA(A) = I and [see IV(9.2)] that h has
limit r(A) at oo. An application of the harmonic function maximumminimum theorem to It shows that this function is identically r(A): so Go (oo, ) = -GAA + r(A).
(18.2)
Conversely, suppose that at is a constant, that A is a measure supported by A, that A(A) = I, and that GA < at with equality quasi everywhere on A. Then we now show that at = r(A) and ). = AA. Observe first that A and AA vanish on polar sets (Section V.11), and we can therefore ignore the subsets of A on which GA # a or GAA # r(A) in the following evaluations: a = fA GA dA.A = fA GAA dpi = r(A).
18. The Robin Constant and Equilibrium Measures Relative to R2 (N = 2)
253
The domination principle now implies that GA = GAA ; so A = A.A. The measure AA is a canonical equilibrium measure for A in the present context.
Right and Left Continuity of the Function - r(-)
The function is an increasing function on the class of nonpolar compact subsets of R2. Now let A. be a monotone sequence of compact nonpolar sets, with limit A. In view of Theorem VII.6, if A. is an increasing sequence and if A is compact, then r(A) if A. is a decreasing sequence, then lim,, r(A) A is nonpolar and lim,,... oo if A is
polar. If A is compact and polar, we therefore define its Robin constant r(A) to be + oo and its logarithmic capacity to be 0.
Strong Subadditivity of - r(-)
Let B be a nonpolar compact subset of R2, and define D = RZ - B. If A is an arbitrary compact subset of D, let vA be the reduction relative to D of GD(oo, ) on A. This reduction is harmonic and bounded on a deleted open neighborhood of the point oo; so (Section V.5) vA has an extension harmonic on the full neighborhood and therefore has a finite limit vA(oo) at oo. The restriction of the function GD(oo, ) - vA to D - A is GD_A(oo, ).
Observe that the set function A r+vA is strongly subadditive [Section VI.3(j)]. It follows that the set function A i--. vA(oo) = -r(A u B) + r(B) and so also the set function A i- - r(A v B) are strongly subadditive on the class of compact subsets of R2 - B. When B shrinks to a point, we find that is strongly subadditive on the class of compact (including polar compact) subsets of R2. Unfortunately the logarithmic capacity function is not strongly subadditive, in fact, not even subadditive. Application of Section VIII.19 According to Section VIII. 19, if D is a deleted open neighborhood of OD, R) =
7) -
on('),
(18.3)
where OD is a positive harmonic function on D, bounded on bounded subsets of D, with boundary limit 0 at quasi every Euclidean boundary point of D. Since the term on the left in (18.3) and the second term on the right both, for fixed., define functions of bounded near oo, the function 4D must have the form OD = log I I + h, where h is harmonic on D and bounded on a deleted neighborhood of oo ; equivalently, OD = GD(oo, ) + h', where h' is harmonic on D, h' is bounded on a deleted neighborhood of oo, and h' has limit 0 at quasi every Euclidean boundary point of D. Moreover h' is
254
I. XIII. Classical Energy and Capacity
bounded on bounded subsets of D because GD(co, ) is, and it follows that h' is bounded; so (Section V.7) h' vanishes identically. Thus WD = GD(oo, ) and
q) -
GD(cc, ),
(18.4)
or in view of the definition of GD, GD(00,
) = IlDIS, G(q,')) -
(18.5)
for every point q of D.
Extension of
to the Class of Analytic Subsets of R2
The set function - r(-) very nearly satisfies the conditions defining a topological precapacity but is not positive. Let A. be a compact nonpolar subset r(A0) defined on the class of of R2, and consider the set function compact supersets of A0. This set function is a positive strongly subadditive set function, and it is easy to modify the topological precapacity extension theorem, using its methods to extend - r(-) + r(Ao), and therefore to the class of analytic supersets of A0. The value -r(A) obtained in this way does not depend on the choice of the compact nonpolar subset A0 of A. Define -r(A) = - oo if A is polar. Since (from Section VI.2) every analytic nonpolar set has a compact nonpolar subset, we have now defined - r(-) on the class of analytic subsets of R2. The set function set function and is regular in the sense that
is now an increasing
-r(A) = sup { -r(B): B c A, B compact}
(18.6)
= inf { - r(B) : A c B, B open}.
The logarithmic capacity set function a-'*) is then also regular in this sense
but as already noted is not strongly subadditive, in fact, not even subadditive.
Equilibrium Measures of Analytic Subsets of R2 Let A be an analytic nonpolar subset of R2, and let A' be a compact nonpolar
subset of A. Define D' = R2 - A' and D = R2 - A. The positive function
(smoothed reduction relative to D') is subharmonic on D and can be extended to be a positive subharmonic function on R2, 0 at quasi every point of A. Denote this extension by GD (oo, ). Suppose first that A is bounded.
18. The Robin Constant and Equilibrium Measures Relative to R2 (N = 2)
255
Then Go (oo, ) is harmonic on a deleted open neighborhood of ao, and if AA is the Riesz measure associated with - GD (oo, ), this measure has compact
support, A(RI) = 1, and there is a harmonic function h on RZ such that Gp (oo, ) = -GAA + h. It can be shown that h is identically r(A). Thus (18.2) is true, and GDAA s r(A) with equality quasi everywhere on A. If A is not bounded, this reasoning must be expanded. In the first place, since GD.(oo, ) is a minimal harmonic function on D' (Section VII. 10), we must suppose (by Theorem XI.3) that A is thin at oo to ensure that is not identically GD.(ao, ). This condition on A can be shown to be necessary and sufficient for A to have finite logarithmic capacity, and then if AA is the Riesz measure associated with -Go (oo, ), it can be shown that AA(RZ) =
1, that (18.2) is true, and that therefore again GAA S r(A) with equality quasi everywhere on A.
Chapter XIV
One-Dimensional Potential Theory
1. Introduction The one-dimensional version of classical potential theory is so special that its discussion has been deferred to this chapter, and much of this theory is so elementary that it will be left to the reader to formulate and justify. A ball in R with center is an open interval with midpoint 1;, and the averages L(u, . S), A(u, , S), and Asu can play the same role when N = 1 as when N > 1, but more direct methods are sometimes clearer. Since an open subset of I8 is a countable union of disjoint intervals, it is usually possible to consider functions defined on intervals.
2. Harmonic, Superharmonic, and Subharmonic Functions A function u defined on a nonempty open subset D of I8 will be called harmonic (superharmonic, subharmonic) if on each component interval of D the function u is finite valued and linear (concave, convex, respectively). Such a function is necessarily continuous.
Every nonempty open subset of I8 except I8 itself supports a positive not identically constant superharmonic function. In the terminology of the multidimensional case F8 is the only non-Greenian open subset of F8 aside from the empty set. In agreement with the definition of a polar set when N > 1, only the empty set is defined as polar when N = 1. That is, the only negligible set in one-dimensional classical potential theory is the empty set.
3. Convergence Theorems It is trivial that the supremum of an upper directed family of harmonic [superharmonic] functions on an open subset D of F8 is harmonic [superharmonic] if finite at a point of each component interval of D. It is just as trivial that the infimum of any family of superharmonic functions on D is superharmonic if finite valued. Thus the Fundamental Convergence Theorem of classical potential theory is both true and trivial when N = 1.
5. The Dirichlet Problem (Euclidean Boundary)
257
If u is superharmonic or subharmonic on D and if B is an open subinterval
of D, relatively compact in D, the function TBu is defined as the function equal to u on D - B and linear on B. If u is superharmonic, the function TB is also superharmonic and is majorized by u. If u is superharmonic on D and has a subharmonic minorant, the function
GMpu exists as in the case N > I and is harmonic. The proof for N > 1 is applicable to the case N = 1. A superharmonic function u on a finite interval D has a subharmonic minorant if and only if the limit of u at each endpoint of D is finite, and in that case GMpu is the restriction to D of the linear function equal at each endpoint of D to the limit of u at that endpoint. A superharmonic function u on the infinite interval ]a, +oo( with a finite has a subharmonic minorant if and only if the limit of u at a is finite and the limit a of u, at + oo is not - oo. In that case GMpu is the restriction to D of the linear function with value at a the limit of u there and with slope a.
4. Smoothness Properties of Superharmonic and Subharmonic Functions We omit the elementary proofs of the following properties of a convex (that is, subharmonic) function u defined on an open interval D. u is continuous, is absolutely continuous on each compact subinterval of D, and has finite or infinite limits at the endpoints of D. (b) u has a right [left] derivate u, [u4] at each point of D. (a)
(c)
The function u, [ul] is monotone increasing and right [left] continuous.
(d)
ui S u and there is equality except at a countable subset of D. At a point of equality the derivative u' (= u; = u;) exists.
In one dimension the Laplacian of a function u is the second derivate u". A function u in C12 (D) is subharmonic if and only if Au Z 0.
5. The Dirichlet Problem (Euclidean Boundary) If D is a finite interval ]a, b[ and if f is a boundary function, it is trivial that the PWB method in the present context yields a solution if and only if f is finite valued and that for finite-valued f, Hf=PD(',f),
PD(S,{a})=b-a'
(5.1)
On the other hand, if the interval D has one finite and one infinite endpoint,
the PWB method yields a solution if and only if the specified boundary
258
I.XIV. One-Dimensional Potential Theory
function f is finite valued, say at the finite endpoint b of D. The PWB solution Hf is then identically fJ, that is, the harmonic measure of {b} is identically 1. If D is not connected but is a proper subset of I8, the PWB method is applied separately to each interval component of D, or equivalently, to D itself, but the Dirichlet problem is not treated for D = 11 because Id is not Greenian. The Dirichlet problem for conditionally harmonic functions is left to the reader.
6. Green Functions When N = I and D is a finite interval ]a, b[, it is natural to define GD as a function on D x D with the property that is harmonic on D and superharmonic on D, with limit 0 at each endpoint of D. These conup to a multiplicative positive constant. Let d1 ditions determine [d2] be the derivative of to the left [right] of . If now uE C121(b), integration by parts yields d11 = u(a)dl - u(b)d2 -
j
d2).
(6.1)
a
To simplify this formula, we specify GD completely by prescribing d1 - d2 = 1, so that
if ry5
,
if11z
,
(6.2)
a) b
-a
and then (6.1) becomes
U(0 =
u) -
16 a
-)u" d11.
(6.3)
The representation (6.3) should be compared with the corresponding representation 1(8.6). Thus if uaC/2'(D) and if u is superharmonic and positive, u is the sum of G MDu and the potential of the measure - Au d11. This example of the Riesz decomposition will be extended to the general case in Section 9.
If D is the interval ]a, + oo [ with a finite, we let b tend to + oo in (6.2) and thereby are led to the definition a) A (q - a).
(6.4)
If D = ] - oo, b[ with b finite, we define GD correspondingly. If D is a union
of two or more disjoint nonempty open intervals,
p) is defined as
259
8. Identification of the Measure Defining a Potential
q) for , q in the same component interval Do of D and defined as 0 for , I in different component intervals. Thus GD is now defined for every Greenian set D.
7. Potentials of Measures If D is Greenian and if µ is a measure of Borel subsets of D, then the potential
GDµ is superharmonic, that is, concave, on each component interval of D on which the potential is finite valued, because q) is concave for each q. We now show that GMDU = 0 when u = GDµ is a superharmonic potential.
It will follow from the Riesz decomposition (Section 9) that conversely a positive superharmonic function u with GMDU = 0 is the potential of a measure. We can assume in the following that D is an interval ]a, b[. Let BB be an increasing sequence of subintervals of D with compact closures in D and union D. Then ra.GDu is a decreasing sequence of superharmonic functions. Moreover, if e B,,, then (7.1)
TB.(GDµ)(b) =
Here n i-1 p5 (, q)) is a decreasing sequence, for n so large that e B,,, with limit 0. Hence T8 GDµ = 0. It follows that u = GDµ has limit 0 at each finite endpoint of D and that if D has an infinite endpoint, the limit of right and left derivatives u; and ui at that endpoint is 0. Hence (see Section 4) GMDU = 0. Alternatively [see the N-dimensional context in Section 111. 1, Observation (a)), the sequence TB u can be proved directly to have limit GMDU.
8. Identification of the Measure Defining a Potential Theorem. If u = GDµ is a superharmonic potential, then dµ = du, in the sense that
u,(/J) - t4(a) _ -µ(]a, fl]) for ]a, P] c D.
(8.1)
In proving the theorem we can suppose that D is an interval ]a, b[ with at least one finite endpoint. If both endpoints are finite, define functions 0, and cbb by
0a(0 =
f(i -
4b(S)
fb(b - q)µ(dq), J4
a < < b.
260
1. XIV. One-Dimensional Potential Theory
Here fs means the integral over ]a, fi]. The functions ¢, and ¢b are finite valued, monotone increasing, and right continuous and satisfy
f (b - l) d4a(q) = f (q - a) dob(q), s
e
a + 62I7I2 2SJ
is a positive parabolic function on d8". Recall that in contrast with this example there is no nonconstant positive harmonic function on 18".
Minimal Parabolic Functions
A positive parabolic function is on an open subset 6 of ft' will be called minimal if every positive parabolic function on d majorized by u is a constant multiple of is. If 1) is 9" or is the lower half-space, it will be shown in Section XVI.8 that the restrictions to b of the functions in Example (a) are minimal and in fact are the only parabolic minimal functions up to constant multiples.
264
I. XV. Parabolic Potential Theory: Basic Facts
EXAMPLE (b). If ii is parabolic on AN and is of the form
the functions f and g satisfy the equations -(r) + [(N - 1)/r] f (r) = cf(r) for r z 0 and g' = cazg/2 on R, and we thereby find the parabolic function ti defined on A' by (CI V),
u(S) = eo'a12 r 2
_ N/2) -
(N- zuz
2
I)(N2)/2J(i.JII)ec2cs/2
p
1(N-z),2(\CI, I)e°=Cs(2
where c is an arbitrary constant, Jk is the Bessel function of the first kind, and !k is the modified Bessel function. The parabolic function u/I-(N/2) can
also be obtained by taking the parabolic function in Example (a) and averaging it over the values of y with 1712 = c. This representation of ti/r(N/2)
is an example of the fact that every positive parabolic function on ftN is an integral over the set of minimal parabolic functions.
3. Coparabolic Polynomials Define the Hermite polynomial Hm...N on RN by am,+
H.,. "N(,I) =
+M N
z
e-1,11=
an(N)mN
(m)
Z 0)
so that if y is a vector in IBN, the Taylor expansion
j'
ao
e
In+rIZ = e In1' C'
y
(1)m, ..
y
(N)IN
n=0 m,+..`+mN=n ml ( ... MN!
m,.
(3.2)
yields (INn,
m
(NV"N
y m1! e z I712 _ YHm,...mN(t1). t ... MN! nOm,+...+mN=n N!
(3.3)
According to (3.1), 11m,..mN = elnl' brl
N (-2n(I)e-171=)
an(I )m, . .
. a'e)'"N
= -2mjHm....mN,
m. =mj-blf ifmj>0,
(3.4)
and this partial derivative vanishes if m1= 0; so (3.4) is valid in all cases when interpreted reasonably. Repeated integration by parts leads to
3. Coparabolic Polynomials
265
e-...rN(7)Hn,.. "N(7)1N(d7) fIRI'H. RN
m,
if m = n,,
!
=10
otherwise.
Thus the sequence of Hermite polynomials is an orthogonal sequence relative to the measure exp (- 7I2)1N(rlq), and (3.3) is the corresponding Fourier series of the left side. jyj2a2t/2) is coparabolic. Define the The function space-time Hermite polynomial ...mN, homogeneous of degree ml + + MN in the variables q"', ... , (11,1112 , by the Taylor expansion ,(IV", )
e-Irl=02hl2 = C'
n==Om,+...+mN=" ml
... y (N)"N H (... MN! m
,
.
.
. MN
(7, t).
The polynomial Hm, ...mN is coparabolic because it is obtained by repeated differentiation of the coparabolic function on the left side of (3.6). If y and q in (3.3) are replaced by (621/2)"2y and -(2621)-112 q, respectively, the left sides of (3.3) and (3.6) become identical; so Hm.mN(q, t) =
2/2 (!_)
= m)
Hm,...mN(-(2621)-1127)
(n
_
2
_ (-721)"eXP2a2t an(I)MI 711Vn, ..
7(N)MN + ,
.
.On
aq(Nr,Nexp
2721
2
(3.7)
.
The term written on the last line is the only term not involving t. The relation (3.4) becomes
m; - by ifmj>0,
(Hm,...mN -
mi= to
ifmj=0,
(3.8)
and (3.5) becomes
I
e-Iel=nc'`Hr,...-N(7+t)H",..."N(n, t)1Nd7) (a2t)n+N/2(2n)NJ2mI !
I
0
... MN!
if m, = n., otherwise.
For each value of t the sequence of space-time Hermite polynomials is an orthgononal sequence in the space variables, and (3.6) is the corresponding Fourier expansion of the left side.
266
I. XV. Parabolic Potential Theory: Basic Facts
A coparabolic polynomial f (# 0) must contain a term not involving t because
M, t) = E r"rf,(7),
rI
0:5 n t < ... < n,,
where f; is a not identically 0 polynomial in the components of 7, and f cannot satisfy the heat equation dual unless nt = 0. In view of (3.7) some linear combination H of space-time Hermite polynomials can be chosen with the same terms not involving t as f, and therefore f - H vanishes identically. Thus every coparabolic polynomial is a linear combination of the space-time Hermite polynomials. EXAMPLE. If N = 1, the successive Hermite and space-time Hermite polynomials are
Ho= 1, Ho = 1,
H3(7)= -8113+ 1211, ...,
Ht(11)= -211, H2 R, (q, t) = 7,
H,(7, t) = 7z - c21,
113(11, t) = 73 - 311o-'t, ... .
4. The Parabolic Green Function of AN Define the function t on 1'8' by
_
Al 7) =
W (2nazt)-"1z exp 2 1ht
if t > 0,
0
ift50.
(4.1)
Note that the time variable is placed first in this notation, as appropriate to the probability interpretation to be given later. The parabolic Green function 6 of 9' is defined on 1'8' x A' (N z 1) by n) = O((, S), (7, t)) = AS - t, - 0).
(4.2)
In more detail, (a) The function
h) is the Green function with pole n for the heat
equation. This function is positive, parabolic on A' - {q}, and (a*)
vanishes below q and in the limit at the point oo. The function G( , ) is the Green function with pole for the adjoint equation. This function is positive, coparabolic on A' - { }, and vanishes above and in the limit at the point oo.
267
5. Maximum-Minimum Parabolic Function Theorem
Observe that when N > 2, there is an intimate connection between the parabolic Green function of iY and the classical Green function of R': o
,s),(n, t))l, (dt) = J 1(s, - n)lt(ds) (4.3)
o zz
= aNI S -
z n) n12-N = aNG(S,
with
_ T(N/2 - 1)
aN -
2Q27rN/2
This relation between G and 6 will be generalized to Greenian subsets of R" in Section XVII.18. The following inequality will be used below. Let D(k) be a (possibly mixed)
partial derivation operator of order k on space variables, and let fi be a positive number. Then if
s) and
ID(k>Q,7)I I - III S c(fl,o)(s -
nI 2
- t)
t)-(N+k-eu2exp - 4oI2(s -
(s > t).
(4.4)
This inequality will be proved for k = 1. The proof in the general case involves more notation but no additional ideas. For k = 1 and s > I acMIT-
IC - oil* 5 (2a) "'a " _(I
_
(2x)-Nao-"-:(f -
r
3C1
20
naexp -I
(4.5)
I1KIf - MI: r+iuiexP -IC - 7I)1
and the right side is majorized by the right side of (4.4) when k = 1. If p is a measure on A", the functions Gµ and FRO defined by
O(, JUN
,G(,1) = I
O(, l)p(d)
(4.6)
.r RN
will be called respectively the potential and copotential of µ on d2;". These definitions will be generalized to potentials and copotentials of measures on an open subset 6 of A" with parabolic Green function Oo.
5. Maximum-Minimum Parabolic Function Theorem This theorem will follow from the superparabolic function minimum theorem (Section 13), but the following direct proof is instructive. Since the function
n') is parabolic on b = 9" - {q} and takes on its minimum value at
268
I.XV. Parabolic Potential Theory: Basic Facts
every point of b below 4, the maximum-minimum parabolic function theorem is necessarily weaker than its harmonic function counterpart. Theorem. Let l; be a point of an open subset D of ft N , and let D(d) be the set of points of D below C relative to D. If ti is a parabolic function on ,6 and if t (e)
is the supremum or infimum of the restriction of u to D(C), then u = ti(C) on
bg). Observation. This theorem when applied to the part of D strictly below an arbitrary horizontal hyperplane implies that if m is the supremum of ti, there is a sequence 0. of boundary points of D with ordinate values strictly less than supo ord ry such that lim lim sup u(C) = M.
n-.D p,f_Rn
Thus if b is an interval, there is a sequence in ,6 converging to a point of the union of lower and lateral boundaries of ,6 such that ti tends to m along this sequence. If u is parabolic on a neighborhood of the closure of the interval A the restriction of ti to b attains its maximum and minimum on the union of lower and lateral boundaries. In proving the theorem we need consider only suprema, and it is sufficient to prove the following result. Let S be a closed line segment in D, not orthogonal to the ordinate axis, and let m be the supremum of ti on the set of points of b below the highest point of S. We prove that ri < m on S if this inequality
is true at the lowest point of S. Let : s) be the upper endpoint of S, and to simplify the notation, let the origin be the lower endpoint of S. If u(0) < in, choose r so that 0 < r < I, so that the set
Do={(ry,t):I,1- Stl - ac, then the function u is the sum of the constant parabolic function a and of a superparabolic potential on AN, the potential Ou of the product measure p = IN x v, with dv = df.
Convergence of Families of Superparabolic and Parabolic Functions
In view of the parabolic Poisson integral for an interval it is clear that if D is an open subset of A', if kcand if ti. is a locally uniformly bounded family of parabolic functions on D, the family of partial derivatives of these functions of order Sk is also a locally uniformly bounded family of para-
bolic functions. It follows that a, is locally uniformly equicontinuous. Furthermore, if a sequence of parabolic functions on D converges locally uniformly, the corresponding sequence of partial derivatives of any pre-. scribed order also converges locally uniformly to the corresponding partial derivative of the limit function. Thus the limit function is also parabolic. An
application of Ascoli's theorem shows that a locally uniformly bounded sequence of parabolic functions has a locally uniformly convergent subsequence.
279
13. Superparabolic Function Minimum Theorem
Let u, be an upward-directed family of superparabolic functions on D with limit u. Since u is an upward-directed family of lower semicontinuous functions, there is (by Theorem 2 of Appendix VIII) an increasing sequence in the family with limit ti. The function to has the superparabolic average property and is therefore superparabolic if it is finite on a dense subset of D; the latter condition is satisfied if every point of D is below, relative to D a point of finiteness of ti. Positive Integral Operations on Superparabolic Functions. Such operations
yield superparabolic functions and yield parabolic functions if the given functions are parabolic. (See Section II.4 for the argument in the classical context.) For example, if ti is superparabolic on an open set D, the functions L(ti, , h), A(ii, , a), and Adu are superparabolic on their domains of definition.
13. Superparabolic Function Minimum Theorem Theorem. Let u be a superparabolic function on an open subset b of if
.
(a)
If ti attains its infimum at a point of b, then u is identically that infimum on the set of points below relative to D.
(b)
The infimum of u is the limit of ti along some sequence of points tending
(c)
to aD. Any lower semicontinuous extension of ti to D attains its infimum on the boundary.
Assertion (a) is an easy consequence of the fact that ti is lower semiconS) c D. Assertions (b) and z A(ti, , S) whenever tinuous and that (c) follow from (a). The dual version of this theorem for subparabolic functions will be called the subparabolic maximum theorem. Observe that Theorem 13 includes Theorem 5.
Application to Functions on Slabs
A nonempty open subset of A' that is either a half-space bounded by a horizontal hyperplane or is the intersection of two such half-spaces will be called a slab. If a slab f) has a lower hyperplane boundary and if a is a superparabolic function on D, then the infimum of ti is the limit of ti along some sequence tending either to the point oo or to a point of the lower hyperplane boundary. To show this we show that if is the part of b strictly below and if is the infimum of u on then there is a sequence i, in tending to the point oo or to a point of the lower boundary of b along which u tends to According to the superparabolic minimum theorem, there is a sequence 0. in tending to a point n of ab(4) along which u tends to we are done. If rl is on the m(4). Unless q is on the upper boundary of
280
1.XV. Parabolic Potential Theory: Basic Facts
upper boundary, then (by the lower semicontinuity of ti) u(q) :9 &;(S), and a trivial variation of the proof of the superparabolic minimum theorem shows
that u is identically m(S) on D(S). Hence there is a sequence rl, with the desired properties.
14. The Operation ig and the Defining Average Properties of Superparabolic Functions We follow the reasoning of Section 11.6 to derive the corresponding results in the present context. (a) If a is superparabolic on D and if B is an interval with closure in D, then ti is locally IN integrable on the intersection with b of any hyperplane
parallel to a coordinate hyperplane, ijti is parabolic on h, and i8ti 5 6. In fact, by lower semicontinuity and local lower boundedness of 6 there is an increasing sequence f. of finite continuous functions of aD with limit u there.
Then PI(B, f) is an increasing sequence of parabolic functions on h with limit l l(h, «), and an application of the superparabolic minimum theorem to u-PI(B, j) on h shows that this difference is positive on B. Hence (from Section 11) ieti is parabolic on h and iBti +
a2IYI2t
2
on the lower half-space. This parabolic function [without the factor (2n)-"] was noted in Section 2 and will be seen in Section XVI.8 to play the same role in the lower half-space for Poisson-integral-type representations and minimal parabolic functions that y) plays in the upper half-space.
17. Extensions of a Parabolic Function Defined on a Cylinder Lemma. Let D be an open nonempty subset of l8" (N,-:! 1), let D = D x ]a, b[
with - oo S a < b < + oo, and let v be a positive parabolic function on D. Then if a < b' < b and if d' is a positive parabolic function on D' = D x ]a, b'[, majorized there by v, the function v' has a positive parabolic extension v" to !, majorized there by v.
Observation. The lemma does not assert that the extension is unique. In the following proof we shall find the maximum extension. Let r be the class of subparabolic minorants of v on b which are majorized on D' by v' and define v" = sup {u: u E r}. If a < b" < b' and if ci is defined
on b by ti
(v' on D x ]a, b"],
1
0
onDx]b",b[,
284
I. XV. Parabolic Potential Theory: Basic Facts
then t E r. Hence 6" is an extension of v'. The class r contains tit v u2 if it contains 61 and tie, and r contains the majorant iBti of ti if tier and if B is an interval relatively compact in D. Thus ti"IB is the limit of an upward-
directed set of parabolic minorants of ti,B; so ti" is parabolic and is the desired extension of v'.
Application to Minimal Parabolic Functions Let b and D' be as in the lemma, and let v be a minimal parabolic function on D. Then 6[6, is minimal on D'. In fact, if v' is a positive parabolic minorant of ti1D., the extension of u' to D provided by the lemma must be proportional
to d; so t' is proportional to vio.. Special Case: Extension of a Bounded Parabolic Function. Let '6 and D' be as
in the lemma. Then an arbitrary bounded parabolic function v' defined on D' has a parabolic extension to D with the same infimum and supremum there as i has on b'. In proving this we can assume that v' has infunum 0. Let y = supb v'. Then the function v - y on b is a parabolic majorant of v' on L5'; so v' has a positive parabolic extension to D, majorized there by y. If D is a finite interval in IB", the preceding result follows easily from our discussion of the parabolic context Poisson integral.
Chapter XVI
Subparabolic, Superparabolic, and Parabolic Functions on a Slab
1. The Parabolic Poisson Integral for a Slab If 6 is the slab 68" x ]0, S[, with 0 < S 5 + co, the restriction to 6 x 6 of ( satisfies the rather vague description of the Green function (;D given in Section XV.7 for smooth regions. It is therefore to be expected from XV(7.3)
that the upper boundary of 6 if b < + oo is a parabolic measure null set and that parabolic measure on the lower boundary is given by uD(4, d7) = o(s,
- 7)1N(d7) =
(7, 0))IN(d7)
[=
s)],
so that if ti is parabolic on 6 with boundary function fin some suitable sense on the lower boundary and if u is appropriately restricted, then
u() = J I(s, - 7)f(7)IN(d7)
[=
s)].
(1.1)
R"
Such representations will be derived below. Moreover we shall see that if p is a suitably restricted charge on R", the Poisson integral
Jer(s. - 7)lt(d7)
}11(i),
[=
s)]
RN
defines a parabolic function on A The integral in (1.1) will be denoted by 61(i), f) and the Poisson integral will be modified in the obvious way when the lower boundary of L) is not the abscissa hyperplane. Theorem. If f is a Lebesgue measurable function on 6l" and if for some S in ]0, + oo]
f
I 1, ,
IN(d7) < +oo
(1.2)
RN
whenever S' < S, then the function ti defined on 15 = 68" x ]0, b[ by (1.1) is parabolic. Moreover, tim sup 6 - C-(G. o)
Jim sup f(7) RNa r{
(C a RN)
(1.3)
286
I XVI. Subparabolic, Superparabolic, and Parabolic Functions on a Slab
and
lira sup ii(c) < D94--ro
lim scup f(ry). neRN.l71_+x
(1.4)
On combining (1.3) with the corresponding inequality for inferior limits we find that u has limitf(C) at (C, 0) if f is continuous at C, and the corresponding specialization of (1.4) is valid. The fact that ti is defined and parabolic on
1) is trivial. Inequality (1.3) is a special case of Theorem I of Appendix VII
because if 0 < 5' < 6 and =
s),
lim
"'W'0
(C-0)
uniformly for ry outside an arbitrary open neighborhood A of C in R'; so if e > 0 and if is sufficiently near (t;, 0),
f.N-A
6(s,
- ry)I f(tl)I1N(dd) 0,a>0);
J
so if A is an open neighborhood of the point co of t 0, Inl--
then for 0 < at < /f, 61 < (2ft2)-', and L1 = 68N x ] 0, b1 [ it follows that
287
2. A Generalized Supcrparabolic Function Inequality
+oo.
lim
Note that fi 5 (2a26)-' because of (1.2). To prove this extension of Theorem 1, observe that by hypothesis there is a constant c > 0 such that f(n) z cexp(/lIgI2) for sufficiently large I4I4 say for Ini -f r. Then
z liminf c f
liminf
D, 3;-+m
D,
1(s,
- PI)exp(f I i 2 - aI I2)IN(dn)
RN (1.7)
because the part of the integral over {In1 5 r} has limit 0 when ICI - +oo. The value of this integral is (1
1 - 2fa s
which tends to + oo when I I - + oo with in D, . The proof of the extension is complete.
2. A Generalized Superparabolic Function Inequality Lemma. Let ti be a positive superparabolic function on the slab 1) = RN x ]0, S[, and define f(g) = lim inf.,., ti(C, s) for C in RN. Then ti z 01(D, f). Let Dk be the slab RN x ] 1/k, S[ fork so large that k > 1/S. Let f. be an increasing sequence of positive functions on IBN, finite valued and continuous
with compact support and with limit the lower semicontinuous function ti(-, 1/k). According to Theorem 1, the function P1(Dk,
is a positive para-
bolic function on Dk with limit f (j) at each point (i;,1 /k) of the lower z 0 on Dk by
boundary of Dk and limit 0 at the point oo; so ii - P1(Dk,
the minimum theorem for superparabolic functions on a slab (Section XV.13). The lemma follows when n -+ oc and then k - oo. Application (a). It follows trivially from Lemma 2 that
s') z f
- s, ' - )ti(4, s)lN(dd),
0 < s < s' < S, (2.1)
RN
and therefore ifs' < t < S,
f
RN
6(t - s', n - ')u(c', s')lN(dd') z
f
e (t - s, n - c)u(
,
s)lN(dc).
(2.2)
JRN
Thus the parabolic average relative to (n, t) of the positive superparabolic function ti on the hyperplane of constant ordinate value s' ( 0, ti is the limit of an increasing sequence of bounded positive parabolic functions.
The map u "f, is a one-to-one linear order-preserving map from the class of parabolic functions satisfying these conditions onto the vector lattice L' (R", l"), and dRj = f dl". As will be noted in Section XVIII.19, the notation introduced in Chapter IX for various classes of functions in classical potential theory is readily adapted to the context of parabolic potential theory. These classes together with their martingale theory counterparts are discussed in Chapter I of Part 3. In this spirit the parabolic functions in Theorem 6(a) are the L'(µ6_) parabolic functions, those in Theorem 6(b) are the D(pA_) parabolic functions, and Theorem 6(b4) asserts that the positive D(µii_) parabolic functions
are quasi bounded. A uniform notation will be used in Chapter I of Part 3 both for classes of functions in the classical and parabolic potential theory contexts and for classes of stochastic processes in the martingale theory context. The assertion of Theorem 6(b4) in all three contexts is covered by Theorem 3.1.5.
To prove Theorem 6, define PI(B(s, t), f) on $(s, t) = R x ]s, t[ by
6(- t, ' -
PI(A(s' t), f)( ', s') =
JS'
R"
Observe that Itil in (a3) and m,(Icil) in (b2) are positive subparabolic functions
with parabolic majorants. It follows that for fixed s > 0 and fixed h(0, s) the parabolic averages PI(B(s, t),
s'),
PI(B(s, t),
s') in
S)
define decreasing functions of t on the interval ]0,s'[. To see this, for example, for the first parabolic average, note that if v is a parabolic majorant of Itil on h, then (by Theorem 5) PI(E(s, t), t)) = ti on E(s, t); so (from Section 2) the function
is an increasing function on ]0,s'[. With the help of this monotoneity result the proof of Theorem 6 becomes so close to that of Theorem 11.14 in its ideas
292
1. XVI. Subparabolic. Superparabolic, and Parabolic Functions on a Slab
that the details will be omitted. Choose 6' with 0 < d' < 6, and for 0 < r < define charges t, and j on R' by z
p,(dq) = ti(q, r)IN(d)l),
ft:(dtl) = u(q, r) exp
1
2a2q r lN(dq)
It is convenient to consider j as a measure on the space I' compactified by a point at infinity, with the infinite singleton ft; null. Instead of showing directly that µ, tends to a limit measure N,; when r -+ 0, it is easier to show that under (a3) the total variation of y is bounded independently of r, that (vague)lim,.oA; = µ' exists, and that (al) is true with 12
tI (dq) = exp 2116, N (dq)
Finally observe that the supremum in (a4) defines a parabolic majorant of Iul on B. Further details of the proof of Theorem 6 are left to the reader.
7. The Parabolic Boundary Limit Theorem A function f on a slab I8N x ] 0, 6[ is said to have normal limit q at the boundary point C : (C, 0) if lim,._o f(C, s) = q. The function is said to have q whenever : parabolic limit q at (C, 0) if limf_S s) - C in a paraboloid of revolution with vertex C and opening upward, that is, whenever
C -+C with liminff.;sI C -
CL-z
> 0. In more sophisticated language, if a
subset A of the upper half-space is called a deleted coparabolic neighborhood
of C whenever for some a > 0 the set A contains the intersection of the paraboloid {s > ajC - C12} with some Euclidean neighborhood of r, then f has parabolic limit q at if f has limit q along the filter of deleted coparabolic neighborhoods. Theorem. Let B = IAN x ] 0, b[ be a slab in OWN, let 6 be a parabolic function
in L (fiB_ ), and let h be a strictly positive parabolic function on B. Then if C is a point of IBN at which the convex variational derivates dlN/d?4 and dIV,;ldN, both exist, the function ti/4 has parabolic limit (dN,;/dN1,)(C) at (C, 0).
Observe that the stated conditions are satisfied at Nh almost every point of IBN. In the more common version of this theorem h - 1; so Nh = IN. In particular, if u e D(Aj_) so that ti = 01(h, f) for some function f on R', the theorem states that ti has parabolic limit f(C) at IN almost every slab boundary point (C, 0).
Theorem 3 of Appendix VII can be applied to prove a modified version of Theorem 7, namely, that under the hypothesis that dlN/dfh and d!1 /dI'4
293
8. Minimal Parabolic Functions on a Slab
both exist as symmetric derivates at , the function ti/h has normal limit (dN,Jd!)(S) at (C,0). The proof of this modification follows the proof of the corresponding result (Theorem I1.15) in the harmonic function context.
In the latter context the Harnack inequality made it possible to go from normal approach to nontangential approach, but this step does not seem possible for general h in the parabolic context. We therefore prove Theorem 7 as an application of Theorem 4 of Appendix VII. In the latter theorem if we set 2
K(s, r) = exp
Za2.s,
6(s) = const s12,
so(a) = 2,
we obtain Theorem 7. Application. We have proved in Section 2 that if ti is a bounded parabolic function on the slab h and if ti has normal limit J ) at IN almost every point (C, 0), then u = PI(B, f ). According to Theorems 6 and 7, the weaker hy-
pothesis that the parabolic function 6 on h is in D(pa_) implies that the parabolic limit, say J(o, exists at IN almost every boundary point (C, 0) and that 6 = l'I(B, f ).
8. Minimal Parabolic Functions on a Slab (a) D = RN x ]0, 6[, 0 < b 5 + co. In view of the Riesz-Herglotz-type representation (Theorem 6) of a positive parabolic function on b, such a function is minimal if and only if it is a positive multiple of the function ( , s) i--. e (s, - C) on b for some point of R'. (b) D = I8N x ] - oo, 0[, the lower half-space. In view of the Appell transformation which takes the upper half-space of l' into the lower half-space, a parabolic function ti on the lower half-space is minimal if and only if it is the Appell transform of a minimal parabolic function for the upper half-space, that is (Section XV.16), if and only if ti is a positive multiple of the function
(.s)i - exp
L(7,
+
aZ1yJ2S1
(8.1)
on the lower half-space, for some point y of Q$N. In particular (y = 0), the positive constant functions are minimal on the lower half-space. It follows (Liouville-type theorem) that a bounded parabolic function on the lower half-space, and therefore surely a bounded parabolic function on ft', is a constant function. Furthermore the representation Theorem 6 transferred to the lower half-space by an Appell transformation yields a representation for a positive parabolic function u on the lower half-space:
294
1.XVI. Subparabolic, Superparabolic, and Parabolic Functions on a Slab
[ + a2IYI2] N(dy),
+
(8.2)
where f is a measure on R' for which exp(-ajyI2)Na(dy) < +oo
(8.3)
fR' for all a > 0. We conclude that every positive parabolic function on the lower half-space is either strictly positive or identically 0 and is monotone increasing in the ordinate variable. We leave to the reader the full formulations of Theorems 6 and 7 in the lower half-space context. (c) 6 =AN. It follows easily from (b) that the minimal positive parabolic functions on A' are the positive multiples of the functions (8.1), now considered on ft', and that every positive parabolic function a on IAN has a representation of the form (8.2) with (8.3) true for all real a. Every positive parabolic function on l'' is either strictly positive or identically 0 and is monotone increasing in the ordinate variable.
Chapter XVII
Parabolic Potential Theory (Continued)
1. Greatest Minorants and Least Majorants If D is a nonempty open subset of off" and if r is a class of functions on D, the greatest subparabolic minorant [least superparabolic majorant] of r, if there is one, is denoted by GMDT [LMT]. For example, if r is a class
of superparabolic functions and if r has a subparabolic minorant then OM j)I- exists and is parabolic. The proof is a translation of that of Theorem 111.2. The corresponding notation in the coparabolic context is OMDr and LMDr. EXAMPLE. Let b be either 119 (N z 1) or an interval in AN. Then
0 for every point in D. In fact, say for when b is an interval, the parabolic minorant in question is positive, is and so has limit 0 at every lateral and lower boundary point of D. This minorant therefore vanishes identically, according to the parabolic function maximum-minimum theorem. More generally it will follow from the Riesz decomposition of a positive superparabolic function on a nonempty open subset b of fr that the parabolic potential of a measure on b if finite on a dense subset of D is superparabolic on l) and has greatest subparabolic minorant 0. majorized by
2. The Parabolic Fundamental Convergence Theorem (Preliminary Version) and the Reduction Operation The proof of the following counterpart of the first version of the Fundamental Convergence Theorem (Theorem 111.3) in the classical context follows the proof of Theorem 111.3 and is therefore omitted.
Theorem. Let T : { tiQ, a e l} be a family of superparabolic functions on an open subset of d8", locally uniformly bounded below, and define u
5 u, lim infii(, ),
rt
(2.1)
296
I XVII Parabolic Potential Theory (Continued)
and
(a) ti is superparabolic. (b) ri on each open set on which ti is superparabolic. u IN,, almost everywhere. (c) (d) There is a countable subfamily of F whose infrmum has smoothing j.
Application : The Natural Order Decomposition Theorem
As application of this simple version of the Fundamental Convergence Theorem in the parabolic context we remark that the classical context Natural Order Decomposition Theorem (Theorem 111.7) translates directly into the parabolic context: If ri, ut, u2 are positive superparabolic functions
on D with u < iut + u2, then there are positive superparabolic functions 5 u2, u = u; + r '. The classical context proof requires only trivial changes. Observe that this decomposition and its proof are also valid for relative superharmonic and superparabolic ti; , ti4 on D for which ti; S iu t , t
functions. Alternatively the decomposition theorem for superharmonic and
superparabolic functions implies trivially the decomposition theorem in the relative contexts.
3. The Parabolic Context Reduction Operations If D is a nonempty open subset of f1N coupled with a boundary aD provided
by a metric compactification, if A c D u aD, and if ti is a positive superparabolic [cosuperparabolic] function on b, the superparabolic [cosuperparabolic] reduction of ti on .4, denoted by R" [R" ], is the infimum of the class of positive superparabolic [cosuperparabolic] functions on b which majorize ri on A n b and near A n 3D, in the sense that each function in the class is to majorize o both on A n D and on the trace on D of some neighborhood of ,4 n D. The smoothed reduction +v R" ["] is superparabolic + [cosuperparabolic] according to Theorem 2 and will sometimes be denoted by ti0" [16 -4]. As in the classical context, it is trivial that A6.4 is the inrimum
of the class of positive superparabolic functions on b which are equal to ti on A n,6 and near A n D. Thus R" = ti on A r),6, and obviously R" S on b. Let C be a point of D, let D4 be the set of points of b strictly below relative to A let ti be a positive superparabolic function on D, and let vS be the restriction of ti to Db. Then if .4 c D, we now prove that Rv (reduction relative to D) is equal on Di to the reduction relative to Di of tii on A n bt.
(This fact implies the truth of the corresponding statement for smoothed reductions.) The point is that roughly the reduction of ti below a point depends only on t below that point and on the part of the target set below that point. To prove the assertion, we need only remark that on the one
3. The Parabolic Context Reduction Operations
297
hand if ti is a positive superparabolic function on D which majorizes ti on A, then the restriction of ti to Dt majorizes t64 on A r),64 and on the other hand if u' is a positive superparabolic function on D{ which majorizes ti{ on .4 n Dc and if ti is a positive superparabolic function on ,6 which majorizes
v on A, then the function ti" equal to ti on b - D4 and to u n ti' on b4 is a positive superparabolic function on D majorizing ti on A with ti" 5 ti on
band ti'"5a'on Ds. The fact that a smoothed reduction R' in the classical context is equal quasi everywhere on A n D to v and that this smoothed reduction is unchanged when A n D is changed by a polar set is considerably weakened in the parabolic context. In fact it will be shown that a smoothed reduction A" is in general equal to to on A only up to a parabolic-semipolar set (to be defined in Section 10) and may be changed if A is changed by a parabolicsemipolar set. This weakening entails that some of the proofs of properties in the classical context cannot be used to prove the corresponding properties in the parabolic context and that it is necessary to change the order of the derivation of the properties common to the two contexts. Just as in the classical context (Section III.4), and k" increase when
A or v increases, the reduction and smoothed reduction operations are subadditive, the function A-4 is parabolic on b - A and equal to As there, and A- = when A is open. Furthermore (see Section 111.5) R" = inf{A" ,,"B: Ana D c A, J) open in DuaD}
= inf
A n 0,6 c A, h open in D u aD}
(3.1)
and
t'`=inf{J 4"B:AnabcE,J3openinbuaJ} = inf {
A BjnD : A n aD c $, f3 open in b u aD}.
(3 . 1 sm)
The counterparts of the other properties listed in Section 111.5 will be listed below in Section 16. EXAMPLE (a). Let .4 be the open upper half-space of A', let A0 be an arbitrary
subset of the abscissa hyperplane, and let v be a positive superparabolic function on ft. The parabolic reduction (relative to A") and smoothed reduction of ti on A u An are trivially v on A u Ao and 0 on the lower halfspace. Hence (by lower semicontinuity of superparabolic functions) the smoothed reduction is ti on the upper half-space and 0 otherwise. The reduction is 0 on off" - (A u,40) because according to Section 16(f) the reduction and smoothed reduction are identical off the reduction set. EXAMPLE (b). If A is a horizontal plane in 1'8", any positive superparabolic
function 6 on A' with ii z I on 4 satisfies the same inequality above 4
298
I.XVII Parabolic Potential Theory (Continued)
(Lemma XVI.2), and it follows that, for reductions relative to A , D 10A is equal to I strictly above A and equal to 0 elsewhere. Thus I 1 D'` J A = 0, and so, unlike the situation in the classical context as given in Section VI.3(h), the smoothed parabolic reduction operation is not always idempotent. [However, according to Section 16(i) this operation is idempotent if A n b is parabolic-fine open.]
4. The Parabolic Green Function Let b be a nonempty open subset of A'(N Z 1), and let be points of f). The parabolic [coparabolic] Green function with pole n [] is defined on
b by
,')-C M,O(
[G1(
(4.1)
It will be shown in this section that, corresponding to the symmetry of the Green function in the classical context, OD = Go. Thus the notation Gd is unnecessary and will not be used in later sections. The function (D will be called the parabolic Green function.
As defined by (4.1), the function tj) is positive and superparabolic on 15, is parabolic on 15 - {i }, and differs from tj) by a continuous function, and p) = 0. Conversely, these conditions uniquely determine ri). The corresponding dual remarks for Go( , ) are omitted. The definition of the parabolic Green function OD is the counterpart of the definition of the classical Green function GD, but no side condition on the domain, depending on the dimensionality, is necessary in the present context because 6 is positive for N z 1. Note that (Section 1) the present definition of Og agrees with that given in Sections XV.4, XV.8, and XVI. I when 15 is AN, an interval or a slab. In particular, OD = 6 when 6 = AN. The properties assinged to O in the smooth region context discussed in Section XV.7 make there the coparabolic Green function with pole , by an easy
(notation of application of the coparabolic maximum theorem to Section XV.7). This is as it should be because, as we shall now show, lip = GD.
Proof that t o = GD. Define 6(-,4) = rj), and let h, be a sequence of intervals with closures in 15 and with the property that each point of 15 has a neighborhood which lies in h. for infinitely many values of n. Define
7) = t8....
t)ib
to obtain a decreasing sequence ti,(-, ri) of superparabolic functions on I) with limit ti(-, q) (cf. the corresponding discussion for the classical context in Section VII.4). Let 'm be a point of 15. The sequence h, can be chosen in
299
4. The Parabolic Green Function
such a way that 4o e $o and that there is a neighborhood of 00 which is either
a subset of h, or of 6 - 1) for each n. For in 6 - {qo} the function O( , ) is coparabolic on a neighborhood of 0, so the functions 60(x, ), ... are also. In fact each of these functions after the first is an integral average of its predecessor, averaged over values of the first argument. Thus is coparabolic on I) and therefore u(4, ) 5 GMDO(, ). Define as the right side of this inequality so that u 5 L. If we had begun with u instead of ri and carried through the dual argument, we would have obtained the reverse inequality, and it follows that u = u; that is,
4)() = GMrid(4, )(4),
OD = GD
(4.2)
The counterpart of the proof in Section VII .4 that u is continuous on D x D proves that ti is continuous on 6 x 1). We shall use the fact (cf. Theorem VII.3) that 6D(4, ) and 4) are bounded outside neighborhoods of their poles. We leave to the reader the easy translation of the extremal properties of Green functions in the classical context (Theorem VII.2) into the parabolic context.
Relativization of the Green Function The reasoning used in the discussion of the Green function can be relativized
with no change in detail, just as in the classical context (Section VII.1) to use an arbitrary nonempty open subset of A' as a reference set instead of A". That is, if 6 and a are nonempty open subsets of dl" with I) c f, then 4) =
4) -
4),
OD(
,)_ (4.1')
on 6 and GM
4)(4) = GMb
-)w-
(4.2')
EXAMPLE. Suppose that D is a half-space of RN, define 1) = D x R, and if ry' a El, denote by 4' the reflection of 4 in ab. Then 4) =
4) - Q, h)
(4.3)
because an application of the superparabolic minimum theorem shows that 4') = on El. More generally, if j = 1, . . . , N successively and if for each j the set Dj is either R or a half-space of R, define N
D) x R, ri i=1
6, = Df x R.
300
1. XVII. Parabolic Potential Theory (Continued)
Then the function
(4, 0 =
s), (h'u, .. , nlN1, t))
fin( , q)
is the product of the Green functions for N = I of Dt , ... , D" written in the respective variables ( ( o h ), s), (,7(l), t)),
... ,
s), (ry1"1, t))
Strict Positivity Set of (;o
n) > 0 if and only if q is in the set bi of points of b strictly below relative to D. To see this, observe first that the inequality 0 S OD 5 6 implies that C;o(, ry') = 0 if ord ry' ord . Next observe that (coparabolic maximumminimum theorem) if il) = 0 and if q' is strictly above p relative to D, then fj') = 0. Since 0 at points arbitrarily close to and strictly below g, it follows that q) > 0 whenever n e Dt. If now we fix
and define u(r) = Gp( , q) when' a AA and define ti(q) = 0 when , e b A. then ti is cosuperparabolic on D and 66(4, ) - u is a positive coparabolic function on D. It follows that this difference vanishes identically; so Dr is the strict positivity set of ), as asserted. Extensions and Contractions of Green Functions
If is a point of b, then 6os is the restriction of 6j5 to Dz x Di. In fact this restriction has the required properties except perhaps the property that 0 when e Dt (or the equivalent dual property). Now if h is a positive coparabolic function on D{ and is a minorant of v(, ), then h = 0 at the points of D; with ordinate values zord 4; so if h is extended to D by defining h = 0 on D - D{, the extended function is a positive coparabolic minorant of C;D( , ) on D and so vanishes identically. Hence h =- 0 on b, so 6m 0, as required. A similar dual argument shows that the
Green function of the set to of points of D with ordinate values strictly greater than ord is the restriction of Gp to Da x to. An observation in the reverse direction will be useful. Suppose that is a finite Euclidean boundary point of b, that every point of D is strictly below and that the part of some open neighborhood a of strictly below is in D. Then for, in b the function O q) has the extension 66,j(-, ry'), parabolic
onDvh-{n}. 5. Potentials
If ,6 is a nonempty open subset of dA' and if p is a measure on D, the functions n)li(dh),
Jd
A6,6 =
Jd
5. Potentials
301
are, respectively, the (Green) potential and copotential of A. Since the properties of copotentials follow trivially from those of potentials, we shall consider only the latter unless the interplay between the two is involved. The potential 06p is lower semicontinuous (Fatou's lemma) and has the super-
parabolic function average property (Fubini's theorem) so C,p is superparabolic on D if to each point n of b corresponds a point of finiteness of (p above n relative to D. If superparabolic, the potential O,,p is parabolic off the closed support of A. If Fi(D) < + oo, the counterpart of an argument in Section IV. I shows that 6,6A is superparabolic on D. In particular suppose that p has compact support A in D, and let a be a neighborhood of A. Since
t;D is bounded on the set (D - E) x A it follows that the superparabolic potential (`. p is bounded on 15 - $ and has limit 0 at every boundary point
r; of D for which limi_i OD(i,,) = 0 when , e A. If 6,611 is a superparabolic potential then 6Md(`iDp = 0 by the counterpart of the proof of the corresponding classical context result (Section IV.3). EXAMPLE (a). Let u be the indicator function of the upper half-space A of f". Then u is a superparabolic function on D = Ift" and is the potential of the measure IN on the abscissa hyperplane. This potential is discontinuous and vanishes on the support of its measure. Thus the domination principle (Theorem V.10) is false in the parabolic context. A trivial variation of this example in which the measure has compact support shows that the Evans-
Vasilesco theorem (Section V.8) is also false in the parabolic context. Versions of the domination principle adapted to the parabolic context will be proved in Section XVIII.16. EXAMPLE (b). Define L5 = R" x ]0, b[ and D' = R" x ] - oo, S[ with 0 < 6
S + co. Recall (Section 4) that ( [OD.] is the restriction of G to b x D [D' x b']. Let 6 be a positive superparabolic function on D with associated Riesz measure i and extend o to v' on D' by setting v' = 0 on D' - D. Then v' is superparabolic and v is the projection on b of the Riesz measure v' associated with v'. For example, if 6 = + co and v =- 1, the function v' is the potential
ti on ft' discussed in Example (a). Now suppose that v is a potential, v = (GDv = G`D.v on D. Since G,y.v = 0 on D' -,6, it follows that v' = OD.v, and
this representation shows that D' - D is v' null. On the other hand, if v is parabolic there is (by Theorem XVI.6) a measure Na on the abscissa hyperplane such that v' = Thus in this case NN is the projection of V on the abscissa hyperplane. We have now proved that whatever the choice of v on D, the extension v' to D' is a potential. One way of phrasing the fact (Theorem XVI.6) that a positive parabolic function on the slab b is given by a Poisson-Stieltjes integral is to state that every such function when extended by 0 to 15' becomes a potential. This fact suggests that one way of deriving the Poisson-Stieltjes representation is to prove directly that the extended function v' is a potential; it is trivial that the corresponding Riesz measure must be supported by the abscissa hyperplane.
302
I.XVII. Parabolic Potential Theory (Continued)
(c). Consider the following potential (p in ft', for which p is
ExAMPL
supported by the surface {(n, t): t = -In14, Inl < 1), fj-,0111 ti(0, 0) is strictly less at the origin than its limit along the abscissa
303
6. The Smoothness of Potentials
hyperplane. This concept of thinness, the counterpart of the classical concept,
will be given a topological interpretation (the parabolic-fine topology) in Section 9.
Dependence of dop on b
(Cf. Sections IV.1 and VII.5.) If b, c t3Z, the difference dD= - 66, on 15, x 6, is a positive function, parabolic in the first argument and coparabolic in the second. If p is a measure on b, whose potential (,D=p is superparabolic, then the difference GD= p - rro, p if defined suitably at the common
infinities of the two potentials is positive and parabolic on 1, .
6. The Smoothness of Potentials The following theorem is the parabolic context analog of Theorem 1.7. In view of the dependence of lop on 6 described at the end of Section 5 the smoothness conclusions of the theorem are applicable in their obvious adaptations to potentials 66p as well as Op. Theorem. Suppose that dp = f dl"+, , where f is 1"+, measurable and is supported by a slab R' x [a, + oo [ with a > - oo. (a)
(b)
If f is bounded, the potential ti = Gp is finite and continuous on and has continuous first partial derivatives with respect to the space variables, given by formal derivation of the integral defining ii. If in (a) f is continuous and satisfies a uniform Lipschitz condition of exponent fi, 0 < fi 5 1, in the space variables, I f(n, s)
s) I s const In - I0,
(6.1)
then 6 has continuous second partial derivatives in the space variables and a continuous first partial derivative in the ordinate variable, given by 021i(S)
=
0
026(S,h
[An, t) -f(
, t)]1"+, (dn) s)
aas )= f',N as
Hence dti = -f
(6.2)
(6.3)
1.XV11. Parabolic Potential Theory (Continued)
304
(n, t))lN(dtl) 5 1, the potential u is bounded under Proof of (a). Since the hypotheses of (a). To prove continuity, observe that if lib is defined by
dt f
ub( ) = f
(6.4)
(n,
the function lib is continuous (dominated convergence theorem), and
u - bl ti al
oi(), then (reduction relative to ft') tiO" = 0 even though A is not parabolic polar. If A is parabolic polar and is a point not in A, there is a positive superparabolic function finite at but identically +oo on A. This result implies, as in the classical context [Section VI.4(c)], that if ti is a positive superparabolic function on an open subset D of I(81P and if At and A2 are subsets of b differing by a parabolic-polar set, then the reductions of a on 4, and A2 coincide off the symmetric difference of these sets, and the smoothed reductions of ti on 4, and A 2 coincide on D. The classical context extension Theorem V.5 translates directly into the parabolic context along with its applications. For example, if ti is subparabolic on D and if A is a closed in b parabolic-polar set, null for the measure associated with ti, then LM ti relative to D and LM a relative to b - A are equal on b - A. It follows that Go = GD_,A on (D - A) x (D - A). The extended superharmonic function minimum theorem (Theorem V.7) becomes (for all N): If ti is a lower-bounded superparabolic function on an open subset D of 11 and if at parabolic quasi every finite point of aD and also at = oo if D is unbounded, lim inf,,.i u(q) c, then iu > c on D. Minimality of
r;)
In the classical context it was proved in Section VII. 10 as an application of an extension theorem that the restriction of GD(C, ) to D - {C} is minimal harmonic, and it then followed easily that for v positive and superharmonic
307
B. Parabolic-Polar Sets
on D, with associated Riesz measure v,
inf
v
= v({t;}).
(8.1)
D-g) GD(S, )
The corresponding argument in the present context shows that the restriction
of
to D - {C} is minimal parabolic and that if r is positive and
superparabolic on D, with associated Riesz measure v, inf
U(n)
= v({ }).
(8.2)
h:GDla.c)>o) 6,6(4,
Similarly the restriction of GD(C, ) to b is minimal coparabolic, and the dual of (8.2) has the obvious formulation. EXAMPLE (a) (Classical-polar versus parabolic-polar sets). Let A be a subset of URN with projection A on the abscissa hyperplane. If A is classical polar,
then A is parabolic polar, and in fact A x R is parabolic polar because a superharmonic function on R' identically + oo on A can be considered as a superparabolic function on 118" (Section XV.15) and as such is identically + oo on A x P. In particular, when N > I a line in ft' parallel to the ordinate axis is parabolic polar. Conversely, it will be proved in Section XVIII.11 that A is classical polar whenever A x P is parabolic polar. EXAMPLE (b). If N Z 1, a singleton is parabolic polar. In fact, if N > 1, this
follows from Example (a). To prove the assertion for N Z 1, suppose that the singleton is {0}, and let p be the measure supported by the set of points J(0,-2 -k) , k z 1) with p { (0, -2-k) } = k-'. The superparabolic potential Gµ is +oo at the origin; so {0} is parabolic polar. It follows that countable subsets of libN are parabolic polar.
EXAMPLE (c). Let A be an IN measurable subset of R, with l r(A) > 0. The set A x {0} is not parabolic polar in L N because a positive superparabolic function on tjN, identically +oo on A x {0}. would be identically +oo on the upper half-space (Lemma XVI.2).
Coparabolic-Polar Sets A coparabolic-polar subset of 9' is defined as a set satisfying conditions (a) and (b) at the beginning of this section with "superparabolic" replaced by "cosuperparabolic." That is, a set A is coparabolic polar if and only if its reflection in the abscissa hyperplane is parabolic polar. The application of this section to coparabolic-polar sets is obvious. It will be proved in Section XVIII.11 that a set is parabolic polar if and only if it is coparabolic polar.
308
1. XVII. Parabolic Potential Theory (Continued)
9. The Parabolic-Fine Topology It will be convenient to introduce the fine topology in the parabolic context at a much earlier stage than in the classical context. In the following discussion otherwise unspecified topological concepts refer to the Euclidean topology.
In the present context there are two fine topologies, the parabolic and the coparabolic. All concepts relative to the parabolic-fine topology are trivially translatable into the corresponding concepts relative to the coparabolic-fine topology by a reflection of U8" in the abscissa hyperplane; so no separate discussion of the coparabolic-fine topology is given. The parabolic-fine topology of AN (N a 1) is defined as the coarsest topology of 68' making every superparabolic function (equivalently, every upper-bounded superparabolic function) on 9" continuous. Since linear functions of the space coordinate functions are parabolic and since the r,, ordinate function is superparabolic, it follows that every Euclidean open interval in Id" is parabolic-fine open. Hence the parabolic-fine topology is at least as fine as the Euclidean topology, and in fact it is finer because there are superparabolic functions discontinuous in the Euclidean topology; for example, the indicator function of the (open) upper half-space is superparabolic and thus is parabolic-fine continuous. The parabolic-fine continuity of this function implies that no point of the abscissa hyperplane is a parabolic-
fine limit point of the upper half-space; that is, the closure of the lower half-space is a parabolic-fine neighborhood of each point of the abscissa hyperplane. This result can be strengthened as follows. Let v be the potential on AN defined in Section 5, Example (c), for which ti = + ao on the upper
half-space but u < + oo otherwise and u has limit + oo at the origin along the abscissa hyperplane. If fi > 6(0), the function v A $ is superparabolic and at the origin is strictly less than its limit there along the closure of the upper half-space. Since t3 A fi is parabolic-fine continuous, it follows that the origin
is not a parabolic-fine limit point of the closure of the upper half-space; that is, the lower half-space is a deleted parabolic-fine neighborhood of the origin and so also of every point of the abscissa hyperplane. This fact implies
that the abscissa hyperplane, in fact each hyperplane parallel to it, has no parabolic-fine limit point even though the hyperplane is not parabolic-polar. Recall however that according to Theorem XI.6, every nonpolar set in the classical context has a fine limit point. A more elegant example showing that the lower half-space is a deleted parabolic-fine neighborhood of each point of the abscissa hyperplane is furnished by the Green function G. According to the dual of Theorem XVIII.l4(f) (whose proof depends on an analysis of reductions not available at this stage), for fixed in (IN, pf lim 6(4,
+ co.
nom{
Thus the set {il: 6(4, n) > c) is a parabolic-fine deleted neighborhood of for every constant c.
309
10. Semipolar Sets
A set will be said to be "parabolic thin" at a point if the point is not a parabolic-fine limit point of the set. Limit concepts relative to the parabolicfine topology will be distinguished by the prefix pf, for example, pf lim sup. The set of parabolic-fine limit points of a set A will be denoted by ADf. An asterisk will be added for the corresponding coparabolic concept : p*f lim sup, AA'f.
The following properties of the parabolic-fine topology are derived in essentially the same way as the corresponding properties of the classical fine topology and the proofs are therefore omitted. (a) The restriction of the parabolic-fine topology to an open subset 1)
of A' is the coarsest topology making continuous every superparabolic function with domain D. (b) The parabolic-f ine topology has a basis of (Euclidean) compact sets. (c) The parabolic-fine topology has the Baire property that the intersec-
tion of a sequence of parabolic-fine open parabolic-fine dense sets is parabolic-fine dense. (d)
If A has limit point 4 then A is parabolic thin at if and only if there is a superparabolic function ti defined on a neighborhood of 4 such that lim inf u(q) >
(9.1)
A a 4-t
A point is a parabolic-fine limit point of a set A if and only if is a Euclidean
limit point of A and if each superparabolic function ii defined on an open neighborhood of has ti(4) as a cluster value at along A. For example, according to XV(14.1) and the example in this section, if A is the complement
of an 1N+, null set, the part of A strictly below a point of ll' has that point as parabolic-fine limit point. Hence the space AN has no parabolic-fine isolated point, and it follows that a nonempty parabolic-fine open subset A of 9' is parabolic-fine dense in itself, A c A'f. If ,4 is a set parabolic thin at and if D is an arbitrary open neighborhood of , there is a potential ti = 6» N satisfying (9.1) for which p has compact support in b and the left side of (9.1) is +oo. Thus a parabolic-polar set has no parabolic-fine limit point. As already noted, the converse is false.
10. Semipolar Sets It was proved in Section XI.6 that in the classical context a set is polar if and only if it has no fine limit point, but it was pointed out in Section 9 that although a parabolic-polar set has no parabolic-fine limit point, the abscissa hyperplane of Il' is not parabolic-polar even though it has no parabolic-fine limit point. The following definition is therefore natural. A subset 4 of 118N will be called parabolic-semipolar [coparabolic-semipolar] if
310
I.XVII. Parabolic Potential Theory (Continued)
A is a countable union of sets each of which has no parabolic-[coparabolic-]
fine limit point. If "parabolic" ["coparabolic"] is omitted here, that is, if the definition is applied in the classical context, a semipolar set is polar so the terminology "semipolar" is not used in the classical context. It will be shown in Section XVIII.12 that a parabolic-semipolar set is necessarily
coparabolic semipolar, and conversely, and that such a set is IN,, null. Trivially a countable union of parabolic-semipolar sets is parabolic semipolar. Thus a countable union of horizontal hyperplanes is parabolic semipolar. Such a union may have parabolic-fine limit points however. A parabolic-semipolar set A is the union of countably many parabolicfine nowhere dense parabolic-fine closed sets; that is, the complement ofA is the intersection of a sequence of parabolic-fine open parabolic-fine dense sets, and (Baire property) the complement of A is therefore parabolic-fine everywhere dense. It follows that if u and ti are superparabolic functions on some open set band if u = ti (or ti 5 v) up to a parabolic-semipolar set, then the relation is true everywhere on D.
11. Preliminary List of Reduction Properties Let I) be an open subset of A', coupled with a boundary aD provided by a metric compactification of D, and let A be a subset of ,6 v aD. In this section we prove certain basic reduction properties, some under restrictions to be removed or weakened later (see Section 16). We shall use repeatedly the fact that in view of the preliminary version of the Fundamental Convergence Theorem in Section 2 an equality or inequality between unsmoothed reductions relative to D is true IN,, almost everywhere on D for the smoothed reductions and is therefore true everywhere on D for the latter. (a) If v is a positive superparabolic function on D, finite valued and continuous at each point of A r),6, then
R"=inf{RB:AcE,Al open in 6v at}.
(11.1)
See the proof of the corresponding fact in the classical context in Section 111.5(e).
(b) If A and h are open subsets of D with A c h and if v is a positive superparabolic function on b, then OllO"QB =
(11.2)
Just as in the classical context (Section VI.4), in view of the fact that the smoothed successive reductions of v on A and h in either order lie between Q 16lAOA and QtoO", it is sufficient to prove that (11.2) is true when A = B.
To prove this, we need only observe that since flvV" = ti on A a parabolic function a on b majorizes 6 on ,4 if and only if u majorizes Q6" on A.
311
1 1. Preliminary List of Reduction Properties
(c) If u and t are positive superparabolic functions on b and if A is an open subset of D, then
R"+
RA +A.4
(11.3)
See the proof of the corresponding fact in the classical context in Section VI.4. Property (c) will be extended in Section 16(g) to cover countable sums of positive superparabolic functions and arbitrary subsets A of b u aD. (d) If v is a finite-valued continuous positive superparabolic function on
D, the set functions A F, R- and A - !r' are strongly subadditive on the class of subsets of D u ab. Property (d) will be extended in Section 16(k) where the restriction that v be finite valued and continuous will be dropped.. To prove (d), we first choose open subsets A and h of D, set v' = Its A Ite, and prove AAuB + AAvB = RA + RB.
(11.4)
The classical context proof of this reduction equality in Section VI.4 needs no change except simplification : "quasi everywhere" is to be replaced by " = v on A n B, so "everywhere." Moreover, when A and h are open, R"$ AA ,j Thus (11.4) implies the validity of the strong subadditivity inequality R ire + R
+RO
(11.5)
when A and B are open subsets of D. In view of (a) this inequality is valid for arbitrary subsets of D u aD, and the corresponding inequality for smoothed reductions, true IN+t almost everywhere on D, must be true everywhere on D. (e) If A, is an increasing sequence of subsets of D with union A and if v is an increasing sequence of finite-valued positive continuous superparabolic functions on b with finite-valued continuous limit 6, then
Iim R"R = I. _a
(11.6)
and the corresponding equation (I 1.6sm) for smoothed reductions is also true. [See Section 16(e) for a stronger property.] It is sufficient to prove (11.6). This limit relation is trivial on any open connected component of b on which v vanishes identically; so we suppose
from now on that
ti is strictly positive. First suppose that each set A is open. Then (11.6) coincides with (I I.6sm) and the limit function v' _ is superparabolic on b, equal to v on A. It follows that 6' z R".
k
The reverse inequality is trivial so (11.6) is true for open sets. In the general case choose in D, c > 0, a > 1, define = A, n If, < and apply (a) to obtain an open superset B; of Aj satisfying
312
1. XVII. Parabolic Potential Theory (Continued)
2 c.
AP(e)
z
hti
a(
) z inf u
"B
U
sup ti A'B
Z
inf u vOAnB(S)s
from which inequality the lemma follows. Just as in the classical context [Section XI.3(a")], we shall need a slight variation of (12.1): if A is parabolic thin at t , then lim
(12.1')
0.
a"
Since sweeping has not yet been treated in the present context, the method of proof in the classical context is not applicable. To prove (12.1'), choose e > 0, let ut be a positive superparabolic function on D with tit(s) _ +oo, and let Bo be a neighborhood of . Then au(S) 1
1
fit
(12.2)
oA ao
infii
sup t3. t
314
I.XV11. Parabolic Potential Theory (Continued)
Apply (12.1) to find ho so small that the first term in the second inequality
is at most t;/2, and with this choice of h, observe that the second term is at most E/2 if h is sufficiently small, since u, is continuous at
.
Parabolic-Fine Limits and Cluster Values
A function ti from a deleted neighborhood of a point of A' into a metric space has parabolic-fine limit [parabolic-fine cluster value] a at if and only if ti has limit a at along a subset of A' which is a deleted parabolicfine neighborhood of [is not parabolic thin at r;]. The proof is a translation
into the parabolic context of that of Theorem XI.9, using the obvious parabolic context version of Lemma XI.8.
13. The Parabolic Fundamental Convergence Theorem Theorem. Let f : { ua, a e ! } be a family of superparabolic functions on an open subset D of IB", locally uniformly bounded below, and define ti = inf.E t 6, Then u < ti,
ti(p) = liminfti(q) = pflimti(q),
r{
(13.1)
n-4
and
(a) u is superparabolic, (b) u = u on each open set on which ei is superparabolic, (c) ti = u except on a parabolic-semipolar set, (d) there is a countable subfamily of r whose infmum has smoothing u.
Conversely, i/',4 is a parabolic-semipolar subset of D, there is a decreasing sequence u, of positive superparabolic,functions on D with limit ti such that
>onA. +
Assertions (a), (b), (d) and the first equation in (13.1) are contained in Theorem 2. In view of (d) it will be assumed in the proof of the direct half of the theorem that ti, is a sequence of parabolic functions, and it can even be assumed, as in the classical context (Section 111.3), that these functions are positive. We now choose r, and r2 with r, < r2, define A,
{ti > r2, r, > ti},
choose in D, and show that the set A = A, ,2 is parabolic thin at . If is not a limit point of A, the assertion is trivial. If C is a limit point of A, then on a sufficiently small neighborhood h of l; we have i+ > u(C) - (r2 - r,)/2 by
315
13, The Parabolic Fundamental Convergence Theorem
lower semicontinuity of ; so ti
Z u>
+r2
2
r'= c
(13 2)
on A n B. Hence (reduction relative to b) u ck ^B and to Z ch ne. Since and therefore in view of Section u(C) < c, we conclude that 1 > 11(f) we conclude that A is parabolic thin at every point i; of D. Since each
function u, is parabolic-fine continuous, the function u is parabolic-fine upper semicontinuous; so ti(C) z pf lim sup ti(q) z pf lim inf ti(q) z q-
y-(
u(O).
(13.3)
If there were a point a in b at which the set;ond and fourth terms in (13.3) were unequal, there would be two numbers, r, and r2, strictly between the values of these terms at e, with r, < r2. Since is parabolic-fine continuous, u the set A,,,2 would not be parabolic thin at C, and consequently there can be no such point C. Thus (13.1) is true. Finally (c) is true because
{u > u} = U ,4
(r, , r2 rational).
(13.4)
Conversely, suppose that ,4 is a parabolic-semipolar subset of D. To prove
the converse half of the theorem, write A = Uo Ak with Akfn D = 0, and let h be a sequence of open subsets of D forming a basis for the Euclidean topology of D. Define A,,,,,= 4knhmn{Rlk^Bm < l} so that (Lemma 12) Akn, C (AA- < A;km},
U Ate, = .4k. M-0
Apply (d) to find a decreasing sequence 6,b of positive superparabolic functions, each at most 1 and equal to I on A,,,,,, with limit t9k,,,m for which AA_ The sequence ti, defined by +a = A
n=
tjk,,,,,2- k-m k.m
has the properties described in the converse half of Theorem 14. In fact ti is a monotone decreasing sequence of positive superparabolic functions with
316
I. XVII. Parabolic Potential Theory (Continued) 2_*_1
6
so up to a parabolic-semipolar set and therefore everywhere on D. Hence v > v on A.
14. Applications of the Fundamental Convergence Theorem to Reductions and to Green Functions Application to Reductions According to the parabolic Fundamental Convergence Theorem, a parabolic reduction R" satisfies the relation A" = A- up to a parabolic semipolar set. In view of the parabolic-fine continuity of superparabolic functions, A. = ti
on (A u.,40) 0 ,6, and this inequality combined with (13.1) implies that R" = Ir,"; = v on A"'r b; the latter fact was proved under stronger hypotheses on v in Section I1(f).
A" _
tl
10
on A, elsewhere,
z 0), then
ord
EXAMPLE. If D = iV and if A = "
t on the upper half-space, 0
elsewhere.
Thus in this example if t; is strictly positive, the set {A" > AA) is the abscissa hyperplane, which is parabolic semipolar but is not parabolic polar.
Application to Green Functions
The counterpart of the argument in Section VII.4 showing by way of the classical context Fundamental Convergence Theorem that the classical context Green function with a given pole has limit 0 at quasi every finite
Euclidean boundary point shows in the parabolic context that d b(-, 0) [O
has limit 0 at every finite Euclidean boundary point except possibly
for those boundary points in some parabolic-semipolar [coparabolicsemipolar] set. (It will be shown in Section XVIII.12 that a subset of A" is parabolic semipolar if and only if it is coparabolic semipolar.) This boundary limit result may be vacuous, however. For example, if b is the upper halfspace, the finite part of the Euclidean boundary is the abscissa hyperplane which is both parabolic-semipolar and coparabolic-semipolar. Actually in this case since 0 when ord 5 ord 4, it is trivial that b(-, q) has limit 0 at every point of the bounding hyperplane.
317
16. Parabolic-Reduction Properties
15. Applications of the Fundamental Convergence Theorem to the Parabolic-Fine Topology Application to the Smoothness of a Parabolic-Semipolar Set
A parabolic-semipolar set is a subset of a Borel semipolar set which is a countable union of Borel sets each of which has no parabolic-fine limit point.
In fact on the one hand according to Theorem 13 (converse assertion), a parabolic-semipolar set is a subset of the exceptional set {u > u} in some application of the theorem, and on the other hand according to the proof of Theorem 13, the exceptional set {u > } in each application is a subset of a countable union (13.4) of Borel sets each of which has no parabolicfine limit point. Character of Av1
The counterpart of the argument (Section XI.6) that in the classical context the set Af is a Euclidean Gd set and that A - Af is polar yields in the parabolic context that 4Pf is a Euclidean Ga set and that A - 4Pf is parabolic semipolar. Although the set Af is fine perfect, the set 4Pf need not be par-
abolic-fine perfect. For example, if A = U i {ord = - I/n}, the set 4Pf is the abscissa hyperplane, which has no parabolic-fine limit point.
Borel Measurability of the Parabolic-Fine Limit Superior Function
If ti is a function from an open subset of A' into A, the function i-. pf lim supo.t a (il) is Borel measurable (and therefore the corresponding inferior limit function is also Borel measurable) because if a e R, the set {
:pflimsup u(q)>a}= R--c
U :ri(d)za+
, =1
Pf
n
is a countable union of Ga sets.
16. Parabolic-Reduction Properties The list of reduction properties in this section includes for completeness some already discussed. The reductions are relative to an open subset I) of A", provided with a boundary 8L) by a metric compactification. The sets on which positive superparabolic functions are reduced are subsets of ,6 v a1), and no further hypotheses not stated explicitly are imposed on either functions or sets. Proofs are given in Section 17 and consist merely
318
l XVII. Parabolic Potential Theory (Continued)
of the reference to the proof in the classical context when the latter proof requires only translation into the present context. We stress that every property in the following list is a property of reductions in the classical context also, in which "(super)parabolic" is to be interpreted as "(super)harmonic" and "semipolar" as "polar." Some of the proofs given in the present context are unnecessarily indirect for the classical context but have the advantage that they are applicable in many general contexts. (a)
If o'=u-R""6,then RA = RA ab + RAnD v
(16.1)
and the corresponding equation (16. I sm) for smoothed reductions is also true. (b)
R"SIi'"S6onA, R" = v on (A u A°J) n D, It" = RA = d on A°f n D, in particular on the parabolic-fine interior
of AnD. R" = A-4 on b when A n 1) is parabolic-fine open.
It" is parabolic on D - A and equal to t there. j "( ) = lim inf . R"(p) = pf lim,_S R"(ry'). (c)
[See also (3.1) and (3.lsm).] If c is finite valued on A n D, then
R" = inf {! : A c h, h n D is parabolic-fine open, B contains a neighborhood of A n 8D1.
(16.2)
If in addition v is continuous at each point of A n,6, the set B in (16.2) can be restricted to be open in D v oD. (d) If A, and AZ differ by a parabolic-polar subset B of b, then R" ' = Ro
onD-Band
(e) If A, is an increasing sequence of subsets of D with union A and if v, is an increasing sequence of positive superparabolic functions on b with superparabolic limit v, then
lim R"R =A-4 ,
(16.3)
and the corresponding equation (16.3sm) for smoothed reductions is also true. If v = v for all n, then (16.3) and (16.3sm) are true for A, an increasing sequence of subsets of D v 8D. (f) R" 5 R" on ,6 with equality on ,6 - A, and also equality on A n D up to a parabolic-semipolar set. (g)
If ti = Ea v is a superparabolic sum of positive superparabolic
functions on D, then
319
16. Parabolic-Reduction Properties
(16.4) 0
and the corresponding equation (16.4sm) for unsmoothed reductions is also true. (h) If ,4 c aD and if v, is a decreasing sequence of positive superparabolic functions on b with limit v, then R=i . (i)
If A c R c D u aD and if A r),6 is parabolic-fine open, then
1 O61"QB = 1v0". If in addition the set Rn D is parabolic-fine open, then
mOB .=H (j)
If A n D and $ n D are parabolic-fine open and if v' = R? A R8,
then R'4"8 + R''6. "B = R" V
V
+ J.
(16.5)
U
(k) The set functions !Y and :. are countably strongly subadditive on
the class of subsets of b u aD. (1) (Strengthening of the assertion in (f) that A' = R'' on D - A.) If e > 0 and if C is a compact subset of b - A, there is a positive superparabolic function ti on b, equal to v on ,4 n b and near ,4 n aD and satisfying the inequality to 5 A" + e on C. (m) If A is a relatively compact subset of D, then 6MDAA = 0; that is, A' is a potential. (n) If v is a potential on D, then RA = RAn° and AA = A'6. (o) If v is finite valued and continuous, then for every point of ,6 the set function is a Choquet capacity on DuaD relative to the class of compact subsets of b u aD. Observation. In the classical context (Section VI.5) the set function
was shown to be a Choquet capacity on D u aD relative to the class of compact subsets of D u aD when v is finite valued. In the present context finite valuedness of v is not sufficient for the validity of (o) according to the following example. Let A be the closure of a ball in R", and let A be the subset A x [0, I/n] of 9". Then A, is a decreasing sequence of compact subsets of f8" with intersection A = A x {0}. Let v be the indicator function of the upper half-space. Then (reductions relative to D = AN) the reduction R vanishes identically, but R"n > 0 on the upper half-space because
according to Lemma XVI.2, the smoothed reduction R"' is at least equal to the parabolic Poisson integral on the upper half-space with boundary function the indicator function of A. (p) If ,4 is an analytic subset of D u 0b, then
R" = sup {i : F c A, 'compact},
(16.6)
320
I. XVII. Parabolic Potential Theory (Continued)
and the corresponding equation (16.6sm) for smoothed reductions is also true. (q)
If u and f, are bounded, then
sup IA- A< supIu - tit,
(16.7)
D
D
and the corresponding inequality (16.7sm) for smoothed reductions is also true. (r) There is a bounded continuous superparabolic potential u* on D for which, for each subset A of b,
Av1nb={A =u*}. (s)
(16.8)
Parabolic context counterpart of Section VI.3(o). To avoid repeti-
tion of complicated inequalities, we refrain from writing this property explicitly. It is the set of reduction inequalities obtained by translating Section VI.3(o) into the parabolic context; that is, "superharmonic" is replaced by "superparabolic," and dots are inserted over set and reduction symbols as required by the parabolic context.
17. Proofs of the Reduction Properties in Section 16 Proof of (a). See the proof of the corresponding property in the classical 0
context in Section 111.5(c).
Proof of (b). The properties listed under (b) have already been proved except for the identification of A; with 9' when A n D is parabolic-fine open. If A is a parabolic-fine open subset of D, the function is a positive superparabolic function equal to v on A according to the third line of (b); so f2" z A , and the reverse inequality is listed on the first line of (b). For general A with A n 1) parabolic-fine open, combine property (a) with the fifth assertion in (b) and the special case just considered to obtain the stated identification. o
Proof of (c). The second assertion was proved in Section 11(a) by referral back to the proof in the classical context. Since superparabolic functions are parabolic-fine continuous, the same proof is applicable to prove the first assertion. o Proof of (d). See Section 8.
0
Proof of (e). If equation (16.3) is true, then (16.3sm) is true up to a parabolic semipolar subset of 6 and therefore is true everywhere on 15 because both
321
17. Proofs of the Reduction Properties in Section 16
sides of (16.3sm) are superparabolic functions. In the following we therefore consider only (16.3). The first assertion of (e) was proved in Section l 1(e) for ti^ and ti finite valued and continuous, using Section 11(a), that is, using the second assertion of (c) of the present section. This proof is applicable as long as 6 is finite valued on ,4 if we use the first assertion of (c) and replace the open
sets in the Section 1 l (e) proof by parabolic-fine open sets. For arbitrary ti,, ti, A^, A with A c D it follows that +.1 =
Jim
Aniv
1/k}. Then ti; is continuous on the compact set e, and we define u, = 2 -k-' ez /sup.ti; .) By countable strong subadditivity (k) Ui°Bk
e
5
E
s RA +
u
2
A
LUk]
on C. The function
t
J
2
324
1. XVII. Parabolic Potential Theory (Continued)
is superparabolic on D, equal to v on A n D and near A n 3D, and is at most R" + s on e, as desired. Proofs of (m) and (n). The proofs follow those of the corresponding classical properties in Sections 111.6 and 111.5, respectively. Note however that the
counterpart ti of the function u in 111.6 should be superparabolic, but not parabolic on any open subset of D; choose for example the potential of a measure supported by a countable dense subset of D. Proof of (o). In view of (e) all that remains to be proved is that
R6 whenever v is finite valued and continuous and A. is a decreasing sequence of compact subsets of D u aD with intersection A. If A c $ and if h is open, then A. c R for sufficiently large n so that RB z lim,,..,, Ru", and therefore in view of (c) R" z lim,,..,, R"4. The latter inequality is actually an equality because the reverse inequality is obviously true.
Proof of (p). Suppose first that v is a finite-valued continuous positive superparabolic function on D. In this case (16.6) is true because according to (o) for each point of D the set function is a Choquet capacity relative to the paving of compact subsets of D u ab. For this choice of d the R'6(4) capacitability of the analytic set A implies (16.6). Equation (16.6sm)
follows on b - A because by (f) the equations (16.1) and (16.6sm) are identical on b - A; equation (16.6sm) on A is deduced by the following argument. According to (d), when e A the left side of (I6.6sm) is unchanged when A is replaced by A the right side is also unchanged in view of (d) and (e) [if 4 e tin (16.6sm), replace tin (16.6sm) by t less each member of a sequence of balls of center and radii tending to 0]. Thus (16.6sm) is reduced to (16.6), and we have proved (p) when v is finite valued and continuous, in particular, when ti is parabolic. Next we prove (p) when v is a
potential. Let v, be an increasing sequence of finite-valued continuous and R" = A"-16' potentials on D with limit v. According to (n) R" 6 = +V so we can apply (e) and the fact that (p) has been shown to be true for v -
to deduce
R,", =AA-6=sup{R$°:neZ+} = sup { R': n e l+, r' c A n D, t compact) = sup { I : C c A n b, ' compact). Since the last supremum is at most the supremum in (16.6) which is itself majorized by the left side of (16.6), equation (16.6) is true. The same argument
involving smoothed reductions yields (16.6 sm) when v is a potential. We have now proved (p) when ti is either parabolic or a potential so (p) is true as stated in view of (g) and the Riesz decomposition of a positive superparabolic function.
17. Proofs of the Reduction Properties in Section 16
325
Proof of (q). The proof follows that of the corresponding classical property in Section VI.4(n). o
Proof of (r). The proof must be more than a translation of the proof of Lemma XI.10 into the parabolic context because the parabolic Dirichlet problem has not yet been treated. The basic method of the proof of Lemma XI.10 will be used however. Let D be a nonempty open subset of RN, let I be an interval with closure in D, and let a [b] be the ordinate value on the lower [upper] face of I. Define s) on D by 1
s) =
s-a
b-a 0
Then ti is superparabolic, and we now show that (reduction relative to D) the function R' is continuous. The function is equal to the smoothed reduction according to the fourth property in (b). The function is lower semicon-
tinuous, and E. 5 ti; so k. is continuous at a point if this inequality is actually an equality at that point. Since R is parabolic and therefore continuous on D - 8I, we need only prove that R,, = ti on 81. On the lower face
of j, 0 5 R 5 ti = 0 so there is equality on this face. If is a boundary point of j not on this face, it will be proved in the next paragraph that I is not parabolic thin at 4; so since k is parabolic-fine continuous and equal to ti on I, there is equality at . (Or, apply the second assertion of (b).) Thus R; is continuous. Let I, be a sequence of intervals with closure in D forming a basic for the topology of b, let tip be defined for Ii as ti was for I, and set (17.4)
The function ti' satisfies the conditions demanded in (r) by a translation into the present context of the corresponding proof in the classical case in Section XI.10.
There remains the proof that each point 4 not on the lower face of I is in Pf. Since (from Section 9) each point of the abscissa hyperplane is a parabolic-fine limit point of the lower half-space, it follows that Pf contains the interior points of the upper face. Finally suppose that 4 is a point of a lateral face, not on the lower face. If N = 1, let 1, be the reflection of j in the bounding side of I containing t . A trivial symmetry argument shows that I, is parabolic thin at if and only if I is. If both are parabolic thin at , their union is also. However, the set I v I, contains all but an 12 null set of the part of some Euclidean neighborhood of strictly below , and so [Section 9(d)] I v it cannot be parabolic thin at . Hence e M. The similar treatment of the case N > 1 is left to the reader. o
326
1.XVI1. Parabolic Potential Theory (Continued)
Proof of (s). The proof follows that of the corresponding inequalities in the classical context in Section VI.4(o).
18. The Classical Context Green Function in Terms of the Parabolic Context Green Function (N z 1) Let D be a nonempty open subset of R", and define f) = D x It The parabolic context Green function 6r, is then invariant under translations of the ordinate axis, CD((, S), (1, t)) = G`DW, 0), (h, t - s)).
It follows that for fixed and t the function (tl, s) '- 6D((4, s), (tl, 1)) is superparabolic on b and is parabolic on D t)} with the canonical isolated t); so s), (q, t)) z OD((n, s), t)). Repeat this intersingularity at
change of space variables to derive equality here, that is, to find that the function (4, i) i-. CD((4, s), (ii, t)) is symmetric. Define the function GD on
RxDxDby
(18.1)
GD(t, , I) = OD((, 0), (17, - t)).
If D = R" or more generally if D is R" less a closed classical context polar set, then (from Section 8, Example (a)] D is A" less a closed parabolic-polar set; so (from Section 8) CD = 6 on 15 and GD(t, ,,) = e (t, - ri). Whatever the choice of D, the function 6D(1, -, -) is symmetric on D x D, and the function 4D(-, -, ry) is superparabolic on D, parabolic on D - ((q, 0)). Moreover Go(t, , ry) 5 6(t. - q). The function GD will be identified in Section 2.IX.17 with the transition density function of Brownian motion in D. Theorem. If D is a Greenian subset of R",
f(N/2- I) a" = 2n""2 a2
(18.2)
if N > 2,
a2 =
1
2na2'
2
at = a2 .
We first verify (18.2) in three special cases. If N > 2 and D = R", then = AN, 6D = 0, and (18.2) follows by direct evaluation. Incidentally it also follows that for any choice of D the integrals in (18.2) converge when N > 2 and ' q. If N = 2 and if D = D+ is a half-plane, let ry be the reflection of ry in OD'. Then (from Section VIII.9) GD.
4) = log
-n l
(18.3)
327
18. Classical Green Function vs Parabolic Green Function
On the other hand, according to (4.3), the function /,,. is given by
J(2aQ2t)-t (ex'2 2a2t
r,D, (t, x,11) =
-ex p -14-g*IZ\ ift>0 2a2t )
(18.4)
if t 2 because f W O((
,s),(,,t))1(di)= +oc
when N < 3. We therefore replace (18.5) by a relative equation (see Section XVII.4)
1 XVII. Parabolic Potential Theory (Continued)
328
o = 0D (c, i) where
n),
(18.7)
ry') is a positive parabolic function. The proof continues as in the
case N > 2, using the truth of the theorem when D = D. When N = 1, there remains only the proof of (18.2) for D a finite interval, and this is carried through just as in the case of bounded open sets when N = 2. Observation. According to Theorem 18, if D is Greenian, if p is a measure s) on D, and if p is the product measure p x 1, on 1) = D x R, then aNGDp(4 )
19. The Quasi-Lindelof Property Theorem. An arbitrary union of parabolic fine open sets differs by a parabolicsemipolar set from some countable subunion.
The proof follows that of the corresponding property in the classical context (Theorem Xl. 11).
Chapter XVIII
The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets
1. Relativization of the Parabolic Context; The PWB Method in this Context Let b be a nonempty open subset of ft', and let h be a strictly positive parabolic function on D. A function v/h on D will be called h-parabolic, b-superparabolic, or h-subparabolic if o is parabolic, superparabolic, or subparabolic, respectively. The notation will be parallel to that in the classical
context, with h omitted when h =- 1. Thus 6m k' "1j. , it, H, ... need no further identification. In the dual context in which h is coparabolic we write
6M4, ltp , t$, 1Y f , ... . The PWB method for solving the first boundary value (Dirichlet) problem for relative harmonic functions translates directly into the parabolic context. As in the classical context, the boundary is that obtained by a metric compactification of the given open subset D of ft'; the boundary of D relative
to the one-point Alexandrov compactification of off" will be called the Euclidean boundary. Upper and lower PWB" solutions H f and ft f4 are defined on D corresponding to a specified boundary function f, and if these solutions
are equal and parabolic, the function f is parabolic h-resolutive with PWB"
solution Hf = M4.. The definitions of h-resolutive boundaries, h-regular boundary points, h-parabolic measure boundary subsets, and so on are translations of the corresponding definitions in the classical context. The properties of h-harmonic measure null sets and of PWB" solutions derived in Sections VIII.5 to VIII.7 go over into the parabolic context with trivial changes in the derivations. The relations between PWB" solutions and reductions (Section VIII.2) are preserved in the present context:
when d is a boundary subset. ExAmPLE (a) (Euclidean boundary). If e D and if A = {il e,015: ord ry' z
ord }, then H
0. In fact, if _ (, s), if
_ (, s - I /n), and if u is
330
1 XVIIi. The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets
the indicator function of the set f0 a b: ord it > s - 1/n}, the function u is in the upper PWB" class on D for the boundary function 1.4; so H 0 when n is so large that a D. Hence H,., (o = 0. EXAMPLE (b). Let D be a Greenian subset of R coupled with a boundary OD
provided by a metric compactification of D, define D = D x I8, and define aD as the boundary of D provided by the compactif ication (D V OD) x 9, where 08 is the two-point compactification of R. A trivial adaptation of
Example (a) shows that the upper boundary (DvaD) x { + oo} is an h-parabolic measure zero subset of aD for every choice of strictly positive parabolic function h on D. Alternatively this follows from the fact that the function ti: s) i-* e' is a positive h-superparabolic function on D for every positive superparabolic function h on D and u has limit +oo at the upper boundary of D. The role of the lower boundary of D is more delicate. We show that the lower boundary is parabolic measure null if D is on one side of a hyperplane of I8". It is sufficient to exhibit a sequence ti, of positive parabolic functions on D with limit 0 and with ti (1') = 0 when ord ry' S -n, and it is therefore sufficient to exhibit such a sequence when D is the half-
space on one side of a hyperplane of R'; we can even assume that the hyperplane is a coordinate hyperplane and that D is the set on which the corresponding coordinate function is strictly positive. Finally we need only consider the case N = 1 because a sequence ti, in this case induces one in the general case. The sequence ti, on b = 0} (N = 1) defined by °` ifs > -n I - (s + n) Jexp2it/i(d2) (s + n) (2/n)''22
tt(, s) _
1
ifs< -n
has the stated properties. (We have simplified the notation by taking a = 1.) It is not difficult to show, although not necessary for present purposes, that u (t, s) is the parabolic measure of {(0, t): t 5 -n}, the Euclidean boundary subset relative to s).
EXAMPLE (c) (Euclidean boundary, h =- 1). Let b be an interval in W. According to Example (a), the upper boundary of D is a parabolic measure null set. We show that aD is parabolic resolutive by showing that every finite-valued continuous boundary function f is parabolic resolutive with Hf = PI(D, f ). Define f, on aD as f except that f, = info f on the upper boundary. Then Hf = Hf because f = f, up to a parabolic measure null set. Since Pl(D, f) is in the upper PWB class on D for f,, it follows that
l l(D,f) >- Hf, = Hf and similarly PI(D,f) 5 Hf; so PI(D,f) = Hf, as asserted.
EXAMPLE (d) (Euclidean boundary, h = 1). Let D be a slab 08'v x ]0,6[ with 0 < 6 < + oo. According to Example (a), the finite part of the upper
1. Relativization of the Parabolic Context; The PWB Method in This Context
331
slab boundary is a parabolic measure null set. We show that the boundary is parabolic resolutive with Hf = PI(D, jo) when f is a finite-valued continuous boundary function and To prove these assertions, observe first that the function fo has limit f(oo) at oo and (from Section XVI.1) if a is either oo or a point of the lower slab boundary the function PI(D, fo) has limit f(C) at t . When b = + oo, the function PI(D, fo) is in both the lower and upper PWB classes on D for f and so can be identified with fff ; when b < + oo, the method used in Example (b) shows that Hf = PI(D, fo) EXAMPLE (e) (Euclidean boundary). (Cf. Section VIII.3.) Let u be h-super-
parabolic on D, and suppose that ti has a finite or infinite limit at every boundary point, thereby defining a boundary function f Suppose also that u is lower bounded and that is upper bounded. Then f is parabolic h-resolutive with Hl = OMoti because ti is in the upper and 6M4-ti is in the lower PWB' class on b for f. Moreover if h =- 1, if ti has a superparabolic extension to a neighborhood of d8" r D, and if this extension has a finite limit at every point of 815, then lim, ft. (q) = whenever the boundary point a is not in some at most parabolic-semipolar set. In fact OMDU can be obtained as the restriction to D of the limit of a decreasing sequence of
locally-lower bounded superparabolic functions on a neighborhood of V - D, each equal to ti outside a compact subset of b, depending on the function. [See the corresponding argument in Section XVII.4 as applied to analyze OMDG(-, it), or see the general discussion of GMD in the classical context in Section 111.1.] According to the parabolic Fundamental Convergence Theorem, the limit function U' majorizes its smoothing, which is parabolic on D, and the two functions are equal up to a parabolic-semipolar set. At a nonexceptional boundary point C of D the function OMDU has limit ti'(C) on approach from within D because u' is lower semicontinuous, u' S U, and U has this limit. Application to the Parabolic Resolutivity of the Euclidean Boundary
The Euclidean boundary of an open subset of d8" is parabolic resolutive, and the set of its finite parabolic irregular boundary points is parabolicsemipolar. In fact this assertion follows from a trivial modification of the proof for N > 2 of Theorem VIII .4, the counterpart in the classical context of this assertion. Observe that the set of finite parabolic irregular boundary points is parabolic-semipolar rather than parabolic-polar. Application to a Representation of Parabolic Green Functions (Euclidean Boundary, h = 1)
If 1 is a point of D, the restriction to aD of 6) (defined as 0 at oo) can be taken as f in Example (e), and it follows that ry') = ry') - Hf.
332
I.XVIII. The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets
More generally. if h is an open subset of b, if it a h, and if f is the restriction to ah of q) (defined as 0 on ah n ab), then OA(', rj) is t;p(-, q) less the PWB solution for h with the boundary function f.
2. h-Parabolic Measure Let D be a nonempty open subset of A' coupled with a boundary provided
by a metric compactification, and let h be a strictly positive parabolic function on D. The development of h-parabolic measure follows that of h-harmonic measure; so the details will be omitted. If A is a subset of aD with an h-parabolic-resolutive indicator function, we define fip(-, A) = H''a, call A a EiD measurable set, and call µo(, A) the h -parabolic measure of A relative to . The set function µv(, -) is a probability measure, and the completion of the restriction of this measure to the class of At measurable Borel sets is an extension of this measure. The h-parabolic measure null sets defined above are the sets A for which µD(-, A) vanishes identically. A boundary function measurable with respect to the a algebra of Nn measurable
sets will be called µ$ measurable, and L'(µo) is defined as the class of µ11 measurable functions from aD into R, with two functions identified when they are equal un almost everywhere (that is, up to an h-parabolic measure null set) for which ! f is is 4D(, -) integrable for every point 4 of D. A function
f from aD into A is h-parabolic resolutive if and only if f is in L' (ti"n), and if so, then Hf =
The work in Sections VlII.8 to VIII.10 goes through in the present context except that even if D is connected, it is not true that a positive parabolic function on D which vanishes at a point must vanish identically; so the L' class for depends on 4; the class of 1o measurable sets and the class L' (14) were defined taking this fact into account.
In the present context the notation tH of Section VIII. II becomes ie. Thus if v' is an h-superparabolic function on b and if h is an open relatively
compact subset of D, the function ikti is h-superparabolic on D and is h-parabolic on h, and &:9 S v, with equality on D - h up to a parabolicsemipolar subset of Oh (Euclidean boundary). In particular, if ti z 0, then t8v = h4U-$. EXAMPLE (a). If b is an interval, we have seen in Section 1 that the Euclidean
boundary is parabolic resolutive, that the upper boundary is a parabolic measure null set, and that µo(-, f) = PI(D, f) so that parabolic measure is given by the densities evaluated in Section XV.9. The inner points of the upper boundary are parabolic irregular, but all other boundary points are parabolic regular according to Section XV.9. EXAMPLE (b). If D = U8' x x ]0, d[ is a slab we have seen in Section I that the Euclidean boundary is parabolic resolutive, that the abscissa hyperplane
3. Parabolic Barriers
333
part of the boundary has parabolic measure 1, and that G(s, b, q)/N(d77)
[=
s)]
for (ry, 0) on the abscissa hyperplane. The points of the abscissa hyperplane
and the point ao are parabolic regular, but all other boundary points are parabolic irregular. It is easy to see (cf. the classical context for D a ball in Section VIII.9) that the Euclidean boundary is parabolic universally resolutive, that the abscissa hyperplane has h-parabolic measure l for every choice
of h, and if (Riesz-Herglotz type representation, Theorem XVI.6), h = PI(D,! .), then µn(4, AO = AS- , h)
s)]
for (rl, 0) on the abscissa hyperplane.
The Support of µD (Euclidean Boundary)
If c e b, define DS = try' e,6: ord ry < ord ). We have seen in Section 1,
Example (a), that µD(, ) is supported by the part of aD strictly below . Moreover, if A is a Borel subset of eD strictly below , then the function A) is the restriction to D4 of p A) because if v [u] is in the upper [lower] PWB class on b for the boundary function 1,; on a D, the restrictions of these functions to D4 are in the corresponding PWB classes on DS for the
restriction of 1,; to oAt. This fact has the following useful implication, in which the roles of A and D are reversed. Let now D be an open subset of A', and let be a boundary point of D. Suppose that every point of D is strictly below 4 and that the part of some open neighborhood h of strictly below lies in D. Then if A is a Borel subset of 8D, the function A) is defined and parabolic on b and has the parabolic extension µD A) to
Dvh.
3. Parabolic Barriers The classical context definitions of weak h-barrier and h-barrier translate directly into the parabolic context, as does the proof that if there is an h-barrier locally at a boundary point, then there is an h-barrier defined on the whole open set in question. The classical context Bouligand theorem (Section VIII. 12) is also true in the present context: If there is a weak parabolic barrier at a Euclidean boundary point a of an open subset D of 1'(N, then there is a parabolic barrier there. To see this, adapt the classical context proof as follows. In view of the fact that the existence of a parabolic barrier
334
I.XVIII. The Parabolic Dirichict Problem, Sweeping, and Exceptional Sets
at C is a local property it is sufficient to show that there is a parabolic barrier
on the trace on b of an open neighborhood of e ; so b can be supposed bounded. Suppose then that D has diameter S < + oo and that ti is a weak parabolic barrier for b at . For notational simplicity assume that C is the origin. Define 4) : it = (ry, t) tll2 + t2 on b, choosing c so large that ¢ is subparabolic. The function 4) is in the lower PWB class on ,6 for the boundary function f: q = (ry, t) i- cI,12 + t2 on D. Hence Hf z 4), and it will be shown
that Rf is a parabolic barrier for b at the origin by showing that Hf has limit 0 there. Let h be an interval containing the origin, of diameter r, let A be a compact subset of b n 6h, and let t/i be the indicator function of (oil - A) n,6 on 6h, so that Aj&, i,1i) is parabolic on h with limit I at every point of aB - A not on the upper boundary. Since the upper boundary of h has parabolic measure 0, there is a positive superparabolic function v on h with v(0) < + oo and with limit + oo at every point of the upper boundary. If uo is in the lower PWB class on ,6 for f, apply the maximum theorem for subparabolic functions to find that uo on B n D is at most (c + I)r2 if Oh does not meet D and is at most (c + l )r2 + (c + 1)6' r
t!
LLinf,4 ti
+ NB(., W)J + rv
if Oh does meet D. Thus in both cases Hf is majorized on B n D by the sum (3.1). This sum has limit
(c + l)r2 + (c + l)62pB(0,,i) + rti(0)
at the origin, and this limit is at most 2(c + 1)r2 + r6(0) if A is sufficiently large. Since r can be made arbitrarily small, the function Hf has limit 0 at the origin, as was to be proved. Theorems VIII.13 and VIII.14 on the relations between regularity and barriers translate directly into the parabolic context.
4. Relations between the Classical Dirichlet Problem and the Parabolic Context Dirichlet Problem Let D be a Greenian subset of R' coupled with a boundary OD by a metric compactification of D. Define D = D x R, and define 0,6 as the boundary provided by the compactification (D u aD) x R, where R is the two-point compactification of R. Let h be a strictly positive harmonic function on D, and define the parabolic function h on D by setting s) = We now consider the Dirichlet problem for h-harmonic functions on D and for hparabolic functions on D. Recall [Section 1, Example (b)] that the upper boundary (D v OD) x { + oo } of D is an h-parabolic measure null set. We suppose that the lower boundary of D is also h-parabolic measure null. This
5. Classical Reductions in the Parabolic Context
335
is true if D is not too large, for example [Section 1, Example (b)], if D is on one side of a hyperplane in R' and h - 1. Let f be an extended real-valued function on OD, and define f on Of) by setting f( ), except that no restriction is imposed on f on the upper and lower boundaries of D. This looseness is convenient, and the actual definition off on the upper and lower boundaries does not affect the PWB" solutions for/. If u is an h-superharmonic function in the upper PWBh class on D for f, the function ti: s) iu( ) is in the upper PWB class on D for f if f is defined as - co on the upper and lower boundaries of D. Hence tl fs) 5 H and similarlyff s) z We conclude that f is h-parabolic resolutive on b whenever f is h-harmonic resolutive on D, and then H f(g, s) = H In particular, if A is a p' measurable subset of 3D, the product set A = A x R is j4 measurable and JD ((g, s), A) = pD( , A). Conversely, if f is Borel measurable and PWBh resolutive and if 8D is h-resolutive, then f is Borel measurable, and we show it is PWB" resolutive. In fact there is nothing to prove if f is bounded, and in the general case PhD(-. I f I) = lim .u M-m
I f I A n) = lim µn(', I f I n n) = µD(', I f I) < + oo.
Suppose in this paragraph that OD is PWB' resolutive and that aD is PWB' resolutive. This hypothesis is satisfied if h = I and aD is the Euclidean boundary. (Note that BI) is not then the Euclidean boundary, but the PWB
resolutivity of aD follows from that of the Euclidean boundary of D and the fact that the upper and lower boundaries of D are parabolic measure null.) If (g, s) is a finite h-regular point of 0,6, the point is an h-regular point of OD. In fact a bounded h-resolutive function f on OD, continuous at , defines a bounded function jon aD as above (setting f - 0 on the upper and lower boundaries). The function f is h-resolutive, bounded, and continuous at so Hf has limit at that is, Hf has limit at . If h - I and if is a finite regular point of OD, then each finite point s) of aD is regular because a weak barrier for D at when considered as a function on D is a weak barrier for D at s).
5. Classical Reductions in the Parabolic Context Let D be a Greenian subset of R', coupled with a boundary provided by a metric compactification, define D x R, and compactify D as in Section 4 to obtain M. Suppose that the lower boundary of b is parabolic measure null; we have seen in Section 1 that this condition is satisfied if D is on one side of a hyperplane of RN. Let v be a positive superharmonic function on D, and define the superharmonic function a on D by setting Let A be a subset of D, and define ,4 = A x R. We now show that if A is analytic, then
336
1 XVIII The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets
R" (S, s) = R"O,
(5.1)
A s) = R O.
(5. l sm)
If A is a compact subset of D, then (from Section VIII.10) R" on D - A is the PWB solution for the set D - A with boundary function equal to 0 on OD r a(D - A) and equal to v on A n a(D - A). Similarly R" on D - A is the PWB solution for the set D - A with boundary function equal to 0 on aD n a(D - A) and equal to ti on A n a(D - A). In view of the relation (Section 4) between the Dirichlet problems for harmonic and for parabolic functions it follows that (5.1) is true for in D - A ; this equation is trivial for 4 in A. If A is an analytic subset of D, then A is an analytic subset of b; so [from Section XVII.16(p)] the value s) can be approximated arbitrarily closely from below by R' s) with E a compact subset of A ; if F is the projection of f on the abscissa hyperplane, the approximation is improved if F is replaced by F x R. Since F is a compact subset of D and since [from Section VI.3(l)] can be approximated arbitrarily closely by RB(A) with B a compact subset of A, it follows that (5.1) is true. Equations
(5.1) and (5.1 sm) are identical when ED - A and the left [right] sides of these equations are equal up to a parabolic-semipolar subset of ,6 [parabolicpolar subset of D]. Since a classical-polar subset of D is the projection of a parabolic-polar subset of b (because a positive superharmonic function on D identically + oo on a set B can be identified with a positive superparabolic
function on b identically +oo on B x R), equation (5.Ism) is true up to a parabolic-semipolar subset of D and therefore is true on D.
Observation. There are three possible approaches to the properties of reductions in the classical context. (1) These properties can be proved using the specific classical context. Such proofs were used in the preceding chapters. Unfortunately some of the most natural proofs cannot be applied in the parabolic context because of the weaker version of the Fundamental Convergence Theorem in the parabolic context and (a related fact) because the domination principle is false in the parabolic context. (2) The reduction properties in the parabolic context can be proved, and then either it can be noted that these proofs are valid in the classical context or (3) 1t can be noted that in view of (5.1) and (5.1sm) the properties of reductions in the classical context can usually be read off from those in the parabolic context, or at least be deduced from them. The choice (1) adopted in this book is inefficient and repetitious but was
made because the proofs in the classical context are thereby clearer and more natural than the more generally usable proofs in choices (2) and (3).
6. Parabolic Regularity of Boundary Points
337
6. Parabolic Regularity of Boundary Points In the following examples b is a nonempty open subset of d8' provided with
its Euclidean boundary. In this section barrier means "parabolic context barrier." Since the existence of a weak barrier for b at a boundary point is a property of b in a neighborhood of o, one can prove that a barrier for b at o exists and thereby prove that o is parabolic regular by exhibiting a weak barrier for the part of D in some neighborhood of o. EXAMPLE (a). If every point of b in some neighborhood of the finite bound-
ary point o is either above the horizontal hyperplane through o or on one side of some other hyperplane through 0, then 0 is a parabolic-regular boundary point. In fact, to construct a local weak barrier for D at , define 0 by
where 4' = 0 is the equation of a hyperplane through o. If the hyperplane is not parallel to the ordinate axis, it can be supposed that a = 1. Under this hypothesis the half-space strictly above the hyperplane is the set {4, > 01, and the restriction of 0 to the part of 1) in an open neighborhood of o is a local weak barrier at o if every point of D in that neighborhood is above the hyperplane. If every point of D in some neighborhood of o is below the hyperplane and if a is a strictly positive number, the function 1i = 1 - exp a4' is strictly positive strictly below the hyperplane and for sufficiently large a is superparabolic there; so the restriction of ti to the part of D in some open
neighborhood of o is a local barrier at o. Finally, if the hyperplane is parallel to the ordinate axis, then a = 0. If the part of D in some open neighborhood of 0 is on one side of this hyperplane, then the restriction of either 0 or - ¢ to this part of b is a local barrier at Observe that if the complement of D includes the part in a neighborhood of 90 of some not horizontal hyperplane through o, then if D contains points arbitrarily near So on both sides of this hyperplane there is a local weak barrier for the part of D on each side of the hyperplane and therefore a local barrier for b itself at o. Such combinations will not be mentioned further. Example (a) implies that the lower and lateral boundary points of an interval are parabolic regular. Conversely, our discussion of the parabolic Poisson integral for an interval can be used to derive the conclusions of Example (a) involving hyperplanes parallel or perpendicular to the ordinate axis. EXAMPLE (b). If o : so) is a boundary point of D with the property that the boundary in some neighborhood of So consists of points with ordinate values 2!s0 and if D contains points arbitrarily near o with ordinate values <so, then o is a parabolic-irregular boundary point of D. In fact (Section 1)
338
1 XVIII The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets
some boundary neighborhood. of o has parabolic measure 0 relative to points of D near but below o; therefore the boundary function values assigned near o are irrelevant to the Dirichlet solution at points of b near but below 0. For example, if N = I and if b is the interior of the simple polygon with vertices (-1,0), (1,0), (1, 1), (0,#), and (-1, 1), the boundary point (0, `-2) is parabolic irregular. Observe that for this choice of D there is no increasing sequence of open sets with parabolic-regular boundaries and with union D, but this lack of parabolic-regular approximation is only a minor inconvenience. Theorem VIII.17 on the approximation of harmonic measure by harmonic measures of boundaries of subsets translates directly into the parabolic context.
EXAMPLE (c). [This example strengthens (a) except when the boundary hyperplane in (a) is horizontal.] Suppose that o: is a boundary point of D, and suppose that there is a ball of center : with S # o such that the ball closure meets D in o but in no other point. Then o is a parabolic-regular boundary point of D. To see this, let 6 be the ball radius, and define u on D by (6.1)
6-P -
where p is a strictly positive number to be chosen below. Then ti > 0, u has limit 0 at 50, to has a strictly positive lower bound outside each neighborhood
of ';o, and if, = (sl, t), z
du(n) = pit - I-p-4 NZ (n -
-
X12
2
Then Oti < 0 sufficiently near 0 if p is large so there is a local barrier at 0.
EXAMPLE (d) (Iterated logarithm criterion for parabolic regularity). [This example strengthens (a) when the boundary hyperplane in (a) is horizontal.] Let I) be the set below the abscissa hyperplane defined by the inequalities 1
-1 < s < O.
12 < 2a21sj logl log lsjj,
(6.3)
We show that the origin is a parabolic-regular boundary point of D by showing that the function c defined by D by t'(S, s) = 110011 -' [1 -
ti( ,s) =
I.sl-112eXp
-1 12
gas
ISIIN+2 /2z
Y
(S, S)], (6.4)
339
6. Parabolic Regularity of Boundary Points
is a barrier for b at the origin. The inequality (6.3) can be written in the form s) < IsII log ISII
.
which makes it clear that ti > 0 on b and that ti has limit 0 at the origin. Furthermore, using the fact that ti is parabolic, we find that
bu(ss)=
-N2
Slog Is1I-'ti(.,s) 0. Then s) > I log ISI I '
` - (logl log IS11)-' > 0
ifs' is sufficiently near 0. Furthermore
bu(ss) _ -1s1 'IloglsiI '-`exp
+ (k - k2)1SI2 + 2o2 1S1
k14---12
I loglsil (log l log lsll )2
Choose s' so near 0 that (I + c) I log ls' lI superparabolic if
Nk
1 +e
2
Iloglsll
exp
(6.8)
-k1412 2a21s1
< Nk/2. The function u is then
340
I.XVIII. The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets
Nk < (k -
Ilog lsll ,
20 IsI
exp
(log I log IsII )z
-k. 2Q IsI
(6.9)
This inequality will be satisfied if
Nk < (k -
(6.10)
2a2Isl
On the other hand, if (6. 10) is not satisfied at right in (6.9) is at least
Iloglsll` exp (log Iloglsll)z
s), the second term on the
-Nk
I -k'
which is at least Nk on D, ifs' is sufficiently near 0, depending on s and k. Thus for any choice of a and k the function v is superparabolic and positive
on A when s' is sufficiently small. Now choose c = (I + e)/k, and define the continuous function f on 0,6, as 1 at the origin and as ti at the other boundary points. The PWB solution Ht on D, for f is the same as the PWB solution for a boundary function differing from f only at a single point; therefore Hf = Hf, when f =f, except at the origin, where f, is defined as 0. Since ti is in the upper PWB class on D, for f,, we conclude that Hf s d and therefore lim Hf(0, s) 5 lim v(0, s) = 0. s-'o
S-0
Since f is continuous with value 1 at the origin, the origin is a parabolicirregular boundary point for D. Moreover c can be made arbitrarily near I by choosing a near 0 and k near 1. It follows that for any choice of c > I and any choice of s' the origin is a parabolic-irregular boundary point of 4. Parabolic Balls A parabolic ball in 9' of radius b and center (gyp, so) is defined as the set
6(so -
(6.11)
Observe that the center of the parabolic ball is a boundary point. The set (6.11) lies strictly below the center and is a solid of revolution with vertical axis through the center. The highest and lowest points of the boundary are, respectively, the center and (gyp, so - b). Example (a) shows that every boundary point of a parabolic ball except possibly the center is parabolic regular, and Example (c) shows that every boundary point of a parabolic
341
8. Sweeping in the Parabolic Context
ball except possibly the center and the lowest boundary point is parabolic regular. The center is a parabolic-irregular boundary point according to Example (e) because the set b, of that example is, for an arbitrary value of c > I and an arbitrary 6 > 0, included in the parabolic ball of center the origin and radius S when s' is sufficiently near 0.
7. Parabolic Regularity in Terms of the Fine Topology Theorem. A finite Euclidean boundary point ( of an open subset D of pN is parabolic regular if and only if C E (AN - D)p
This theorem is the parabolic context version of Theorem XI. 12, and the proof is omitted because the only change needed in the proof of that theorem to make it applicable in the present context is to change "polar" to "parabolic semipolar."
EXAMPLE. If D is a parabolic ball (Section 6), the center is a parabolicirregular boundary point; so D is a deleted parabolic-fine neighborhood of the center. Similarly the other examples of parabolic-regular and parabolicirregular boundary points in Section 6 have simple interpretations in the parabolic-fine topology.
8. Sweeping in the Parabolic Context The sweeping operation defined in the classical context in Section X.1 becomes a pair of linked operations in the present context. If b is a nonempty open subset of ftN, A c D, and if GDµ [zGD] is a superparabolic [cosuperparabolic] potential, the smoothed reduction %A_4 CuAAOA] is also a potential, namely that of a "swept/measure" u0" [iµ0"]
Gvu6" =
JAQ-4 = µ0"'Vp.
(8.1)
The swept measures are supported by A n D, and it will be shown in Section 13 that if A is a Borel set, then Oµ0" [Oµ0"] is supported by the in general
smaller parabolic-fine [coparabolic-fine] closure of A in D. Since GojO" is unaffected when A is changed by a parabolic-polar set, N 0" is also unaffected by such a change in A. Dually Ou0A is unaffected when A is changed by a coparabolic-polar set. It will be shown in Section 1 I that a set is parabolic
polar if and only if it is coparabolic polar. The notation of Section X.1 is adapted to the present context by writing So(5, ) for the probability measure on D supported by and by defining bb and b6 by
342
1
XVII1. The Parabolic DirichletFt Problem, Sweeping, and Exceptional Sets
sp\>') _ dd(S.')DA,
Vp(S,') =
(8.2)
These swept measures are determined uniquely by
r. 7)) ,')DA(il) = p(S. VU(', xx
GD(', J)UA(S) _
(8.3)
and the reasoning in the classical case (Section X.1) shows that do 5 1 and
36 f 'i0vdA = f ACDdv = +oo D
unless i = 0. Hence (a)
D
(c). On the other hand, if A is not parabolic polar,
the superparabolic function j 1 " does not vanish identically; so [from Section XVII.16(p)] the set 4 has a compact subset h for which the function
348
I.XV111. The Parabolic Dinchlet Problem, Sweeping, and Exceptional Sets
1 [s does not vanish identically. This function is a bounded superparabolic potential with associated Riesz measure supported by A. Since (c) implies that this potential vanishes identically, (c) (a). Thus (a) p(c); so dually (a*) p (c*), and since (a) = (as), the proof of Theorem 11 is complete. 0
Application to the Projection of a Parabolic-Polar Set
According to Section XVII.8, Example (a), if A is a classical-polar subset of 68", then A x68 is a parabolic-polar subset of G`8". Conversely, we now
prove that if A = A x R is a parabolic-polar subset of f8", then A is a classical-polar subset of P". It is sufficient to prove this assertion for A bounded and A a G° set, in which case A is also a Ga set. Hence (from the application in Section VI.2) if A is not polar and if D is a Greenian superset of A, there is a measure tt on D supported by A and making G0µ strictly positive bounded and continuous. However (from Section XVII.18), G0µ considered as a function on D = D x P is a bounded superparabolic poten-
tial, GDµ = GDp with µ = (p x /,)/a" supported by A. Hence A is not parabolic polar, contrary to hypothesis.
12. Parabolic- and Coparabolic-Semipolar Sets Theorem. The following four conditions on a subset A of a nonempty open subset D of A8" are equivalent :
(a) A is parabolic semipolar. (a*) ,4 is coparabolic semipolar. (b) [(b*)] (If A is Borel) the only measure v on D supported by A and with GDv [vOD] finite valued and continuous is the null measure.
Observation (a). In connection with (b) and (b*) recall (from Section XVII.15) that a parabolic-semipolar set has a parabolic-semipolar Borel superset: the dual assertion for coparabolic-semipolar sets follows trivially. In view of this observation it is no restriction on generality in the following proof to assume that A is a Borel set. Observation (b). As the following proof shows, "finite-valued" in (b) and (b*) can be replaced by "bounded." Observation (c). Condition (b) of the theorem implies that A is IN,, null because if A is a bounded Borel /"+, nonnull subset of 9" and if v is the projection of l"+, on A, then (by Theorem XVII.6) the potential GDv is bounded and continuous. Proof. (a) =::-(b*) Let v bea measure on Dsupported by a compact parabolicsemipolar subset 4 of b, and suppose that vOD is finite valued and contin-
12. Parabolic- and Coparabolic-Semipolar Sets
349
uous. Recall that a set is exceptional for the parabolic context Fundamental Convergence Theorem in the sense of the converse statement of Theorem XVI1.13 if and only if the set is parabolic semipolar. Now the proof of the classical Fundamental Convergence Theorem (Theorem VI.1) when translated into the present context shows that a measure v supported by a compact subset A of D is necessarily the null measure if (1) A is exceptional for the parabolic context Fundamental Convergence Theorem and if (2) vOD is finite valued and continuous. Hence v is null in the present context. More
generally suppose that i is a measure supported by a Borel parabolicsemipolar subset A of D, with VGp finite valued and continuous, and let Ao be a compact subset of A. If vo is the projection of v on Ao, then doO, and
(v - vo)G, are finite valued and lower semicontinuous with a continuous sum; so VIGD is finite valued and continuous, and we have just proved that therefore ivo is null. Since this is true for all A0, we conclude that v is null, as was to be proved. (b*) (a*) It is sufficient to prove that a Borel subset A of b is co-
parabolic semipolar if every finite-valued continuous potential id,, of a measure v supported by A vanishes identically. Actually we shall only use this implication when vCo is bounded and continuous. Let * u = *µGp be a bounded continuous cosuperparabolic potential satisfying the dual property of that satisfied by ti* in Section XVII.16(r). The cosuperparabolic potential *p0"Go is lower semicontinuous, < *ti, and continuous at every point where there is equality, in particular, at every point of A°f n A
If µ, is the projection of * µ 0 ' on a compact subset A, of A°'f n A, the potentials µ,Gp and µ,)G, are lower semicontinuous with continuous sum at each point of A,, and so both are continuous at such a point. Since µ, GD is coparabolic and therefore continuous on b - A, , it follows that µ,G6 is bounded and continuous on D; so (b*) implies that µ, = 0. Thus vanishes on compact subsets of 4°'f n A and therefore vanishes on this set; that is,
f6
SD(4, A°*f n A) *µ(d4) = 0.
Thus the positive superparabolic function 4°'f n,4) vanishes u almost everywhere on D, certainly on a dense subset of 15, and therefore vanishes identically on D. It follows (by Theorem 11) that the set A°f n'4 is parabolic polar and so coparabolic polar, and therefore the union A of the coparabolicpolar set 4"f n,4 and the coparabolic-semipolar set A - A°*f is coparabolic semipolar, as was to be proved. (a*) (b) _ (a) These implications are dual to the already proved implications (a) = (b*) (a*) and are therefore true. The set of these implications implies the truth of Theorem 12.
350
I.XVIII. The Parabolic Dinchlet Problem, Sweeping, and Exceptional Sets
13. The Support of a Swept Measure Let 1) = A", and let A be the abscissa hyperplane. Then A°J = 4°'f = 0, and the measures So( , ) and are supported by A. The situation in the classical context (Section XI.14 and XI.18) is quite different in that the sweeping operation is self-dual and that li (g, ) is supported by A°f. Furthermore, although A°f and 4°'f are Borel sets, the sets A u Aaf and A u A°'f, which are involved in Theorem 13 below and its dual, are not necessarily Borel sets. For example, if A is an arbitrary subset of the abscissa hyperplane of AN, then ,4 u Aof = A u A°'f = A. For this reason in the following counterpart of Theorem XI.14 the set A is supposed Borel. Recall that if 6 the measure associated with u) = BtiD" is not Qµ0" but
Theorem. Let D be a nonempty open subset of A', let A be a Borel subset of b, and let be a point of D. (a)
I f e*f, then
(b)
If is a measure on D, then the measure Oft" [in particular, the
(c)
measure 5 , )] is supported by (A u A"f) n D. If A is parabolic-fine dense in itself and if N is supported by A, then
{i }) = I.
µJµ"
Proof. (a) If eAAf, then c5 (, I according to Theorem 10(b). (b) Suppose that v and v' are positive superparabolic functions on D and that v' < v, with equality on A. Then °
A
=
=
so for every point in D the measure b
o'):
6) =
(13.1)
is supported by the set {ti = v'}.
Now let v be a decreasing sequence of positive superparabolic functions
on D chosen (Choquet topological lemma) so that limn..,o v = v. has smoothing DvQ". After replacing v by v A v if necessary, we can suppose that v < z with equality on A. It follows that the measure is supported by {v = v) for all n and therefore is supported by the intersection (ti,, = v) of these sets. Thus S is supported by the set A u [{v. = v) n (b - A)].
Suppose that C is a compact subset of D - A. According to Section XVII.16(l), the sequence v, can be chosen in such a way that um = ivO" on C. Hence, if S is fixed, the projection on D - A of the measure is
supported by the set
ti} n (D - A). Finally, if v is chosen as the
function ti' defined in Section XVII. 16(r), it follows that 8",(, ) is supported
by (A u A°f) n D. The evaluation of in (8.9) shows that this swept measure is also supported by (A u A"') n 1). (c)
Follows trivially from Theorem 13(b).
351
14. Internal Limit Theorem; Smoothness of Superparabolic Functions
14. An Internal Limit Theorem; The Coparabolic-Fine Topology Smoothness of Superparabolic Functions According to Sections II and 12 a subset of C'8" is, respectively, coparabolic polar or coparabolic semipolar if and only if it is parabolic polar or parabolic
semipolar; we shall write "parabolic" in both cases from now on. This self-duality is less deep than the relations between parabolic and coparabolic
concepts displayed in the following parabolic context counterpart of Theorem XI.4. If b is an open subset of ll' and if i and It are positive superparabolic functions on b, the function ti = i/h is defined in the obvious way when the ratio is neither 0/0 nor + oo/+ oo. The function ti is thereby defined parabolic quasi everywhere on the strict positivity set of h and is also defined on the i strict positivity subset of the h zero set. We shall see that it is important to allow h to vanish. Let v,; and vg be respectively the Riesz measures associated with i and h. The singular component of v6 relative to vi, will be denoted by i,6, in particular, vL = v,; if h is parabolic. If 0 is a function defined
on a subset of D and if e D, def ine 0*(C) = p*f lim,i_ O(q) if this limit exists.
The Zero Set of h
If is in the zero set Z of h, then every point of b below relative to b is also in Z, and in particular, if it is the horizontal hyperplane containing C, the set Z contains the open in it connected component of it n b containing C. It follows that the trace in D of the Euclidean boundary of Z consists of countably many such components and that if C is a point of such a boundary component, h > 0 on the part strictly above C of a sufficiently small Euclidean neighborhood of C. Theorem. (a)
The function u* is defined
(al) vc + vi almost everywhere on ,6 and (a2)
Parabolic quasi everywhere on the strict positivity set of i + lt.
(b) u* < + oo vi, almost everywhere on D and also parabolic quasi everywhere on the strict positivity set of h. (c) Let v,;i and vii be the respective projections of i6 and vti on the zero set Z of h. Then at vti almost every point of Z, u« = dv, = dvi,
(d)
dvi,i
Let t be a parabolic-polar subset of D. Then
(dl) (14.1) is true Ji, almost everywhere on t. (d2) h * = + oc vk almost everywhere on F.
(d3) 0 = + oo 0 almost everywhere on P.
(14.1)
352
I.XVIII The Parabolic Dinchlet Problem. Sweeping, and Exceptional Sets
(e) The function u* is coparabolic fuse continuous, and the set where u : ci is parabolic semipolar. (f) In particular, if (E D,
p*flim v(n) a--c
=
inf
C)
v(n)
16:anc4.6>01
(14.2) C)
(g) Let be a point of D, let F be a subset of ,6 with Euclidean topology limit point S, and suppose that ord it > ord C when l a F. Then if t is coparabolic thin at C, there is a superparabolic potential v on D, strictly positive on
a Euclidean neighborhood of C, for which FlimcG°(n)
_ +oo.
(14.3)
C)
Conversely, if there is a positive superparabolic function v on D, satisfying (14.3), the set P is coparabolic thin at C. Observation. Since ti and h are positive superparabolic functions but are otherwise unrestricted, (d2) implies that v* = + oo v,; almost everywhere on
F. Other obvious implications of Theorem 14 can be obtained by trivial manipulations. For example, (b) implies, after an interchange of v and h, that u* > 0 v,, almost everywhere on D. Note that (d2) is the parabolic context counterpart of Theorem V.11.
Proof of (f). Assertion (f) is a special case both of (c) and (dl). Its direct proof is a translation of that of Theorem XI.4(c), but we give the translation because a direct proof is so much easier than the proof of the general case and because the direct proof illustrates the adaptation of the Theorem XI.4(c) proof technique to the present context. The equality of the first and second terms on the right side of (14.2) was pointed out in Section XVII.8. It can be supposed, replacing v by v - 6,;({C })GD(, C), that v,;((C) ) = 0. Under this condition, unless (14.2) is true, there is a strictly positive number b such that the set h = 10: v(ri) > bGD(rl,C)} is not coparabolic thin at C. Apply the dual version of Theorem 10 to obtain
v(,) >_ v (q) 2- bGv(, )OB(q) = bGa(1, )l B() = bOD(f, t). contrary to the hypothesis that 616,6(-, 1) has infimum 0.
Proof of (g). The proof is the counterpart of that of Theorem XI.4(d) and is omitted.
Proof of (al). If 0 < a < b, define
A = {6< ah},
B = {v z bh},
G = .4p*f n $°*f n 6, ,
(14.4)
353
14 Internal Limit Theorem; Smoothness of Superparabolic Functions
and denote by v,,t the projection of vw on e. Following the proof of Theorem
XI.4(a), we find that in the reduction notation of Section XVII.16(s) as translated from the classical context [see Section VI.3(o)] the smoothed reductions h,;g, h,iaAa, ... all majorize ODvk. According to the parabolic context version of VI(3.12), the sum of these smoothed reductions is at most
[e A (bh)]/(b - a); so v. = 0, that is, v,,(O) = 0. Now according to the analysis of the zero set Z of h given at the beginning of this section, a point
of D must be either a Euclidean interior point of k or a coparabolic-fine interior point of b - Z. The Euclidean interior of k is vi, null and every coparabolic-fine interior point of D - 2 at which 0 does not exist is in e for some rational pair a, b. It follows that t exists v,, almost everywhere on D. Apply this result to h/c to find that u* also exists v,; almost everywhere on D.
Proof of (a2). In the context of Theorem XI.4 the fact that u* exists quasi everywhere on D is a triviality, and in fact u* = u quasi everywhere on D, but in the present context assertion (a2) that u* exists parabolic quasi everywhere on the strict positivity set of h + v is by no means trivial. Since (a2) is a local assertion, we can assume that h is bounded below on D by a strictly positive number fl. Let ho = C'DVtio be a potential majorized by fi, with measure i'Fo supported by the set e defined in (14.4). Then
hOAA+hoABAB+ ... 0 if and only if ,i e,64 the function K has
domain 6 x A. For q in di the function K(q, ) is superparabolic on 1), parabolic on 1) - {ry'}, with value I at C. The classical Martin boundary treat-
ment suggests that the Martin boundary points to be introduced should correspond to limit functions of the family {IC(q, ), rj a fit} when q in ,6t tends
to the Euclidean boundary 8b of b. This procedure will only yield Martin boundary points of 6 below C; more precisely this procedure will yield a boundary for I). A second new complication is the one-sided nature of Harnack's parabolic context inequality, which is relied on to bound K(h, ) as n tends to a!. In fact this inequality, based on the normalization K(4, C) = 1, only bounds K(ry', ) strictly below C. Thus a limit function of K(, ) as it - 8d cannot be defined at C so the normalization of such a function at C requires a new formulation. This problem does not arise in the classical context. These difficulties will be at least partially surmounted in two ways, with-
364
I.XIX. The Martin Boundary in the Parabolic Context
out renouncing the Martin approach. One way is based on a Martin point set pair, as defined in Section XVIII.17; the other way is based on a Martin measure set pair, to be defined in the next section.
2. The Martin Functions of Martin Point Set and Measure Set Pairs The Martin Function of a Martin Point Set Pair The Martin function k of a Martin point set pair (C,15) is defined on b x b by
K(ii' 6 =
o
(2.1)
PdD(b' o
For fixed q the function K(q, -) is superparabolic on 15, parabolic on ,6 and has the normalization "K(r), a;) = 1, that is, K(ry', -) has parabolic-fine limit I at . This normalization displays a limitation of the approach to
Martin boundaries by way of Martin point set pairs: every positive superparabolic function u to be considered on b will have to satisfy the condition °fti(t) < + oo. In particular, the minimal parabolic functions on D and the
positive parabolic functions on b represented by the parabolic context Martin representation will have to satisfy this condition. If D' is an arbitrary nonempty open subset of A" and if e b', the pair b) is a flat Martin point set pair to which Martin boundary theory based on Martin point set pairs can be applied. In some cases we shall see that we can then vary to obtain a Martin-type boundary for d itself and a Martintype representation of an arbitrary positive parabolic function on D'.
Martin Measure Set Pairs and Their Martin Functions Let b be a nonempty open subset of ", and let v be a measure on D. Then the pair (v, b) will be called a Martin measure set pair if the following two conditions are satisfied :
MMS(1) The copotential v0D is finite valued and continuous on the MMS(2)
complement of a compact subset o, of 15. If e b, there is a point strictly above relative to D such that no neighborhood of ' is v null; that is, 4' is in the minimal closed in ,6 support of v.
Observe that condition MMS(2) implies that v cannot have compact support in D. It is trivial that to every nonempty open subset ,6 of ft' corresponds a measure v such that (v, D) is a Martin measure set pair, and in fact measures
2. The Martin Functions of Martin Point Set and Measure Set Pairs
365
can be chosen for which v( is finite valued and continuous on b (so Q,, = 0)
and D is itself the smallest closed in b support of v. Since C,,(4, ) > 0 on AI condition MMS(2) implies that v( > 0 on D. The Martin function K for the pair (iv, b) is defined on D x D by K(q, ) _
(= 0 at the infinities of
D('
, q)
(2.2)
v,(
For fixed q with id M) < +oo the function
K(q, -) is superparabolic on D, parabolic on D - {ry' }, and has the normalization f DK(q, )dv =L This normalization displays a limitation of the approach
to Martin boundaries by way of measure set pairs: every positive superparabolic function ti to be considered on D will have to satisfy the condition fti u dv < + oo. In particular, the minimal parabolic functions on D and the
positive parabolic functions on b represented by the parabolic context Martin representation will have to satisfy this condition. Every positive parabolic function on D satisfies this condition, however, for a suitable choice of v, depending on the specified function. Observe that for a Martin measure set pair (i,,6) the function u f v u dv from the space of positive superparabolic functions on D, with the topology of pointwise convergence, into A' is lower semicontinuous (Fatou's lemma), as is (Section XVIII.17) the function u --. °fr (r) for ({, b) a Martin point set pair. The fact that there is lower semicontinuity here rather than continuity as in the corresponding classical context is a complicating feature of the parabolic context Martin boundary constructions. Although the approaches to the parabolic context Martin boundary by way either of Martin point pairs or set pairs involve restrictions on the classes of positive parabolic functions representable by the Martin representation, in certain cases (for example, if the domain ,6 involved is a slab), we shall see that the Martin representation leads to a representation of every positive parabolic function on D.
Generalization of Martin Point and Measure Set Pairs
It is clear that if D is an arbitrary nonempty open subset of AN and if we assign a measure v on the union of D with the set of parabolic-irregular boundary points of b, and if we suppose that v has suitable properties, we can combine the Martin boundary developments based on Martin point set and measure set pairs. We shall not need this refinement, however. Admissible Superparabolic Functions
If (C,,6) [(v,,6)] is a Martin point [measure] set pair we shall call a positive superparabolic function u on 6 admissible if °Jti(C) < + oo [16 u dv < + J.
366
1.XIX. The Martin Boundary in the Parabolic Context
If h is an admissible strictly positive parabolic function on D, a function ti/h on b will be called admissible if ti is the difference between two admissible
positive superparabolic functions on D.
3. The Martin Space DM Theorem. If (c', b) [(v,,6)] is a Martin Point [measure] set pair there is a unique up to homeomorphisms metrizable compactification DM of D with the following properties: (a)
The Martin function k has an extension (also denoted by K) to DM x D[(DM - o;,) x b] for which, when C e D, the function is finite valued and continuous on DM [D - o;].
(b)
K(n t ,
)=
IC(rl2 ,
) if and only if n t = i12
The set 8MD = DM - D will be called the Martin boundary of (C', D) or (v, b) as the case may be. This theorem implies (see the discussion in the corresponding classical context of Theorem XII.3) that K(il, ) is a positive parabolic function on D when i is a Martin boundary point. The proof follows that of Theorem XII.3, but observe that DM is defined in terms of a specified pair (S', D) or (v,,6) so k is uniquely determined. Thus ,6 itself has not been assigned a Martin boundary. Let f be a strictly positive continuous function on D, lv,.t integrable over D, and for h and ii2 in D define d(i1 1 i2) = I [I A
)I]l( )lx+t(d )
0
to obtain a metric on D compatible with the Euclidean topology of D. The
proof of the theorem then follows that of Theorem XII.3 [with a minor variation for (i,,6) when 6i # 0] and will be omitted. The appropriate versions of the Harnack inequality have been proved for (c', b) in Section XVIII.17 and for (i,,6) in Section XV.11, and in view of these versions C e a'D implies that IC(t , ) = K(C, ) locally uniformly on b and that a sequence ry'. of points of b converges to a Martin boundary point C if and
) = K(C, ) locally uniformly on D. The lower semionly if continuity property of our normalizations (Section 2) implies that for a in amb (C',D):
(v,D):
-fk ,C') s I; JI(C.)dv 1.
(3.1)
We shall need more general inequalities. Suppose that A is a finite measure on DM. supported by DM - fr; in the (i,15) case. Then the function
4. Preparatory Material for the Parabolic Context Martin Representation Theorem
367
JpM K(C, )A(dC) is superparabolic on 1), parabolic if A is supported by eMb, and we now show that
b):
Pf(J
K(C, )A(dC))
(0
= L.-
rrg(C, C')A(dC) 5 , (bm) (3.2)
(v, b):
f i(d4) D
J DM
k(C, C)A(dC) 5 . (,6m),
with equality if and only if A is supported by the set of points C for which there is equality in the corresponding inequality under (3.1). Inequality (3.2) for (t, b), with equality under the stated condition, follows trivially from (3.1) for (v,1)). To analyze (3.2) for (C',,6) observe (see Section XVIII.17, but note that C there is replaced here by C') that if h is a Euclidean open subset of t) with every Euclidean boundary point except C' in 1) and if h is a parabolic-fine deleted neighborhood of C', then PfNe (C',
f
s z(bM). (3.3)
k(C, -)A(dC)) =
DM
pM
If we order the class of sets 1} by inclusion, then (from Section XVIII.17) when h shrinks to {, the first term of (3.3) increases to the first term of (3.2) for (C', tl) and the integrand in the second term of (3.3) increases to Pf'(C, C'). Thus (3.2) for (C',,6) is true if integration of the second term of (3.3) to the limit when h shrinks to C can be justified; on the latter point it need only be pointed out that (Fatou's lemma) the function C' Pfµ$(C', K(C, )) is lower semicontinuous. Finally, it is trivial that there is equality in (3.2) for (C', b) if A is supported by the set of points C for which there is equality in (3.1) for (C',1)).
The Zero Set of
i) > 0 if and only if it is strictly below relative to 6, it follows that if 7, is a sequence of points in d with limit a Martin boundary point C and if the numerical sequence ord, . has limit a, Since (from Section XVII.4)
then K(C. ) = 0 when ord C 5 at.
4. Preparatory Material for the Parabolic Context Martin Representation Theorem In this section we shall derive the parabolic context counterpart of Lemma XII.4. The latter lemma was the key lemma in the derivation of the Martin representation theorem, and it will be clear from the following proof that the reasoning in the derivation of that theorem can be translated without diffi-
368
1 XIX. The Martin Boundary in the Parabolic Context
culty into the parabolic context. We shall therefore omit the proofs of the counterparts of the other results in Sections XII.4 to XII.8 leading to the Martin representation theorem, but we shall state the basic results in the parabolic context for ease in later reference.
Lemma (Parabolic Counterpart of Lemma XII.4). Let (c','6) [(f,,,6)] be a Martin point [measure] set pair with Martin function K. If ti is a positive admissible parabolic function on ,6 and if A is a subset of b [with t n aA = 0] there is a measure A, on DM, supported by the boundary of A relative to D"', for which (reduction relative to b) QuOA
=
(4.1) DM
and
D):
1 uV (C'): (4.2)
(v, D):
d'
AA(DM)=
fib
In particular, there is a measure A,;(= A,;o) supported by aMD for which
f
,i =
IC(e, )2;,(dC)
(4.3)
J BM0
and
PJu( ');
(c', D):
(4.4)
Judy.
(',D):
A measure ,, on aMD satisfying (4.3) must also satisfy (4.5)
?MD
for every Bore! subset t of aMD.
We treat first the (z',15) case. If A is relatively compact in A then jtiD" is the potential GDA' of a measure j.' supported by 0,4; so a(dd)
Q A = ODA = J r'A
(4.6)
5. Minimal Parabolic Functions and Their Poles
369
and by (3.2) PfO, A(C")
=
fk(C, )A(dd) _ ,(aA).
P f.4
If A is not relatively compact in b, let A be the intersection of A with the nth
member of an increasing sequence of relatively compact open subsets of b with union D. According to what has just been proved, the potential WA. can be represented in the form (4.7)
f,5.1
where 1 is a measure supported by aA and 2n(bm) = PfNOAflg) 5 "fW (O-
(4.8)
The sequence 2 is a bounded sequence of measures on the compact space D". If A is the limit of a vaguely convergent subsequence, then 2 is supported by the boundary of A relative to D", and (4.7) yields (4.1); equality (4.2) for
W, D) follows from (4.8) and the lower semicontinuity of the operator ti -Pfv(e'), operating in this case on an increasing sequence of positive superparabolic functions. The proof that (4.3) implies (4.4) is carried through by a direct translation of the corresponding implication proof in Theorem XII.4. In the (i,D) case A(dd) in (4.6) is and we go on as in the (r', D) case.
5. Minimal Parabolic Functions and Their Poles Poles of Parabolic Functions If ,6 is a nonempty open subset of 9" provided with a boundary by a metric compactification, a boundary point is said to be a Pole of a positive parabolic function h on D if (reduction relative to D) Uh1(0 = h. If h is minimal parabolic on D and is not identically 0, either is a pole of h or bhOul = 0 because the parabolic reduction operation on a boundary subset is idempotent. The elementary properties of poles in the classical context (Section X.7) are valid in the present context with no change in derivation.
Minimal Martin Boundary Points
A Martin boundary point
of ((,b) [(i,,,6)] will be called minimal if its
associated parabolic function k(C, ) is minimal with
370
I.XIX The Martin Boundary in the Parabolic Context
h=I
I
f D
K(C, ') dv
=
(5.1)
11
and the set of minimal boundary points will be denoted by 8;'D. Observe that the counterpart of (5.1) in the classical context is trivially true for each function associated with a Martin boundary point but that in the present context one Martin boundary point, say Co, associated with the identically vanishing function, is associated with a minimal function but is not a minimal boundary point. According to the following counterpart of Theorem XII.5,
however, every minimal parabolic function which is admissible and not identically 0 is a constant multiple of for some uniquely determined minimal Martin boundary point C. The proof of this theorem follows that of Theorem XII.5 and is omitted.
Theorem (Parabolic Counterpart of Theorem XII.5). Let (r', D) [(i,,6)] be a Martin point [measure] set pair with Martin function K determining a Martin boundary aMD. (a)
Every minimal parabolic function on ,6 has a pole on 8"'D. If a Martin boundary point is the pole of a positive *0 admissible parabolic
function ti, then is the only pole of ti, ti = const K(C, ), and is a minimal Martin boundary point. In particular, if ri is a minimal Martin boundary point, the function k(,, ) has pole ry'. (b) IfC is a minimal Martin boundary point and if A is a set of minimal Martin boundary points, then is either K(C, ) or 0 according
6. The Set of Nonminimal Martin Boundary Points We leave to the reader the easy formulation and proof of the parabolic counterpart of Lemma XII.6. In the parabolic context Theorem XII.7 becomes the following.
Theorem. If b is a nonempty open subset of 98', then in the context of either
Martin point or Martin measure set pairs for D the set A of nonminimal Martin boundary points is an F, set, and for every positive admissible parabolic function h on b (reduction relative to D), hQ" 0. In particular, if 4 is strictly positive the set A is h -parabolic measure null.
The proof follows that of Theorem XII.7. We also leave to the reader the application of Theorem 6 to prove the parabolic context counterpart of Lemma XII.8. The embarrassing possibility in this context that h may not be strictly positive can be treated by proving the desired result first for h + E with c > 0 and then letting a tend to 0.
371
8. Martin Boundary of a Slab b = R' x ]0, d[ with 0 < b 5 + co
7. The Martin Representation in the Parabolic Context Let D be a nonempty open subset of d8". We shall use the notation corres-
ponding to that of the classical context Martin representation theorem (Theorem XII.9). Thus "Sm refers to the class of differences v/h = (v1 - v2)/h,
where v, is a positive parabolic function on D and h is a strictly positive parabolic function on D, omitted from the notation if h =- 1. The following is a parabolic version of the classical Martin representation theorem. Theorem. Let (C', D) [(i,,6)] be a Martin point [measure] set pair determining a Martin function k and Martin boundary a"'D. (a)
To each admissible function v in!;. corresponds a unique finite-valued signed measure MM on aMD, supported by the minimal Martin boundary amD, positive if v is, and satisfying v
=f
Jvdv].
vfv(C')
C=
aMb
(7.1)
(b)
If h is a strictly positive admissible parabolic function on D, the correspondence v/h .-. M,; is an isomorphism between the vector lattice
(c)
of admissible functions in 45,,, and the vector lattice of finite signed measures on a",6 supported by aMD. An admissible function v/h in "5,,, is quasi bounded or singular if and only if k6 is, respectively, absolutely continuous or singular relative to A. In the quasi-bounded case v=
K(C, ) M;, Mi(dC)
(7.2)
J aM»
The proof follows that of Theorem XII.9 with slight variations.
8. Martin Boundary of a Slab D = RN x ] 0, 8[ with
0<S. (9.1)
limK(ry,
(b)
If In I(1 - t)-' -' + oo, then /t(j,)
0.
The Martin boundary therefore consists of a point y corresponding to each point y of 08", with K(y, ) given by the right side of (9.1), and a point 0 with
/t(0, ) - 0. Since the context here reduces to that in Section 8 under an Appell transformation taking the lower into the upper half-plane, we go no
further except to remark that b has many parabolic irregular boundary points but that the upper half-space has a quite different configuration, which suggests the interest of a more careful analysis than we have given of regularity at co in the parabolic context. Martin Boundaries for U8' Martin point set pairs cannot assign a boundary to Q;8", but if a measure v is chosen on Ll to make (v, A") a Martin measure set pair (for example, if v is chosen in such a way that vG is cosuperparabolic on d8", that v is supported by a set whose intersection with the slab R' x ] - oo, a[ is bounded for each a, and that the smallest closed support of v has points with arbitrarily large ordinate values), then we obtain a Martin boundary for (v, A'). This boundary consists of a point 0 corresponding to the identically vanishing parabolic function and of points y obtained like the corresponding Martin boundary points of the lower half-space obtained above. It was shown in Section XVI.8 as an application of the Appell transformation to the known properties of parabolic functions on a slab that a positive parabolic function on 9" is minimal if and only if it is a positive multiple of
a function
-+exp( + a2lyl2s/2) and that a positive parabolic
375
10. The Martin Boundary of t5 = ]0, + oo [ x ] - oo, b[
function on @8" can be represented in a unique way as an integral XVI(8.2) over the class of minimal parabolic functions. The reader is invited to derive these results as applications of the Martin boundary theory developed in this section.
10. The Martin Boundary of b= ]0, +[ x ] - oo, S [ This Martin boundary has features not present in those discussed in Sections
8 and 9. To simplify the notation, we have taken N = 1, and we leave to the reader the formulations for N > 1. Suppose first that S < + co. Let ' be a strictly positive number, and define 4' = (c', b). Then (c', I)) is a Martin point set pair, and if K is the Martin function for this pair,
6b(4,0=As-t, K(q, 4) =
e(s -
7)-Os S),
- 7) - G(s - t, + 7)
n = (7, t). (10.1)
B(b-7)-G(b-t,+7)
Corresponding to each point T of R we introduce the symbol T' and for in the right half-plane ]0, + co [ x I8 define _ z
K(r
)=
a(2n)112(s - T) 3/2 exp 2az(s
r)
ifs > r
(10.2)
if sST.
0
Corresponding to each point y of - U8+ we introduce the symbol y" and for in the right half-plane define (e -Y4 - &4) exp
}
zazs if y < 0,
(10.3)
if y=0. The functions K(r', ) and K(},", ) are positive and parabolic on the right half-plane. The limit functions of k are the following: (a)
When q - i = (0, r) with r < S, lim K(tq, ) = K(r', ) n-r
(b)
K(r', ')
on b.
(10.4)
on D.
(10.5)
Whent - -ocand 7/t-.a', 50, lim K(7', ) = K(` ')
376
I.XIX. The Martin Boundary in the Parabolic Context
Observe that lim
K(T'' )
_
= K(0", K(0',
for e D.
If n = (q,t)ed and either (cl) I-y +oo and q/(1 + jtl)-i +oo or
(c)
(c2) t - 6 with no restriction on the varying of ?I, then lim K(i1, ) - 0.
The Martin space of (c', /)) is therefore D compactified by adjunction of
the set {0} x ] - oc, 6[ identified with the set {i : - ao < T < S}, the set {! ": ; E - P+ }, and a single point 0: k(T', ) and K(y", ) are defined respectively by the right sides of (10.4) and (10.5); K(0, ) 0; y in b tends to i or or 0 in the Martin topology if and only if the respective conditions (a) or (b) or (c) are satisfied. Next we show that every Martin boundary point r' is minimal. According to Theorem 5, it is sufficient to show that each function K(T', ) has pole r'. Let A be the set of all Martin boundary points except the
point T'. If c > 0, the function identically equal to t on £ is positive and superparabolic and majorizes K(T', ) on a deleted Martin neighborhood of each point of A. Hence (reductions relative to the Martin space) K(T', ) QA
-0;so
k(T',
k(T', ) OiMD
To. The boundary set {T': r 5 To} is then N/, null, and the discussion needs only trivial changes.
Part 2
Probabilistic Counterpart of Part 1
Chapter I
Fundamental Concepts of Probability
1. Adapted Families of Functions on Measurable Spaces (a) Filtrations of a Measurable Space. Let (fl, F) be a measurable space, and let (I, 5) be a linearly ordered set. A filtration of (f2, F) is a map t - F(t)
from I into the class of sub a algebras of IF, increasing in the sense that is called a filtered s 5 t implies that F(s) a .9r(t). The triple (0, F, measurable space.
If (52, .F, fl-)) is a filtered measurable space and if the index set I is an interval in l ordered by 5, define F+(t) = n,,,F(s) for t in I to get a is said to with F(t) c .F+(t) for tin I. The filtration filtration is necessarily be right continuous if F(-) = In particular, right continuous. (b) Families of Functions. Let {x(t), t e I) be a family, indexed by a parameter set I, of functions from a space f2 into a "state space" S2'. The value of the function x(t) at the point w of 0 will be denoted by x(t, w); the function w) : t r- x(t, CO) is called x(t) will also be denoted by x(t, ). The function the sample function determined by co. The family x(-) of functions on 0 is described as continuous, or right continuous, or ... if the sample functions
are all continuous, or right continuous, or .... for a state space and parameter set structured to make the description meaningful. (c) Adapted Families of Functions. If {x(t), tell is a family of measurable into a measurable functions from a filtered measure space (S2, space (12', -W') the family is said to be adapted to if for each tin the parameter set I the function x(t) is measurable from the measurable space
(0,9(t)) into the measurable space (0%9'). The notation will always imply adaptedness unless the contrary is stated. If { x(t), t e 1) is a family of measurable functions from a measurable space (Q, F) into and if I is linearly ordered, define ,9r0(t) = a measurable space
,9r{x(s),s 5 t} for t in I (see Appendix IV.1 for the notation used here) making to obtain a filtration an adapted family. The filtration go(-) is minimal for adaptedness in the sense that if (F(t), to l) is a filtration with respect to which is adapted, then #0(t) c
F(t)fortin1.
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2.1. Fundamental Concepts of Probability
Filtered measurable spaces and their adapted families of functions provide a mathematical formalism modeling certain physical ideas. A measurable space (Q, 37) is a mathematical model of the set of possible events in some physical context, together with a distinguished class of compound events. If I is a subset of R, a filtration {.fi(t), t e 11 of (Q, F) is a mathematical model for the flow of events in time. Each pair (t, w) represents a possible outcome of an experiment at time t, and .fi(t) represents the class of compound events observable before or at time t. The value x(t, w) of a function x(t, -) at co models the value of some observable at the outcome (t, w), and the function x(t, ) itself is therefore incorporated in F (t) in the sense that this function is supposed .F(t) measurable; that is, is an adapted process. The Measurable Sets of a Topological Measurable Space If a measurable space is given as a topological space, the a algebra of measur-
able sets will always be the a algebra of Borel subsets of the space unless some other a algebra is specified. In particular, the state space R means the measurable space (R, .(R)).
2. Progressive Measurability If I is a subinterval or singleton of R, we denote by(/) the class of Bore] subsets of /. If I is a singleton, R (l) consists of that singleton and the empty
set. A family {x(t), t e 1) of functions from a measurable space (5),.x) into a measurable space (0', .I') is called measurable if the function (t, (0) r+
x(t, w) is measurable from the measurable space (I x Q,. (I) x .F) into the measurable space (0',.F'). This definition will frequently not be strong enough, however, when we deal with an adapted family {x(t),.F(t), teR+}. In fact in dealing with an adapted family one frequently wishes to deal with a subfamily of the form {x(t), .F(t), I < c} and wishes the analysis of this subfamily, in particular, the measurability of the subfamily, to depend only on properties involving parameter values in the interval [0, c]. Unfortunately even if 9 = YE R+ .fi(t), the measurability of the original family
does not imply that the subfamilies of the stated form are measurable. The following definition is formulated precisely to provide this measurability. Suppose then that {x(t), ,fi(t), t n R+ } is an adapted family of functions. A subset A of R+ x 0 will be called progressively measurable if A n ([0, c] x Q)
e 9([0, c]) x F(c) when c >_ 0. The class of progressively measurable sets is a a algebra. The adapted family is called progressively measurable if is measurable from the space R+ x 0 coupled with the the function
389
2. Progressive Measurability
or algebra of progressively measurable sets into (a', .5'), equivalently, if x for c z 0 the restriction of the function x .5(c)) into (0', 5'). (In this phraseology the c] x ([0, condition that a set A be progressively measurable is equivalent to the condition that the adapted family { le(t, ), .fi(t), t e 68+ } with state space (R+,.V(R+)) be progressively measurable. Observe that even without the prior hypothesis of adaptedness the progressive measurability condition implies adaptedness. We leave to the reader the easy proof that if {x(t), .fi(t), t e 18+ } is progressively measurable, then for b > 0 the family {x(b + t),
5(b + t), to R+} is also progressively measurable. It is immediate that if is progressively measurable and if f is a measurable function from the state space (f2', .') into the measurable space then
If
is also progressively measurable, with state space
Alternative Characterization of Progressive Measurability
In the preceding context define a new filtration by setting .5-(0) = .5(0) and .5- (t) = Y 0, and consider the following conditions on the family {x(t), to R+} of functions from (Q,JF) into (0', Jr): (a)
(a') (b)
The family is adapted to is right continuous. For c > 0 the restriction of the family
to the set [0, c[ is
R([0, c[) x F- (c) measurable. Observe that the pair (a'), (b) implies (a). We now show that is progressively measurable if and only if (a) and (b) are satisfied. To prove
that (a) and (b) together imply progressive measurability, we need only
observe that the subsets [0, c[ x 0 and {c} x 0 of [0, c] x 0 have the latter set as union and are in R Q0, c]) x .5(c) and that under (a) and (b) the function is measurable on both subsets and so also on their union, relative to .([0,c]) x JIF(c). Conversely, progressive measurability implies (a), and another piecing together of measurable functions shows that progressive measurability also implies (b). EXAMPLE. Let (x(t), .fi(t), t e R+ } be an adapted family whose state space is Polish. Then the family is progressively measurable if the sample functions
are all right continuous or all left continuous. For example, in the right continuous case define
n
1 5j 5n, n / if(j - 1) nC 0. Then the restriction of x F(c) x measurable and so the progressive measurability condition is satisfied when c > 0. The condition when c = 0 is satisfied because the family is adapted. The left continuous case is treated similarly. Observe that if whenever e > 0 the restriction of the family to the
set [0, c] is 4[0, c] x ,F(c + e) measurable, then (b) is satisfied. In fact this condition implies that for 0 < S < c the restriction of this family to [0,C - 6] is 91([O, c - S]) x ., (c - 21
..([0, c[) x F-(c
measurable, and this yields (b). In other words the adapted process is progressively measurable if the process {x(t), F(t + e), to I8+} is progressively measurable for every e > 0.
3. Random Variables A probability space is a triple (S2, JF, P) consisting of a measurable space (U..x) and a measure P on F with P{O} = 1. It will frequently be unnecessary to specify 0 or F, but it will always be supposed even though not mentioned that P is complete, that is, that subsets of P null sets in F are also in F. A random variable is a measurable function from a probability space into a measurable "state" space. If the state space is topological, its measurable sets unless otherwise specified will be the Borel sets. The distribution of a random variable x is defined as the measure y on the class of measurable
sets of the state space induced by x: u(A) = P{xeA}. The integral of a real-valued random variable x will be denoted by E{x}, and the integral of x over a set A will sometimes be denoted by E{x; Al.
Sometimes the context is expanded to allow measures P other than probability measures, even to allow P{i2} = + oc, but if so, the expansion is mentioned explicitly. We follow the common abuse of notation in measure theory, in which the same notation is used for equivalence classes of measurable functions under the equivalence relation of almost everywhere (a.e.) equality, or "almost sure" (a.s.) equality in the slang of probability, as for their members. The proper interpretation will be clear from the context. Thus the essential infimum ess info z, of a family of random variables (Appendix IV.9) is determined only up to a null set and properly speaking is an equivalence class, but the language and the notation will refer either to the equivalence class or to one of its members, as indicated by the context. The members of such an equivalence class will sometimes be called versions of the class.
4. Conditional Expectations
391
4. Conditional Expectations If x is a real integrable random variable on a probability space or a real random variable bounded on one side, and if St is a a algebra of measurable
sets, the conditional expectation E{xIS } is the uniquely determined up to null sets random variable which is Sr measurable and satisfies E{E{xjS=};A} = E{x;A}
for AeSr.
(4.1)
The notation E{xlF } always refers to any one of the possible choices of the conditional expectation rather than to the equivalence class. Unless the contrary is stated, it will always be assumed in discussing this conditional expectation that x is integrable. The Radon-Nikodym theorem ensures the existence and uniqueness up to null sets of conditional expectations, and the following six properties follow easily from the defining properties. (a) E{ yjF } = y a.s. if y is 9 measurable, in particular, if y is a constant function. (b) If y is bounded and 9 measurable, E{xylS } = yE{xjSr } a.s. (c) E{x + yjSr } = E{x!F } + E{ yjSF } a.s.
(d) x5 y a.s. implies that E{xjg } 5 E{yjF } a.s. and almost sure equality in the first inequality implies almost sure equality in the second.
These properties imply that the map x"E{xjS } is a positive linear idempotent transformation from L' into L', of norm 1. (el) If Ae.Sand if x=ya.s.on A,then E{xjS=}=E{yf.F}a.s.on A. (e2) If A E Sr r I and if the a algebras Sr and I have the same trace on A (that is, if a subset of A is in 9 if and only if the subset is in !fl, then E{xj5} = E{xlg} a.s. on A. (f) If S= c 4, then E{E{xlSt}14} = E{E{xl§}I5} = E{xI.4r} as.
(4.2)
The next properties are less immediate, and their proofs will be sketched. (g) If x is independent of Sz (that is, if x is independent of every Sr measurable random variable), E{xIF } = E{x} a.s. In fact, if y is the indicator function of a set A in Sr, the random variables x, y are independent and
E{x; A} = E{xy} = E{x} E{ y} = E{E{x} ; A}
as.
(4.3)
(h) Jensen's inequality for conditional expectations. If f is a convex function and if x and fix) are integrable, then
f[E{xIS=}] S E{ f(,c)I5} 'a.s. In fact, if
z a + b for all ,
(4.4)
392
2.1. Fundamental Concepts of Probability
E{ f(x)IF} z aE{xl.*T} + b a.s.,
(4.5)
and since a convex function is the supremum of all the linear functions it majorizes, and even the supremum of a properly chosen countable subset of those linear functions, (4.5) implies (4.4). If 3F contains only the space and the empty set, (4.4) reduces to Jensen's inequality. According to (h), if pal, the transformation takes LP into itself. In particular, let 9R be the space of square integrable F measur-
able random variables. The transformation maps LZ onto 0, and it is easy to verify that x - E{xl.F } is orthogonal to 0, so that the transformation acting on LZ is the orthogonal projection onto R. (i) If x is integrable, the family of all conditional expectations of x is uniformly integrable. In proving this it can be assumed that x is positive (or replace x by lxl), and then if we write x, for E{xl.F } and if c is a strictly positive constant,
E{ x,, ; x, > c} = E{x; x, > c},
P{x,, > c} 5
E xs = Mx c
c (4.6)
The second relation ensures that, for sufficiently large c, P{x, > c} is uniformly small as .F varies and the left side of the first relation must then also be uniformly small for large c as F varies, and this property is precisely the uniform integrability property. Alternative Proof The following less direct proof displays a technique which has many applications. Since x is integrable, this function constitutes a uniformly integrable class; so there is a uniform integrability test function 0 (Appendix V) for which E{b(I xl) } < + oo. Apply Jensen's inequality (h) above for conditional expectations to obtain
E{1(E{Ixl IF})) s E(E{D(Ixl)IF} = E{'D(Ixl)}. Since the left side is uniformly bounded as .F varies, the family of conditional expectations of x is uniformly integrable.
More generally a trivial extension of either proof shows that the class of conditional expectations of the random variables of a uniformly integrable family is uniformly integrable. (j) If {x(t), t e J } is a family of integrable random variables and is directed upward neglecting null sets, then (whether or not ess sup,. I x(t) is integrable)
393
5. Conditional Expectation Continuity Theorem
ess sup E{x(t)lF } = E {ess sup x(t)IF} rel
rcl
a.s.
(4.7)
111
This fact for integrals, that is, in the present context for expectations, is proved in Appendix IV.9. Since the family (E{x(t)I5 }, t e 1} is directed upward neglecting null sets, the result for integrals yields, when A E .F, E {esssuPE{x(t)Is } ; A } = sup E{x(t); A} = E {esEsup x(t); A} , (4.8)
and this equation is equivalent to (4.7) if we choose the essential supremum on the left, as we can, to be .F measurable.
The corresponding result for downward-directed families is deduced by replacing x(t) by -x(t). Specialization. If x is an integrable random variable and if is a family In particular, of random variables, is defined as is a finite set of random variables, say x(1), ... , x(n), it follows if (Appendix IV.3) that there is a Borel measurable function 0 on R" such that
E{xlx(l),... ,x(n)} = 4,[x(1), ... ,x(n)]
a.s.
(4.9)
5. Conditional Expectation Continuity Theorem Theorem. Let {,9=(n), n e l+ } [ {F(n), n e -V}] be a filtration of a probability space (0, .F, P), and define F( + oo) =Yo .fi(n) [JF(- oo) = n°_. 5(n)]. Then if x is an integrable random variable on f2, it follows that in both the almost everywhere and Lt senses
lim E{xl.fln)} = E{xlfl+oo)}
ft-M
(5 . 1)
in the forward case and
lim E{xIF(n)} = E{xl#'(-oo)}
(5.2)
in the backward case.
This theorem has applications in surprisingly many apparently unrelated contexts. It is a martingale convergence theorem and as such will be derived again (Sections 111.14 and 111.17), but an independent proof will be given
here to avert suspicion of circular reasoning. To simplify the notation, define x(n) = E{xl S(n)} for n in 1+ or in -Z+ according to the context.
394
2.1. Fundamental Concepts of Probability
To prove (5.1), choose integers j < k < ! < m and real numbers a < b, and define
v=i{min nf{n21:x(n)Zb},l [j, k, !; r] =
/snsk
x(n) S a; v = J
(5.3)
[j, k;1, m] = 5min x(n) 5 a; max x(n) z b = U [j, k, !; r], 15n5m
jSnSk
r=t
[a; b] _ {liminfx(n) < a, lim sup x(n) z b }
.
01-Go
Then x(r) z b on [ j, k, !; r]. so
f
J
x dP=
f J,k.l;rl bdP
x(r)dPZ
lJ,k.l;r1
Ij.k;l.w,1
(5.4)
=bP{[j,k;l,m]}. Observe that if "5 a" had been " < a" or " Z b" had been "> b" in the above definitions, the change would carry over throughout with no change
in the reasoning, and a trivial continuity argument for varying a and b shows that here and in further reasoning below the inequalities allowing equality imply those prescribing strict inequality and conversely. In view of this fact (5.4) yields, when successively m -+ + oo, ! -+ + oo, k --+ + oo,
j-4 +oo, xdP> bP{[a;b]}.
(5.5)
If the triplex, a, b is replaced by - x, -b, -a in (5.5), this inequality becomes
xdP < aP{[a;b]},
(5.6)
Lb, and we conclude that P{[a;b]} = 0. The equality diverges} =
U
[a; b]
a to P{x(t) = x(to)} = 1.
(5.8)
To see this when is increasing, observe that (Appendix IV.2) there is a countable subset J of 1 such that x(oo) is YFj G(t) measurable, and we leave to the reader the proof that (5.8) is true with to any upper order bound of J. The following argument is more instructive and shows that to exists satisfying (5.8) whether is increasing or decreasing. An order-preserving map 0 will be exhibited in Section 111.4, involving only elementary inequalities, taking I into a subset I' of R, with the property that ¢(s) = 4)(t) if and only if P{x(s) = x(t) } = 1. The set I' must have a last element t' because I' has a cofinal sequence s;, and if s, is a sequence in I with 4)(s) = s.' and if t is an order upper bound of s,, then 4)(t) = t' is the last element of 1'. Each element to of 46-(t') satisfies (5.8).
6. Fatou's Lemma for Conditional Expectations Lemma. Define 1, as in the generalization of Theorem 5, and let {x(t), t E I } be a family of positive extended real-valued random variables. Then
essliminfE{x(t)J4(t)} z E{essliminfx(t)1f(oo)} a.s. In particular, if I =
Z+,
(6.1)
397
7. Dominated Convergence Theorem for Conditional Expectations
liminfE{x(n)I'4(n)} z E{liminfx(n)Il(oo)}
a.s.
(6.1')
For every k in Z+ and sin I the left side of (6.1) is almost surely at least
essliminfE{k n essinfx(r)I5(t)} = E{k n es2infx(r)I#(co)} a.s. r,> S s it
(6.2)
Inequality (6.1) follows from the fact that the right side of (6.2) increases as
k and s increase and has the essential limit E{essliminf,rx(t)JI(co)} according to (4.7).
7. Dominated Convergence Theorem for Conditional Expectations Theorem. Let {9(n), n e Z + } be a monotone sequence of a algebras of meaz 4(n) surable subsets of a probability space, and denote by V(oo) either
or n.,4(n) according as W(-) is monotone increasing or decreasing. Let {x(n), n e Z + } be a sequence of random variables on the space, with almost Ix(n)j) < + oo. Then sure limit x(oo) and
lim E{x(n)I4(n)} = E{x(oo)I3(oo)} a.s.
(7.1)
To prove this theorem, apply Fatou's lemma for conditional expectations to the sequences {Msuy Ix(m)I + x(n), neZ+},
{sup Ix(m)I - x(n), neZ+}.
Application. Let Jr be a a algebra of measurable sets of a probability space. Let x and v be random variables on the space with x independent of y measurable with respect to F. Let j be a bounded Borel measurable function on 682, and define f, on 68 by f, (?I) = E{ f(.r, q)}. Then f, is Borel measurable, and E{ f(x,,v)I.fl = f, (y) almost surely. To prove this assertion, let r be the class of bounded Borel measurable functions f for which this assertion is true. The class r includes the functions of the form q) r--# g and h bounded Borel measurable functions on R because in this case f, (i) = E{g(x) }h(ry), and by Section 4(b) and (g)
E{g(x)h(y)I.,F} = E{g(x)I,}h(y) =fl(y)
a.e.
(7.2)
The class r is linear and is closed under bounded pointwise convergence in view of the dominated convergence theorem for conditional expectations. Hence r is the class of bounded Borel measurable functions on 682, as was to be proved.
2.1. Fundamental Concepts of Probability
398
8. Stochastic Processes, "Evanescent," "Indistinguishable," "Standard Modification," "Nearly" Let (S2, .,F, P) be a probability space, and let I be an arbitrary set (to be used as the parameter set of a family of random variables). A subset A of I x d2 is called evanescent if there is a set A in F for which P{A} = 0 and A C I x A. A subset of an evanescent set is evanescent, and a countable union of evanescent sets is evanescent. For many purposes evanescent subsets of I x 0 are
counterparts of the polar subsets of the state space in classical potential theory, and we shall accordingly describe a relation true on I x 0 up to an evanescent set as true quasi everywhere.
A stochastic process on (Q, 337, P, 1) is a family of random variables {x(t), t e I } with a common state space, defined on (0, S=, P), with parameter
set 1. The random variable x(t) of this process has the value x(t, (.0) at the point w of S2, and the process thus defines a function from I x .0 into the state space of the process, that is, the state space of the random variables.
with the property that each function x(t) = x(t, ) is measurable. Two with common probability space, parameter set, and processes and quasi everywhere on state space are called indistinguishable if
I x f2, that is, if P{w: x(t,w) = y(t,(9) for all t} = 1.
(8.1)
Two subsets of I x S2 are called indistinguishable if their indicator functions determine indistinguishable processes, that is, if the sets differ by an evaneswith common probability space, and cent set. Two processes parameter set, and state space are said to be standard modifications of each other if
P{w: x(t,w) =Y(t,(0)) = 1
(8.2)
for all t. The adjective "standard" is sometimes omitted. Sums of processes, functions of a process, and so on are defined in the obvious way in terms of the random variables of the processes.
The concept of an adapted family {x(t), .fi(t), t e 1) of functions on a measurable space was defined in Section 1. If the measurable space is a probability space, the family becomes an adapted stochastic process, and the above notation always implies adaptedness unless the contrary is stated. A stochastic process indistinguishable from one having a specified char-
acter will sometimes be said to have nearly that character, and the corresponding description will be used for subsets of I x 0. Thus a nearly progressively measurable process is a process indistinguishable from a progressively measurable process.
8. "Evanescent," "Indistinguishable," "Standard Modification," "Nearly"
399
Normalizing Hypotheses on Filtrations Let {.f(t), t e 1 } be a filtration of a measure space, and suppose that I has a first point, denoted by 0. If the given measure is complete, as is our convention for probability measure spaces, the hypothesis that .F(0) contains the null sets implies that the restriction of the given measure to each a algebra .f(t) is complete, but the converse is false. In practically every investigation involving a filtration the parameter set has a first element, and it is possible
without loss of generality to suppose that the first a algebra contains the null sets, at the cost of enlarging each a algebra .fi(t) to the a algebra generated by .fi(t) and the null sets. If the parameter set is it is usually also possible without loss of generality to replace by the right continuous filtration OB+,
Progressively Measurable Stochastic Processes If {x(t), .f(t), t e I8+ } is a progressively measurable stochastic process every sample function w) is necessarily Borel measurable. If this process is only nearly progressively measurable, it need not be progressively measurable, but almost every sample function is Borel measurable. For example, define a realvalued process with w) arbitrary for w in some null set of a probability space, define co) = 0 otherwise, and define .fi(t) for all t as the or algebra consisting of the null sets and their complements. The process fl-)) is then indistinguishable from the identically 0 progressively measurable process but is progressively measurable only in trivial cases. It is nevertheless true that most theorems involving progressively measurable processes are valid with unimportant modifications for nearly progressively measurable processes.
Almost Surely [Right] [[Left]] Continuous Processes and Progressive Measurability If {x(t), t e R + } is a stochastic process with a topological state space the process will be called [almost surely] continuous or right continuous or left continuous if [almost every] sample function has the stated property. If to R+} is almost surely [right] [[left]] continuous and adapted and if .F(0) contains the null sets the sample function (0) can be redefined, say by setting x(t, (o) = x(0, (o) for all t when w is in the exceptional null set,
to obtain a [right] [[left]] continuous process adapted to Jr(-) and indistinguishable from the given process. If in addition the state space is Polish, this modified process is progressively measurable according to Section 2. It would be consistent with our conventions to call an almost surely right
or left continuous process nearly right or left continuous, but the almost surely terminology is the customary one.
400
2.1. Fundamental Concepts of Probability
Stochastic Processes in the Essential Order
The class of extended real-valued processes with a common probability space and parameter set is ordered by the essential order in which is a majorant of x(), denoted by or z if P{x(t) 5 y(t)} = 1 5 for all t. This order is thus an ordering of equivalence classes in which two processes are identified if each is a standard modification of the other. The usual inexact language in which a "process" may actually be an equivalence class will be used when there is no danger of confusion.
Essential Order Infima and Suprema
If r : {.r,(-),ac-J} is a family of stochastic processes with a common probability space, common parameter set, and extended real-valued random variables, the essential order infimum of the family, denoted by ess inf,E j or ess inf r, is the process whose tth random variable is ess in f , x,(t) [chosen to be .F(t) measurable if r is specified as a family of processes adapted to a common filtration The process ess sup r is defined and denoted dually. These processes are uniquely determined up to standard modifications and therefore are unique in the essential order. LP Bounded and Uniformly Integrable Processes
A stochastic process is said to be L° bounded if the family of its random variables is L" bounded and is said to be uniformly integrable if the family of its random variables is uniformly integrable.
Sample Function Integrals If {x(t), .fi(t), t E 98' } is a progressively measurable stochastic process with state space the extended reals, the process sample functions are Borel measurable. If the process random variables are positive, the sample function integrals are well defined, that is, meaningful, and if y(t) = fo x(s)1, (ds), that is is, if for each point to of the probability space on which the process defined y(t. to) = It x(s, w)1, (ds), then the process { y(-),.F(-)) is progres-
sively measurable (Fubini's theorem). This integral process is also well defined and progressively measurable if instead of supposing that the random variables are positive, it is supposed that the process sample functions are 1, absolutely integrable over finite intervals. If the process {x(-), is not supposed progressively measurable but merely indistinguishable from a
progressively measurable process, the above remarks are modified in the obvious way.
9. The Hitting of Sets and Progressive Measurability
401
9. The Hitting of Sets and Progressive Measurability Theorem. Let (f ,.F, P; .fi(t), t e R) be a filtered probability space, suppose that the restriction of P to .fi(t) is complete for all t, and let c be a strictly positive constant. (a)
If A" is a subset of R x 0, analytic over the class of progressively measurable sets, then the projection of A" n ([0, c] x (2) on 0 is in 3F(c).
be a progressively measurable stochastic process Let on (S2, P, 5), with an arbitrary state space (X, .1). (b) If A' is a subset of P+ x X analytic over i(1) x T. then
(co: (t,x(t,w))eA' for some t 5 c}e.F(c). (c)
(9.1)
If A is a subset of X analytic over L", then (co; x(t, co) e A for some t S c} E .5(c).
(9.2)
Assertion (c) is a special case of (b) because in (b) the set A' can be chosen as [0, c] x A. Assertion (b) is a special case of (a) by the following argument. The map (t, co) t-- (t, x(t, (o)) from the measurable space R' x S2 coupled with the class of progressively measurable sets into the measurable space (R x X,
R(68`) x L") is measurable because the inverse image of any product set [0, b] x Xo with X. in LI is a progressively measurable set. Hence (Appendix
Theorem 1.8) the inverse image of any set A' analytic over R(R) x .I" is a set A" analytic over the class of progressively measurable sets. Finally the projection of A" n ([0, c] x i2) is then the set in (9.1); so (a) implies (b). To prove (a), observe that by Theorem 7 of Appendix I the projection of A" on i2 is analytic over 5(c) and therefore is in 5(c) by Lusin's theorem (Theorem 4 of Appendix II because the restriction of P to JW(c) is a complete measure.
Variant of the Theorem In (a) the projection of A" n ([0, c[ x S2) on i2 and in (b) and (c) the sets in (9.1) and (9.2) with "t < c" replaced by "t < c" are all in w- (c) = Y«.f(t). The method of proof of the theorem yields this variant, or the variant can be deduced by an application of the theorem with c replaced by c - 1 In, n z 1. Generalization
If in (b) and (c) the process is supposed merely indistinguishable from a progressively measurable process, the conclusion remains valid if 5(0) contains the P null sets. This added condition implies the validity of the condi-
402
2.1. Fundamental Concepts of Probability
tion already imposed, that the restriction of P to .f(t) be complete, because P as always is itself supposed complete. Change of Origin The analysis of the hitting of a set in the parameter interval [0, c] or [0, c[ is
valid with trivial changes for hitting in an interval [b, c] or [b, c[ with
0 0, if A = {w: (t, x(t, (!))) E A' for some t < c},
(12.3)
then A c ,F(c) and A e.?(c). Moreover since A = t-1(A) (notation of Section 10), it follows that P{A} = P{A}. Thus P{A} depends on the finite-dimensional distributions and the right P) and continuity hypothesis but not otherwise on the choice of In particular, if A is an analytic subset of X and A' = [0, c] x A, it follows that the probability of the set and if A is defined similarly for
407
13. Measurability versus Progressive Measurablity of Stochastic Processes
{w:.x(t,w)eA for some t < c}
(12.4)
depends only on the finite-dimensional distributions and the right continuity hypothesis. is supposed only almost surely right continuous, we If the process
can omit the P null subset of C corresponding to non-right continuous sample functions to get the same results as in the right continuous case.
If in (12.3) and (12.4) the restriction on t is t < c instead oft < c, all of the above results remain valid, in fact can be slightly sharpened since the sets in (12.3) and (12.4) are then in F - (c).
13. Measurability versus Progressive Measurablity of Stochastic Processes Let (12, .F, P) be a probability space. We shall need a definition of convergence in measure for random variables from this space into IB when the random variables may have the values ± oo with strictly positive probability. In this context we shall say that a sequence x, of random variables converges in measure to x if the sequence arctan x, converges in measure to x in the usual sense, equivalently, if
lim E{arctan x - arctan
0.
The metric
d(x, ') = E(I arctan x - arctan y }
(13.1)
on the space S of extended real-valued random variables has the property that a sequence converges in the metric if and only if the sequence converges in measure in the above sense. This metric, together with the identification of two random variables whenever they are equal almost everywhere, makes S a complete metric space. Observe that if a sequence in S converges in the metric (13.1), then some subsequence of the corresponding random variables converges almost everywhere, in the topology of A. There is therefore a sequence c. of strictly positive numbers such that if d(x, x x almost everywhere. We use here the standard abuse of language, leaving it to the reader to decide when a symbol represents a function on S2 and when it represents a point of S, that is, an equivalence class of functions. Observe that if we wished to treat only random variables bounded in modulus by 1, the L' metric could be used instead of the metric (13.1). In fact one way to carry out the work of this section is to replace each random variable x by (2/it) arctan x and thereby make all random variables bounded in modulus by 1.
408
2.1. Fundamental Concepts of Probability
In the present context a process {x(t), t e R') with state space 68 defines a function fi'x(.) : t r- x(t) from IB+ into S. This map taking a process into a
function is one to one and onto if two processes are identified when they are standard modifications of each other. Theorem. Let (12, . , P) be a probability space, and let {x(t), t e l l } be a process on the space, with state space A. (a)
(b) (c)
The process has a measurable standard modification if and only if the function fl., is Bore! measurable. Suppose that the probability space is provided with a right continuous filtration (.F (t), t e R+ } for which .5(0) contains the null sets. If the process is adapted and measurable, it has a progressively measurable standard modification. If x 12 relative to the completion of the measure 1, x P, there is a progressively measurable process such that
P{x(t) = x'(t)} = I for 1, almost every t.
(13.2)
is a measurable process, then fix(.) is Bore! measurable. Proof of (a). (i) If is determined by a function (t, w))-+ Since each measurable process ne+} x(t,w) and since pointwise convergence of a sequence implies pointwise convergence of the corresponding sequence { fx1(.)' n e 71+ },
it is sufficient to show that if
has the form n
x(t, w) _ Y_ ct 1AJ(t)1 At(w) I
with At, ... , A. disjoint Borel subsets of I8+ and Ate., then #,(., is Borel measurable. This Borel measurability is trivial. has a measurable standard modi(ii) If fix).) is Borel measurable, then
fication. To show this, it is sufficient to show that the class of functions from OB+ into S corresponding to processes which have measurable standard
modifications includes the continuous functions and is closed under convergence. Suppose first that fx,.) is continuous. Define Ok(t) =.12-k
for (j - l)2-k < t < j2-k (j= 1,2 ....).
Since fl., is uniformly continuous on compact intervals, there are positive integers k, < k2 < ... such that (see the beginning of this section) d[x(s),x(t)] < c if Is - tI < 2-k and s v t < n; so for each fixed tin O8+ lim x[4k.(t)] = x(t) a.s.
(13.3)
13. Measurability versus Progressive Measurability of Stochastic Processes
409
If we define x'(t) = lim sup, x[4kn(t)] the process
is a measurable n e Z } is a sequence standard modification of Next suppose that of processes on (52,,x', P) with the property that each function is Borel has a measurable standard modification, and that measurable, that limn- flan,.) = P.,.) exists in the S metric on 08+ ; that is, limn-m x(t) has been chosen to be exists in measure for all t. We can suppose that by measurable. Define min {k: d [x(t), xk(t)] < The set
{t: mn(t) =j) is a Borel subset of R, and for each t d[x(t), xmn(t)] < en;
so
x(t) a.s.
urn
and
(13.4)
The process defined by (1, w) -s
xj(t, (j)) lAnj(1)
is measurable because it is the product of two measurable processes, so is measurable. The process summing over j we find that the process lim is therefore a measurable standard modification of and the proof of (a) is complete. o
Proof of (b). There is no loss of generality in proving (b) if we suppose, S 1, and we shall assume that by that all processes are bounded in this way throughout the proof of (b). then corresponds a process To each process after replacing
x '(t) = E{x(t)I.F(t)}.
Recall our convention that x°(t) is a version of the indicated conditional is therefore determined only up to a standard expectation and that is adapted and is a standard modification modification. The process if the latter process is adapted. To prove (b), it is sufficient to show of is measurable, in particular, whenever /3X,., is a Borel that whenever has a progressively measurable measurable function, the process standard modification. It is therefore sufficient to show (i) that such a if it exists when /x,.) is continuous and (ii) that modification exists for such a modification exists for and
if it exists for each process of the sequence /3r,.,.
410
2.1. Fundamental Concepts of Probability
Proof of (b)(i). If fist.) is continuous, the conditional expectation continuity theorem can be applied to (13.3) to obtain, in view of the right continuity of
E{x(t) (t}} = x°(t). a.s.
lim
(13.5)
When 2-"^ < e, the nth term on the left defines a process adapted to and progressively measurable relative to the filtration ,f( + e). Hence if the limit in (13.5) is replaced by the limit superior, the resulting process is progressively measurable relative to .IF(- + e) for all strictly positive e and so (Section 1.2) is progressively measurable relative to and this process is a standard modification of Proof of (b)(ii). Apply the conditional expectation continuity theorem to (13.4) to obtain lira E{x.,(,)(t)1F
x°(t)
a.s.,
(13.6)
and repeat the reasoning just used, thereby showing that the process defined when the limit in (13.6) is replaced by the limit superior is a progressively measurable standard modification of The proof of (b) is now complete.
Proof of (c). If
x
is a function
(t, ,,)--. x1(t, w) on e8+ x f? which is measurable relative to -4(R) x F, that is, for which is a measurable process, such that x P almost everywhere. Hence (Fubini)
P(x(t) = x,(t)} = 1 (for 1, a.e. t). According to (b). the process x, has a progressively measurable standard modification and this is the process demanded by (c).
14. Predictable Families of Functions Let (i2, ; fi(t), t e 1 } be a filtered measurable space. The concept of a predictable function family strengthens that of an adapted family in a way suggested by the name.
Discrete Parameter Case, I = Z'
A sequence {x(n); F(n), n c- Z' } of functions from Q into a measurable if x(0) is .F(0) measurable and space is called predictable (relative to if x(n) is F (n - 1) measurable for n > 0.
14 Predictable Families of Functions
411
Continuous Parameter Case, I = R' Consider the class of adapted families {x(t); .fi(t), t e R' } of functions from 0 into R for which every sample function w) is left continuous, that is, is left continuous. (Left continuity at 0 is vacuously for which the family
satisfied by every function on R'.) The smallest a algebra of subsets of R' x Il making the function (t, w) t -x(t, w) measurable for every such left continuous family is called the predictable a algebra, and the sets in this or algebra are called predictable. A family { y(t), t e R' } of functions from fl into an arbitrary measurable space is called predictable if the function (1, w) i-+y(t, co) on the space R' x 0 coupled with its predictable a algebra
is measurable. A predictable family is necessarily adapted to the given filtration. Since left continuity of a function family with state space (R, R(R)) implies progressive measurability (Section 2), the predictable sets are pro-
gressively measurable, and a predictable function family is progressively measurable. The predictability definition implies that if is predictable, if a > 0, and if x,(t) = x((t - a) v 0), then the family is also predictable. EXAMPLE (Continuous parameter case). By definition of predictability a left
continuous adapted family with state space (R, M(R)) is predictable. More generally a left continuous adapted family with state space a metric space coupled with its Borel sets is predictable. In fact, whenever f is a continuous function from the state space into R, the family (f is predictable, and therefore the set {(t,(o): x(t,w)eA} is a predictable set whenever A is a Bore] subset of the state space; so is predictable. Generators of the Predictable a Algebra
If 0 5 a < b 5 + x and if A e F(a), the subset ]a, b] x A of R' x () if b < + x or ]a, + oo [ x A if b = + x is a predictable set, as is the product set {0} x An when A,E.F(0). The class of finite unions of sets in the class r of product sets of these two types is an algebra, and we now show that the a algebra I generated by r is the predictable a algebra. To see this, observe first that if x(t) = 1,,l(t)4 with 0 an JF(0) measurable function from 0 into R, or if 0 5 a < b and if x(r) = 11,.bl(t)o with 0 an .f(a) measurable function from fl into R, then the family is 4 measurable. Second, observe that if { is a left continuous adapted family of functions with state space (R,-V(R)) and if 4,'
y;,(t) = y(0)I1oi(t) + Y y((i -
1)2-")11(J-1)2-".J2 -"10),
i=t
then the family y.(-) is If measurable because it has just been shown that each summand family is. Finally lim, so is I measurable,
412
2.1. Fundamental Concepts of Probability
and therefore I is the predictable a algebra, as asserted. The class of sets analytic over the paving r therefore (Theorem 4 of Appendix I) includes the predictable a algebra (so is the same as the class of analytic sets over the predictable a algebra). For some purposes it is convenient to replace the generating family r by a generating family r' involving compact parameter intervals. Define r as the class of product sets [a, b] x A with 0 5 a < b, A e.F(0) if a = 0
and AE U, 0. The a algebra generated by r' includes the predictable a algebra because
]a, b] x A= U a+ 1, bl x A =t L
n
JJ
[a < b, A e .te(a)], (14.1)
{0} x A= n x0,11 x A n=t L
n
Here the sets on the left are in r and those on the right are in the a algebra generated by r'. Conversely, just as trivial a representation of r' sets in terms of predictable sets shows that the a algebra of predictable sets includes
r'. Thus the a algebra generated by r' is the predictable a algebra. The representation (14.1) shows that the class of analytic sets over r' includes the predictable sets. We leave to the reader the similar trivial argument showing that if 0 5 a < b 5 + oc, the predictable a algebra is the a algebra generated by the class r" of product sets [a, b[ x A with A e ,T(0) if a = 0 and Ae U, .(t) if a > 0. The class of finite unions of these product sets is an algebra. The class of analytic sets over r" includes the predictable sets. The predictable a algebra is the smallest a algebra making the function (o) measurable whenever is an adapted continuous (t, (n) family with state space (l8,.i(R)). To prove this, we need show only that if G" is a a algebra of subsets of R' x Q making every such family measurable, then G" contains r. This inclusion follows from the fact that the indicator function of a compact interval of I8' is the limit of a decreasing sequence of continuous functions on hr vanishing outside an arbitrarily small neighborhood of the interval.
Chapter II
Optional Times and Associated Concepts
1. The Context of Optional Times Let (Q, 4r ; .fi(t), t e 1) be a filtered measurable space. If I does not have a last element, extend I to I u { + oo }, where + oo is a new element ordered
after every element of I, and define .F(+oo) as any sub a algebra of S containing Y_ S and T is .F(S) measurable, then T is optional.
EXAMPLE (bl). If S is optional and a is a positive constant, then S + a is optional. EXAMPLE (b2). If S is optional define [S]" by
[S].
j2-"
when (j - 1)2-":5 S <j2-n,
+oo when S= +cc.
Then [S]" >_ S, with strict inequality where S is finite valued, and the sequence [S]. is a monotone decreasing sequence of optional times with limit S. (c)
If S and T are optional, then S v T and S A Tare optional.
If S and T are optional and S< T, then F(S) c .F(T); if S < T, then ,F(S+) c .F(T). (d)
In fact, if AE.F(S) and S < T, then
An{T_ 0. (3.1)
Let F, be the class of intersections with { T5 c} of .f(c) sets, and consider the three measurable spaces
(;T 0: (t,w))EA"}
(tel).
(4.1)
Here and below the infimum of the empty set is defined as +oo. If {x(t), t c- I) is a family of functions on SZ with arbitrary state space (X,.X) and if A' is a subset of I x X, the hitting time of A' (by { [t, x(t)], t E 1) ) is defined as
inf{t>0: [1,x(t))eA'}
(te1).
(4.2)
Finally, if A is a subset of X, the hitting time of A (by {x(t), t e l }) is defined as
inf{1 > 0: x(t,w)eA}
(tel).
(4.3)
In each case the entry time of the set in question is defined in the same way as the hitting time except that "t > 0" is replaced by "t >_ 0." Observe that If A' = I x A, the hitting and entry times of A' reduce to those of A and that if A" = { (t, uw): [t, x(t, (o)] a A", the hitting and entry times of A" reduce to those of A'. The analysis of measurability of hitting and entry times is trivial when I= ll+. For example, if A e" , it is clear that the entry and hitting times of are measurable and in fact are optional for From now on A by we shall therefore assume that I = H. Observe that if T' [T"] is the entry [hitting] time of A" and if c > 0, then
{T' < c} = {a): (t,w)EA" for some t < c}, {T" < c} = U 10): (t, (0) E A" for some t Ern, cl },
and the corresponding observation is valid for the entry and hitting time of A' and A. The evaluations in (4.4) and the corresponding ones for A' and A make it easy to deduce optionality of entry and hitting times in a suitable context, as follows.
Theorem. Let (f),f,P;.f(t),teH+) be a filtered probability space, and suppose that for each t the restriction of'P to .f (1) is a complete measure. (a)
If A" is a subset of H+ x S2, analytic over the class of progressively measurable sets, the entry and hitting times of A" are optional for
420
2.11. Optional Times and Associated Concepts
(b) Let
be a progressively measurable process on the proba-
bility space, with an arbitrary state space (X, fl, let A' be a subset of 68+ x X, analytic over V(I8+) x X, and let A be a subset of X analytic over X. Then the entry and hitting times of A' and A by { [t, x(t)], t e OB + } and {x(t), t e R+ }, respectively, are optional for In particular, if the state space is Polish and if the process is right continuous, then the distributions of these times depend on A', A, and the finite-dimensional distributions of together with the assumption of right continuity but do not depend on the choice of the probability space.
To prove the theorem, recall that the sets on the right in (4.4) and the corresponding ones for A' and A were shown in Section 1.9 to be in PF(c); so the entry and hitting times are optional for The last assertion of (b) is trivial because the probability of hitting and entry in the stated contexts, by time c, does not depend on the choice of probability space according to Section 1.9.
Generalization
For n < + oo let T. be the nth entry time of A"; that is, inf it > 0: (t, w) a A" for at least n values of t}.
Then T, is the entry time of A" and under the hypotheses of (a) is optional for If n < + on and if T. is optional, the stochastic interval T,,, + oo Q is progressively measurable. Hence is the entry time of the progressively measurable set A" n ] T,,, + oo Q and so is optional. It follows that T, , T2,. .
.
and therefore also T,,, = lim, T. are optional. The corresponding assertion is true for A' and A. Further, if .f(0) contains the null sets, part (a) of the theorem as just generalized is valid if A" differs by an evanescent set from one as described in (a), and part (b) of the theorem as just generalized is valid if is indistinguishable from a progressively measurable process.
Last Hitting Times
We use the notation introduced at the beginning of this section. The last hitting time of A" by { (t, w), t e 1) is defined as
S" = sup It > 0: (1,w)eA"}
(4.5)
under the convention that this supremum is 0 of the set in question is empty.
The last hitting times of A' and A, in the respective contexts of (4.2) and
421
S. Application to Continuity Properties of Sample Functions
(4.3), are defined in the obvious way. If c > 0, let S(c) be the hitting time of A" by { (c + t, w), t e R ) . The function S" is not optional except in trivial cases, but {S" > c} = {S(c) < +oo} so that measurability problems for last hitting times can be reduced to dual problems for hitting times.
5. Application to Continuity Properties of Sample Functions Let {x(t), .fi(t), t e R+ } be a process on (S2, .F, P) with a Polish state space. Let be a metric for the state space, define the strict right oscillation w) at t by 0,* Q, w) of the sample function
O!(t, w) = lim
sup
a-o t 0.
The process {x(-), ,F(-)) is a martingale and
E{x(n)} = 1,
lim x(n) = 0
R-W
a.s.
This process cannot be uniformly integrable because the sequence is not L' convergent to 0, although it is almost surely convergent to 0. The process is a right closable supermartingale, right closable by the constant random variable 0, for example. The process is not a right closable submartingale and hence not a right closable martingale because if x right closes this submartingale, then E{xl,F(n)} z x(n) almost surely, and this inequality implies uniform integrability of because [Section 1.4(i)] the family of conditional expectations of an integrable random variable is uniformly integrable.
3. Elementary Properties (Arbitrary Simply Ordered Parameter Set) Proofs are omitted if trivial. (a) If is a supermartingale, the function decreasing, and if s 5 t, 5 t2,
E{x(t2)J.F(s)} 5 E{x(t,)J.fls)}
a.s.
is
(3.1)
The proof of the second assertion is simply a manipulation of conditional expectations :
436
2.111. Elements of Martingale Theory
E(x(t2)I,F(s)} = E{E{x(12)IF(t0}IF(s)} s E{x(tl)I.9T(s)) as. (3.2)
(b)
If for each n e l the process
is a positive super-
martingale and ifs < t, then lim inf x (s) Z linm inf E{x (t) I.fls)) Z E {liminfxn(t) IF(s) } a.s., (3.3)
is a supermartingale if its random so that the process {lim inf,... variables are integrable. (c) If is a submartingale and if f is an increasing convex
function with E{ f [x(t)] } < + oo for all t, the process {f is a submartingale. If the first process is a martingale, f need not be increasing
for the conclusion to hold. The analogs of these properties in classical potential theory are in Section 1.11.9. To prove the first one, for example, apply Jensen's inequality for conditional expectations to deduce, when
s i}, then A 1 c- 9(i) when i equality yields
T-1
k
Jn
[x(T) - x(S)]dP = Y 1=0
Y [x(i + 1) - .x(i)] dP Al i=1 (6.2)
k
zi
i
[x(i + 1) - x(i)] dP < 0,
I
n.J,
and there is equality if the given process is a martingale. Thus the integrated
version of the desired supermartingale inequality or martingale equality is true.
Observation. According to this theorem if T. is an increasing sequence of bounded optional times the process {x(T,), .I(T.) } is a supermartingale, or a martingale if the original process is a martingale. Application to Right Closable Submartingales and the Class D
Let {x(t), .r (t), t e 1) be a positive right closable, equivalently, uniformly integrable according to Section 3(e), submartingale, right closable by a positive random variable x. In view of Theorem 6 as applied to -,x,
x(7)< E{xj.F(T)} a.s.
(6.6)
if T is optional, taking on finitely many values, and therefore (6.6) is true if T is optional and countably valued. Hence is a class D process because [Section 1.4(i)] the family of conditional expectations of x is uniformly
integrable. Thus there is a uniform integrability test function 'D and a constant c such that
E{b[x(T)]} < c
(6.7)
whenever T is a countably valued optional time. In particular, if I = OB+, if is almost surely right continuous, and if S is an arbitrary optional time, (6.7) can be applied to [S] as defined in Section II, Example (b2), and (n -+ co) (6.7) is therefore true for T = S. Thus this submartingale is
440
2.111. Elements of Martingale Theory
in the class D in the strong sense that the family {x(T): T optional} is uniformly integrable. In particular, a right closable, equivalently, uniformly integrable, martingale { y(t), .fi(t), t e I } is a class D process [set x(t) _
'(')I]; if I = 68+ and if the martingale is almost surely right continuous, the family {y(T): T optional} is uniformly integrable. I
7. Optional Sampling Theorem for Right Closed Processes If the process in Theorem 6 is right closed, the optional times need not be bounded according to the following theorem. Theorem. If {x(n), ,fln), n e Z` } } is a supermartingale and if T (< + x) is optional, then x(T) is integrable. If S (< + ac) is also optional with S:5 T, then (6.1) is true, and there is equality if the process is a martingale.
If the process
a martingale, x(T)
is
is
integrable and x(T) =
E{x(+x)J,F(T)} according to Section 2; so (6.1) with equality reduces to
x(S) = E{E{x(+x)I,F(T)}I.F(S)}
a.s.
(7.1)
which is true because : (S) e .F(T). If the process is a supermartingale,
x(n) = [x(n) - E{x(+oc)j,F(n)}] + E{x(+x)I.F(n),'
a.s.
(7.2)
is the sum of a positive supermartingale with last element 0 and a so that martingale, both relative to Thus it is sufficient to prove the theorem for a positive supermartingale with last element 0. Under this hypothesis
define T. = T A n, S. = S A n, and observe that according to Theorem 6 E{x(0)} _> E{x(T)}. that E{x(0)} >_ E{x(T)}. Thus x(T) is integrable. Again according to Theorem 6, if m < n, X(Sm) Z E{x(T;)IF(Sm)}
a.s.,
(7.3)
a.s.
(7.4)
and when n - x this inequality becomes x(SS) > E{X(T)I.F(Sm)}
by Fatou's lemma for conditional expectations. Now A e F(S) implies that Am = A r {S < m} e,F(Sm); so (7.4) implies that
Jx(S)dP jx(T)dP. f" m
This inequality when m
oo yields the integrated version of (6.1).
(7.5)
441
7. Optional Sampling Theorem for Right Closed Processes
EXAMPLE. If (x(n),.F(n),ne7L+} is a positive supermartingale, the theorem
is applicable because the process can be closed on the right by defining x(+oo) = 0. This is the most used example. More generally the theorem is applicable according to Section 3(f) to a supermartingale {x(n), .f(n), n e 7L+ } which majorizes a uniformly integrable submartingale.
Generalization. Let {x(t),.f(t), t e 1 } be a supermartingale with an arbi-
trary linearly ordered parameter set I for which there is a first and last element. If T is a countably valued optional time (with values in 1), then we shall now prove that x(T) is integrable. Moreover, if S is also a countably valued optional time with S < T, then we shall prove that (6.1) is true and that there is almost sure equality if the process is a martingale. The proof
follows that of Theorem 7. Denote by 0 the first element of I and by oo the last element of /. Then just as in the proof of Theorem 7 the martingale case is covered by Section 2, and only the case of a positive supermartingale with last random variable 0 needs further examination. Let r, be the sequence of parameter values taken on by S and T, define n
on U {T = r;}, i=o
T. _ oc
on U {T = rj}, j=n+I
and define S similarly in terms of S. Then Sn and T, are optional, take on only finitely many values, and Sn < Tn. Theorem 6 is applicable to S. . . . . rn, oo are involved. An application of Theorem 6 to the pair (0, Tn) of optional times shows that E{x(T,) } < E{ x(0) 1, so when n - + co, we find that x(T) is integrable. An application of Theorem 6 to the pair (Sn, Tn) shows that
and T n because only the parameter values r 0 .
x(T,)dP < f
x(T,) dP = fs"
x(S,) dP
JISn=rjl
=rj)
IS=ri )
so (n-+x) f
x(T) dP
a} < sup E{x(t)} t,A
tEA
a positive supermartingale]. (9.2')
If the process is left closed, by x(a), inf E{x(t) } = E{x(a) }
IEA
sup E{x(t) } = E{x(a) }
a submartingale] a supermartingale].
tEA
If the process is right closed, by x(b),
sup E{x(t) v 01 = E{x(b) v 0}
a submartingale].
Moreover in this right closed case the derivation of the first inequality in (9.1') yields the in general smaller right-hand side E{x(b); sup,EAx(t) Z a}. If the parameter set of the submartingale or supermartingale is uncountable, (9.1') and (9.2') remain true if "sup" and "inf" on the left are replaced
444
2.111. Elements of Martingale Theory
by "ess sup" and "essinf" respectively, because these essential bounds are equal almost everywhere to ordinary bounds over suitably chosen countable parameter subsets. On the other hand, if the process parameter set A is an interval and if the process is almost surely right continuous, then .(9.1') and (9.2') are correct as written since replacing A in these inequalities by a countable dense subset including the right-hand endpoint of A if A is right closed changes neither the left nor the right side.
10. Conditional Maximal Inequalities The inequalities in Section 9 can be made more precise by conversion to conditional inequalities. For example, let {x(j), .'(j), 0 5 j 5 n} be a submartingale, and suppose that A e 5(0). An application of the first inequality in (9.1) to the submartingale { 1,x(j), 9(j), 05 j5 n j leads to the integrated version of
aP{maxx(j) z al.F(0)} < E{x(n)zI. (0)} S E{x(n) v 05(0)} a.s., jSn
(10.1)
where z is the indicator function of the set {max;s x(j) >_ a}. The other inequalities in Section 9 can be extended similarly. Such extensions to conditional inequalities will be omitted from now on.
11. An L" Inequality for Submartingale Suprema Theorem. If {x(t), t e 1 } is a positive right closed submartingale, closed by the random variable x, if A is a countable subset of I, if p > 1, and if 1 /p + 1 /q = 1, then
E{supx(t)P} < qPE{xP}. tEA
According to Section 9,
aP{supx(t) >_ a} 5 E{x; supx(t) >_ a), teA
(11.2)
teA
and it is therefore sufficient to drop the martingale theory context and to show that if x and y are positive random variables satisfying
aP{y>_a}5E{x;y>_a}, then
(11.3)
445
12. Crossings
E{ yP} < qPE{xP}.
(11.4)
Now E{ yP} = p f dP f n
a
y(w)
o
5p Lip-2da
I
P{ y z a}aP-' da
ap-' da = p o
xdP=p f xdP f Jn
y'Iw)
aP-2da (11.5)
Jo
= qE{xyP-'} 5 gE'IP{xP}E'w{ yP} If E{ yP} < + oo, this inequality yields (11.4). Otherwise, replace y by y A k, for which (11.3) is still true, thereby deduce (11.4) for y n k, and let k - + oo.
A common application is to processes with parameter set 0, 1, ... , +oo ordered as indicated. In this case one chooses A = 1. If I is uncountable, (11.1) is true with A replaced by 1 and "sup" replaced by "ess sup" because there is a countable subset to of I with the property that ess sup, E, x(t) _ sup,E,ox(t) almost everywhere. If I is a right closed interval of R and if the given process is supposed almost surely right continuous, then (11.1) is true
with A = I because if A is a dense subset of I including the right-hand endpoint, then sup,,, x(t) = sup, E A x(t) almost everywhere. Modification of Theorem 11 if the Submartingale Is Not Right Closed
.F(-)) is a submartingale Observe that in Theorem 11 the process if x(t)P is integrable for all t; so the function t)-+E{x(t)"} is monotone in-
creasing. If sup,E, E{x(t)P} = lima E{x(t)P} < + oo, the process x() is uniformly integrable because the pth power is a uniform integrability test function when p > 1. Hence [Section 3(e)] the submartingale {x(-), .F(-)) is right closable. The closability argument in Section 3(e) can be interpreted to imply that x(t) has a weak limit x in L'(Q, Y,E, P) when t increases and
that x closes the submartingale We shall see in Theorem 15 that if I= Z+, it is even true that x(n) = x exists almost surely and in L. Then by Fatou's lemma E{xP} < +oo; so by (11.1) the submartingale is also uniformly integrable and therefore is right closed by xP; moreover (dominated convergence theorem) x(n) = x in the LP sense. We leave to the reader the easy extension of these results to other parameter sets.
12. Crossings Let f, g, h be functions from a linearly ordered set I into R, with g 5 h. The number Dn [ f; g, h] of downcrossings of [g, h] by f is defined as the supremum of the values of the positive integer k for which there exist t,
Tk_, : x(j)
(12.6)
SkAn.
As usual the minimum of the empty set of numbers is defined as + oo. The sequence T. is an increasing sequence of optional times for Jr(-).
Proof of (12.2). In this proof we abbreviate Dn[x(-); x'(-), x"(-)] to Dn. In the first place if j is odd,
E{x(Tj) - x(Tj+,)} = E{x(Tj) - x(Tj+,); Tj < n}
(12.7)
> E{x"(Tj+1) - x(T,+1); Tj < n}
E{x"
j+ 1_ E J
T
j+1}
2
2
- E{x(n);Tj -oo. See Section 15 for further discussion of the convergence of a right closable supermartingale. This inequality shows that
Generalization to Arbitrary Parameter Sets Let {x(t), t E 1 }
be an L' bounded supermartingale martingale or sub-
martingale with an arbitrary linearly ordered parameter set I except that I has no last element. If I is countable, the method of proof of Theorem 13 is applicable to yield the fact that lim,t x(t) exists and is integrable. If I is not countable, ess lima x(t) exists and is integrable because we can assume (Section 4) that I is a subset of R ordered by in which case I has a cofinal sequence and (Appendix Theorem IV. 10) the stated result follows from the convergence result for countable I.
Continuous Parameter Case
is a supermartingale martingale or submarSuppose that the process tingale with parameter set a subinterval I of R whose right-hand endpoint a (< + oo) is not in 1. If the process is L' bounded, we have just seen that exists almost surely when t tends to a along the rationals, and this fact implies, if the process is known to be almost surely right continuous,
that this limit exists almost surely when t tends to a unrestrictedly. This is the typical application of the L' bounded martingale theory convergence theorem in the continuous parameter case, and it will be shown in Chapter IV that the hypothesis of almost sure right continuity is not very restrictive.
14. Convergence of a Uniformly Integrable Martingale Let (0, F, P, .f(n), n e Z) be a filtered probability space, and define .f (+ oo) = Y0 .f(n). Recall from Section 3 that a martingale is uniformly integrable if and only if it is right closable, so that the most general uniformly integrable martingale relative to the above unextended filtration is a sequence {x(n), F(n), n E Z+ } with x(n) = E{xl.f (n) }, where x closes the martingale.
In this context define x(+oo) = E{xl.fi(+oo)}. If p >_ 1, then (submartingale inequality if the expectations are finite) E{Ix(0)I°} <E{lx(1)l°} < 5 E{ l xl p}, so that the martingale is L' bounded (as is implied by the given uniform integrability without invoking martingale theory), and in general if the last expectation is finite, the martingale is L° bounded. The following martingale theorem, in which p > 1 and the preceding notation
2.111. Elements of Martingale Theory
452
is used, contains the forward conditional expectation continuity theorem proved directly in Section I.S. Theorem. If {x(n), (n), n E 7L+ } is a uniformly integrable martingale right
closable by x, then lim, x(n) = x(+ oc) almost surely, and also in the L° metric whenever xc L° [equivalent) v, for p >I whenever < + oo ]. The process
[x(0),F(0)], .
. .
, [x(+co),.f(+00)], [x,.F]
E{Ix(n)I P}
(14.1)
is a martingale.
is L' bounded, Theorem 13 is applicable, and under the given Since uniform integrability hypothesis the almost everywhere convergence assured
by Theorem 13 implies L' convergence. Furthermore, if AE,F(n), then (martingale equality) E{x(m); A} = E{x(n); A} = E{x; A}
if m Z n,
(14.2)
and when m - cc. this equation yields, in view of the L' convergence of
E{ lim x(m); Al = E{x; A}. M_m
(14.3)
This equality holds for A in the algebra Uo ,f(n) and therefore holds in the generated a algebra fl+ oo) ; so this limit random variable is x(+ oo). It is trivial that the process (14.1) is a martingale. If p > I and if E{Ix(n)IP} < +co, then (Fatou's lemma) E{lx(+oo)IP} < +oo. If p> I and if E{Ix(+oo)IP} < +co, then (Theorem 11 applied to Ix(n) - x(+oo)IP} < +co, and lx(n)IP} < +oo. So converges to x(+oo) therefore (dominated convergence) the sequence in the LP metric.
Extension to Other Parameter Sets
is a uniformly integrable martingale, right closable by Suppose that x, for which the parameter set I is either (a) a countable dense subset of an open interval ]0, a[ or (b) the interval ]0, a[. Theorem 14 (and its proof with trivial modifications) remains valid in both these contexts when suitably
rephrased. More specifically, in (a) lim,tax(t) exists almost surely and in
L'; in (b) esslim,t.x(t) exists, and the limit also exists in V. Moreover in (b) if the process is almost surely right continuous, the essential limit is
also an almost sure limit. In both cases the limit is almost surely E{xtY,E,.F(t)}. The extension when xeLP is obvious.
15. Forward Convergence of a Right Closable Supermartingale
453
Application to Approximation Theory
Let {x(t), t E I } be an arbitrary infinite collection of random variables on
some probability space, and if J c ],define f(J) = f{x(t), t EJ}. Then if x is an f(I) measurable and integrable random variable, we shall now show that there is an increasing sequence J, of finite subsets of I such that
limE{xl.F(J,)}=E{x19lUJnl}=x a.s. and in L'.
(14.4)
Since (Section I.4) E{xl.f(J,)} is equal almost surely to a Bore] measurable function of the finitely many random variables with indices in in, equation (14.4) exhibits a canonical way to approximate x by functions of finitely many of the random variables. In fact let x be as stated, and (Appendix
IV.2) choose J countable and so large that x is f(J) measurable. If Jn in (14.4) is chosen to be the set of first n members of J in some enumeration, (14.4) reduces to a special case of the conditional expectation continuity theorem, that is, a special case of Theorem 14. Observe that if x is an arbitrary
integrable random variable on the given probability space, (14.4) should be replaced by
lim E{xj.f(Jn) } = E{xI.f I U Jn l} = E{xj.f(!)} a.s. and in L'. rm \o f)))
(14.5)
15. Forward Convergence of a Right Closable Supermartingale a right closable supermartingale, limn_,, x(n) = x(+ oo) exists (finite) almost surely, and if x right closes the supermartingale the process, (14.1) is a supermartingale. Theorem. If {x(n), .f(n), n E 7L+ } is
It was pointed out in Section 13 that Theorem 13 is applicable to right closable supermartingales. Hence only the second assertion of the theorem requires proof. Since the process is the sum of a positive supermartingale and a right closable martingale according to
x(n) = [x(n) - E{xI.F(n)}] + E{xj.F(n)},
(15.1)
and since the corresponding right closure result for martingales was proved in Section 14, it can be supposed from now on that the given supermartingale is positive. Apply Fatou's lemma for conditional expectations to get
E{_r(+oo)I,F(n)} < liminfE{x(m)j.f(n)} < x(n) a.s.,
(15.2)
454
2.111. Elements of Martingale Theory
which shows that.r(+ x) coupled with F(+ oo) = Yo .f(n) right closes the given supermartingale. Finally
E;x4.f(+x)
limE{xj,f(n)} 5limx(n)=.r(+x) a.s.,
(15.3)
and therefore (14.1) is a supermartingale in the present context.
As the negative of Example (d) in Section 2 shows, an L' bounded supermartingale need not be right closable.
Extension to Other Parameter Sets
The rephrasing of Theorem 15 for a process with a parameter set which is either an interval ]0, a[ or a countable dense subset of this interval is similar to the corresponding rephrasing in Sections 13 and 14.
16. Backward Convergence of a Martingale The fact that the parameter set Z .... - I. 0 has a last element makes backward martingale theory convergence theorems stronger than forward ones It will be convenient to treat the martingale case first. :
Theorem (Backward Half of the Conditional Expectation Continuity Theorem Already Proved Directly in Section 1.5). Suppose that {x(n),_-fln), nE/ ; isamartingale,that is,x(n) = E;x(0)1.f(n), a.s. Then limn-_, x(n) _ E; r(0)in" , as. and in L' norm.
The process is uniformly integrable because it consists of conditional expectations of a random variable x(O) [Section 1.4(i)]. Just as in the proof of Theorem 13 the downcrossing inequality, easier here because the process is right closed, together with Fatou's lemma implies that there is an almost sure limit which is integrable. To identify this limit, which we denote by
.V(- x ), choose A in F( - x) = n°_ ,, f(n). The martingale property and the uniform integrability of the process yield
1x(_dP= lim
JA v(n)dP
1^v(0)0.
(16.1)
and this equality identifies x(- x) as the stated conditional expectation. The convergence is L' convergence because the process is uniformly integrable.
The rephrasing of Theorem 16 for other parameter sets follows that of the forward convergence theorems and is left to the reader.
18. The r Operator
455
17. Backward Convergence of a Supermartingale Theorem. Suppose that {x(n), ,`i=(n), n E z- } is a supermartingale. Then (a)
E{x(0)} 5 E{x(-1)} 5
.
-' 15 + oo.
(b) limN, x(n) = x(- oo) exists almost surely; x(- 00) A 0 is (c)
in-
tegrable; - oo < x(- oo) 5 + oo almost surely. If l < + oo, then x(- oo) is almost surely finite and left closes the supermartingale, the supermartingale is uniformly integrable, and the convergence in (b) is in L' as well as almost sure.
The example x(n) _ - n shows that the limit in (b) may be identically + x. Part (a) is trivial and is stated only for orientation. Just as in the proof of
Theorem 13 but more easily here because the process is right closed, the downcrossing inequality implies that there is an almost sure backward limit x(- oo). Apply Fatou's lemma and the supermartingale inequality to obtain E{x(-oo) A 0) Z lim E{x(n) A 0) -a E{x(0) A 01,
(17.1)
which implies that x(- o0) A 0 is integrable. In view of Theorem 16 and the fact that the process can be written as the suet of a positive supermartingale and a uniformly integrable martingale according to
x(n) = [x(n) - E{x(0)j.F(n)}] + E{x(0)1.fln)},
(17.2)
it is sufficient to prove the present theorem for a positive supermartingale, and positivity will be assumed from now on. If I < +oo, the process is L' bounded; so x(- oo) is integrable. Finally, according to Fatou's lemma,
E{x(-oo)) 51, whereas for every c the process
A c is a bounded
supermartingale; so for all n,
E{x(-00) A c} = lim E{x(m) n c} Z E(x(n) A c).
(17.3)
This inequality yields E(x(-oo)) Z E{x(n)} and thereby E{x(-oo)} = 1. Thus there is equality in Fatou's lemma which implies uniform integrability of the process.
The rephrasing of Theorem 17 for other parameter sets is left to the reader (see Section 14).
18. The T Operator This probabilistic operator is the counterpart of the r operator in classical potential theory. Let be a stochastic process with state space (68, R(R)) and with an arbitrary linearly ordered parameter set I. It is
456
2.111. Elements of Martingale Theory
supposed that the process random variables are integrable. Let S and T be countably valued optional times with values in 1 and with S:5 T. Suppose writing zsTx(t) that x(T) is integrable, and define the process instead Of TSTX(') (t), by
_ x(t) on IS > t} v IT < t}, TSTY(t) -1E{x(T)IF(t)} on {S5 t 5 T}.
(18.1)
Observe that
TSTx(t) = E(x(T v t)I.flt)}
a.s. on {S:5 t).
(18.2)
is adapted to and TSTx(T) = x(T) almost surely The process is a martingale by Section 2, Example (a). Furthermore the process between times S and T in the sense that the process {TSTX((S V 1) A T), .F((S V t) A T), t e 1}
is a martingale; in fact TSTx((S v t) A T) = E{x(T)LF((S v 1) A T)}
a.s.
is a supermartingale or In the most important applications submartingale. Suppose, for example, that this process is a supermartingale. If T is order bounded above and below, it follows from the supermartingale
optional sampling theorem that x(T) is integrable and that P{TSTx(t) < x(t) } = 1 for all t, that is, in the essential order. Furthermore 5 is then also a supermartingale (and a martingale the process between times S and Tin the above sense). In fact, if s < t, then E{tSTx(t)IF(s)} < E{x(t)I.F(s)} 5 x(s) = TSTx(s) a.e. on IS > s} and, almost everywhere on {S < s}, E{TSTx(t)I.F(s)} = E{1{sss)TSTx(t)LF(s))
= E{l1sss1E{x(T v t)I.flt)}I.fls)} = E{x(T v t)I.F(s)} = E{E{x(T v t)I.F(T v s)}I.F(s)} _< E{x(T v s)I.9r(s)} = TSTX(s)
Define TT by (18.1) with all references to S deleted. If T is order bounded
is a supermartingale [submartingale], above and below, and if is also one, is an essential order minorant [majorant] of x(), then
19. The Natural Order Decomposition Theorem for Supermartingales
457
and is a martingale up to time Tin the sense that the process
{trx(T A t),.F(T A t),tE1} is a martingale. Observe that The operation
if I has a first element a. on processes is the martingale theory coun-
terpart of the classical potential theory operation u i-- rBU on functions, as defined in Section 1.11. 1 for B a ball and in Section I.VIII.I I for B an open set (in both cases relatively compact in the domain of u). In most probabilistic applications I = R+, the processes involved are almost surely right continuous, and the optional times S and T need not be countably valued. The adaptation to this context will be given in Section IV. 14.
19. The Natural Order Decomposition Theorem for Supermartingales In the following theorem and its proof all the supermartingales have the same (arbitrary) parameter set and filtration. be positive supermartingales, and suppose that that is, for each parameter value the 5 x1 indicated inequality is true almost surely. Then there are positive supermartingales and for which (essential order) 5
Theorem. Let (essential order)
X10 + The proof is a translation of that of Theorem 1.111.7, a particularly easy example of the translation of potential theoretic reasoning into the corresponding probabilistic context. Choose x; as the essential order infimum of the class of positive supermartingales for which 5 and then choose as the essential order infimum of the class of positive supermartingales Then for which x(-) :g x(-) + S x,(-) + x2 and for any parameter value t, x(-) 2 r X(')] < X'10 + for parameter values z t and is a positive supermartingale. Hence this process majorizes and it follows that It is easy to check that the bracketed process is the same as x'2
z If this inequality is evaluated at a parameter value s < t, we find that is a positive supermartingale, majorized (essential order) by impossible unless there is equality. The proof is complete.
458
2111. Elements of Martingale- Theory
See Section V.5 for Theorem 19 as modified for right continuous processes in the continuous parameter case.
20. The Operators LM and GM Let r: be a class of stochastic processes with a common linearly ordered parameter set, on a common probability space, and adapted to a common filtration If in the essential order of processes adapted to there is a least supermartingale majorant [greatest submartingale minorant] of r, this majorant [minorant] will be denoted by LMr [GM r]. Strictly speaking LMr, for example, is an equivalence class, and we shall sometimes describe its members as its versions. The notation LMr can refer either to the equivalence class or to one of its versions, but the intended meaning will always be clear from the context.
Theorem. If r is a class of supermartingales and has an essential order submartingale minorant, then GMr exists and is a martingale.
The proof of this theorem is formally identical with that of its classical counterpart Theorem 1.111.2. The class ro of submartingale essential order v with minorants of r contains and is therefore directed upward in the essential order with essential order supremum (limit), say
an essential order minorant of r. We prove the theorem is a martingale. For every parameter value t, in ro,
by showing that in r, every every
s
5 t,x(-) S
in the essential order. Thus and the essential order supremum of ro is the same as that of e ro}. For every t the latter class on the parameter set _j
(22.1)
To prove the theorem, define y(j) as the right side of (22.1), and observe (set T - j) that y(j) >- z(j) almost surely. Next observe that for fixed j the class of conditional expectations in (22.1) is directed upward. In fact if S and
T are optional and 2j. define U = T or U = S according as the inequality
E{z(T)j.flj)} z E{z(S)jf(j)}
(22.2)
is true or false. Then U is optional and E{z(U)j.(j)} is almost surely the maximum of the conditional expectations in (22.2). Next observe that f( ); is a supermartingale: so In fact, if j>0 and if the optional time sequence S. is chosen with Sk z j for all k and is also chosen to make the sequence monotone increasing with limit t'(j). then
EjT(I)I.f(j- I)} = limE{E{z(Sk)IF(j)}I (j- 1)} k-m = lim E{z(S5)j9(j - 1) } 5 T(j - I) k-M
Finally, if
a.s.
r and if T is optional and >-j, then
z'(j) ? E{z'(T)I/ (j)} ? E{z(T)j. (j)',
a.s.:
so :'(j) >_ r(j) almost surely; that is, essinf r. as asserted. We shall not pursue the general case far enough to obtain a probability counterpart to Theorem I.XI.20 but proceed at once to the probability counterpart of classical potential theory reductions. Application to Reductions
Let A be a subset of Z' x C for which the process j -+ IA(j, ) is adapted to .fl-). and let If be a positive supermartingale. Define R A.,, also to be denoted by llA , as R.(.,,0,. Then according to Theorem 22. if we
define I&(+ c)=0.
R.,(j)=esssup {F.{z(T)l,t(T)j.(j)},Toptional, and we show further that if T; is the entry time >-j of A, that is, T (w) = min {k >_ j: (k, w) E A },
a.s.
(22.3)
461
23. Application to the Crossing Inequalities
thin
R= .,(j) = E{z(T)I(j)} as.
(22.4)
To prove (22.4), observe that if T is optional and zj, then IA(T) 5 1JT A T) < IA(T;), and observe that IA(T A Tj) = 0 when T < Tj; so
E{z(T)1A(T)I.I(j)} <E{E{z(T)I,F(T A Tj)}I,t(T A T)I,f(j)}
(22.5)
< E{z(T A T)IA(T A Tj)I-flj)} < E{z(T)Iflj)} a.s. Thus the supremum in (22.4) is attained when T = T j, as asserted.
Specialization to the Parameter Set 7L
If the parameter set is 1, instead of Z', the corresponding analysis is easily carried through, or we can define ,F(j) = f (n) and z(j) = z(n) for j > it to reduce the context to that of the parameter set V. Equations (22. I ), (22.4), and (22.5) remain true under the convention that all processes are extended to vanish at the parameter value + ce.
23. Application to the Crossing Inequalities We suppose that we are in the context of the preceding section with parameter set Z., and we shall use the notation ze,('), ze,e2('). . respectively, for be adapted processes on the Q el..... Let OcI, 0 given filtered probability space. Suppose that 0 and that are positive supermartingales; that is, we are in the and context of Theorem 12. In Section 12 we defined Dn as the number of times an sample sequence proceeds from above to below Define
A = Ici,(o):
Z 1 , ,x(j,(o) S x'(j,w)}.
B= {(j,(u): je7L,; ,x(j,w) z x"(j,(u)). It is easy to check using (22.1) that (almost surely)
yie(O) > E{y'(n); Dn[x('); x'(-), x"(-)] Z 1IF(0)}; (23.1) °A 8Aa(0) Z E{}' (n);
x'(,),x"(,)] Z 21-F(O));
462
2.111. Elements of Martingale Theory
etc., with equality if
is a martingale. Now the iterated reduction in-
equalities in Section 1.VI.3(o) were proved in the context of classical potential theory using elementary reduction properties valid in the present context. Hence (almost surely)
.5
.v (') + ABA(') + YAaAEA(') +
(23.2)
and
Yee(') +YAaAe(.) +
5
S
A
(23.3)
In view of (23.1) inequality (23.3) implies
S x(0) n x"(0) a.s.,
E{y'(n)
(23.4)
which yields (12.2) on integration. (See the remarks in Section 10 on conditional inequalities.) In particular let a and b be positive constants with a < b, let be a positive supermartingale, and set ay(-), in (23.3) to obtain the almost sure inequality {
Y.4$(') +
A by(')
+ ... S
b-a
(23.5)
The corresponding particularization of (23.4) can be obtained by the same
substitution or by summation and integration in (23.5). The martingale theory translation of I . VI(3.11) is the sequence b by(-)
5 ()YAe. ...
(23.6)
of almost sure inequalities. On integration these yield (12.4). The point is that the classical context iterated reduction inequalities in 1.VI.6(o) remain
valid when translated into the martingale theory context, and that the translated inequalities yield, on integration, the crossing inequalities proved directly in Section 12.
Chapter IV
Basic Properties of Continuous Parameter Supermartingales
1. Continuity Properties The continuity properties of continuous parameter supermartingales derived
in this section are of course also valid for continuous parameter submartingales and martingales. Recall that a process is said to be [almost surely] right continuous if [almost] every sample function is right continuous and the process is said to be [almost surely] right continuous with left limits
if [almost] every sample function is right continuous and has a left limit at every point. The usual smoothness hypothesis to be imposed on a continuous para-
meter supermartingale is almost sure right continuity. Moreover the following remark shows that it is possible to change the reference filtration of an almost surely right continuous supermartingale to obtain a new one to which the supermartingale is adapted, which is right continuous, and for which each a algebra contains the null sets. Let {x(t), .F(t), t e R+ } be an as the almost surely right continuous supermartingale, and define is a supera algebra generated by .F(t) and the null sets. Then is right continuous, .F,+(0) conmartingale. Moreover the filtration is a supermartingale because ifs' < t, tains the null sets, and (x(-),.Fl" E{x(t)j.3F,(s')} 5 x(s')
a.s.,
and when s' Is sequentially, we obtain the supermartingale inequality for in view of the conditional expectation continuity the process theorem. In the following theorem convergence in measure has the usual definition, not the special definition for R valued random variables needed in Section 1.13.
Theorem. Let {x(t), F(t), t e R' } be a supermartingale, and suppose that .r(0) contains the null sets. Let B [B+] be the set of values oft in R' for which it isfalse that lim,.., x(s) = x(t) [lim,l, x(s) = x(t)] in the sense of convergence in measure, and let A be a countable dense subset of R+.
464
(a) (b)
2.IV. Basic Properties of Continuous Parameter Supermartingales
The set B is countable. The restriction to A of almost every
sample function is bounded on the trace on A of every compact interval, has left and right limits
at every point of R', and fort in R+ - B', P { lim x(s) = e 3slr
(c)
x(t) = 1. J
If t e B, define y(t) = x(t) ; if t e R' - B+, define y(t) =
x(s) where this limit exists, and define y(t) arbitrarily where this limit does not exist. For all t define x(t) = lim4asl,x(s) where this limit exists, and define x(t) arbitrarily where this limit does not exist. Then .F +(-))are supermartingales, (c l) The processes {y(-), martingales if is a martingale. (c2) Almost every sample function and almost every sample function are bounded on compact intervals of R+ sample function has right and left limits at (c3) Almost every
every point of R+ and is right continuous at every point of R+ - B+. Almost every sample function is right continuous and has a left limit at every point of R+. The processes are L' bounded on compact intervals
ofR+. (c4)
is a standard modification of indistinThe process guishable from if and only if B+ is empty, as is true if is a martingale and
Proof of (a) and (b). If to > 0, the restriction to A n [0, to] of almost every process sample function is bounded, according to the inequalities 111(9.1') applied to - x(-). If such a restriction does not have a left and right limit at every point of [0, to[, there must be a pair of rational numbers rt , r2 with rt < r2 such that the sample function restriction to A n [0, to[ has infinitely many downcrossings of [r1 , r2]. Since the downcrossing inequality 111(12.5) yields
5
E{x(0) A r2 - x(to) A r2)
r2 - r t
(1.1)
the number of downcrossings is almost surely finite simultaneously for all rational pairs rt, r2 and values of to. Therefore the restriction to A of almost every sample function has one-sided limits x(t-) and x(t+) for every t. Moreover, since
E{!x(t)j} = E{x(t)} - 2E{x(t) A 01 < E{x(0)} - 2E(x(to) A 0)
(t s to), (1.2)
465
1. Continuity Properties
an application of Fatou's lemma shows that these one-sided limits are are Lt bounded on integrable and that the processes compact intervals of W. Now consider the space of almost surely finitevalued random variables, identifying two random variables if they are equal almost surely, and define the distance between the random variables x and y as E{ I x - Y J A 1). Convergence of a sequence in this metric is equivalent
to convergence in measure of the corresponding sequence of random variables. Let 4(t) be the point of this metric space corresponding to the random variable x(t). Observe that if At is a second countable dense subset of 68+, the results proved above for A when applied to A u At show that the left- and right-hand sample function limits are almost surely the same for A as for A t . Among other things, this fact implies that when s tends sequen-
tially and strictly monotonely to t, the random variable x(s) tends almost surely to x(t-) or x(t+) depending on the direction of approach; so the function ¢ on R has left and right limits at all points. If e > 0, the set of points t at which the oscillation of 0 is at least a is closed and can have no finite limit point; so this set is finite in each compact interval, and it follows that the set of discontinuities of 0, which is the set B of the theorem, is countable. Moreover the definitions of B and B+ can now be strengthened: tefR+ - B [teER+ - B+] if and only if whenever A is a countable subset of R with [left] limit point t, it follows that A
lim x(s) = x(t)
[Alim x(s) = x(t)]
a.s.
13
is a martingale, if .F(-) is right continuous, Proof of (cl)-(c4). If and if 0:5 s' < s < t, the almost sure equality x(s') = E{x(t)jF(s')} combined with the conditional expectation continuity theorem yields x(s+) = E(x(t)j.fls)} = x(s) almost surely when s' tends sequentially and monotonely to s. Thus in this case B+ is empty. To prove the rest of (cl)-(c4), observe that the process is a standard modification of and is is a because .F(0) contains the null sets. Hence adapted to supermartingale, a martingale if is a martingale. If we choose A to be a superset of B, an application of (a) and (b) shows that (c l)-(c4) are is an almost surely right continuous true for y(-) and that adapted process with left limits. If B+ is empty, x(-) is adapted to
Furthermore P{x(t) = x(t)} = 1 if and only if teR+ - B. To prove that is a supermartingale, which implies that this process is a mar-
tingale if
is [apply the first result to
suppose that
05s<s' 0, and define
T"=inf{t: a"(t)-a(t)>E}. Then T is an increasing sequence of optional times. Let T. = lim"_. T. Since a"(t) = a(t) = I for sufficiently large t almost everywhere on { T < + oo }, and for t in I", the evaluation
470
2.IV. Basic Properties of Continuous Parameter Supermartmgales
an(t) = P{T < (k + l)2-"I.flt)} implies that lim,..,, an(t) = a(t) = 0 almost everywhere on {T = +oo} (by the conditional expectation continuity theorem), it is sufficient to prove
that T. = + oo almost surely. Now E{an(Tn)llr +co)} _ k =O
E{a((k+ 1)2-")I.W(T")}dP
f
lTne Ink)
_ Y_
k_OJ {TneInk)
f k_0
a((k + 1)2-")dP 2.7)
cc
of. To reduce this fact to what we have already proved, define z.(1) for t > 0 as xx(a + t) on Z.({x(a) < /3} and as 0 elsewhere. Apply (a) to the increasing sequence 1),.9r(a + t), t e Q8+ } of supermartingales to find that z according to what we have proved already is an almost surely right continuous supermartingale and that therefore almost every sample function is bounded on compact intervals. The remaining part of (b) follows.
Elementary Proof of Almost Sure Right Continuity of
is natural, The proof given above of almost sure right continuity of but it is of interest to exhibit an elementary proof that does not involve the relatively deep theorems on hitting probabilities. The following proof does not even need the supermartingale convergence theorems. We can is right continuous, that F(0) assume as in the proof already given that contains the null sets, that each process x
and that
is positive and right continuous,
is bounded. Define x(+ oo) = x (+ oo) = 0. Observe first
475
4. Increasing Sequences of Supermartingales
that if S. is a sequence of optional times and if A = S. = 0}, then Ae.,F(O), and an application of the right lower semicontinuity of sample functions and of the supermartingale inequality at optional times [valid for because it is valid for yields E{x(0);A}
Ejliminfx(S,);Ar 0 and define
x (t) - x(0) > -21, T = lim sup T,,. I
n-cO
The time T is optional because up to a null set
T. z c} = sup I x (r) - x(0)I < 2
(r rational, c > 0)
and therefore T is also optional. Furthermore T is almost surely strictly positive because otherwise, there is a set of strictly positive probability
on which lim, T. = 0 and E
I x.(T.) - x(0)I z 2
(4.4)
for sufficiently large n. Now under (4.4) either x(0) >- e/2 for sufficiently large n, which leads to lim inf,
x
x(0) e/2 and thereby contradicts (4.3) or under (4.4), for every k, x(0) < -e/2 for infinitely many values of n, in which case xk(0) x(0) < -e/2, which is impossible for large k. Thus Tis almost surely strictly
x
positive. Furthermore I x (t) - x(0)I < e/2 on [0, 1x(l) - x(0)I < e/2 on [0, T[, and if we set
and so almost surely
r(t) = lim sup Ix(s) - x(t)I, 6-0 r<s a. Conversely, if (7.2,1= R+) is true and if is a bounded almost surely right continuous martingale, apply (7.2,1= R+) with z = y(a)
and then let a - + cc to derive the condition (6.2,1= R+) that
be
natural. Proof of (a3) for I = R +. According to Section 6, if either A(-) is almost surely
continuous or if there is a predictable optional time T such that A(t) = cltr,+ml(t), then is natural. Thus to show that is natural if nearly is predictable, it is sufficient to show that (dropping evanescent sets) if predictable, right continuous with left limits, and if all its sample functions
486
2 IV. Basic Properties of Continuous Parameter Supermartingales
are increasing, then can be written as a countable sum of processes of these two types. Choose fl > 0 and define To = 0 and
T",1=inf{t>T":A(t)-A(t-)2! P),
n>0.
The process A(- -) is predictable by left continuity; so the process A(-) A( -) is predictable, and therefore the set {(t,w): A(t,w) - A(1 -,co) z fi} Is predictable. For n > I the nth entry time of this set is T". Hence (Section 11.9) T. is predictable; so (Section 11.7) the process AT"(-) = lrT".+, is predictable. The process
Y
"=o
is a predictable increasing process for which the sample function discontinuities of magnitude z fl are decreased by ft. First choose jJ = 1, thereby decreasing jumps >- I by 1, then apply the same procedure, but with fi = #, to the above difference process, then with fl = I to the new difference process, and so on, to obtain an evaluation of as a sum of the desired type. Conversely, suppose that is natural. We show first that does not
charge any totally inaccessible optional time T, that is, A(T) = A(T-) almost everywhere where T < + oo. To see this, define
]k2-",(k + I)2-"], and observe that
E{A(T) - A(T-); T < +oo} = lim "~ k=0
[A((k + 1)2-") - A(k2-")] dP; ITs
(7.6)
so it is sufficient to show that the right side vanishes. The sum in (7.6) is equal to E{ IITS(k+1)2-"1[A((k + 1)2-") - A(k2-")] k=0
-
IITSk2-";[A((k + 1)2-") - A(k2-")]},
which by (7.2, / = R) is equal to k
EJ
P{ T< (k + 1)2
dA(t)J-k EI f !"5
J
(7.7)
and as in Section 2, the first sum in (7.7) can be written in the form E{f'o a"(t-)dA(t)}, and this If we define the interval I"k and the processes
487
7 Natural versus Predictable Increasing Processes (I = Z' or R')
expectation, by the uniform convergence of the sequence Section 2. has limit (n
proved in
oo)
E{ f, a(t-)dA(1)} = E{ I J
Ilr.+xt(I)dA(t)}
Jo
(7.8)
1
If now T. is defined by
T"
(k + 1)2-" when Te /,;k, when T = + oo, +00
then T is a decreasing sequence of optional times with limit T and T. > T almost everywhere where Tis finite valued and not equal to a dyadic rational
number. Since the distribution function on R of a totally inaccessible optional time is continuous. T. > T almost surely where T is finite. The second sum in (7.7) is equal to
'
E! If llr +a,t(r)dA(r)}, which has limit (n - x,) the right side of (7.8); so asserted.
does not charge T. as
In the proof of (a3) it was shown that if nearly predictable can be expressed as a countable sum of increasing processes of which one is almost
with T and surely continuous and the others have the form is to be predictable. That reasoning in the present context, in which proved nearly predictable and in which we have just proved that charges no totally inaccessible optional time, leads to the same expression for except that as first defined each optional time Tis accessible. In fact we need only point out that the discussion in the proof of (a3) leads to optional times T charged by and therefore without totally inaccessible components. is Since is nearly predictable if T is accessible (Section 11.12), nearly predictable if natural.
Proof (bl)
If
and / = /', then E{ '(k + 1)[x(k + 1) - x(k)] } = E{ i'(k + 1)E{x(k + 1) - x(k)I,flk) } } = 0
because v(k + 1) is .F(k) measurable and {x(-),.fl-)) is a martingale. Hence (b I) is true when I = l' . If I = R' and if { is a bounded left continuous process, define
488
2 IV Basic Properties of Continuous Parameter Supermartingales
1,(0) =1'(0), y"(t) = E y(k2-")t]k2-".(k+111-n1(t)
if t > 0,
k=0
so that the process { v"(k2-"), ,F(k2-"), k e 71 + } is predictable and therefore by what we have just proved
E{ If zl;,(t)dA(t)}=E{J"y"(t)dB(t)}. 0
(7.9)
o
When n -' oc this equation yields (7.3,1= R+). Thus the latter equation is true if is bounded and left continuous, and since the class of bounded predictable processes on O+ is the smallest class of bounded processes closed
under bounded monotone convergence and containing the left continuous adapted processes, (b1) is true. (b2) We give the proof for 1= OB+; the proof for I = 7L+ is similar but less deep. Suppose first that A(+ oo) and B(+ oc) are bounded, and combine (7.1,1= ff8+) with (7.3,1= IB+) to find that E{ y(+ oc) [A(+ oo) - B(+ oo)] } = 0 whenever (y(-), .F(-) I is a bounded almost surely right continuous pre-
dictable martingale. If now we choose we find that A(+ or,) = B(+oc) almost surely. Hence (martingale equality) for each value of t it follows that A(t) = B(t) almost surely; so by the almost sure right continuity of and these processes must be indistinguishable. In the general case we need only find an increasing sequence S, of optional
times for which lim"..,,S. = +oc almost surely and for which A (S.) and B(S") are almost surely bounded for each n. In fact, if the result for bounded
processes is applied to the pair A(S" A ), B(S" A ), it follows that these processes are indistinguishable so and are. To find a sequence S., observe that for n > 0 the entry time T. of the predictable set {(t,w): A(t,w) + B(t,w) >- n}
is predictable (Theorem 11.9), and if T", is a sequence of optional times announcing T", then A(T.) + B(T",) < n; so we can choose
S"=T1.V... VT"".
8. Generation of Supermartingale Potentials by Increasing Processes in the Discrete Parameter Case Theorem. Every supermartingale potential {x(n), .f(n), n e71 + } is a quasibounded class D supermartingale and is generated by a unique up to standard modification, that is, unique up to indistinguishability, increasing predictable L' bounded process.
9. An Inequality for Predictable Increasing Processes
489
In view of Section 6 it is necessary only to show that there is a unique generating process as described. Given a supermartingale potential {x(n), .fi(n), n e 71 + } define an increasing predictable process by A(O) = 0,
A(n) _ Y [x(m) - E{x(m + l )j. '(m)}] if n > 0,
(8.1)
M=O
and define A(+co) = sup,
A(n). Then E{A(+oo)} = E{x(0)} and
E{A(+oo) - A(n)Ifln)} = Y E{[x(m) - E(x(m + 1)I9(m)}]I f(n)} mn = x(n)a.s.;
(8.2)
is generated by is any superMoreover, if martingale potential generated by an increasing predictable process it is immediate that so
x(m) - E{x(m + 1)I.9(m)} = A(m + 1) - A(m) a.s.
(8.3)
is uniquely from which (8.1) follows as an almost sure equality; so determined up to a standard modification. [Uniqueness here also follows from Theorem 7(b).] Decomposition of a Submartingale If {A(n), 9(n), n e 71 + } is a predictable increasing process and if { y(n), .fi(n),
ne71+} is a martingale, the process is a submartingale. Conversely, if {x(n),9(n),ne71+} is a submartingale and if A (n) is defined by (8.1) with then replaced by is a predictable increasing process and where is a martingale. It is left to the reader to check that is Lt bounded if and only if and are. This decomposition of a submartingale makes it possible to deduce the almost sure convergence of an L' bounded discrete parameter submartingale from the almost sure convergence of an L' bounded discrete parameter martingale.
9. An Inequality for Predictable Increasing Processes Lemma. Let {x(n), .F(n), n e 71 + } be a supermartingale potential generated by the predictable increasing process A (n). Let c be a strictly positive constant, and define the optional time Tc by
if A(+oo)Sc T`-{inf{j:A(j+1)>c} ifA(+oo)>c. +oo
490
2.1V. Basic Properties of Continuous Parameter Supermartingales
Then
J A(+m)>c)
A(+co)dP53
fA(+o)>c/2) x(Tn)dP.
(9.1)
By definition of T, A(T,) 5 c and {A(+oo) > c) _ {T, < +co} eF(TT) so that cP{A(+oo) > c} 5 2 I
[A(+oo) - A(TcJ2)]dP (9.2)
52I
x(7 12) dP,
and this inequality combines with the supermartingale inequality to yield
f
x(Tc)dP+cP{A(+oo)>c}
A(+oo)dP5 I A(+ao)>c)
(A(+ao)>c)
,3f
x(T,,2) dP,
A(+ao)>c/2)
as was to be proved.
10. Generation of Supermartingale Potentials by Increasing Processes for Arbitrary Parameter Sets Let (0, .F, P; .fi(t), t e I) be a filtered probability space. It is supposed that the linearly ordered parameter set I has a first but not a last point. Theorem. The following conditions on a supermartingale potential {x( ), are equivalent: (a)
The process is generated by an increasing L' bounded adapted process
for each tin I,
x(t) = E{A(+co) - A(t)I9r(t)}
= E{A(+oo)jF(t)} - A(t) a.s. (b) (c)
(10.1)
The process is in the class D. The process is quasi-bounded.
Observe that when I= Z+, this theorem does not contain Theorem 8 because in that theorem the increasing process is predictable; in fact this
491
10. Generation of Supermartingale Potentials by Increasing Processes
predictability determines the increasing process in Theorem 8 up to a standard modification. Uniqueness cannot be asserted in the present theorem. Moreover, according to Theorem 8, a supermartingale potential for the parameter set Z+ is necessarily quasi-bounded and in the class D, whereas there is an example in Section IX.6 below for the parameter set I8+ of a continuous supermartingale potential which has neither property. In the following proof we denote'(` I .fi(t) by F(+ co). Proof. (a) (b) and (a) (c) See Section 6 where these implications are derived without the adaptedness hypothesis on (b) (a) Assume first that I is a bounded subset of G2+ and includes 0. The random variable x(t) can be identified with a point of Lt (Q, .F(+ oo), P). This function has a left [right] limit at each point of the closure !of I which is a limit point of I from the left [right] according to the supermartingale convergence theorems and the condition that the given process is in D, so is uniformly integrable. According to an elementary theorem on functions from a subset of R into a metric space, the set Io of points t of !for which two of the members x(t -), x(t), x(t+) of Lt (12, 9(+ co), P) exist but are not the same is countable. Let J be a countable subset of I, dense in 1, including 0, !o, and every point of ! which is not a bilateral limit point of 1, and let J
be a finite subset of J, including 0, with Jo c Jt c
, Uo J. = J. The
restriction of to J. is a positive supermartingale to which Theorem 8 as trivialized for finite parameter sets can be applied. (Theorem 8 is stated for the parameter set Z+ but can be applied to a finite parameter set, say 0, . . . , k, by defining the process random variables to vanish identically at parameter values strictly greater than k.) According to Theorem 8, there is a process {A (t), t e for which A (0) = 0, A&) is increasing and is predictable, and if we define co) as the value of A.(-) at the maximum parameter value in J,,,
x(t) =
Aa(t)
a.s., teJ,,.
(10.2)
Moreover for c >r( 0,
2E{x(0)},
2f 5
(10.3)
and according to` Lemma 9, there is an optional time T (depending on n and c) with values in J,,, for which
x(T)dP.
3 f1An(+oo)>e)
(10.4)
fA,(+oo)>c/2j
In view of (10.3), limr.,m P{A (+ co) > c/2} = 0 uniformly as n varies; so by
492
2.IV. Basic Properties of Continuous Parameter Supermartingales
the class D property of x(-) the right side of (10.4) has limit 0 when c - oo, uniformly as n varies, and it follows that the sequence A,(+ oo) is uniformly integrable. Hence some subsequence A.(+oo) converges weakly in L'(f , .F(+oo),P) to a limit random variable A(+oo), and consequently for each parameter value t in I the sequence E{A, (+oo)19(t)} converges weakly in
L'(1,.4z(t),P) to E{A(+oo)I.F(t)}. From (10.2) if teJ, the sequence A,(t) also converges weakly in L' (i2, .fi(t), P) to a positive F(t) measurable random variable A(t), with A(0) = 0. Furthermore (10.1) is true for t in J, and A(-) is an increasing process on J. By definition of J each point t in I - J is a bilateral limit point of J, and x(t) = x(t+) = x(t-) almost surely if the one-sided limits are defined either as L' limits or as sequential almost sure limits. Hence for t in I - J,
x(t) = E((A(+oo)j.f(t+)} - A(t+)
=E{A(+oo).F(s)A(t-) as. l
(10.5)
I-
$- t. The almost sure supermartingale inequality
y(t) > E{y(S A T")13 (t)}t
yields (n - x) y(t) z E{y(S-)Ifs t} u {T < t}, on {S:!9 t 5 T}.
(14.1)
Here the conditional expectations are to be chosen to make rSTx(-) almost surely right continuous (Section 2). Obviously rSTx(T) = x(T) almost surely. is a supermartingale. It follows from Theorem Suppose now that (essential order); so by almost 3 that x(T) is integrable and that 5 sure right continuity this inequality is true quasi everywhere on R` x 0. The argument in Section 111.18 translated into the present context shows -fl')) is a supermartingale, a martingale on QS, T]J, in the that sense that the process
{tSrx((S V t) A T),.F((S v t) A T), to R' } a submartingale are clear. is a martingale. The dual results for Observe that if is a positive supermartingale and if we define
x(+ oo) = 0, then we need not restrict S and T to be finite valued in the preceding discussion.
The Operators LM and GM
is an almost It will be sufficient to discuss GM. Suppose that surely right continuous supermartingale which has an essential order submartingale minorant. Then (from Section III.20) one version of the mar-
15. Potential Theory on R' x )
501
is For each positive integer k the sequence .F(t), t < k} is, for n >_ k, a decreasing sequence of almost surely right continuous martingales. An application of Theorem 4 shows
tingale
n -.
that one version of is an almost surely right continuous martingale, and in the present continuous parameter context we shall accept only such versions. More generally, if r is a class of almost surely right continuous supermartingales adapted to for which GM r exists, we now show that GMr has an almost surely right continuous version. Such a version is the only one we shall accept; it can be modified trivially to yield a right continuous version, if desired. To prove the assertion, observe first that we can assume that r is directed downward, at the price of adjoining to r the finite minima of its members; the enlargement of r does not change GMr. The r} is a downward-directed class of almost surely right class {GM continuous martingales with essential order infimum and limit GMr, and according to Section 111.5, there is a decreasing sequence of these mar-
tingales whose limit is a version of GMr. An application of Theorem 4 shows that this version is almost surely right continuous.
15. Potential Theory on R' x 0 Let (0, .f, P; ff(t), t E R+) be a filtered probability space for which fl-) is right continuous and F(0) contains the null sets. Many of the results we have obtained suggest that one can construct a theory on tl + x S2 which is formally parallel in many ways to parabolic potential theory and therefore also to classical potential theory, in which almost surely right continuous supermartingales play the role of superparabolic functions. The nomenclature, for example, Fundamental Convergence Theorem, GM and r operators, has been chosen with this in mind. In later chapters when we compose superparabolic functions with space-time Brownian motion to get supermartingales, it will become clear that the relation between a suitably constructed potential theory on 08+ x fI and parabolic potential theory is more than just a formal resemblance. Probability theory and potential theory developed independently of each other aside from interrelations between random walks and Brownian motion with solutions of parabolic and elliptic differential equations, until on the one hand axiomatics of generalized potential theory were devised and on the other hand the theory of Markov stochastic processes led to a generalized potential theory in which potential theoretic ideas were defined in terms of probabilistic ones. In both theories generalized versions of superparabolic functions lay at the base of the subject, and under suitable conditions it was shown that the axiomatic potential theory could be generated by Markov
process theory. It then became possible to prove theorems in axiomatic potential theory by probabilistic methods. We shall discuss in later chapters the Brownian motion process which generates classical potential theory and
502
2.IV. Basic Properties of Continuous Parameter Supermartmgales
the space-time Brownian motion process which generates parabolic potential theory.
Although probabilistic methods have frequently been used to derive potential theoretic results, derivations in the reverse direction have been rare. To illustrate the possibilities and to give insight into the whole subject, we shall use the ideas and methods of potential theory to analyze reductions in the context of potential theory on 98' x Q. We shall need a few properties of the fine topology on this space.
16. The Fine Topology of R x i2 Let M. .F, P;.f(t), to R') be a filtered probability space for which is right continuous and .r(0) contains the null sets. The fine topology of R' x fi is by definition the smallest topology on R' x 0 making every right continuous supermartingale into a (fine topology) continuous process in the sense that the function (1, w) -' x(t, w) is to be continuous in the fine topology. The set of fine limit points of a set A will be denoted by
A. EXAMPLE (a). If A is a P null subset of 0 and if x(-, -) is the indicator function
of A = R' x A, the process
is a right continuous martingale;
so A is both fine open and fine closed.
as the indicator function of the stochastic interval 10, nJ. Then {x(-),.F(-)} is a right continuous EXAMPLE (b). Let T be optional and define
supermartingale; so both stochastic intervals TO, T[ and I T, + oo Q are fine open and fine closed. If S and Tare optional with S < T, we conclude that
the stochastic interval IS, TQ = 10, 71 n IS, + oo Q is fine open and fine
closed. If we choose T. = T + I/n, we find that the stochastic interval IS, TJ = f i, IS, T,[ is fine closed. In particular, the graph [Sj is fine closed. If S is optional, if A c .F(S), and if S,, = S on A but S = + oo otherwise, the stochastic interval QSA, TAI = (R' x A) n IS, TQ is fine open and fine closed. Observe that the intersection of this stochastic interval with another one (R' x A) n QS', T'[ of the same type is
(R' xAnA')nISvS',TAT'Q, also of this type, and that S v S' is identically constant if S and S' are.
A Base for the Fine Topology. The class of sets (stochastic intervals) (R' x A) n [a, TQ with a a positive constant and A c F(a) is a base for the fine topology because if { is a right continuous process, not necessarily a supermartingale, and if T, is the entry time of the set
503
16. The Fine Topology of R' x S2
{ (s, w) : s > t, x(s, (0) S c), then
{(t, w): x(t,w) > c} = U. (R
x {co: x(t,w) > c)) r) it, T,Q.
IER
If {wo} is a P null singleton and if toe 08', the set (R+ x {wo}) n Qto, to + e[
is a fine neighborhood of (to, wo) for every e > 0, and the set of these fine neighborhoods for all E > 0 is a base for the fine neighborhoods of (to, wo). Hence a subset A of l8' x f2 has (to, wo) as a fine limit point if and only if A contains points (t,,, wo) with t > to, t = to. Thus if all 0 singletons are P null the fine topology is the direct product of the discrete topology on fl and the one sided Euclidean topology on l8+ in which a base for the topology is the class of left closed intervals. Evanescent Sets and the Fine Topology
The fine closure of an evanescent set is evanescent because an evanescent set is a subset of an evanescent set of the form R+ x A with P{A} = 0 and [from Example (a) above] this product set is fine closed. Almost Thinness at an Optional Time If T is optional, call a subset A of [I x i2 almost thin at T if the set [TD n Af is evanescent. Then an evanescent set A is almost thin at every optional time.
A progressively measurable set A almost thin at every optional time T is semipolar if all Sl singletons are null, according to the following argument. If T, is the entry time of A, then IT,I c A up to an evanescent set, and the hitting time T2 of A after T that is, the second entry time of A, is almost surely strictly larger than T, on { T, < + oo }. Continuing in this way, we find an increasing sequence T. of optional times such that T. < almost
everywhere on {T < +oo}, and c A up to an evanescent set. Going on by transfinite induction, the next optional time is the first entry time >_ T,,, .... If a is a countable ordinal for which T. has been defined, either T,+, = + ac almost surely, in which case A is semipolar because Ac IT up to an evanescent set, or P{T,+, > T,} > 0; so E{arctan T,+,} > E{arctan T,}. It follows that the second possibility can occur only countably many times,
504
2 I V Basic Properties of Continuous Parameter Supermartingales
that is, there is some countable ordinal a with T,+, = + x almost surely, and we have just seen that then A must be semipolar.
17. Potential Theory Reductions in a Continuous Parameter Probability Context (See Section 111.22 for the corresponding discrete parameter theory.) Let (S .. , P: f (t), t e 08 +) be a filtered probability space for which 9(-) is right continuous and .y(0) contains the null sets. All processes below are be a positive process, and denote adapted processes on this space. Let by r the class of almost surely right continuous supermartingale pointwise It is supposed that r is not empty. The class r' is directed majorants of
downward, and we define R_,., = ess inf r (= either an equivalence class under identification of indistinguishable processes or a member of the class,
as indicated by the context). Observe that the equivalence class R,(.) is unchanged if r is defined as the class of almost surely right continuous supermartingales which are quasi everywhere majorants of or if "almost surely" is omitted here. Define R_i.,(+x) = :(+ x) = 0. According to the Fundamental Convergence Theorem (Theorem 5), almost every sample function of the supermartingale R_,., has right and left limits at all points, and if R") is defined as the right smoothing of R2(.), then R=V) < R,(., quasi everywhere. For i c- R + define
r(t)=esssup{E{z(T)Jf(t)}: Toptional, >_ t);
(17.1)
this random variable is determined only up to a null set. The method of is a positive supermartingale and a minorant of r. which in the essential order is both a majorant of in the essential order; Thus no matter how v(t) is chosen, R=,., > that is, P; R_,.1(t) ? Y(i) > z(t) J = I for all t. According to Section 1 there is a countable parameter set B such that r(t) can be chosen for each t in such sample function has right and left limits at all a way that almost every points, and is right continuous except possibly at points of B. Choose Y(1) quasi everywhere. If R_ in this way. With this choice Y, (t) is defined by the right side of (17.1) except that the optional time T is supposed strictly greater than t, then r+ is a supermartingale, r+ (t) < r(t) almost surely, and t' J t sequentially implies that proof of Theorem 111.22 shows that
t'
:5y(-) quasi every(t) = vu +) and thereby have is almost surely right lower semicontinuous. where. In particular, if and therefore quasi everywhere; so Y, and are indistinguishable quasi everywhere. We conclude that
18. Reduction Properties
505
almost surely right continuous supermartingales. Unfortunately we shall not be able to make this lower semicontinuity hypothesis in the following application to reductions, in which the process corresponding to above where is now an almost surely right continuous is the product supermartingale and A is a subset of 08+ x Q.
Application to Reductions (on the Above Filtered Probability Space) Let A be a subset of R+ x 0 for which t - IA (t) is a process adapted to and let be a positive almost surely right continuous supermartingale. Define the reduction R (.) as the supermartingale R=,.)IA,.). The right smoothing of the reduction will be denoted by RA() or by According to Theorem 5, the process RA.) is determined up to indistinguishability by the following conditions: The reduction is a progressively measurable process. Whenever is in the class f of almost surely right continuous positive supermartingales quasi everywhere on A, the inequality is true quasi everywhere on 08+ x fl. z If is a process satisfying the preceding conditions, then Z quasi everywhere on 68+ x Q. RA.
is a supermartingale with almost sure Furthermore a reduced process right and left limits, coinciding quasi everywhere with the respective right and left limits of its right smoothing R=( and Rz quasi 5 everywhere, with equality up to a semipolar set. Finally, a suitably chosen decreasing sequence of members of r' has as limit a version of the reduction. We shall see that many reduction properties in the present context are
close counterparts of reduction properties in the context of classical and parabolic potential theory. In one respect the present context is simpler than those earlier ones. The fact that superharmonic and parabolic functions can have infinities complicates their theory, whereas almost every sample function of an almost surely right continuous supermartingale is finite valued with finite left limits.
18. Reduction Properties The context is that of Section 17. We consider reductions of a positive almost on subsets of 68+ x Q. surely right continuous supermartingale Reduction properties in this context are counterparts of reduction properties in parabolic potential theory as listed in Sections 1.XVII.I I and 1.XVII.16 except that for simplicity we have not introduced counterparts of reductions
on boundary sets. The following properties are listed as far as possible in the order of their parabolic counterparts, and their counterparts are always
506
21 V. Basic Properties of Continuous Parameter Supermartingales
referred to. Proofs are given or referred to in Section 19. We leave to the reader the translation of the listed properties to the context of a countable parameter set. (a) [See Section 1.XVII.16(b).]
RA (t+) = Rz .,(t+), RA. (t-) = R A.,(t-) q.e.;
z(t) z Ri.)t) z R i.,(t+) q.e.; Ra., = R= .,
up to a semipolar set;
R" .) = Ra., q.e. if A differs from h by an evanescent set. (b)
[See Section I.XVII.16(b).]
q.e.on AnAf; R".) = R" =z(-) q.e. on ,4f, in particular, q.e. on the fine interior of A; R" = RA(.) q.e. when A is fine open. (c) [See Section 1.XVII.16(b).] If S and T are optional times with S 5 T and if IS, T[ r),4 is evanescent, then R; ., = R .) quasi everywhere on QS, TQ, and Ra is an almost surely right continuous martingale on [S, T] in the sense that the process
(R=.,((S v t) A T),.. ((S v t) A T),teUB+}
(18.1)
is an almost surely right continuous martingale. (d) [See Sections 1.XVII.11(a) and 1.XVII.16(c).]
R (.) = ess'inf (R8 ): A c , E progressively measurable and fine open} q.e.; (18.2)
one version of the right-hand side is the infimum when a runs through some decreasing sequence of fine-open supersets of A. In particular, if all almost
surely right continuous supermartingales on the given space are almost surely lower semicontinuous, then whenever is almost surely continuous, the set 1) in (18.2) can be further restricted to have open t cross sections for fixed co. (e) [See Sections 1.XVII.11(b) and 1.XVII.16(i).] If A c $ and if A is
are indistinguishfine open, then the supermartingales OB and able. If 1) is also fine open, these supermartingales are indistinguishable from flz(i 0 P".
18. Reduction Properties
507
(f) [See Section 1.XVII.16(f ).] If A is indistinguishable from a progressively measurable set, then R ) = R".) quasi everywhere on (R+ x S2) - A. (g) [See Sections 1.XVII.11(d) and 1.XVII.16(k).] The set functions A i R 1.1 and A -4 R' are strongly subadditive in the sense that quasi everywhere on R+ x 0 both
R :8 + Ri :)B < R= + Ra.,
(18.3)
and the corresponding inequality (18.sm) for smoothed reductions are true. The exceptional evanescent set depends on A, B. [See (i) for countable strong subadditivity.] (h) [See Sections 1.XVII.11(e) and 1.XVII.16(e).] If A, is an increasing sequence of subsets of Q8+ x f2 with union A and if is an increasing sequence of almost surely right continuous positive supermartingales with limit then quasi everywhere on I8+ x 0 both (18.4)
lim RTry.) = Rz .) n-M
and the corresponding equation (18.4sm) for smoothed reductions are true. (i) [See Section 1.XVII.16(k).] The set functions A)- Ra.) and A) -+R ) are countably strongly subadditive in the sense that if A. and B are sequences of subsets of R+ x 0 with An c Bn, then quasi everywhere on R+ x S both RU-1" +
RU'-"- + R' An< RU'
0
8n
0
R:r)
(18.5)
and the corresponding inequality (18.5sm) for smoothed reductions are true. (j) [See Section 1.XVII.16(g).] If n e 7L+ } is a sequence of almost surely right continuous positive supermartingales, if Eo (almost
surely right continuous by Theorem 4) is a supermartingale, that is, if Eo E{zn(0) } < + oo, then quasi everywhere on 68+ x D both R1r
R./
(18.6)
0
and the corresponding equation (I 8.6sm) for smoothed reductions are true. (k) [See Section 1.XVII.16(m).] If S is a finite optional time and if
A = [0, S], then GM R".) = 0 quasi everywhere (equivalently, R=) is a supermartingale potential) if either S is bounded or S is unrestricted and is uniformly integrable. (1)
[See Section 1.XVII.16(o).] Suppose that (S2, ,
has the
property that every almost surely right continuous supermartingale is almost is an almost surely surely lower semicontinuous, and suppose that continuous class D positive supermartingale. Then the set function 4F-* R",.,
508
2 IV. Basic Properties of Continuous Parameter Supermartingales
is a Choquet capacity on R' x Q relative to the class I', of progressively measurable sets whose t cross sections for fixed co are compact, in the sense
that
AcB=t,-0:5R,.,5Re.,
(18.7')
q.e.
A IA ---- R("iIR(., q.e. R A^>IR;.,
(18.7") q.e.
(18.7'")
(m) [See Section 1.XVII.16(p).] Suppose that (f2, has the property that every almost surely right continuous supermartingale is almost
surely lower semicontinuous, and suppose that
is a positive almost
surely continuous supermartingale. Let r, be the class of predictable product sets defined in Section 1.14, and let r, be the class of finite unions of r' sets. Then if A is analytic over the class of predictable sets, there is an increasing sequence E. of r,, subsets of ,4 for which quasi everywhere on R' x Q both
limR=' =R'4
(18.8)
and the corresponding equation (I8.8.sm) for smoothed reductions are true. (n) [See Section 1.XVII.16(q).] If are bounded positive and almost surely right continuous supermartingales, then sup' IRA., - R (.) 1 5 sup'
I,
(18.9)
where sup' means the supremum up to evanescent sets, and the corresponding inequality (18.9sm) is true for smoothed reductions. be (o) [See Section 1.XVII.16(s).] Let {x( ), () } and { y( ),
almost surely right continuous positive supermartingales, let a and b be positive numbers with 0 5 a < b, and define A = ((f, (9): x(t,w) 5 ay(t,w)},
B= {(t,w): x(t, w) z by(t,w)}.
r.
theiterated reducDenote . . tion and downcrossing inequalities 1I1(23.2)-111(23.6) are true in the present context, as are also the other martingale theory translations of the classical
context reduction inequalities in Section 1.VI.3(o). Moreover these inequalities are true for the parameter restricted to an interval. In particular a, b] refers to downcrossings on the parameter interval if [0, t], then (almost surely) for t e R', E{
5 x(0)b
[b. (0)]
(18.10)
19. Proofs of the Reduction Properties in Section 18
509
19. Proofs of the Reduction Properties in Section 18 (a) The properties listed in Section 18(a) were all derived in Section 17. (b) The first two lines in Section 18(b) follow easily from two facts: (i)
There is a decreasing sequence of right continuous supermartingales on A and so also on Af, and tending quasi everywhere equal to to RA =).).
(ii)
Each fine neighborhood of a point (to, (DO) contains a set of the form
[to, to + e] x (co.) with e > 0.
To prove the third line, observe that R".) = R .) quasi everywhere on A; so R" is a version of the unsmoothed reduction. +=(-)
(c) To prove Section 18(c), observe (see Section 14) that if T is a finitevalued optional time and if (x.(-), n e 1+ } is a decreasing sequence of almost
surely right continuous positive supermartingales with quasi everywhere limit R A.), then Tsrx is an almost surely right continuous positive martingale on IS, majorizes quasi everywhere on A, and TSTR .) is indistinguishable x.(-) quasi everywhere. Hence from R.) and is (by Theorem 4) an almost surely right continuous martingale on QS, T] and so is equal quasi everywhere on [S, T to its right smoothing. If Tis not finite valued, the same argument is applicable without any change if we define R-4. )(+oo) = 0 and allow T to be infinite valued in the definition of Tsr. The needed properties of the operator TST remain 'valid under this extension. (d) The first assertion of Section 18(d), in which R4. is indistinguishable from RB.), follows from the fact that, assuming as we can in this proof that is right continuous, if is a right continuous positive supermartingale essential order majorant of on A and if c > 0, then e is an essential order majorant of on a progressively measurable fine-open neighbor-
hood of A. If every almost surely right continuous supermartingale on the probability space is almost surely lower semicontinuous "fine open" in the preceding argument can be replaced by "open t cross sections for fixed w." The right-hand side of (18.2) is a countable infimum according to Theorem 5. (e) See the proof of Section 18(e) in the parabolic context [Section I.XVII.11(b)]. (f) To prove the equality of reduction and smoothed reduction quasi everywhere on A = (R' x f2) - A, observe first that the strict inequality set is not only semipolar but (see the proof of the Fundamental Convergence Theorem) is, up to an evanescent set, a countable union Uo [S. T of optional
time graphs. It is to be shown that, for each value of n, the set A = {w: [S.((9), w] eh) is null. This set, the projection on f2 of the set [S D r )A,
because, for c z 0, the set A n {S S c}, the projection on f Z of [S ] n h n ([0, c] x fa), is in .F(c). Define an optional time S by setting S, = S on A and S = + oo elsewhere. If T. is the time of is in
510
21V Basic Properties of Continuous Parameter Supermartingales
(r, (o) entry into the nearly progressively measurable set [S ] r).4, then { T. = S.' < + co } c Af up to a null set ; it then follows from (b) that T. > S almost everywhere on A. Since the set [S,;, Tn[ n A is evanescent Rz it follows from (c) that R= almost everywhere on A; this equality implies that A is null. (g)
To prove strong subadditivity of the reduction operator and its
smoothing, suppose first that A and h are fine open, in which case (18.3sm) reduces to (18.3), and define z'(-) = R= .) A RB.). Then [see the proof of the corresponding potential theory equality I.VI(3.4) and 1.XVII(11.4)] q.e.
RZ iB + Ri t.e = R A.) + RB.)
(19.1)
This equality implies inequality (18.3) for fine-open A and $ and in view of (18.2) thereby implies (18.3) for all A and E. Inequality (18.3sm) follows trivially from (18.3). (h) The proof of (18.4) follows the proof of the parabolic potential theory equation 1.XVII(11.6). If each set An is fine open, (I8.4sm) reduces to (18.4) and follows from the fact that on the one hand (quasi everywhere)
the limit on the left side of (18.4) exists and is majorized by the right side and on the other hand the limit process is almost surely right continuous on A and so majorizes (Theorem 4) and (quasi everywhere) majorizes the right side on I8+ x Q. If the sets involved are not fine open, let T be a be the finite valued positive Y20JF'(r) measurable function, and let indicator function of the set 5 k}. Then E{R=j.,.)(T)xk(T)} S E{RB.,(T)xk(T)} < k. Choose e > 0, at > 1, define An = An n
and apply (18.2) to find a fine-open superset $; of A satisfying E{R=()(T)xk(T)} + 2-'e.
(19.2)
The sequence A° is an increasing sequence with union A, and we define A = Uo hj A. By strong subadditivity of reductions E{Rs Uo81(T)xk(T)} S E{R ")(T)xk(T)} +
-Ej
R'4J(T)xt(T)} 0
111
5 E(R")(T)xk(T)} + 2e.
(19.3)
511
19. Proofs of the Reduction Properties in Section 18
Since (18.4) is true when the sets involved are fine open, (19.3) yields (when
n-4oo) E{R ".)(T)xk(T)} 5 E{R".)(T)xk(T)} S li m E{R ")(T)xk(T)) + 2e R-W
(19.4)
5 a lint E{ R=.1.)(T)xk(T)} + 2e. -m
Hence E{R A.)(T)xk(T)} S lint
(19.5)
n-W
and the reverse inequality is trivial. Since there is monotone convergence of integrands in (19.5), we conclude that for every k !im R=1.)(T)xk(T) = R=.)(flxk(T)
as.
The set Ck: (R'4.)(-,
li m
has the property that its intersection with the graph of Thas a null projection on 0, whatever the choice of T. It follows (from Theorem 11.8) that Ok is evanescent and that therefore (18.4) is true. Then (18.4sm) is true up to a semipolar set and since both sides of this equation are almost surely right continuous processes the two sides must be indistinguishable. (i)
Inequality (18.5) and (18.5sm) follow from strong subadditivity
combined with (18.4) and (18.4sm). (j) To prove (18.6) for two summands, it is sufficient to give the proof for A fine open, in which case the proof of the classical. context counterpart I . VI (4.2) translates easily into one valid in the present context. It then follows from (18.2) that (18.6) is true for arbitrary A and finitely many summands
and therefore by (18.4) for countably many summands. Finally (18.6sm) has now been shown to be true up to a semipolar set, and since both sides of (18.6sm) are almost surely right continuous processes, the two sides must be indistinguishable. (k) If A = Q0, SD and if A = QS, + oo Q, it is trivial that R-4., = 0 quasi everywhere on A. Hence the martingale GMR= vanishes quasi everywhere if S is bounded. If S is not bounded
limE{R= (t)} = lira [(S> z(t)dP, .
and this limit is 0 if potential.
(19.6)
ri
is uniformly integrable; so R'. is a supermartingale
512 (I)
2 IV Basic Properties of Continuous Parameter Supermartingales
Assertion (18.7') is trivial, and (18.7") is included in Section 18(h).
To prove (18.7"'), suppose first that A = 0. Then if T. is the entry time of A,,, the sequence T, is increasing. U" { T = + oo } = D almost surely. and by Section 18(c) the process R= .) is an almost surely right continuous process on f 0, T,11. Furthermore, if we define z(+ tx) = 0. RV)
t}. The process
defined by
(20.1)
514
2 W. Basic Properties of Continuous Parameter Supermartingales
E{z(S)j.9'(t)}
zo(t) = z(t) 0
if t < S
if S:5 t < T
if +oo>t>T
is a positive right continuous supermartingale if the conditional expectations are defined suitably (Theorem 2), is equal to z(-) on A, and is a quasi everywhere minorant of v(-). Hence is a version of RA.).
EXAMPLE (b). If A is as in Example (a) and B = QS, T}, then any positive right continuous supermartingale majorizing on h majorizes R z(.)
quasi everywhere and satisfies (20.1). Moreover for s > 0 the process go is also a positive right continuous supermartingale majorizing z(-) on B. It follows that quasi everywhere
E{z(S)j.f(t)} ift<S, R".,(t) = z(t) if S T,
RB =R-4 quasi everywhere, and RB < RB., on QTD, a semipolar set.
EXAMPLE (c). If 0 < a < h and if A e 9(0) when a = 0 but A E when a > 0. then [a, b] x A = RS, Tl with S = a on A and S = + co otherwise, T = h on A and T = + oo otherwise, so the reduction evaluation in Example (b) is applicable. Theorem. Suppose that (S2, . , j F(-), P) has the property that every almost .surely right continuous supermartingale is almost surely lower semicontinuous, and .suppose that is an almost .surely continuous positive supermartingale. Let A he predictable , and let T,' [T,] he the entry [hitting] time ol'A n Q t, + cc Then for each t > 0
RA.,(t) = E{z(T,')j.(t)}
a.s.
(20.2)
R" (t) = E{z(T,)j.f(t);
a.s.
(20.2sm)
It is sufficient to prove this theorem for bounded since (20.2) and (20.2sm) are true for A n when ne7 Z. Moreover the truth if true for of (20.2) implies that of (20.2sm) by the following argument. Since RA is the smoothing of R= ,, there is a countable subset of O8+ such that for t not
in this set R" (t) = R".,(t) up to a null set which depends on t. Now the process T, is the smoothing of T. T, = limsy, T,. so there is a countable subset of 08' such that for t not in this set T, = T,' up to a null set which depends on t. Thus, if (20.2) is true, it follows that (20.2sm) is also true for t
515
21. The Energy of a Supermartingale Potential
not in some countable set. Finally if s > 0 and if t decreases sequentially to s, not hitting the exceptional countable set, (20.2) yields (20.2sm) for s, in view
of the almost sure right continuity of the left side and the dominated convergence theorem for conditional expectations as applied on the right. bounded. Let A be a sequence of sets We proceed to prove (20.2) for for which (20.2) is true. To prove (20.2) for arbitrary predictable A, it is sufficient to prove the following two assertions. (i) (ii)
If Ao c A, c .. , and U',4,, = A, then (20.2) is true. If A o A, , and no A = A, and if each set A is progressively measurable and has compact t cross sections for fixed w, then (20.2) is true.
In fact, if F, is the class of finite unions of the sets of the type considered in Example (c), it follows from (ii) that (20.2) is true for countable intersections of ro sets, and therefore by (i) combined with Section 18(m) it follows that (20.2) is true for every predictable set A. To prove (i) and (ii), let S be the T,'. , and entry time of A n QT, + co Q. In (i) So, >- S1, >_
On the one hand, lim, R") = R i.) quasi everywhere by Section 18(h). On the other hand (dominated convergence theorem for conditional expectations),
E{z(T,)I.f(t)}
lim
n-M
a.s.
(20.3)
so (20.2) for A becomes (20.2) for A when n - oo. In (ii) So, < S
- x(t) almost surely, simultaneously for all t. Theorem. Let be a right continuous filtration of a probability space, and suppose that ,(0) contains the null sets. (a)
A nearly predictable almost surely right continuous supermartingale
{x( ), 9rO} is almost surely lower semicontinuous, almost surely continuous if the process is a martingale. If in (a) is right closable it can be written as a sum of processes adapted to .f nearly predictable, for which sample function is monotone increasing, constant (bl) Almost every except for the - x(-) sample function jumps, and x'(0) = 0. (b2) is an almost surely continuous supermartingale and is right closable. (c) If every. IF(-) optional time is predictable and if .F(-) is predictable, then every almost surely right continuous supermartingale is nearly predictable and therefore is almost surely lower semicontinuous, almost surely continuous if the process is a martingale. (b)
Proof of (a). Since it is sufficient to prove (a) for x( A n) for all n > 0, we can assume that is a right closable supermartingale, or a right closable
martingale if the process is a martingale. The process {x( - ),() }
is
almost surely left continuous and therefore nearly predictable so the process
x( - ),
is nearly predictable, and it follows that when c > 0,
the set
H, = {(t,(o). x(t,uw) - x(t-,w) z c}
519
23. Supermartingale Decompositions and Discontinuities
is a nearly predictable set. Furthermore the graph of the entry time T of H. is in H, up to an evanescent set ; so (by Theorem 11.9) T is nearly predictable.
Hence x(T-) ;?.t x(T) almost everywhere where T < + oo according to Section 22(b), and this is impossible unless T = + oo almost surely. We is a (right conclude that the supermartingale assertion of (a) is true. If shows that the and closable) martingale, this result applied to martingale assertion of (a) is true. It will be convenient to prove (c) before (b).
is right closable as a supermartingale. Proof of (c). Suppose first that If n E Z+ and k e 1, define Tok = 0, and if T.-, k is already defined, define Tnk = +oo if T. -I k = +oo and Tnk = inf {t > Tn_, k: 2k < Jx(t) - x(t-)I 5 2k+' }
otherwise.
Then Tnk is optional, therefore predictable by hypothesis, and therefore by Section 22(b) x(Tnk-) > x(Tnk) when n > 0 and Tnk < +oo. Define Jnk = x(Tnk -) - x(Tnk) when Tnk < + co and Jnk = 0 otherwise, and define a right by continuous process xnk(t) = JkIITks0
(=0 when Tnk = +oo) for I e P+.
is right closable it is right closable by lim, x(t) which we denote Since by x(+ oo ). According to Section 22, Jnk Z 0 almost surely, the process
here and below], the is nearly predictable [the reference filtration is process is a right closable (by x(+oo) + Jnk) almost surely right continuous supermartingale, and E{Jnk} < E{x(0) - x(+oo)}. If the jump and if we define (summing over all processes are added successively to
n and k) J = EJnk and x'(t) = Exnk, then E{J} s E{x(0) - x(+oo)}. It follows that J and x'(t) are almost surely finite. The process is nearly predictable, and its sample functions are almost all right continuous except for positive jumps Jnk at Tnk. The process
is almost surely
continuous and therefore is nearly predictable, and we conclude that (supposed right closable) is nearly predictable. The supermartingale is right closable by x(+oo) + J. If is not right closable, this result implies that x( A n) is nearly predictable for all strictly positive n; so is nearly predictable. was obtained satisfyProof of (b). In the proof of (c) a decomposition of ing (bl) and (b2), under the hypothesis that was a right closable almost
surely continuous supermartingale satisfying certain conditions on the reference filtration A glance at the proof shows that wherever these conditions on
are used, near predictability of
suffices.
Chapter V
Lattices and Related Classes of Stochastic Processes
1. Conventions; The Essential Order In this chapter certain stochastic process classes which arise naturally in martingale theory will be discussed. These classes and their relations with the identically named classes in Chapter IX of Part I will be studied in later chapters. See Appendix III for the lattice theory to be used. All stochastic process concepts in this chapter are relative to a specified filtered probability space (0_F, P; .fi(t), t e I), where I is a linearly ordered set, arbitrary unless specifically limited. As always the probability measure is supposed complete; in addition it is supposed that each a algebra .fi(t) contains the null sets. The stochastic processes considered are adapted to and have state space (A, .(68)). Recall from Section 1.8 that in the essential order stochastic processes are grouped into equivalence classes by identifying processes which are is an essential order standard modifications of each other and a process if majorant of
P(y(t) z x(t)) = 1,
tel.
Recall further that the essential order infimum ess inf r of a set r of stochastic processes, that is, the essential order infimum of their equivalence classes, can be obtained as follows. If we write e r to mean that the equivalence is a version of an equivaclass containing is in r, in other words that
lence class in r (see the remark on the abuse of notation in Section 1.1), then ess inf r is the equivalence class consisting of all the versions of
(essinfx(t),tel}. x(.
r
Recall that the essential infimum of a family of random variables is determined only up to a null set. The class of processes in the essential order is a complete lattice for I a singleton and therefore for arbitrary /.
2. LM
521
Is a Submartingale
when
The Continuous Parameter Context
This is the context in which I = R', is right continuous, .F(0) contains the null sets, and the processes to be classified will be almost surely right continuous. Recall that two almost surely right continuous processes on the
parameter set R' which are standard modifications of each other are indistinguishable, that is, equal quasi everywhere on 0 x R'.
when
2.
Is a Submartingale
(See Section 1.IX.2 for the potential theory counterpart of this section.) Let be a submartingale with an arbitrary linearly ordered
parameter set I having a first element. If S and T are countably valued 5 optional times, upper bounded in I, with S:5 T, then 5 TTX(') in the essential order. According to the dual of the Special case in Section
has no essential order supermartingale majorant and lim,tE{x(s)} = +oo, or has an essential order supermartingale major111.20, either
ant, LM
is a martingale,
exists, the process { LM
LM x(-) = esslim
(2.1)
es
up to a standard modification, and E{LM x(t)} = lim E{x(s)} = su?E{x(s)}. st
(2.1')
SC-
(As usual we write LM x(t) for [LM (t).) In view of one of the forms of the submartingale sampling theorem (Theorem 111.7) the supremum and directed limit sup (E{x(S) } : S optional, countably valued, bounded) is equal to the supremum in (2.1'). Thus LM of expectations of the random variable class
(2.2)
if and only if the set
{x(S): S optional, countably valued, bounded}
(2.3)
is bounded.
If is a positive submartingale, the existence of LM is equivalent to the L' boundedness of the random variable class (2.3) and in this case a trivial argument shows that S in (2.3) need not be bounded in 1. Moreover the hypothesis that I has a first element can be dropped because in any case the process is left closable by the random variable 0 coupled with
the trivial a algebra (0, i2). In particular, suppose that the positive sub-
522
2.V. Lattices and Related Classes of Stochastic Processes
martingale is uniformly integrable; that is, suppose that there is a uniform integrability test function 0 for which sup E{m[x(s)] } < + oo.
(2.4)
BE
In this case LM exists, and, since {'F[x(-)] is a positive submartingale, (2.4) is true if and only if LM exists. We now show that then LMt[LM also exists and LM 'F[LM
LM
(2.5)
up to a standard modification. To see this observe that up to standard modifications
E{4[LM x(s)]J.'(-)} = E{'F[ess limE{x(s')I,F(s)}]I. S't
S E{ess lim E{t[x(s')]J _fls)} IF(-)} +' T
= E{ Take the essential limit as s increases to find that the left-hand side of (2.5)
is an essential order minorant of the right-hand side. The reverse order relation is trivial.
Continuous Parameter Context Recall from Section IV. 14 that in the continuous parameter context (defined in Section 1) if {x(-), is an almost surely right continuous submartingale for which LM x(-) exists, this martingale majorant can be chosen to be right continuous; the notation LM x(-) will always refer to an almost surely right continuous version. Moreover, if _fl-)) is a positive almost surely right continuous submartingale, not only is the existence of LM equivalent to the finiteness of
sup (E{x(S)}: S optional (< +oo) countably valued},
(2.6)
as already stated in the general context, but the restriction that S be countably valued is unnecessary. In fact if the supremum in (2.6) is c, let T be an arbi-
trary optional time except that T< +oo. Then if [T], is defined as in Section 11.2, Example (b2), it follows from Fatou's lemma that
E{x(T)} S liminfE{x([T]n)} 5 c. n_ .O
If an integrable random variable x(+oo) can be adjoined to right close the submartingale, as is possible if the submartingale is uniformly integrable
3 Uniformly Integrable Positive Submartingalcs
523
[Section II1.3(e)], then this argument is valid for an arbitrary optional time T < + x : so in this case S in (2.6) can be an entirely unrestricted optional time.
3. Uniformly Integrable Positive Submartingales In Section 1.iX.3 a class D(,4 _) of functions on a Greenian subset of 118' was defined, and Theorem 1.IX.3 treated functions u for which u e D(µ"1)_ ).
The applications in view were to positive h-subharmonic functions and to h-harmonic functions. The probabilistic counterparts of class D(µD_) functions on a specified set are the class D stochastic processes on a probability space, relative to a given parameter set and filtration. as defined in Section 11.11. The following theorem is the probabilistic counterpart of Theorem i.IX.3, and the applications in view are to positive submartingales and to martingales. All the processes in the theorem are defined on the same filtered probability space. Theorem. Let 1, be a stochastic process with state space and an arbitrary linear/' ordered parameter set. /J { j x(-) 1, ,9r is a submartingale, the following conditions are equivalent : a D.
(a)
(b) (c)
is uniformly integrable. There is a uniform integrabilitp test Junction N for which the submartingale has a martingale essential order majorant.
(d) (If {x( is a martingale) xt(') - x2('), where is a positive martingale, in D, and therefore wti/brmlp integrable. (e) The submartingale { I 1, 9(-)' is right closable.
Moreover, if {LM
satisfies (a) and (b), and if O satin/ies (c). then the martingale is uniformly integrable, LM (D[LM
(3.1)
LM
up to a standard modification, and, if
is a martingale. each process
in (d) can be chosen so that the submartingale {m[x;( )], t O} has a martingale essential order majorant.
If here is identified with in Section 111.2 and Section 2. the present theorem follows; if {x( ),() } is a martingale, one representation of with the properties stated in (e) and the final assertion of the theorem is LM
[LM
(See the corresponding discussion in the proof of Theorem I.IX.3.)
(3.2)
524
2 V. Latuces and Related Classes of Stochastic Processes
Theorem 3 versus Theorem 1.1X.3. The parallelism between these two theorems is obvious except that no potential theory counterpart of Theorem 3(e) has been suggested. To find such a counterpart, suppose that Jul is a class D(pD_) h-subharmonic function on a connected Greenian subset D
of R", as in Theorem 1.IX.3, and define v = LMplul, a class D(4_) hharmonic function according to that theorem. The function v - Jul is an h-potential because
GM,(v
- Jul) = v - LMoIui = 0;
so (Theorem 1.XII.18) the function v - Jul has u' almost everywhere minimal-fine boundary limit 0 on the Martin boundary 0"D. According to Theorem I.XII.19, a quasi-bounded h-harmonic function on D is the PWB" solution Hj for some h-resolutive boundary function f on a' D, and Hf has f as its p' almost everywhere minimal-fine Martin boundary limit function. Moreover we shall show (Theorem 3.1.5) that an h-harmonic function on D is quasi bounded if and only if it is in D(/D_ ). Hence there is an h-resolutive function f on a' D which is the µD almost everywhere minimal-fine boundary
limit function of both v and Jul, and Jul S Hf = v. The µo measurable and
integral Martin boundary functions f,, that is, the h-resolutive Martin boundary functions f,, for which f, z f are potential theory counterparts of the random variables which close the submartingale { in Theorem 3(e).
Continuous Parameter Context
In this context, as already noted in Section 2, when in Theorem 3 is supposed almost surely right continuous, all the least majorants in the theorem can be supposed almost surely right continuous and in view of (3.2) each process x;(-) in Theorem 3(e) can be chosen to be almost surely right continuous and to satisfy the last assertion of the theorem.
4. L" Bounded Stochastic Processes (p >_ 1) The following theorem is in the context of Theorem 3 and when p > I is merely a specialization of that theorem. Recall that a process {x(t), t e 1) is called L" bounded if sup,E,E{ix(t)I"} < +oo. Theorem. Let
be a stochastic process with state space (R, 91(R))
and an arbitrary linearly ordered parameter set, and suppose that p z 1. If fl-)) is a submartingale, the following conditions are equivalent: (a) (b)
is L" bounded.
The submartingale majorant.
has a martingale essential order
5. The Lattices ('S=, 5), ('S', 5). (S=, 5), (S'. 5)
(c)
(If
525
is a martingale);
where
F(-)) is a positive LP bounded martingale. (d)
(If p> 1); the submartingale
is right closable by a
random variable in LP.
Moreover, if p z I and if
satisfies (a) and (b), then the martingale
is LP bounded,
(4.1)
up to a stochastic modification, and if (x(-), F(-)) is a martingale, each process has a martingale in (c) can be chosen so that the submartingale {x f essential order majorant.
The proof is left to the reader because whatever is not already covered by Theorem 3 with D(s) = sP follows easily from the discussion in Section 2. Observe that assertion (c) of the present theorem with p > 1 is slightly
stronger than Theorem 3(d) with 0(s) = sP. In fact in Theorem 3(d) it is and if has a martingale not asserted that if essential order majorant, then has a martingale essential order majorant. However, for cD(s) = sP and p z I this assertion is true because then
I x(')I P S [xl(') + X2(')]' S 2P-'[x1(') +
Continuous Parameter Context In this context (see the corresponding remarks in Section 3) if is almost surely right continuous, all the other processes involved in Theorem 4 can also be supposed almost surely right continuous. Observation (Recall the definition of LP in Section 11.11.). It is trivial that a process in LP is LP bounded. Conversely (context of Theorem 4) if is
LP bounded with p > I and if is a submartingale, it follows that a LP. In fact it is sufficient to prove that if T is an optional time with finitely many values then E{Ix(T)IP} 5 sup,.,E{Ix(t)I"}, and this inequality is true because if s is the maximum value of T in the parameter set order then E{Ix(T)IP} S E{Ix(s)IP).
5. The Lattices ('Si, 5), ('S+, 5), (Si, 5), (S+1 5) (See the corresponding potential theory lattices in Section 1.IX.5.)
The Lattice CS', :9) We assume, as described in Section 1, a specified filtered probability space
with an arbitrary linearly ordered parameter set. Denote by ('Si, 5) the
526
2 V Lattices and Related Classes of Stochastic Processes
lattice, in the essential order, of those stochastic process equivalence classes under standard modification which contain supermartingales having positive supermartingale essential order majorants. Recall the essential order nota-
tion 5, z, =, V, A. Let r be a subset of 'S'. If the set ro of supermartingales in the r equivalence classes has an essential order minorant in 'Si, then (from Section 111.5) the equivalence class containing the versions of ess inf ro is Ar. It follows that ('Si, 5) is a conditionally complete lattice, but observe that if r is a subset of 'Si with an essential order majorant in 'Si, then Vr is not necessarily the equivalence class containing the essential suprema of the class of supermartingales in the equivalence classes of r but is the essential order infimum of the class of essential order 'S' majorants of IF.
The careful language used above is correct but inconvenient, and we shall frequently follow the usual incorrect but convenient abuse of language in which, for example, ro above is identified with r and Ar is described as the essential infimtun of r even though r is not a set of processes but a set of equivalence classes and ess inf ro is not an equivalence class but a process.
If r c 'Si and if r has an essential order minorant in 'Si, then (Section
111.5) Ar = Art for some countable subset rt of r. If r c 'St and if r has an essential order majorant in 'S', then Vr = Vrt for some countable subset rt of r. In fact, if r is directed upward in the essential order, then v r = ess sup r, and the assertion was proved in Section III.5 for essential suprema. If r is not directed upward, apply this result in the directed case to the set of 'S' suprema of finite subsets of F. In the following, we sometimes prime V and A when referring to 'S*. The Lattice ('S', 5)
The sublattice ('S, 5) of ('S5, S) consists of the 'St equivalence classes containing positive supermartingales.
The Continuous Parameter Context: (Si, 5), (S', 5) Observe that if l = Wan equivalence class in 'St contains a right continuous
supermartingale if the class contains an almost surely right continuous supermartingale and that two almost surely right continuous supermartingales in the same equivalence class are indistinguishable, that is, are equal
quasi everywhere on U8' x 0, or in our other terminology are equal up to an evanescent subset of this product space. If define
x(t) = liminfx(t) Then (Section IV.1) the process supermartingale, and
(r rational).
(5.1)
is an almost surely right continuous
527
5. The Lattices ('S*, 5), ('S', S), (St, 5), (S', 5)
P{x(1) = x(1)} = 1
except possibly for a countable set of values of t. Moreover the limit inferior
in (5.1) is an almost sure limit for each t. Let St be the set of equivalence classes of almost surely right continuous supermartingales, under the relation
of indistinguishability. Then St can be imbedded in 'St (in the present continuous parameter context) in an obvious way. If r is a subset of St and if we denote by 'r the subset of 'S± consisting of the equivalence classes of the latter set which contain those of r, we have noted above that if AT exists there is a sequence {x n e 7L+ } in r such that the process is almost surely right upper determines AT. The process in the essential order, and there may be strict semicontinuous so 5 is in the equivalence class of the maximal inequality. The process essential order St minorant of IF. Thus, if we denote by (St, S) the set St in the essential order, this set becomes a conditionally complete lattice, for which we shall use the order symbols :5. 2--, V, A, but the St order infima and suprema are not inherited from the natural imbedding in ('St, 5). The
argument just given together with the analysis of ('St, 5) shows that if
r c St then the (St, 5) infimum [supremum] of r, if it exists, is the infimum [supremum] of a countable subset of r. If r is countable and if ro is a set of supermartingales consisting of one member from each equivalence class in r, then the equivalence class nr has as one member the process the pointwise infimum of 170, and (Theorem IV.4) if r is directed for upward and is bounded above in (S', --5), the equivalence class VF has as one member the pointwise supremum of rt,.
The Lattice (S+, S)
In the continuous parameter context the sublattice (S+, 5) of (St, S) consists of the St equivalence classes containing positive right continuous supermartingales. The Natural Decomposition in the Continuous Parameter Context The natural decomposition theorem (Section 111. 19) is valid in the continuous
parameter context, in which all the supermartingales in the theorem are almost surely right continuous. In fact, if are almost surely right continuous positive supermartingales and if up to an evanescent set, then according to the natural decomposition theorem of Section I11.19 there are positive supermartingales x, such that
5 x&),
x, 0 + x'2()
528
2.V. Lattices and Related Classes of Stochastic Processes
up to a standard modification, and it follows that X;(')
x,('),
x(') = +'(') +
q.e.
6. The Vector Lattices ('S, -_n}, Then T, = + oo because almost every almost surely right continuous supermartingale sample function is bounded on compact intervals. Observe )e S+ with r,.x(t) < n for t < T. and that with r(t) < n that
535
13. Quasimartingales (Continuous Parameter Context)
for t < T. and y(t) = 0 for t Z T. Thus so
x(), and
is bounded;
0 quasi everywhere, that is,
x(T A t) = x(t) =
(12.2)
almost everywhere on the set {T. > t}. Since the first and third terms in (12.2) are trivially equal almost everywhere on the set {T 5 t) [because this set is in .F(t) and the function x(T A t) is .fi(t) measurable], the process is a local martingale. Conversely, suppose (12.1) is a martingale. Hence is a local martingale, so that there is an increasing e S, and that that
sequence T. of finite optional times with almost sure limit + oo such that each process (12.1) is a martingale. If is in S+ and is a bounded specific then z(T A -)S x(T A ); so z(T A ) is a martinorder minorant of gale with a bound independent of n. The martingale equality E{z(0)} = E{z(T A t)} yields E{z(0)} = E{z(t)} in the limit, and therefore 0 quasi everywhere; so is singular, it follows that Since as was to be proved. Adaptation to the Parameter Set Z+
Under the obvious definition of a local martingale with parameter set I+ a trivial adaptation of the preceding proof shows that the counterpart of Theorem 12 for the parameter set Z+ is true.
13. Quasimartingales (Continuous Parameter Context) In the continuous parameter context (see Section 1) quasimartingales have been defined variously. We shall call an almost surely right continuous a quasimartingale if it is L' bounded and if adapted process -,, +oo implies that there is a constant c for which 0 = t0 < tt
i, manipulation of conditional expectations yields 1-I
y(ti) - E{ y(t;)I-,`(t,)} z Y I E{x(tk) - x(tk+t)I ,F(ti)} k=i
(13.5)
z E{x(ti) - x(tt)I,F(ti)} According to (13.5), the process { y(t,) Thus the positive processes
{Y(t),.f(t.)},
a.s.
.F(t.) } is a supermartingale.
{Y(t.) -
are supermartingales. We have already noted that E{ y(0)} 5 c. The reader by a supermartingale can readily verify that adding points to t replaces on the original sequence t.. Let {y"(j2-"),.F(j2-"), which majorizes when t, = { j 2-", j e Z+ } . For tin Q Z+, the version of j e Z+ } be, for n e and n so large that y"(t) is defined, y"(t) increases almost surely as n increases. to obtain positive supermartingales Define
{z(t),."(t), teQ),
{z(t) - x(t),.F(t), teQ}.
Apply Section IV.1 again to find that the positive right limit processes z( are almost surely right continuous supermartingales relative to on the parameter set R+. The desired representation of is x()
= z() - [z() - x()}.
Observation. Theorem 13 asserts that the quasimartingale x() has a representation x() = x,() - x2(), where x.() is a positive almost surely right continuous supermartingale. There is some interest in minimizing x, () and x2(). An indirect but interesting way is to observe that the class of pairs
(x,(), x2()) with the stated properties has the property that if (x'1(), x2()) and (x,,xz) are in the class, then (x1() A x;(), x2() A x2()) is in the class. Thus the set of first components and the set of second components are both directed downward, and it is not difficult to show that the properly smoothed version of the pair of essential infima of these two sets is the desired minimal
pair. A more direct method suggested by the classical discussion of the positive and negative variations of a function of bounded variation is to
538
2.V. Lattices and Related Classes of Stochastic Processes
replace the discussion of by the two series
as defined by (13.2) by the processes defined
Y [E{x(Ik) - x(tk+l)I',F(I,)} v 0],
k2j
- kzj Y [E{x(tk) - x(tk+l)I'f(t)}
A 0].
The Parameter Set 7L+
A process {x(n), ,fi(n), n e Z+ } is called a quasimartingale if it is L' bounded
and if
YE{IE{x(k) - x(k + 1)1.F(k)}I} < +w. 0
An adapted process is a quasimartingale if and only if it is the difference between two positive supermartingales. The proof follows that in the continuous parameter case, with obvious simplifications.
Chapter VI
Markov Processes
1. The Markov Property Let
be a stochastic process with state space (X, 9t") on a filtered
probability space (f2, .F, P; .fi(t), t E 1). The process is called a Markov process if when s < 1 and A E ", then
P{x(t)EAI.,F(s)} = P{x(t)EAIx(s)} a.s.
(1.1)
Define W(s) = .F{x(r), r >_ s}. It will now be shown that (1.1) implies
E{zl,F(s)} = E{zJx(s)}
(1.1')
a.s.
whenever z is a function from 0 into I8 which is #(s) measurable and is either integrable or positive. If s < t and z = with A in (, equation (1.1') reduces to (1.1). The validity of (1.1) or the equivalent (1.1') will be referred to as the Markov property. To prove that (1.1') is true under (1.1) it is sufficient according to the usual approximation procedure to prove (1.1') for z the indicator function of a set in V(s). Since this a algebra is generated by the algebra of finite disjoint unions of sets of the form {x(tt) e Aj, j < n},
with to = s
0);
(1.20)
x
here s = 0 yields the version of (1.18) in the present context, since u(0, ) is the initial distribution.
2. Choice of Filtration Let {x(-), .F(-)) be a Markov process with an arbitrary linearly ordered parameter set
mo(t) = .flx(s), s s t}, mo(t) = a algebra generated by .mo(t) and the null sets, .moo =
F'(t) = a algebra generated by .4z(t) and the null sets.
The larger the a algebras of the filtration the more significant is the assertion that has the Markov property. The minimal choice of filtration to which is adapted is operating on both sides of (1.1) with E{-I9o(s)} shows that is Markovian. In the other direction it is trivial that is Markovian. Thus
.lo(t) c F(t) c F'(t),
.
(t) c F'(t),
(2.1)
and the filtrations AO(-), some of which may be identical, all make Markovian. The following lemma will be used in Section 7.
Lemma. If .f(t) c .tea for some parameter value t, then F'(t) c .ryro(t).
To prove the lemma, consider the class r of bounded random variables y for which
E{ylflt)} = E{yl. o}
a.s.
(2.2)
If y, is bounded and .mo(t) measurable, and if y2 is bounded and is measurable with respect to the a algebra .9` generated by the null sets and .F{x(s), s >- t}, then YIY2 e I- because
E{y,Y2I. (t)} =YiE(Y2IF(t)) =YiE{YsIx(t)} =Y1E{Y2I9o(t)} (2.3) = E{YIY2I.'o(t)} a.s.
3. Integral Parameter Markov Processes with Stationary Transition Probabilities
545
Since r is a linear class which is closed under bounded almost everywhere convergence, r includes the bounded random variables measurable with respect to .moo (t) Y.F' ; that is, r includes the bounded .F, measurable random
variables. In particular, if y is bounded and ,F'(t) measurable, the left side of (2.2) is equal almost surely to y, and (2.2) therefore implies that y is .mo(t) measurable; that is, .F'(t) a .mo(t), as was to be proved.
3. Integral Parameter Markov Processes with Stationary Transition Probabilities Let (X, 3l) be a measurable space, let p be a probability measure on X, and let A) F- p(s, A) be a stochastic kernel with state space (X, X) (Appendix VI.1). Let {x(n), .fi(n), n e Z+ } be a Markov process with initial distribution p and stationary transition probabilities, with transition function p (Section 1) so that
1)eAIF(n)} =p(x(n),A) a.s.,
PP,{x(A)EA} = µ(A),
Pa{x(j)eAjj5 n} = j'(do)dt) ,
(3.1)
jP(.i ,(3.2)
and i n particular,
P{{x(j)EAj,j 5 n} = lAp() fA,
(3.2')
f.n Abusing language somewhat we shall describe this Markov process from
now on as a Markov process with stationary transition function p. The evaluation (3.2) determines the finite-dimensional distributions of the process and thereby determines P. on .Or{x(n), n E Z+ }, because the latter a algebra
is generated by the algebra Un o.F{x(m), m 5 n}. Conversely, as noted in Section 1, to a specified measurable state space, initial distribution, and transition function corresponds a Markov process {.i(n), .fi(n), n e Z+ } with the specified initial distribution and transition function, under a slight restriction on the state space. Here the random variables of the process are defined on the space ' of functions from Z+ into X, i(n) is the function value at n, and (3.2) is now a definition. Since the parameter space is Z +, the Tulcea generalization of the Kolmogorov extension theorem shows that no restriction need be imposed on the measurable states ace. We shall sketch the construction of the process. Each point w of is a sequence t , ...) of points of X, i(n, d)) = c,,, and .4 (n) = JJIF{z(0), ... , z(n) }, .4g = Y i(n) = {x(n), n e Z } . The measure P is defined on I as the extension to .I of the measure on finite-dimensional product sets determined by setting P{z(j) E Aj, j 5 n} equal to the right side of (3.2). (The Tulcea
g
546
2.V1. Markov Processes
proof that the extension exists is omitted.) This construction has the advantage that the space (!ft, A) is defined without reference top orp, an advantage illustrated by the fact that if Jr contains the singletons, the function g F-. {R} for A in is 3C measurable and for arbitrary p PN{R} = f
(3.3)
X
because these assertions are true when A is a finite-dimensional product set. If is an integral parameter Markov process on a probability space (S2, JF, with the above state space and transition function, the map 0: w [x 0, w), x(l, w), ... ] is measurable from (S2, .fl into (L'2, and
P,, = 0' (P,) in the sense that ¢-' (,fi(n)) = .F{x(j), j < n}} c 5(n), -`(
.F{x(j),je7L+} c 9, and
P{R} for Ac.?. In view of
these facts it is usually correct as well as convenient to prove theorems for discrete parameter Markov processes by proving them for the special ones as just defined on (n, A). Arbitrary Initial Measures
Observe that the preceding discussion remains valid if p(X) # 1, which implies that P{S2}
1. To avoid pathology, we shall always assume that y is the sum of a sequence of finite measures, but we allow p to be infinite valued. All that is needed is the acceptance of the idea that in probability theory the measure on the space on which random variables are defined need not be a probability measure; that is, P{Q) need not be 1! Whenever P may not be a probability measure, this fact will always be mentioned, however.
Substochastic Transition Functions If p is substochastic, the state space (X. fl can be enlarged (Appendix VI.I )
to a state space (X,,X-) by adjoining an absorbing state a, a "trap", to obtain a stochastic transition function extending p to (Xe', do). The adjoined point a is absorbing in the sense that if a sample path reaches the point, the path stays there from then on. The first time Tthat a is reached is an optional
time, the "lifetime" of the process, equal to + oo for a path that never reaches a. If convenient, the state a can now be dropped, so that to any initial distribution corresponds a process
for which
P{x(n) is defined} = P{x(n)eX} < I. In discussing processes with substochastic transition functions it is always to be understood in writing x(n) when the trap point has been dropped that
547
4. Application of Martingale Theory to Discrete Parameter Markov Processes
there is an unwritten convention limiting the context to the set on which x(n) is defined, the set on which n is strictly less than the process lifetime.
Theorem (Strong Markov Property in the Discrete Parameter Context). Let {x(n), .f(n), n e7L+ } be a Markov process with stationary transition function p, and let S be afinite-valued optional time. Then {x(S + n), .'F(S + n), n e IL+ }
is a Markov process with transition function p and initial distribution the distribution of x(S).
Since S + n is optional as well as S, all that must be proved is that if A e. ", then
P{x(S+ l)eAJ.F(S)} =p(x(S),A) as.
(3.4)
Since {S = n} e.F(S), the left side of (3.4) is equal almost everywhere on {S = n} to P{x(n + 1) e A jx(n) }, which itself is almost surely p(x(n), A), as was to be proved.
If S is identically constant in (3.4), this equation is equivalent to the Markov property of as stated in terms of a transition function. The fact that this relation holds when S is optional is called the strong Markov property. Extension of Theorem 3
A slightly more general version of (3.4) is obtained by observing that the proof of (3.4) also shows more generally that if S is not necessarily finite valued, then (3.4) is true almost everywhere on the set {S < + oo }. Still more has lifetime S', if A C- X, and if S < S' generally, if p is substochastic, if almost surely, then the proof of (3.4) shows that this relation is true almost
everywhere on the set {S < S'}. A still more general version of Theorem 3 in which S + I is replaced by a random variable measurable relative to .F(S) is useful in the continuous parameter context and in that context will be proved in Section 9. The reader is invited to formulate and prove the counterpart of Theorem 9 in the present context.
4. Application of Martingale Theory to Discrete Parameter Markov Processes Let (X,.") be a measurable state space, and let {x(n),ne7L+} be a Markov process with this state space, from a point of X. It is supposed that the process has stationary transition probabilities, with a transition function p. Call a function u from X into A superharmonic [harmonic] [ [subharmonic] ] relative to p if u is measurable, if I uI) < + oo, and if u) < u .
548
2.V1. Markov Processes
[p(-, u) = u] [ [p(-, u) Z u] ]. A trivial computation shows that the composed
process {u[x(n)],n >_ k) is a supermartingale [martingale] [[submartingale]] relative to with ,fi(n) _ 3F , . . . , x } if u is superharmonic [harmonic] [[subharmonic]] and if E{ju[x(n)]j) is finite for nzk. Observe that this expectation is finite when n = 0 if a is finite at the initial point of the process and that this expectation is finite by hypothesis when n = 1, and therefore if u is superharmonic and positive, this expectation will be finite when n > 1. According to the martingale theory convergence Theorem 111.13, we conclude that u[x(n)] exists and is finite almost surely if u is positive and superharmonic. EXAMPLE (Boundary limit theorems in potential theory). Let D be an open Greenian subset of RN, and for each point of D let be an open relatively compact subset of D containing . Define A) = MDISI(b A).
Here µD14, is harmonic measure; so this measure is supported by It is way that A) is Borel measurdepends on in supposed that able when A is a Bore] subset of D. The classical superharmonic, harmonic, subharmonic functions are, respectively, superharmonic, harmonic, sub-
harmonic relative to this transition function. The same procedure in the context of parabolic potential theory makes the superparabolic, parabolic, subparabolic functions, respectively, superharmonic, harmonic, subharmonic relative to a transition function defined in terms of parabolic measure. Observe that in this case the sample sequences move in the direction of decreasing ordinate values. We return to the classical context, and to simplify the situation we make the hypothesis that D is bounded. We shall choose in two ways. First let D. be an increasing sequence of open relatively compact subsets of D Dk for k = min {n: E with union D and with D c D.,,. Define Then p as defined above is a stochastic transition function, and the corresfrom a point x(0) of D determines sample ponding Markov process
sequences which run through successive boundaries of D. toward 8D. Furthermore each coordinate function u of RN is harmonic and bounded on u[x(n)] D; so is a bounded martingale, and it follows that x(n) exists almost surely; that is, almost exists almost surely. Hence sequence path converges to a boundary point of D. Next we apply every
this technique to a different choice of
This time let e be a strictly
C A (] - 8DI/2)), so that positive number, and choose To µa4,0, -), that is, is the uniform probability distribution on A) is Borel measurable when A prove that with this choice of function f) is continuous when is a Borel subset of D, it is sufficient to prove that f) follows from .f is continuous and bounded on D, and this continuity of
4. Application of Martingale Theory to Discrete Parameter Markov Processes
549
the fact that is the unweighted average off on and that the radius of varies continuously with . With this choice of the fact that x(oo) exists almost surely is proved as before, and the limit must be on OD because I x(n + 1) - x(n)I = e n
Ix(n) 2 aDI
a.s.
The distribution of x(oo) is a measure pD[x(0), ] on OD. It will be shown this measure is the harmonic measure later that for both choices of PD[x(0), ], and in fact we shall show much more than this, but now we verify this assertion only for aD regular. Let 0 be a continuous function from OD into R. Then H,, is a bounded harmonic function on D with boundary limit function 0; so is a bounded martingale closed (Theorem 111. 14) by its almost sure limit H,[x(oo)]. Hence MD[x(o), O] = E{o[x(oo)] { = Hm[x(0)] = uD[x(0). m],
and since 0 is arbitrary, µD = µD, as asserted. if u is a positive superharmonic function on D, For both choices of the composed process is a positive supermartingale except that the parameter value 0 should be omitted if u[x(0)] = +eo. Thus according to the martingale theory convergence Theorem 111.1 3, a positive superharmonic function has a finite limit on almost every path to the boundary; this is a Fatou-type boundary limit theorem. Observe that for the second choice of when a is small, the paths are sequences whose successive points are close together, and this fact suggests that the above results are valid for suitable continuous paths instead of sequences. This is true: what we have proved is valid when properly interpreted for Brownian motion continuous
paths from a point of D to the boundary. In fact for both of the above choices of D(5) an
sequence can be identified with a sequence of points
tending to a boundary point of D along a Brownian motion path to the point. The probabilistic evaluation of harmonic measure as a Brownian motion hitting distribution, the boundary limit theorem for positive superharmonic functions, and the supermartingale significance of the composition of a positive supermartingale with Brownian motion are parts of Theorem IX.7. The identification of harmonic measure with a Brownian motion hitting distribution is made in Theorem IX.13(a). The latter identification carries with it the fact that the hitting distribution on is µD(4)0, ) which implies that for each choice of D(4) above a sequence with the assigned distribution can be obtained by choosing x(1) as the first hitting place on (D(x(0)) by a Brownian motion path from x(0), x(2) as the first hitting place thereafter on (D(x(1)) and so on. In the parabolic context the discussion is changed only in that "harmonic" is
replaced by "parabolic," and "Brownian motion" by "space-time
550
2.VI. Markov Processes
Brownian motion." If h [h] is a strictly positive harmonic [parabolic] func-
tion, harmonic [parabolic] measure can be replaced by h-harmonic [hparabolic] measure throughout the discussion to derive corresponding results in relative contexts. The continuous path processes for which the results are valid are the conditional [space-time] Brownian motion processes
discussed in Chapter X, where the counterparts of the classical context results are proved.
5. Continuous Parameter Markov Processes with Stationary Transition Probabilities Let (X, I) be a measurable space, let P be a probability measure on X, and let (t, , A) "p(1, g, A) be a stochastic continuous parameter transition func-
tion (Appendix VI.3). Let {x(t),F(1),te98+} be a Markov process with initial distribution u and stationary transition probabilities, specifically with transition function p (Section 1), so that p satisfies the Chapman-Kolmogorov equation : for A e X, s > 0, t > 0, (5.1)
p(s + t, , A) = J P(1, 7, A)P(s, , d01), X
andfors-0,1>0, Pµ{x(0)EA}=µ(A),
a.s.
(5.2)
We shall describe this Markov process from now on as a Markov process with stationary transition function p. If 0 = to < .. < 1, and A;E3-,
AA EA,,j c} for I an interval and u a Borel measurable
(b)
function from X into A. E, (sup,. t u[x(r)] } if [notation of (a)] u -a 0 or if this expectation with u replaced by Jul is finite for all .
(c)
P{{limsup,tru[x(t)] > c} for u as in (a) and for T the hitting time of an analytic subset of X by
(d) E E {lim sup,tT u[x(t)] } if [notation of (c)] u 2 O OT if this expectation
with u replaced by Jul is finite for all . If for every point of the state space there is a continuous process with initial point and the given transition function p, then all the work in this section can also be carried through with Sl the space of continuous functions from R+ into X.
The Hitting of an F. Set We now change some of the notation in the right continuous context we are
treating. Let A be an analytic subset of X, let I be a compact parameter interval, let be a right continuous Markov process from g with the given transition function p, and recall that the space on which may depend on c. Consider the function u--. P {x,(t) e A for some t in 11.
is defined
(6.3)
According to the discussion we have given, the function (6.3) is universally measurable. We shall now show, under further hypotheses imposed on A and the transition function that the function (6.3) is even Borel measurable and that the proof of this Borel measurability does not need the theory of
analytic sets. This fact means that some later results in this book can be obtained without the (unpalatable to some readers) invocation of analytic sets. Abstention from analytic set theory makes stochastic process theory
554
2 VI. Markov Processes
much less elegant and sadly incomplete, however. Observe that if is as above and if !' is a countable subset of parameter values, the function
BHP{x,(t)eA for some tin I'}
(6.4)
is Borel measurable. In particular, if A is open and if !' is a countable dense
subset of I including both endpoints, the functions in (6.3) and (6.4) are identical. Now suppose that for each the process can be supposed continuous. In this context we shall now show that the function (6.3) is Borel measurable when A is an F, set. It is sufficient to prove this Borel measurability when A is closed. Let B. be for n z I the open set of points at distance < 1/n from A. Then A is hit by in the parameter interval I if and only if B. is hit by in the parameter interval I for every n. Hence the function (6.3) is the limit as n -+ o0 of (6.3) with A replaced by B.; so the function (6.3) is Borel measurable. Observe that if above is supposed merely almost surely right continuous rather than right continuous, or almost surely continuous rather than
continuous, then the conclusions remain valid and the reasoning is unchanged aside from the acknowledgement of exceptional null sets.
7. Continuous Parameter Markov Processes: Lifetimes and Trap Points Recall that a function defined on a subset A of a measurable space is said to be measurable if A is measurable and if the function is measurable with respect to the class of measurable subsets of A. Let (t, , A) -p(t, , A) be a substochastic stationary transition function with state space (X, .Y). The Chapman-Kolmogorov equation implies that the function i i-+p(t, , X) is monotone decreasing on ]0, + oc [ for all c. We assume throughout this discussion that lim,.o p(t, , X) = 1, which implies that the function X) is right continuous on ]0, + x [ for all S. Generalizing Section 5 slightly, define a Markov process with state space (X,. "), parameter set initial probability distribution µ, and stationary substochastic transition function p as an adapted process {x(t),.F (t), t e 08+ } satisfying (for 0< s < t and A e 1) R',
P{x(0)eA} =µ(A),
P{x(t)eAI F(s)} = p(t - s, x(s), A) a.s.
(7.1)
The process random variables need not be defined on the whole space; so "a.s." means "almost everywhere where x(s) is defined." In any expression involving a random variable of the process the added condition that the random variable is defined is to be understood and will sometimes be written explicitly. Thus the set {x(t) e X } is the domain of definition of x(t). With
7. Continuous Parameter Markov Processes. Lifetimes and Trap Points
555
this convention (5.3) remains valid. According to (7.1) and the ChapmanKolmogorov equation, the random variable x(O) is defined almost surely, and the function t c - P{x(t) e X), equal to Jxp(t, c, X)p(dc) when t > 0, is monotone decreasing and right continuous on R+. If 0 < s < t, the random variable x(s) is defined almost everywhere where x(t) is defined because
P{x(s)eX,x(t)eX}= `p(dc) J'(t X
=
x
f
(7.2)
P{x(t)eX}.
X
If there is a positive random variable S (S + oo) such that x(t) is defined if and only if t < S, the random variable S is called the process lifetime. Integral parameter process lifetimes were discussed in Section 3. The existence of a process lifetime is not much of a restriction. In fact without the hypothesis of the existence of a process lifetime define S by
S((v) = sup {r: r is rational, x(r)eX} so that S is almost surely strictly positive and
P{x(t)EX} = P{S z t} = P{S > t}. Define y(t) on the set IS > t} [which differs from the domain of x(t) by a null set] as x(t) and leave y(t) undefined otherwise. Then the process is Markovian with transition function p, initial distribution p, {
lifetime S and is a standard modification of If is a Markov process with substochastic transition function p and lifetime T, enlarge the state space by adjoining a single point a and
extend p to be stochastic on the enlarged space, making a an absorbing point (Appendix VI.3). If now x(t) is defined as a fort > T, the extended process is Markovian with transition function the extension of p. Thus results for Markov processes with stochastic transition functions can be applied to those with substochastic transition functions. Going in the opposite direction, if p is a substochastic transition function with Polish state space X and if p is a probability distribution on the state
space, there is a Markov process with transition function p and initial distribution p according to the following argument. Adjoin a point a to X as an isolated point of X u a, and extend p by making a an absorbing point. There is then a Markov process with the extended state space, the extended transition function, and the given initial distribution (supported by X). If i(t) is replaced by its restriction x(t) to the set {i(t) e X }, the process .0(-)) will have the properties described at the beginning of this section and, if desired, can be modified to have a lifetime, as discussed above.
556
2.V1. Markov Processes
A a Algebra Equality for Almost Surely Continuous Markov Processes We shall need the following intuitively obvious fact. Let be an almost surely continuous Markov process from a point . It is supposed that the state space is a Polish locally compact but not compact space. Let S be the process lifetime. A trap point is adjoined as described above. Let { be the adjoined point of the Alexandrov one-point compactification of the state
space. It is supposed that x(S-) = C almost surely. Fort in R+ let flt) be the a algebra generated by the null sets of the x(-) probability space and .F {x(s), s 5 t}, and define -9 = YE R. ,fi(t). Let B. bean increasing sequence
of open relatively compact subsets of the state space, containing , with union the state space, and let S. be the hitting time of 8B1 by x(-). Then S. = S almost surely. Define 9 = Y.Ez+ _*(S.). Then we show that F =,#. The inclusion I c F is known from Section I.12. To prove the reverse inclusion, we prove that x(t) is I measurable for each t. In doing this it is no restriction to identify the process trap point with . Then we c 9(S.) c 9 measurable and need only observe that x(t A S.) is F(t A that x(t) = liM.-. x(t n almost surely.
8. Right Continuity of Markov Process Filtrations; A Zero-One (0-1) Law In some contexts the desirable right continuity of Markov process filtrations is intrinsic, as illustrated by the following theorem. Theorem. Let {x(-), bean almost surely right continuous Markov process with Polish state space and with transition function p. Suppose that whenever
t > 0 and f is a bounded continuous function on the state space, the function p(t, , f) is continuous, or at least the process p(t, x(-), f) is almost surely right continuous. Then (a) The process .F '(-))is Markovian with transition function p. (b) If for all t z 0 the a algebra .fi(t) is generated by the null sets and
.F{x(s),s < t}, then Observation. If the state space is locally compact and second countable, it will be clear from the proof of the theorem that f in the hypotheses need only be continuous with compact support. The theorem and the following proof are valid even when p is substochastic. For f as in the theorem the Markov property in the form
Elf [x(s' + t)] I.fls') } = p(t, x(s'), f) a.s. (t > 0) implies when s' I s > 0 sequentially that
(8.1)
557
9. Strong Markov Property
E{ f [x(s + t)]j.F+(s)} = p(t, x(s),j)
a.s.
(8.2)
in view of the hypotheses of the theorem and the dominated convergence theorem for conditional expectations. Thus (a) is true. Under the added hypothesis of (b), Lemma 2 implies that ,F+(t) a .fi(t); the reverse inclusion is trivial ; so (b) is true.
A 0- 1 Law. Suppose under the hypotheses of (b) in the theorem that the
Markov process is one from an initial point. In this case the a algebra F(0) = .F(0+) consists of the null sets and their complements. In intuitive phraseology this evaluation of F(0+) states that every sample function asymptotic property as the parameter value tends to 0 is either almost surely of true or almost surely false. For example, if T is the hitting time by
an analytic subset of the state space, then P{T = 0} must be either 0 or I. See Section VII.6 for a discussion of this 0-1 law and its application in the context of Brownian motion.
9. Strong Markov Property The following continuous parameter version of the strong Markov property indicates possible strengthening of the discrete parameter version in Section 3, but we shall not need such a strengthening. Recall (from Sections II.1 and then .F+(T) _ ,F(T+) and 11.2) that if T is optional for a filtration
that if S is an .F(T) measurable random variable and if S z T, then S is is a process with parameter set R and state optional. Recall also that if space the extended reals, and if 0 5 S S + oo, then the notation x(S) refers to the function defined on the set {S < + oo) as x(S) and defined as 0 elsewhere. We also adopt the convention that if p is a continuous parameter transition function, thenp(0, , A) = 1A for A a measurable state space subset. Theorem. Let p be a continuous parameter substochastic transition function with a Polish state space, and suppose that p has the following properties :
If is a point of the state space, there is a right continuous Markov process with transition function p and initial point . (b) If f is a bounded continuous function on the state space (or merely continuous with compact support if the state space is locally compact and second countable) and if t > 0, the function p(t, -J) is continuous, or at least the process { p(t, xr(s), f ), s e R + } is almost surely right continuous for each process in (a). (a)
Then if (x(-), .F(-)) is a right continuous Markov process with transition function p, if S and T are optional times with S z T, if S is .F(T) measurable, and if A is a Borel subset of the state space,
P{x(S)eAI -,F(T+)} =p(S- T,x(T),A) a.e. on {S< +oo}. (9.1)
558
2.V1. Markov Processes
In particular, when S = T + t with t a positive constant,
P{x(T+ t)eA13Z(T+)} = p(t,x(T),A) a.e. on IT < +x}. (9.2) Observation (1). Condition (a) implies that for f as in (b) the function f) is right continuous on 68' because if is chosen as in (a), the function t i- p(t, S, f) = E{ f [x1(t)] } is right continuous. Observation (2). If.fr(t) _ 9(T + t), the theorem implies that the process is a Markov process on { T < + oo } with transition func{x(T + ), .F '(-)) T tion p and initial distribution the distribution of x(T). To see this, first note that the process in question is adapted because (Section 11.3) the function
x(T + t) is 9(T + t) measurable, and second note that if to Z 0 and if the pair (T, T + t) in (9.2) is replaced by (T + to, T + to + t), then (9.2) becomes the Markov property for the translated process; that is,
P{x(T + to + t)eAI.IFT(to)} = p(t,x(T + to), A)
a.e. on {T < +o }. (9.3)
Observation (3). If S = t v Twith t a strictly positive constant, (9.1) yields
P{x(t)eAI.9 (T+)} = p(t - T,x(T),A) a.e. on {T< t}.
(9.4)
Observation (4). Hypothesis (a) is not weakened if is supposed only almost surely right continuous because then there is also a right continuous process with the other stated properties. In the same vein the following is is an almost surely an alternative conclusion to the theorem: if right continuous Markov process with transition function p, if .F(0) contains the null sets, if S and Tare optional times with S z T, if S is .F(T) measurable, and if A is a Borel subset of the state space, then (9.1) is true. In fact under these hypotheses the process is indistinguishable from a
right continuous process also adapted to and the theorem can be applied to the latter process. In proving the theorem it is no restriction (Section 7) to assume that p is stochastic. We can also assume without loss of generality that _'F(0) contains the null sets because (Section 1.8) fl-) can be extended if necessary to make this so. Furthermore it can be assumed that Jr(-) is right continuous is Markovian with transibecause (by Theorem 8) the process tion function p. It will be convenient to prove (9.1) in the equivalent form
9'
E{ f[x(S)]J.-F(T)} = p(S- T, x(T),f) a.e. on IS < +oo} when .f is as described in (b).
(9.5)
9. Strong Markov Property
The Special Case (9.2)
If S = T + t, equation (9.2) will be proved by proving first that
E{.f[x([T]" + t)]I' ([T].)} =p(t,x([T]n),f) a.e. on {[T]" < +oo (9.6)
That is we prove that for A in . ([T]")
f[x([T]n+t)]dP=
J Iv [Tln 0).
The Hunt theory has been presented in varying degrees of generality. For example, the theory includes classical potential theory on Greenian subsets of R' with regular boundaries if the following hypotheses are imposed. Hl. The state space is a locally compact Hausdorff space satisfying the
second separability condition, coupled with its Borel sets (but
561
10. Probabilistic Potential Theory; Excessive Functions
sometimes it is necessary for adequate generality to make 1 the class of universally measurable sets over the Borel sets).
-J) takes the Banach space of continH2. The transformation uous functions f on X with compact support into itself, and this transformation has the identity as strong limit when t - 0.
We shall discuss various aspects of Hunt theory below, as needed. A 0-excessive function is called excessive. An a-excessive function is also fl-excessive for fi > a. If pa(t, , ) = e `p(t, , ), then pQ is a substochastic continuous parameter transition function, and a function is fl-excessive relative to pa if and only if the function is (a + fl)-excessive relative top. If p is made stochastic by adjunction of a trap point to the state space, as described in Section 7 an a-excessive function before the trap is adjoined remains so if defined as 0 at the trap point. A simple argument shows that the limit of an increasing sequence of a-excessive functions is a-excessive.
If it is supposed that p(-, , A) is R(R') x X measurable when A is in X, then the role of a Green function is taken by the kernel Gz with state space (X, X) and positive parameter a, defined by A) = f"O e
A)!1(dt).
(10.2)
If f is a positive measurable function on X its a -potential is the measurable function on X defined by
jetP(tf)li(dO.
(10.3)
0
This a-potential is a-excessive because J
e°'p(s,ff)lt(ds) 5
(10.4)
and there is equality in the limit when t - 0. It will be shown in Section IX.17 that if the state space is a Greenian subset D of RN coupled with its Borel subsets and if p is the transition function of the Brownian motion process in D, then d7) = const
7)1N(d7)
Thus the Hunt potential of a function f becomes the classical potential of the measure f dlN up to a multiplicative constant. A corresponding result will be proved relating parabolic potential theory will space-time Brownian motion. The analysis of a-excessive functions and a-potentials for the transition
562
2.VI. Markov Processes
function p is the same as the analysis of excessive functions and 0-potentials for the transition function p,.
Invariant Excessive Functions If u is not only a-excessive but
e-°'p(l, , u) = u
(10.5)
for all t > 0, then u is called invariant a-excessive. For example, if v is aexcessive, the function lim, a "p(t, , v) is invariant a-excessive. If u is a-excessive and if (10.5) is true for some strictly positive t, then u is invariant
a-excessive. In fact on the one hand (10.5) is true for 21 when true for i because if true for t, then
px(2t, , u) = pa(t, .pa(t, , u)) =pa(t, , u) = u,
and on the other hand by monotoneity of the function u) equation (lb.5) is true for t < .s if true for s. The concept of an a-excessive function links Hunt potential theory with martingale theory because the condition (10.1a) is precisely the condition that if is a Markov process with transition function p. then the process {e-a`u[x(t)], ,fi(t), t e l `} is a supermartingale on the parameter set for which the process random variables have finite expectations. The condition (10.5) for an invariant a-excessive function u is the condition that this supermartingale be a martingale on this parameter set. The following lemma will be useful. Lemma. If a is a-excessive and if
u=n
'
o
1/n
e-'sp(.s, , u A n)!1(ds),
n e Z',
then the sequence u is an increasing sequence of bounded a-excessive functions, with limit u.
At the expense of replacing p by p, we can assume that x = 0. Obviously
un t. Thus the condition that v be excessive, namely that p(t, , v) increase to as t decreases to 0, has a simple probabilistic interpretation in this special case. Suppose that a > 0, and define the substochastic transition function pa
by p (t, , A) = e-"p(t, , A). Suppose that u =_ I so that
E{e-aT }.
Let S be a positive random variable defined on the same probability space as x,(-), independent of with distribution density (relative to /t ) t - ae-" on R. Then the probability that an path hits A at a strictly positive time but before time S is Equivalently, v(D is the probability that a path from of an almost surely right continuous Markov process with transition function pa hits A at a strictly positive time.
566
2.V1 Markov Processes
13. Conditioned Markov Processes Let (X, X) be a measurable space, let p be a stationary substochastic transition function with state space (X, X), and let h be a strictly positive excessive function for p. The hypothesis of strict positivity is unnecessary but avoids some technicalities and is satisfied in the applications to be made. Define
X"={h 0}
(15.1)
is excessive (Section 12). If A is closed and if 4 e X - A, the process to T4 > t }, is Markovian with killed at time T., that is, the restriction of state space X - A and transition function p, given by
p,(r,q,B)=P,x,,(t)eB,Try> t;.
(15.2)
Next suppose that A is not necessarily closed but that h is strictly positive, and define p"(t, S, dq) = p(t, 4, dq)h(q)/h(S). Consider killed at L, except that the parameter value 0 is dropped; that is, consider x 4C defined for each If 0 < t, < .. < t., t > 0 as the restriction to ; L4 > t} of
15. Killed Markov Processes
569
P{x4L,(tf)Edgt,j 5 n) = P{xr(t1)Edryt,j 5 n;LS >
(15.3)
= p(t , dtl,) ... p(t This evaluation has a simple interpretation:
nn-1,
is Markovian with
P{xCL,(t) E drt} = p(t, , dry)h(n)
(15.4)
probabilities can be and stationary transition function p". That is, computed by computing probabilities for a Markov process from , with transition function p", and then multiplying the probabilities obtained by the constant Thus killing a Markov process at the last hitting time of a set leads to a conditioned Markov process in the sense of Section 13. The process killed at L, in the present context is trivially right continuous, verifying a property of conditioned Markov processes derived in Section 13. Observe that p" may be substochastic even if p is stochastic and that the distribution of xeL;(t) may not be a probability measure: P { xcL{(t) e X } = p(t, c, h) ;
when t tends to 0, the right-hand side tends (increasing) to the limit
(15.5)
h().
Chapter VII
Brownian Motion
1. Processes with Independent Increments and State Space 9 (a) Discrete Parameter Case Let y(0), y(1), ... be mutually independent random variables with distribu-
tions qo, q1, ... , respectively, on R", and define x(n) = 2:oy(j), F(n) = .F {y(0), ... , y(n)} = 9 {x(0), ... ,x(n)}. If the transition function p with (A - ), where A is a Borel subset is the process Markovian with initial distribution qo and successive transition functions po, p...... In fact according to the application in Section 1.7, state space R" is defined by
A) =
of R" and A - is the translation of A by
P{x(n + 1)eAJ.f(n)) = P{x(n) + y(n + 1)eAJ.fln)}
=
x(n)) =
A)
a.s.
If 0 5 n1
0 the set of increments
x(t) - x(s): r > s} is independent of .'(s). The two Brownian motion definitions are equivalent in the following
2. Brownian Motion
573
is a Markov process satisfying sense. According to Section 1, if BM 1 *-BM3*, then conditions BM 1-BM4 are also satisfied, and conversely, and if F(t) is defined as F(w(s), if BM 1-BM4 are satisfied by a process S:5 t), then satisfies BMI*-BM3*. and Throughout this book a "Brownian motion" specified as not otherwise delimited will mean a Markov process satisfying BM I *-BM3* with no further unstated hypotheses imposed on the filtration
All sample functions of a Brownian motion can be made continuous without affecting BMI-BM3 or BMI*-BM2* by redefining the discontinis a uous sample functions to make them continuous. Observe that if Brownian motion from a point, then the Ncomponent processes are mutually independent one-dimensional Brownian motions with the same variance parameter as is a Brownian motion with variance parameter a2 and if to > 0, If the process {w(to + t), t e R+} is a Brownian motion with the same variance and with initial distribution the distribution of w(to). The parameter as process {w(t/a2), to R+} is a Brownian motion with variance parameter I; the processes {[w(t) - w(0)]/a, to R+} and {w(t/a2) - w(0), t e R+} are both Brownian motions from the origin with variance parameter 1. Brownian Motion with the Parameter Set R
A Brownian motion with the parameter set R is defined as a process (w(t), ,fi(t), t e R} which satisfies BMI *-BM3* except that the parameter set is R instead of R+ and that it is no longer supposed that the filtered measure space on which the process is defined is a probability space or even that the probability value assigned to the whole space is finite. More precisely it is supposed that there is for each t in R a measure p(t, ) of Borel sets, finite for compact sets, with 0 < p(t, R") 5 + oo, which is the distribution of w(t), and probabilities are then defined as usual for a Markov process, in terms of p and the transition density assigned in BM2*. Then for s e R, t > 0, A E R(R),
P(s, di) I t (t, ,
J
R"
P(s + t, A).
(2.1)
JJA
R") is identically some strictly positive constant fi (5 + oo). If s is replaced in (2.1) by so - t, we find that ,a-"t'"'2 1"(A) z p(so, A), and Hence
(t -+ + oo) it follows that fi = + oo. Thus the context imposes the condition
that the absolute probability distribution p(sqq, ) be an infinite-valued measure for all so. If we defneti(ry, t) for (n, t) a R" by
a(a, , n)P(t - a, dt)
ti(n, t) = J R"
(2.2)
574
2 Vii. Brownian Motion
with a > 0, it follows from (2.1) and the Chapman-Kolmogorov equation that ti does not depend on the choice of a. The function ti(-, t) is a version of the density dp(t, )/dIN of the w(t) distribution and s),1(t, , n)1N(d4) = u(n, s + t)
u(4, s)1N(d4) = + co, JaN
(2.3)
RN
when t > 0. Since the function (n, PI) = 6(t, - tr) is parabolic on the function ti, which is Borel measurable and is finite valued 9N on a dense subset of ftN, has the parabolic average property and so is parabolic on fit'. Under our hypotheses when seR the process {w(s + t), teR+} is s). The distribua Brownian motion with initial distribution of IN density
tion of each increment x(t2) - x(t,) with t, # tZ is an infinite-valued measure. Since the process is Markovian, the reversed time process is also Markovian, and a version of the reverse transition density relative to IN is easily calculated: the density of a transition from at time s to r at time
t<sis
ti(,i, t)G(s - t, 17, ) (2.4)
(Recall from Section I.XVI.8 that a positive parabolic function on f l' is either strictly positive or identically zero.) In terms of ordinary probabilities a version of the distribution of {w(so + t), t c- R') conditioned by w(so) = o is that of a Brownian motion from go, and the distribution of {w(so - t), t e R+ } under the same condition is that of a Markov process from o with transition density determined by (2.4). The special case of stationarity is important: if p(t,-) = IN for all t, that is, if ti = 1, the process forward and backward transition densities are the same. In this case the processes { w(so + t), t e f+ } and { w(so - t), t E R+ } are both Brownian motions with initial distribution 1N.
Existence of Brownian Motions (Parameter Set R+)
The conditions BM I -BM2 are conditions for a Markov process with state space RN and with a specified transition function. Hence (Section VI.5) such a process exists for an arbitrary choice of initial probability distribution µ, for example, as a canonical process on the space of all functions from P+ into RN. A useful additional fact is that since the distribution of the process in question is the integral with respect to p(dd) of the distribution of the process with initial point , it follows that there is such a process when p is an arbitrary measure on R', finite on compact sets, and not identically 0. has a In Section 3 we shall show that any process satisfying
standard modification satisfying BM3*. Thus Brownian motions with parameter set R+ exist.
2. Brownian Motion
575
Existence of Brownian Motions (Parameter Set R) We show that if ti is an arbitrary not identically vanishing positive parabolic function on 0, then z
P{x Z a} < a exp 2az
(3.1)
In fact the left side of (3.1) is
2 d - a} = P{ y < -a} for all a (equivalently, for all positive co. A finite
577
3. Continuity of Brownian Paths
sum of symmetrically distributed mutually independent random variables is itself symmetrically distributed. (b) If y(1), . .. , y(n) are mutually independent symmetrically distributed random variables and if x(k) = E; y(i), then (3.3)
P { max x(k) >- a } < 2P{x(n) >- a}. k5n
Let T= min { j: x(j) >- a}. Then the set {T=j} is independent of x(n) - x(j), and since the latter random variable is symmetrically distributed, n
n
1
i
P{maxx(k)>-a}=>P{T=j} P{T=j,x(n)-x(j)>-0) k!gn
(3.4)
< 2P{x(n) > a}. (c)
If N = 1 and if
satisfies the defining conditions BM I -BM3 of
a Brownian motion,
P (sup [x(r) - x(0)] >- at } < 2P{x(t) - x(0) >- a}
(r rational).
r5I
(3.5)
According to (b) this inequality is true if the supremum on the left side is replaced by the supremum for only finitely many values of r:5 t. The inequality is therefore true as stated except possibly for the countably many values of a which are discontinuities of the distribution function of the supremum on the left. The inequality is true without exception because both sides of (3.5) define left continuous functions of a.
By symmetry, if x(r) - x(0) is replaced in (3.5) by its absolute value, the right side of (3.5) should be doubled.
Theorem. If is a process with the finite-dimensional distributions of a Brownian motion, it has a standard modification with continuous sample junctions.
We can assume in the proof that the variance parameter is 1 and that N = 1. Applying (c) above, if m > 0, n > 0 and if r and s are rational, mn-1
P{
sup
0<s-r< 11n
Ix(s) - x(r) I > 2a J< Y P { j=0
S5m
s0<s-u/p
Ix(s) - x(j/n)I >a l
n52/re
= mnP { sup Ix(s) - x(0)I >- a J< 4mnP{x(2/n) - x(0) >- a} s 52/n
< 8mn'/Za-' exp -a2n 4
(3.6)
578
2.VII. Brownian Motion
oo, the restriction to the rationals of almost every sample function is uniformly continuous on [O,m[. In other words the restriction to the rationals of co) for w not in some null set coincides with a continuous function on R'. Since Since the last term has limit 0 when n
IimE{[x(t) - x(s)]2} = 0, x, if
is a standard modification of and has continuous sample functions is defined, say, to be identically 0 on the exceptional null w set.
Canonical Brownian Motion in I8"
According to our conventions a Brownian motion in R' is any stochastic process satisfying certain distribution and continuity conditions. For some purposes it is convenient to have a canonical Brownian motion, however, for a specified initial distribution and variance parameter. Suppose then that an initial distribution p and variance parameter a2 are specified. Let f2 be the space of all continuous functions tb from 68' into R', define v(t, w) = co(t) and define .A = {w t), t e R'). Then (Section 1.10) a probability measure p can be defined on . making a Brownian motion with initial distribution p and variance parameter a2. The process will be described as the canonical Brownian motion for the specified u and a2. In the language of Section I.10 is the canonical process associated with every Brownian
motion with the specified initial distribution and variance parameter. In accordance with our conventions, P is to be completed. The measure Y is a measure on the space of continuous functions and is commonly described as "Wiener measure" because Wiener was the first to construct this measure rigorously. It is a matter of taste and convenience whether in discussing a canonical Brownian motion from a point of R" we further restrict 6 by the condition dh(0) = 5: the choice will be specified when it matters. We shall not limit ourselves to canonical Brownian motions because noncanonical ones arise in many contexts. The distributions of the interesting functions defined on a Brownian motion with given initial distribution and variance parameter do not depend on the choice of Brownian motion. For example, the probability that a Brownian path from a point hits a specified analytic
set when the parameter varies in a specified interval may depend on the variance parameter but (Section 1.12) not on the choice of Brownian motion.
4. Brownian Motion Filtrations According to Section VI.9, a Brownian motion has the strong Markov property if F(0) contains the null sets of the probability space, and is continuous instead of merely almost this condition is unnecessary if
4. Brownian Motion Filtrations
579
surely continuous. If f, (t) is the a algebra generated by ,F(t) and the null is still a Brownian motion. Since s < s' < t sets, the process the random variable implies that w(t) - w(s') is independent of w(t) - w(s) is independent of .F; (s), and this independence implies that () } is a Brownian motion. Thus there is no loss of the process J w(-), F(-)) the a algebra generality in assuming that for a Brownian motion that .5(0) contains the null sets of the probability space and fl-) = is right continuous. In particular, if (w(-), .F(-)) is a Brownian motion
is,
with .fi(t) = .5{w(s), .s < t} and if .5,(t) is the a algebra generated by 5(t) .IF, and the null sets, then { is a Brownian motion, and (by Theorem VI.8) is right continuous. In view of the properties of Brownian motions the entry and hitting times of analytic subsets of the state space are optional for a Brownian motion { whenever. (0) contains the null sets of the probability space and is right continuous. The space-time Brownian motions have the strong Markov property and the preceding entry and hitting time property under the stated conditions on the Brownian motions involved. .fl-)} be a Brownian motion in RN with ,fi(t) generated by S{w(s), s >_ t} and the null sets. Define 5(+co) = YER fi(t).
Theorem. Let (a) (b)
Every optional time is predictable. is predictable; in fact, if T is an increasing sequence of optional
times with limit T then .5(T) = Yo 5(7,). (c)
is Every almost surely right continuous supermartingale nearly predictable, almost surely lower semicontinuous, almost surely continuous if the process is a martingale.
Observation (1). In view of the martingale continuity properties proved in Section IV. 1, assertion (c) for martingales is equivalent to the assertion determined by a that every martingale {x(.), with the filtration Brownian motion as stated in the theorem has an almost surely continuous standard modification.
Observation (2). In (c) for supermartingales if the supermartingale is right closable, Theorem IV.23(c) together with assertions (a) and (b) of the present theorem implies that the sample function discontinuities can be separated out to make a jump process. Proof of (c) in the martingale case. A trivial argument shows that it is sufficient
to prove that if x is an integrable .5(+oo) measurable random variable, then x(t) = E(xI,flt) } can be defined for t < + oo to make a continuous process. In proving (s) we shall use repeatedly the fact that if x. is a sequence of random variables with L' limit x and if (*) is true for x and its martingale is true for x and (x.(-), fl-)), then In fact, going to a subsequence if necessary, it can be assumed that E{Ix,,,., - x.1} < 2-" which implies
2.VI1. Brownian Motion
580
x,,()I]
[submartingale maximal inequality applied to I
P{suplx"+l(t)-x"(t)I zn-2} t2,
neglecting null sets. Thus (*) is true for this choice of x. More generally an only slightly less trivial calculation shows that (*) is true if "
X =fo[w(0)] F1 J[W(ti) - w(ti-r )] 1
where f, is continuous on R' with limit 0 at oo and 0 S to < .. < t". It follows (Stone-Weierstrass theorem) that (*) is true for the random variable
x =f [W(0), w(tt) - w(to), ... , w(t") - w(t"-1)]
for f continuous on 08"'"+t) with limit 0 at oo and therefore if x is an integrable random variable which is a Borel measurable function of w(0), w(t 1) - w(to), ... , w(t") equivalently, if x is an integrable random variable which is a Borel measurable function of tv(0), iv(to), ... , w(t"). To finish the proof, we need only remark that according to Section 111. 14 every F(+ co) measurable and integrable function x can be approximated arbitrarily closely in L' by random variables of this special type. Proof of (a). To prove that an optional time T is predictable, it suffices to prove that T A b is predictable for every positive constant b, and so it suffices to prove that every bounded optional time T is predictable. Define
x(t)=E{T- TA 'I (')}=E(TI-flt)}-Tnt, The process
0St5+oo.
(4.3)
is a supermartingale, and in view of what we have
just proved the conditional expectations in (4.3) can be chosen to make
5 Elementary Properties of the Brownian Transition Density and Brownian Motion
581
this supermartingale almost surely continuous. Moreover (Theorem IV.2) x(T) = 0 almost surely. For n z I define T. = inf {t: x(t) 5 l/n} to obtain an increasing sequence of optional times for which almost surely T.5 T, 1/n almost everywhere on this set. and T. < Ton {T > 0} because Tn, then almost everywhere on IT > 0} Furthermore, if T' =
0=x(T') =E{T- T A T'jF(T')}; so T = T' almost surely. Thus T, announces T if we ignore exceptional null sets which can be eliminated by trivial modifications of T, .
Proof of (b). To prove that each set in F(T) is in Yo .F(T ), let x be the indicator function of a set in .4=(T), and recall that we have proved above a continuous that we can define x(t) = E{xI.F(t)} fort 5 +oo to make process. Apply the conditional expectation continuity theorem and the martingale optional sampling Theorem IV.2 to obtain
E{xlY.,F(T,)} = lim E{xl.F(T,)} = Iim x(T,) = x p
n-m
n-M
a.s.;
(4.4)
so x is Yo .F(T,) measurable, as was to be proved.
Proof of (c) in the supermartingale case. According to Theorem IV.23. assertion (c) is implied by (a) and (b).
5. Elementary Properties of the Brownian Transition Density and Brownian Motion (a) The transition density (t, , ry) - t (t, , rl) = 41(1, q - ) satisfies the
heat equation
and in fact G(s - t, , q) =
s), (,, t)). This relation between Brownian motion transition density and the parabolic Green function of ft N will be extended to the corresponding relation between the transition density for Brownian motion in an open set D and the parabolic Green function of D x R in Section IX.17. As we shall see in cases related to the Dirichlet problem for parabolic functions many Brownian motion probabilities can be evaluated in terms of solutions of the heat equation on the relevant domains with the appropriate boundary conditions or, if there is stationarity in time, can be evaluated in terms of solutions of Laplace's equation with appropriate boundary conditions.
582
2.VII. Brownian Motion
(b) A token of the intimate relation between Brownian motion and classical potential theory is the fact that [see I .XVII(18.2)] to G(t, , ry)1, (dt) is a constant multiple of the Green function of i8" when N > 2. This relation between Brownian transition densities and Green functions will be extended
to transition densities for Brownian motions in Green domains in Section IX. 17.
(c)
A trivial calculation shows that if
is a Brownian motion, then
lim,..,o P{I w(t)I > c} = I for every c > 0, that is, lim,. Iw(t)) = +oo in measure. (d) The part of almost every Brownian path corresponding to a specified parameter interval has infinite length. To prove this, it is sufficient to show that for one-dimensional Brownian motion from 0 with variance parameter l the part of almost every Brownian sample function corresponding to
the parameter interval [0, b] is of unbounded variation, for every b > 0. To see this, define s = E' I w(jb/n) - w((j - 1)b/n)l, and note
E"xp {el -Iw
n
Since the distribution of n'12jw(6/n)j is independent of n,
njw(b/n)I =
+ao in measure, so that lim,,. E{exp(-sp)} = 0, and since s increases with n to the sample function total variation on [0, b] when n oo along the integral powers of 2, almost every sample function has infinite total variation on [0, b], as asserted. (e)
(N = 1) Let
be a Brownian motion from the origin, with
variance parameter a2. Then E{w(s)w(t)) = a2(s A t) because if t > s, E{w(s)w(t)) = E{ [w(t) - w(s)]w(s)} + E{w(s)2} = a2s.
(5.3)
Conversely, if w(-) is a process with Gaussian finite-dimensional distributions having zero means and if E{w(s)w(t)} = a2(s A t) for some strictly positive
constant a2, has the finite-dimensional distributions of a Brownian motion from the origin with variance parameter a2. (f) If is a Brownian motion from the origin, define t > 0,
Y(t) = 0
ift=0.
An application of (e) to the one-dimensional component processes of Moreshows that has the same finite-dimensional distributions as over the process sample functions are almost all continuous on ]0, + oo [, has and lim,.o y(t) = 0 a.s. if t -+ 0 along a countable dense set because this property. Hence this limit relation is true when t - 0 unrestrictedly. Thus is Brownian motion, and the relation between and shows that
583
6. The Zero-One Law for Brownian Motion
properties of Brownian motion for large parameter values can be translated into properties for small parameter values and conversely. For example, continuity of Brownian motion at 0 implies that lim,.o tw(1 /t) = 0 almost surely, that is, lim
t-.m
w(t) I
= 0 a.s.
(5.4)
This result is a version of the strong law of large numbers in the continuous parameter context, as applied to Brownian motion.
6. The Zero-One Law for Brownian Motion If the sets of a a algebra of measurable sets of a measure space are all null sets or the complements of null sets the a algebra is called trivial. Theorem. If is a Brownian motion from a point, the a algebras 9, and F2 generated, respectively, by the null sets and
n .F{w(s), s< t}, t
n .97 {w(s), s >_ t}
(6.1)
t>o
are trivial.
It can be assumed that the initial point of the Brownian motion is the origin. If A c- f, and if t > 0, the set A is independent of the class of differences {w(s2) - w(s,), t < s, < s2) and therefore is independent of the a algebra .f{w(s2) - w(s,), 0 < s, < s2} which, since w(0+) = 0 almost surely, together with the class of null sets generates a a algebra including .F,. . Thus A is independent of itself and so has probability either 0 or 1. Apply this result to the Brownian motion { tw(1 /t), t e I8+ } [see Section 5(f)] to derive the triviality of .F2.
Observation. Since a Brownian motion process satisfies the conditions of is enlarged if necessary Theorem VI.8(b) after the associated filtration to have .`1v(0) contain the null sets, it follows from Section VI.8 that for a Brownian motion from a point the a algebra .F, is trivial. This proof was not used above because a direct proof is more elementary and just as short.
The Zero-One Law for Space-Time Brownian Motion In view of the definition of space-time Brownian motion in terms of Brownian
motion the direct translation of Theorem 6 into the space-time Brownian motion context is true.
2.V11. Brownian Motion
584
Applications. In the following applications is a Brownian motion in R" from the origin, and TA is the hitting time of a set A by
(a) N = I. According to Section 5(c), lim,.., Iw(t)I = +co in measure: so if c > 0. it follows that P { w (t) > c} > 1/4 for sufficiently large t. Hence P; w(n) > c for infinitely many values of n in 7L' } > y. Since the condition in the braces defines an .F set, the probability in question must be 0 or 1. and so is 1. and therefore lim sup,- w(t) = + oo almost surely because c is arbitrary. Similarly or by symmetry the inferior limit is almost
surely - x. Thus almost every path hits each point of R at arbitrarily large parameter values. In particular, almost every path hits 0 at arbitrarily large parameter values and therefore also [Section 5(f)] hits 0 at arbitrarily small strictly positive parameter values. We shall see in Section IX.5 that when N > I, almost no Brownian path in R" ever hits a specified point at a strictly positive parameter value. (b) According to Theorem 6, if A is an analytic subset of R". the value of P; T" = 0} must be either 0 or 1. The following examples exploit this fact. The dichotomy suggests that a topology of R" [A'] can be defined by making a point a limit point of a set A if the hitting time of every analytic superset A, of A by a [space-time] Brownian motion from the point is almost surely I. We shall see in Section IX.15 that this can be done (even with A, open in the Euclidean topology) and that the topology so defined is the [parabolic] fine topology already defined in Section I.XI.I [Section I.XVII.9]. From now on in this section the remarks relating probability results to fine topology results will always be for the classical context, but the corresponding remarks for the parabolic context will also be true. (c) Let A be an open right circular cone in R" with vertex the origin. Then P{ w(t) E A; is independent oft > 0 and is proportional to the cone central angle. Hence, if t. is a sequence of strictly positive parameter values with limit 0.
P;
E A for infinitely many values of n}
(6.2)
is strictly positive, and it follows from (b) that P{T" = 0} = 1. Actually Theorem 6 implies the stronger result that the probability in (6.2) is 1. We conclude that if r(t,o,) is the point of B(0, 1) hit by the ray from the origin through w(t, (o), then for almost every co the sequence y(1,, w) is dense in B(0, 1).
Cluster Values along Arcs Let u be a function from a metric space D into a metric space D' with distance function d'. Recall that if 0 is a function from an interval ]0, i5[ into D, with C = O(]0. (Jr), and if the limit 0(0 +) exists, then the closed possibly empty
585
6. The Zero-One Law for Brownian Motion
set n,>o u(4(]0, t[))- is the "cluster set," that is, the set of limiting values, of u at ¢(0+) along C. A point q' is in this cluster set if and only if lim infd'(q', r_o
0.
In (d)-(f) below, u is a Borel measurable function from W into a Polish space D' with metric d'. (d) If D' = dB, we weaken the hypotheses on u by demanding only that the set Ar = {u > c} be analytic for all c in R. Under this hypothesis
{ lim sup u [w(t)] > b } = U { T"r = 0} a .f, r-+o
(r rational) ;
r> b
so lim sup,.o u[w(t)] is .ft measurable and therefore almost surely constant. In particular, if A is an analytic subset of D and if u = 1", the set A, is analytic
for all c and we find that P{T" = 0) is 0 or 1, as already observed in (b). This special case illustrates the fact that Bore] measurability of u is sometimes
too strong a hypothesis. If u is supposed Borel measurable, both the above limit superior and the corresponding limit inferior are almost surely constant. We shall see in Section IX. 15 that the constant values here are, respectively, the fine topology superior and inferior limits of u at the origin. (e) A point q' of D' is a cluster value of u either along almost no or along almost every path back to the origin because according to (d) the probability
P{liminfd'(q',u[w(t)]) = 0) r-o
is either 0 or 1. Let A' be the set of points q' for which this probability is 1. We shall see in Section IX.15 that q' is in A' if and only if q' is a fine topology limit value of u at the origin. (f) The set A' defined in (e) is closed and is the cluster set of u along almost every path back to the origin and if D' is compact, P{lim d'(A', w(t)) = 01 = 1. r-o
(6.3)
In fact A' is obviously closed so if q' a D' - A', there is a neighborhood B' of
of so small that if B = u-' (B'), then P{ TB' = 0) = 0. The set D' - A' is therefore a countable union Uj Bj of open sets Bj with this property. It follows that neglecting a null set of paths, the cluster set of u along each path back to the origin is a subset of A'. Furthermore, if A o' is a countable dense subset of A', the cluster set of u on almost every path back to the origin includes Ao and therefore includes A' and so is A'. The set A' may be empty, but if D is compact, A' cannot be empty, and (6.3) is true because if A" is an open neighborhood of A', a finite subunion of U; B; covers D' - A".
586
2.VII. Brownian Motion
(g) According to Theorem 6 the value of P{lim,.,c u[w(t)] exists) is
either 0 or 1. If this probability is 1, then the limit is almost surely constant, that is, a single point of D'. In fact the set A' discussed in (f) is a singleton in this case: so this assertion follows from (f) if D' is compact. If D' is not compact, the assertion follows from the fact that the limit random variable
is measurable from a trivial o algebra into a separable metric space and therefore is almost surely constant. It will be shown in Section IX. 15 that lim,. u[w(t)] = of almost surely if and only if f limf.,0 u() = ry'.
7. Tied Down Brownian Motion Let
be a Brownian motion on ir' from and let n be any point of R' < s < I the joint density of ww(s1), ... ,
(possibly ). If 0 < s, < given that w{(t) = 01 is
(7.1)
,CR-1 ,CJ
on the paraThis Gaussian joint density is that of a Markov process meter interval [0, t], with initial value , value >I at t, and transition density (from t;, at time s, to S2 at time s2, with s, < s2 < t) "(S2 - S1, C1, C2) C2)
01 - S21 C21 0
(7.2)
C(t-st,C1,PW
The joint density (7.1) is also that of a Markov process reversed in time on [0, t] with initial value, at time t and value at time 0 and transition density (from C2 at time S2 to C, at time s, , with 0 < s, < s2)
Osi, , CJt'(s2 -
s1, Ct, C2)
(7.3)
Either way the N component processes are mutually independent.
When N = 1. the transition density (7.2) is Gaussian with mean and variance, respectively, C1 (r - s2) + 17(s2 - si)
r-S,
Q2[(s2
t-S,
s2)]
The random variable w; (s) is (for 0 < s:5 t) Gaussian. with mean and variance, respectively,
tr,(1 -s)+qs
o 2s(t-s)
I
t
587
8. Andre Reflection Principle
The joint distribution of ww,,(st), ww,,(s2) is Gaussian with means and variances as just evaluated and covariance u2st(t - s2)/t.
Suppose that N = I and that t is specified, and consider the process process is almost defined by y(s) = wo(s) - swo(t)/t, 0S s < t. The surely continuous, and its finite-dimensional joint distributions are Gaussian with zero means and - s2)
E{Y(st)Y(s2)} = a2s1(t
(7.4)
s) + qs]/t, 0 < s < t, the and processes If y*(s) = ww,(s) have the same finite-dimensional joint distributions, and it follows that there is a choice of that is, a process with the specified finite-dimensional distributions, with continuous sample functions having values at time 0 will refer to such a and q at time 1. In the following the notation process on RN except that the continuity condition may be weakened to almost sure continuity. This process is sometimes called a Brownian bridge. If 0 is a Denote expectations and probabilities for w,, by
< s" < t, bounded Borel measurable function on (RN)" and if 05 s, < define x, = 4[we(st), ... , ww(s")] and x,,, = 4[xwn(st), ..., ww,,(s")]. These and are random variables defined on the probability spaces of processes respectively. Then E4',, {x,,} = E4
x4 G(1 - S., w (sn), q)
(7.5)
The reader is invited to generalize this equality to a larger class of pairs (xc, x4',,).
finite-dimensional distributions when time is By definition the reversed become the distributions. In particular, and probabilities are the same for events that are invariant under time reversal, for example, the event: a sample path enters a specified set at some time in [0, t]. That is, Pc',, = Pq, on the class of such symmetric events.
8. Andre Reflection Principle Let be a Brownian motion in R from the origin. Let a be a strictly positive constant, and let T be the hitting time of of by Then according to the strong Markov property of Brownian motion [see VI(9.4)] if A is a Bore] subset of R and if A' is the reflection of A in the point a,
P{w(t)eAl.,F(T)} = $z(t - T, a, 1)11(dn) = Jt(t - T,a,q)11(dp) = P{w(t)eA'J.F(T)}
a.e. on {T < t}.
(8.1)
2.VII. Brownian Motion
588
In particular, if A = [a, +oo[, the value of each integral is Z . Ignore the last two terms in (8.1) in this case and integrate over {T :!g t} to obtain
P{x(t)>_a,T- a, T > t} = 0, we find that P{ sup w(s) Z a} = 2P{w(t) z a};
(8.2)
s5I
that is, there is equality in (3.5). Since the left side of (8.2) is P{T < t}, the distribution of T is absolutely continuous relative to IN, with density given by z
(Density of the distribution of T) =
(2no2)-t'2at-312
exp - --
.
(8.3)
To derive a refinement of (8.2), suppose that A in (8.1) is a Borel subset of ] - oo, a[ and integrate over (T< t} to obtain
P{w(t)eA, T< t} = P{w(t)eA', T< t}.
(8.4)
Since the term on the right is trivially equal to P{w(t)eA'}, we have proved P {sup w(s) > a, w(t) a drl ) = G(t, 2a - n)h (dn)
,sl
(n < a),
(8.5)
which implies that P { sup w(s) < a, w(t) a do } = [e (t, n) - 6(t, 2a - n)] 1 t (dn)
(n < a).
(8.6)
s51
Since the distributions involved here are continuous, "7I when SS < + xc, the integrand in (9.5) is bounded uniformly for all t, in D, and
n in a compact subset of D. Hence GD(t, 4, ) is continuous on D and is obviously the only continuous density for the transition probability of Brownian motion in D. The Chapman-Kolmogorov density equation /D(s + t. . S ) =
s)/N(dq)
(S > 0, t > 0)
(9.6)
D
is true for all , C in D in view of the following facts: (a) (b)
The left side varies continuously with C. The right side varies continuously with C, because the integrand does and the integrand is majorized by the integrable function
fD(s, ,
)(2nvzt)-Niz
591
9. Brownian Motion in an Open Set (N 2 I)
(c)
The equation is true for IN almost every C when is fixed because GD is a transition density for Brownian motion in D.
In view of the inequality GD 5 G it follows from (9.6) that GD(t, , ) is a
continuous bounded function on D x D for each value of t. Moreover, according to the analysis of the integrand in (9.5), the expectation on the right in (9.5) is for fixed q a bounded function of
t) on D x ]0, + oo [.
Potential Theory Significance of eD
Let D be a nonempty open subset of R", define D = D x R, and let (ID be the parabolic Green function of D. The function ((c, s), (q, t)) F_. GD((, s), (q, 1)) is a function of s - t, , q, and in Section I.XVII.18 a function GD was defined by GD(s - t, , q) = ObW, s), (q, t)).
(9.7)
In Section IX.17 we shall identify GD as defined by (9.7) with the Brownian
transition density function GD defined in this section. The probabilistic evaluation of parabolic measure to be made in Section IX. 13 makes the expectation in (9.5) the solution Hf of the parabolic Dirichlet problem on b with f the restriction to ad (Euclidean boundary) of e(-, , q). In this light (9.5) can be interpreted as the Dirichlet solution construction of (ID described
in Section I.XVIII.I. EXAMPLE. Let D be a half-space of RN, define b = D x R, and if ,i E U, denote by q' the reflection on 1 in aD. Then [Section 1.XVII(4.3)] rl) = Q, q) - O(, q'); that is, if GD satisfies (9.7), if s) and t), and if q' is the reflection in R" of q in OD, GD(t, c, q) = G(t, , q) - G(t, a, q') = G(t, q - ) - G(t, q' - ).
(9.8)
Observe that according to (8.8), in which = 0, equation (9.8) is true with the probability interpretation of GD as a Brownian transition density function. Canonical Brownian Motion in an Open Set
Let D be a nonempty open subset of R". Adjoin a point a to D, defining the topology of D u a so that D v a is the one-point compactification of D. Let t) be the class of all functions m from R+ into D u a with the following properties :
cu(s) = a implies that w(t) = a for t > s.
592
2.VII. Brownian Motion
The function tv is continuous on the interval [0, . (d)) [, where S(to) = inf {s > 0: to(s) = 8} and there is a left limit (Euclidean topology of RN) at Si(tu) when 9((o) < + oo. Define w(t, d)) = tu(t), fi(t) = .F {w(s),s 5 t},, .F{w(s),sc- Rr}. If P .4 is a Brownian is the completion of a measure on , under which will be said to be a canonical Brownian motion in D, the process motion in D. If D = RN this definition yields a canonical Brownian motion
in RN as already defined in Section 3. One way to obtain a canonical Brownian motion in D is to operate on a canonical Brownian motion in RN as follows. If S is the hitting time of the Euclidean boundary of D by the Brownian motion, change every sample function value to 0 at every parameter value at least equal to S.
10. Space-Time Brownian Motion in an Open Set If D is an open subset of AN, a space-[co]time Brownian motion in D is a space-[co]time Brownian motion with initial distribution supported by D, killed at the hitting time of DD. The killed process is Markovian with substochastic transition function p given by
P{wt(t)EA,t < 94},
4=
(10.1)
where w
is a space-[co]time Brownian motion from and 94 is the hitting time of ab. From now on only the space-time Brownian motion is considered unless the contrary is stated. We shall use the space-time notation introduced in Section 2. The measure p(t, 4, ) is supported by D,_ and the set function A s - p(t, , A x is - t}) for A a Borel subset of D5_, is majorized by
Af- f
A
so the first set function is absolutely continuous relative to IN and is therefore the integral of a density. A unique choice of this density will now be made. Following the reasoning in the Brownian motion case, let : s) and, : (q, t)
be points of b, and if s > t let n) be the probability that a Brownian bridge w has the property that the space-time bridge
r],0 < r 5 t} from
to q never hits 3D. Define
JOs 0
07)I(4, 0 ifs > t,
if s s2 > The condition that an extended real-valued positive universally measurable function ii on D be excessive for space-time Brownian motion in D is the validity of the inequality
I
5 t}(SI)
(sI > s2)
(10.5)
together with equality in the limit when .s2 T s, . If in addition there is equality in (10.5) whenever .s, > s2, the function ti is invariant for space-time Brownian motion.
The transition density (o of space-cotime Brownian motion is defined by dualizing the preceding discussion and satisfies the equation
In(n, )=fn(
(10.6)
in view of the remarks on time reversal in a Brownian bridge at the end of Section 7.
The Function
and I n( 1)
Let D be a nonempty open subset of 08" (N z 1), and define D = D x R. Equation (9.6) combined with the inequality In 5 f. implies that for fixed is excessive for space-time Brownian motion , in D the function in b, invariant excessive for space-time Brownian motion in b - '(?1. 0)), continuous and 0 at the points of D - {(q, 0)) with ordinate values 0, and consider the following events. Each parameter value t, below depends on the sample function. QaO':
Oho':
ON QahQ: QhahQ:
it-,(-) first meets of at a and does so before time t. iv,(-) first meets 81 at b and does so before time t.
There exists it with 0 < t, < t and w,(t,) = h. There exist t,, t, with 0 < t, < t2 < t and ww(t,) = a, w,(12) = b. There exist t, I. 12, 13 with 0 < t, < 12 < 13 < t and ww(i,) = h. w4(12) = a, µ',(t3) = b
and so on. Then ObO' = ObO - OaO' n OhO, OaO' n ObO = OabO - ObO' n OabO.
Oho' n QabO = ObabO - QaO' n QhahQ,
so that OhO' = OhO - QabO v QhahQ - QabahO u
.
(11.2)
The probabilities of these events are easily calculated using the reflection principle. For example. P;OabO} is the probability that a path reaches h by time t after having hit a, or, reflecting the path in a at the hitting time
of a, and settings=h - a P {QahQ} = P{minwjs) 0, a formal application of Taylor's theorem leads to 0 = E{{u[w(t + 6),t + 6]Iw(s), s < t} - u[w(t), t]
=b
2 etil + 0(62)
= 6&i + 0(62),
(13.1)
59t
2 VII Brownian Motion
where the partial derivatives are evaluated at [w(t), r]. Thus it is plausible that ti must he coparaholic and that conversely the process tt[tr( ), ] is a martingale in some local sense if iu is coparabolic. Equivalently, u compounded with space-time [cotime] Brownian motion is a martingale in some local sense if and only if ii is parabolic [coparabolic]. In particular (if u is a function of the space variable), a function u on R" composed with Brownian motion is a martingale in some local sense if and only if u is harmonic.
The results suggested in this section will be formulated rigorously and proved in later chapters
Chapter VIII
The Ito Integral
1. Notation be a Brownian motion in IJB, defined on some probability is right continuous and that ,F(0) space (Q, .F, P). It is supposed that contains the null sets. The Ito integral !{, ¢ dw will be defined for stochastic processes in the space IF of not necessarily adapted to F(-) real processes Let
{ Ib(t), t c l
process
} with the following property: there is a progressively measurable for which depending on P{ a,: f jk(s,ctw)j2ds
< +ao} = I
(1.1)
0
for all finite t and P{4(t) = ar(t)} = 1
(1.2)
for /, almost every t. In this chapter ds refers to Lebesgue measure /, . For economy in later references absolute value signs are used in (1.1) and similar contexts because the present discussion will be extended in Section 7 to cover
vector processes and processes with complex state spaces. Observe that a
process indistinguishable from one in r is itself in r. Let /, x P be the completed indicated product measure on R' x 0, defined on the completion of R (R') X . relative to this product measure. According to Section 1.13, if {4 (), is an extended real-valued adapted process and if the function is /, x P measurable, then this process is in IF if (1.1) is satisfied. of r whenever We metrize r by identifying two members and (1.2) is true for /, almost every t, thereby making r into a space of equivalence classes, and by defining (abbreviating the notation) 2-kE J I A
r dist (0, 0) _ k=1
1
012 Ifo k
10 _
ds lizl
.
(1.3)
)J}}
are progressively measurable, and we are adopting the and usual convention : the right-hand side of (1.3) is the distance between the Here
2.Vl1I. The Ito Integral
equivalence classes containing
and vi(-). In the following the reader is asked to judge from the context whether notation like refers to an individual process or to an equivalence class. It is also left to the reader to verify the fact, which we shall not use, that r is complete in its metric. The subset of r for which (1.1) is true with t = + oo will be denoted by , and the subset of T for which
i
E! fo
lcI2ds1 < +ao
metric
for f sometimes preferable to that induced will be denoted by r'2. A by the r metric is the F metric in which (1.3) is strengthened to
I dist(0,t)=El A
ltrz). I0-IpI2ds{[J
(1.4)
A r2 metric sometimes preferable to that induced by the r or r metric is the L2 metric in which (1.3) and (1.4) are strengthened to
r'2dist(.0,Eu21jo )= . 1.0 -,I2ds}.
(1.5)
The spaces f and r2 are complete in their metrics. There is convergence of n eV) to in the r metric if and only if a sequence
plim f n-
(1.6)
o
for all finite t, and there is convergence in the I' metric if and only if (1.6) are chosen to be progressively is true for t = + oc. Here 0.(-) and measurable members of their equivalence classes, in t or r as required. There are sometimes formal advantages in having members of r defined at the parameter value + oo. This can be effected for our purposes by defining
¢(+oo) arbitrarily, by defining ffl(+oo) = F, and leaving the r, F and r2 metrics unchanged.
Observe that if 0 is a bounded member of r, then is in I' or i' or r'2 if is. A useful special case is ii(-) = 1sT, defined as the indicator function of the stochastic interval IS, TD. As usual the definition of an integral is first given for a simple linear class of integrands. In the present context this class, a subclass of 1'2, is the class r'o each of whose equivalence classes contains a process defined as follows. There is a finite set 0 < t, < < tk < + oo and corresponding bounded random variables f, , ._f, f, such that f is F(:) measurable and that
2. The Size of ro
601
if tSt,,
0
if t;_, < 1:5 1,, 2 < j < k,
0(1) =
(1.7)
if t > tk.
0
A function 0 in ro has many representations (1.7) because additional partition points can be adjoined without changing adjusting f as required. If and in ro are to be considered simultaneously, partition points can be adjoined if necessary to obtain representations of and with the same partition. Using this fact it is obvious that ro is an algebra. Observation on predictability of Ito Integral integrands. The process defined by (1.7) is adapted, left continuous, and therefore predictable. The
following lemma states that ro is dense in the sets r, r, r2, and a trivial adaptation of the proof shows that each equivalence class of each of these sets contains a predictable member. We shall not use this fact.
2. The Size of ro Lemma. The set ro is dense in r, r, and F. in the metrics of these spaces.
It is sufficient to consider progressively measurable members of these spaces. Suppose first that is a progressively measurable member of r, and define
Then lim,, .
>V(t)
ifl41(t)lSnandt5n,
0
otherwise.
44') in the r metric. Next define
ift< m n. The process is progressively measurable, I4'. 1 < n, a ro, and lim,,,-. in the r metric because there is bounded 1, almost everywhere sample function convergence. This completes the proof of the
lemma for the r metric and the proof of the lemma is concluded by the observation that for 4(') in r or 1-2 the convergence assertions in the above proof are also valid in the metric of the space f or r2 as the case may be.
602
2.V11I. The Ito Integral
Refinement. A trivial computation involving the Schwarz inequality shows
that a.s. 0
0
Thus an arbitrary element of r, r, or r2 is the limit, in the metric of its space, of a sequence {¢ n z 0} in I'0 satisfying almost surely
JII2ds fI2ds 0
(2.1)
0
simultaneously for all t.
3. Properties of the Ito Integral oo if If 0 05t E r], the integral x(t) = J 0dw will be defined uniquely up to a null set. Moreover it will be shown that a version of x(t) can be chosen for each tin such a way that the map 0(-) is a linear continuous map from IF [I'] [[r2]] into a metric
space r' [r] [[I-Z]] of processes defined as follows. The space I" is the space of almost surely continuous adapted processes metrized into a complete metric space (in which indistinguishable processes are identified with each other) by
r' dist
Y 2-kE{ I A sup Ix(t) - y(t)I }.
(3.1)
rsk
1
The space F" is the subset of r' for which almost surely lim, x(t) = x(+ oo) exists and is finite. The I" metric is defined by strengthening (3.1) to
I' dist
E{ I A sup IX(f) - y(t) 1). rc+m
(3.2)
The space r2 is the subset of I"' for which E{sup, I.
is a martingale; so if 0< s < t and if s, t are in
Hence
(4.2)
it
follows that E{x(t)I.fls)} = x(s) almost surely. Since any pair of parameter is a martingale. values can be adjoined to the set it follows that This martingale is L2 bounded, hence has orthogonal increments, and
E{Ix(+oo)I2} =62EI f °°I4(s)I2ds}.
(4.3)
0
If q() is defined by (1.7) and
by (3.10)
E{ ym(r)I'F(i _1)}
I = y,,(0)
ifj = 1,
a.s.
y,,(tt-1)E{exp [I--t (K'(tt)
(4.4)
2 (tt - tt t)fJ tJ
- w(tt_1))]I_F(tt_1)) exp 1
a.s.
ifj > 1.
Now f_, and w(l) - w(tt_,) are mutually independent, and whenever z is a normally distributed random variable with mean 0 and variance a, E{exp(az)} = exp(a2(x/2). Hence the right side of (4.4) reduces to y'y(tt_,). is a martingale, and it follows repeating an argument Thus { just used that {y,,(-). F(-)) is a martingale.
2.VIII. The Ito Integral
606
5. The Stochastic Integral for an Integrand Process in r Suppose that c f,. Then (Section 4) the process { y, defined by (3.10) is a positive martingale, and if c5 > 0 and b > 0, the submartingale maximal inequality 111(9.1') yields
PJsup
1 +?z
I
6
r_h ,Iii
2
_
j+ 2
JJ42ds}
0+ 262
rsh I.(')
f h tae 1 5 2e- ua.
(5.3)
We conclude that if
ne Z' } is a sequence in 1' with limit 0 in the r metric, then the sequence { J' 4 dtv, n e Z i has limit 0 in the r, metric.
That is, the linear map 0 -+J / dw from r0 in the f metric into the metric space r' is uniformly continuous. It follows that there is a unique uniformly continuous linear extension of this map, from the r metric closure
of r,, that is, from r, into F'. We accept this extension as the definition of the integral O ¢ dw; this integral then satisfies Section 3(h).
Furthermore the map q5"Ja 4 dw takes r into F'. and if we apply the reasoning just used but this time with the f metric on F00 and the T' metric on the image of CO, we find, setting b = + oe in (5.2), that the map is uniformly continuous. Hence this map is uniformly continuous from T into F'. Finally we consider r, in the IF2 metric. Since 110 0 dwl is a submartingale for 4 in f;,, the L2 maximal inequality 111(11.1) for the continuous parameter context is applicable and yields E{sup I f odidwl2}
t, } that
(0
iftl2.
(6.3)
This process is in 1,0, and the definition of the integral for an integrand process in r70 makes (3.4) trivial in this case. in F, let (c4) To prove (3.4) for arbitrary in F0 with limit in the F metric, and define to (c3),
J
n e Z') be a sequence $o 0 dw. According
608
2.VIlI. The Ito Integral
and (3.4) follows because in view of the boundedness of T, lira n-n;
IoT(')On(') = lor(')O(')
in the r metric; so
plimJx(T)-xn(T)I- T if necessary, that T = + oo and that is in f. In view of the metric properties of the map 0 dw it is sufficient to prove (3.6) with S replaced by [S]n and with in To, given by (1.7), and adding points to t in (1.7) if necessary, we can suppose that t includes the values {j2-",jeZ+} in the interval [0. tk]. Finally in view of the linearity of the above map we can even suppose that k = 2. The process is now in ro and has a representation of the form (1.7) with k = 2, and the computation verifying (3.6) is trivial. Proof of Section 3(f). Suppose first that S=_ 0, T=- + oo and that (3.7) is true. Then is in r2. Let be a sequence in r0 with limit in the r2 metric, and define xn(t) = f o 0. dw. Then lim"-,, in the r2 metric; in particular, (t < +00).
lim E{ Jx(t) - x"(t)12} = 0
Since the process {xn(t), ,f(t), t < + oo } is a martingale, it follows that the limit process is a martingale. Moreover E{ Jx(+oo)12}
= a2E
r Jo
I0I2dsl J)
because this equation is valid for and 4"(-). In the general case of we find Section 3(f) if we apply what we have just proved to [see (3.5)] that
{x(Tnt)-x(Snt),.I'(/),t t} for to R+. Then f satisfies M, and f(t) 5 r if and only if f(r + s) > t whenever s > 0. Moreover f =,f Finally f [ f(r)] _ r if r is not a point of a constancy interval off, and Jj f(r)] = r if r is not a point of a constancy interval off that is, if r is not a discontinuity point off. Increasing Families of Optional Times and Their Inverses
Let { flt), t e R+} be a filtration of a probability space, right continuous and with ."(0) containing the null sets. Let IS,, t e R+} be a right continuous
process whose random variables are finite-valued optional times for 'fl-) and whose sample functions satisfy condition M. Define an inverse process S. by the above procedure, so that m
S,(cu) = inf {r: S,(w) > t},
{S, 5 r) =n {Sr+im > t}.
(9.1)
The process IS., -fl-)) is an adapted process whose sample functions satisfy
condition M. The filtration is right continuous, and 9(0) contains the null sets. Moreover {.S 5 r} e 9(r) in view of (9.1) and the right continuity of Hence each random variable S, is optional for Furthermore .fi(t) c .*(S,) for all t; that is, if A e9(t), then A n {S, 5 a} e F(a) for all a, equivalently,
Ann{S,+,M>t}n{S, 5P}e.,F(P)
(9.2)
for all at and P. In fact (9.2) is trivial if t > fi, whereas if t 5 fi, all the sets in (9.2) are in .f(fi). A Change of Variables in the Ito Integral
be a one-dimensional Brownian motion with variance Let parameter o2, let {0(-), 9(-)) be a progressively measurable process in the class r (Section 1), and suppose that {S, t e R } is a process as above whose sample functions satisfy
616
2.VIII. The 116 Integral
JS,lo(s)12ds
(9.3) 0
for all tin IB+. Define iv(t) =
rosI
0 dw, J
and observe that E{w(t)2} = att. Theorem. The process { iv(
is a one-dimensional Brownian motion with
variance parameter a2. Define J o 0 dt+w. To prove that {ti (),() } is a Brownian motion with variance parameter a2, it is sufficient to show that for a < f3 the condi-
tional distribution of x(SS) - x(S,) given .f(S,) is Gaussian with mean 0 and variance a2(fl - a). Thus in view of the characteristic function of the normal distribution it is sufficient to show that for each complex constant y 2
E ex
[x S
xS
.F S
ex
a2(2 - a)
a.s.;
(9.4)
that is, it is sufficient to show that, in the notation of (3.10), E{ yrm(SS)I.
(SS) } _ yrm(S,)
as.
In other words the problem is to show that the process { is a martingale. Now according to Section 3(k), y4los", the process {yy(
{YY4,(Sp A '),.F(')}
is a martingale, right closable by y, ,,(S,,), and therefore uniformly integrable. Hence (martingale sampling theorem) the martingale equality at times t = S. ,F(-)) is a martingale, and t = + oo is satisfied; that is, the process { as was to be proved.
Extension to N > I
If
is a Brownian motion in 1 N with variance parameter a2 and
if ¢ is a progressively measurable process with state space 08 N, in the (vector)
space t, satisfying (9.3), the theorem remains true with no change in proof: is a one-dimensional Brownian motion with variance the process { parameter a2. Observe that the theorem remains true if null exceptional sets are allowed, that is, if only almost every sample function satisfies condition M and if (9.3)
617
9. A Change of Variables
is only true almost everywhere for each t [which implies that (9.3) is true almost surely simultaneously for all t.J JA(t)
_ Application (a). If N z 1 and if 101 = I in Theorem 9, then S, = t, .F(t), and w(t) = Yo 0 dw. We leave it to the reader to show that in this case `, 41 div = Jo (i(,4,) dw Jo
a.s.
(9.5)
0
whenever * e t (state space R) and (0¢) is the scalar multiple of the vector 0 by 0. More generally, if M(t) is for each t in R' an N x N orthogonal are matrix-valued random variable and if the component processes of in r, then the vector process {Jo M dw, is an N-dimensional Brownian motion with parameter value o2.
be a real vector process in r, and
Application (b). (N,2: 1) Let define 00(t, w)
(N-u2
if 100(t'(0)l 0- 0'
if
.. , N-112)
0.
w(t) = J' I0dw. 0
Then 1; so according to Application (a), the process one-dimensional Brownian motion with variance parameter o2 and
JIoId=Jodw r
r
o,
0
is a
a.s.
Thus every vector stochastic integral can be represented as a one-dimensional stochastic integral with a positive integrand.
Application (c). (N z 1) Let be a real vector process in r, define 1; 0 dw, and define T. by (6.6). Suppose that T. is almost surely finite valued for all a. Then according to Theorem 9, the process is a Brownian motion. In other words the integral process is a Brownian motion under a change of time. If 7, is not almost surely finite valued for all a, that is, if to 10 12 ds is not almost surely + co, define
k(t) = Tka = inf
it
I
:
2 J 0r
(fi(t)
(1,...,1) rkl2 ds Z a},
if t:5 k,
ift>k, xk(t) = Jo, Ok dw.
618
2.VI11. The Ito Integral
According to these definitions and Theorem 9, the process
is a Brownian motion and coincides with parameter values < a.
on the set {Ts < k} for
10. The Role of Brownian Motion Increments be a Brownian motion in RN. Fix t > 0, define
Let
Alt) = Wla)(f2-nt) - 1V(a)((I
- 1)2 "t),
omitting the superscript when N = 1, and for f a complex-valued function
on R' x R define "t],
Jj,, =f[wV(j2 fn (s) =
if (j - 1)2t < S 5 j2 -"t,) -> 1?
fu- t )n
ifs=0.
IfTw(O), 0]
Lemma. When f is continuous and finite valued Jim 2"
p
Y fUOj''Ago) =
"~m j-1
az
if a = fl,
r f [w(s), s] ds Jo
if a # fl.
0
10.1
(
)
If f is bounded these limits are also L2 limits.
To simplify the notation, take a = 1. Assume first that jf < c. Then z"
2
f[w(s),s]ds - Y_ fu-11n2-"t
= lim E 1I r [.Jttv(s). s] -.f (s)] dsl2 J.
(10.2)
J
limtEi If[w(s),s]-f(s)l2dr}=0.
-W
o
Thus to prove (10.1) for a = fi in the L2 limit sense for f bounded it is sufficient to prove lim E11j= : fu_t)n(0;n'z - 2 "t)12 }=o.
(10.3)
10. The Role of Brownian Motion Increments
619
When the indicated square is multiplied out, each term has the form tah - 2 n t)(Akn (a)Z - 2-RI), f(j-1)nik-1)n(ejn
j 0 and a
partition n: to=0 b. 0(t)
¢R(t) _
Then
is in r if
is almost surely finite valued on it, and
J0dw = j 4(tj-()[w(t1) - W(t1-1)1 Theorem. (a) If the restriction to [0, b] of almost every is bounded and 1, almost everywhere continuous, then b
plim 6(°)-O
(b)
fo
sample function
b
0dw.
&xdw=
(11.2)
fo
If E{ Ic(t)I2) < + co fort 5 b and if lim Efl4(t) - O(s)I2) = 0,
(11.3)
S-0
oss.rsb
then (11.2) is true as an L2 limit. Proof. (a) Under the hypotheses of (a) the usual reasoning in the discussion of the Riemann integral yields b
lim 6(m)- O
1¢R-0I2ds=0 a.s.;
fo
so lim j(,.o r dist (¢R, 0) = 0, and since the map 0 dw is continuous in the (T, n pair of metrics, it follows that (11.2) is true and even that
rs
(11.4) o
621
12. Ito's Lemma
(b) Apply Section 3(f) to find
f 1.0. dW 121 = IIJ¢d/+ -
E
UzEll I.0 -.0.12T
o
o
(11.5)
5 azb sup E{14(t) - 4(s)Iz}. It-zIsb(.1
Oss,tsb
Therefore (b) is true, and in fact (11.4) is true as an LZ limit because the map Jo 0 dw is continuous in the (I'z, I'z) pair of metrics. o Integration by Parts
If
is in t and if the restriction to the interval [0, b] of almost every sample function is right continuous and of bounded variation, then
f
Odw = w(b)4,(b) - w(a)4(a) - f bwdo a.s.,
(11.6)
Jo
o
where for almost every point co of the basic measure space the integral on the right is the Riemann-Stieltjes integral J w(s, w) d,4(s, w). In fact the hypotheses of Theorem 11 (a) are satisfied, and the sum in (11.1) is equal to n
(11.7)
w(b)O(b) - w(a)O(a) - Y w(t) [O(t) - 0(t;-1)], !=1
which almost surely yields the right side of (11.6) when S(ic) -+ 0.
12. Ito's Lemma Let {
9(-)) be a Brownian motion in R', and let (q, t) -. u(q, t) be a
continuous function on RN x R' for which the derivatives
_ uO
au
azu
au u,=anu,'
at '
uri=aq(;,aqu,
exist and are continuous. Before proving Ito's lemma we prove an important special case: simultaneously for all t e R', N
u[w(t), t] - u[w(0), 0] =
a=1
o
N
or
=
a2 N f0t
r
uo ds + Y, 1 ua dw(a)(s) + 2 Y Au ds +
u. ds
a=1
o r
f ua dw(a)(s) a.s. a=1 Jo
(12.1)
622
2.VI11. The Ito Integral
The argument in each integrand is [w(s), s]. This relation is commonly written in the form du = du dt + Y ua dw(a',
(12.1')
a=1
and from now on relations like (12.1) will sometimes be written in the corresponding differential form. It is sufficient to prove (12.1) to be valid for fixed t because each side of (12.1) is the tth random variable of an almost surely continuous process. Fix t and denotej2-"t by sj". so that 2n
u[w(t),I] - u[w(0),0] = Y {u[w('sjn),sjn] - u[W(sjn),S(J-I)n]j j=1
(1221
2"
+ E {u[W(sjn)'S1/- lln] - u[w(Sti- lln) S(i- 1)n] If
-
j=I
The jth term in the first sum is u0[w(sj"), sj"]2-"t up to an error which for each continuous Brownian motion sample function is uniformly o(2-") as j varies and n -+ x. Hence the first sum in (12.2) has the first integral in (12.1) as almost sure limit when n - oo. The jth term in the second sum in (12.2) is, if ej(n) = Nrlal(sjn) - W(a1(S(j-1).)'
N
N
ua9[w(s(j-I,),s(j-!),,]A1")ej(n)
2
a-I
(12.3)
a.p=1
up to an error which for each continuous Brownian motion sample function is uniformly n(Ea=1 A('j.)') as j varies and n - oc,. According to Lemma 10 and Theorem 1 1 , when n -+ x, the sum over j = 1, ... , 2" of the terms in (12.3)
has as limit in measure the sum of the second and third sums in (12.1). Finally the summed error o(EaN=, 11" Aj ) is o(1) for almost every Brownian
motion sample function because the inside sum has almost sure limit cr2t when n -" r_ by Levy's result (10.7). Localization
If, more specially, u is a function of class C(21 on 1 N and we consider u equation (12.1) reduces to
composed with
0,2
u[N (t)] - u[w(0)] =
2
N
Duds + Y f') uadw(a)(s) 0
x=1
More generally, if u is only defined and of class L(21 on an open subset D of I8N and if P{It'(0)eD} = 1, let S be the hitting time of 8D by Then
623
12. Ito's Lemma
(12.1) is almost surely valid for t < S in the sense that if Do is an arbitrary open relatively compact subset of D and if S0 is the hitting time of aD0 by then (12.4) is almost surely true on {w(0)E Do} if i is replaced by t A So. To see this, let u0 be a L(2)(R') extension of the restriction of u to Do. Then
an application of (12.4) to u0 with t replaced by t n So gives the desired result. We now turn to Ito's lemma. Let u and
be defined as at the beginning
of this section. Let M be a strictly positive integer, and for I < m 5 M and 1 < n < N let and Wm be processes in r except that the integrability requirement for Y'm is weakened: it is supposed only that almost every sample function is integrable over finite intervals. Define x"'(.)] by
x(m)(t) = x(m)(O)
N
r
,{{,
Wmndw(n) +
+ Y, n=1
0md1,
fo
J0
where x(m)(0) is an arbitrary flO) measurable random variable. Ito's Lemma. Under the stated conditions, for u: t I-+u[x(t), t],
2 M
Y
du = u0 dt + Ymum dx(m) +
dx(m) dx("),
(12.5)
2 m.n=1
m=1
where dx(m) dx(") is to be interpreted as the formal product under the convention (dt)2 = dt dw(") = 0,
dw(m) dw(n) = a2bmn di
so that
j
dx(m) dx(") = a2
0. W,,//,naA
a=1
To simplify the notation, we give the proof for M = N = 1. That is, dx(t) = 0 dw(t) + /i dt, u = u[x(t), t], and (12.5) reduces to z
du = uo+u1O+ 2u11O2)dt+u14.dw.
(12.6)
It is sufficient to prove the lemma for ¢ and i/i in I70, given by representations of the form (1.7) with a common partition :
0(t)=''(t)=0 00) =4-1' 0(t) _ P(t) = 0
0(t)=gg_1
if t I.F(s) } = E{exp } NZ = exp a2(t - s) 2 a.s.
(1.3)
t ()} is a stochastic process for which M3 is a martinConversely, if gale whenever the components of y are purely imaginary, then the process has the Brownian motion finite-dimensional distribu-
tions with variance constant a2. In fact, if s < t. the martingale equality (with y = ifJ and /3 real)
E{exPli 2 the harmonic measure evaluation (I.8) generalizes to PIw,(SS)EAi
From this it follows that if f is a resolutive boundary function, the PWB solution Hf is given by
Hf(S) =
(1.9)
Corresponding results will be proved for PWB' solutions. The evaluation (1.8) can also be derived by solving the appropriate differential equation problem. The rigorous argument is the following, once it has been shown that the left side of (1.8) defines a harmonic function of S, that is, a linear function. Let be a Brownian motion from 0, so that S + is a Brownian motion from S and the measure space does not depend on . A So + iv(-) path (with
a < 5o < b) which hits b before a will do so if the path is translated by S - So With S > So : so the function -- P ps,4(S4) = b} is an increasing function of S. Since almost every is,(-) path is strictly positive at some parameter
values arbitrarily near 0, this increasing function has limit I at b, and a similar argument shows that the limit is 0 at a. Hence the function must be given by the right side of (1.8). This trivial example illustrates some of the problems involved in a rigorous justification of the evaluation of Brownian motion probabilities. Application of M2. Let D be a bounded open subset of R", and define Sy as above. Equating expectations of the random variables of M2 with q= 0 at times 0 and S{ A t yields 0 = E{ I w4(S4 A 1)12 - Na2(S4 A i)},
(1.10)
No2E{S } = Eflws(S4)J2}.
(1.11)
and therefore
In the special case N = 1. D = ]a, b[ this equation reduces to
630
2.IX. Brownian Motion and Martingale Theory
a2E{Se} = ( - a)(b - ),
(1.12)
in view of the distribution of wf(SE) already evaluated. Thus the function i-+E{Sf} is the solution of the equation a2 Au = -2 with boundary limit function 0. For N > I and bounded D the martingale equality (1.11) becomes, if we accept (1.8') (to be proved in Section 13),
a2NE(S4)=f p
-
(1.12')
J aD
As in the case N = I, the function - E{S4} is the solution of the equation
a2NAu = -2 with the boundary limit function 0 in the sense that a2NE{S4} + is the harmonic function which is the PWB solution of the harmonic function Dirichlet problem with boundary function >f F--.Ir1I2.
Application of M3. Let N = 1, let be a Brownian motion from the origin, choose a > 0, and let T be the hitting time of {a} by The density of the distribution of T is given by VII (8.3) and will now be derived again using the martingale M3. The optional time T is almost surely finite because
according to the application of MI above the probability that a Brownian path from 0 hits a before -c (with c > 0) is c/(c + a), which has limit 1 when c + oo. Equating expectations of the random variables of M3 at times 0 and T A t yields 1 = E{exp [YW(TA t) - y2a2
T2 A tl
D.
(1.13)
Take y real and strictly positive, so that the integrand is at most exp (ya). When t - + oo (1.13) becomes
I = E{exp Lya - y2a2 2J },
(1.14)
E exp [_Y2a22]}=exP(_Ya).
(1.15)
that is
This equation specifies the distribution of Tby way of its Laplace transform.
2. Coparabolic Polynomials and Martingale Theory It was shown in Section VIII.13 that the composition of a [co] parabolic polynomial with space-[co] time Brownian motion from a point is a martingale. We shall establish this result again in this section without using the
631
2. Coparabolic Polynomials and Martingale Theory
Ito integral and incidentally gain insight into the space-time Hermite polynomials. Since every space-time coparabolic polynomial is a linear combination of space-time Hermite polynomials, it is sufficient to consider these as defined in Section 1.XV.3 by (N)mN
(1)m,
ao
n,, ...mN(7),
Y_ Y_ n=o m,+- +mN=n m(1 ... mNl
(2.1)
and to compose these polynomials with space-cotime Brownian motion from the origin. That is, we consider [w(t), 1],.f(t), t e R+} with a Brownian motion in l from the origin, and we shall prove that the process { Hm ... mN .F(-)) is a martingale.
To prove this fact, define yr(t) = exp [ - jyj2a21/2] as in VIII (3.10) so that (1)m
(N) N
} m1(... MN! N
yr(t) _
and by 1.XV(3.9) (if n = m1 +
H.,...mN[u'(t)]
(2.2)
+ mN)
rN[i'(t)]Hnl.. nN[t;(t)]}
E{Rrn
1(a2t)n+NI2(27r)N/2m11 ... m,!
if m =
i0
otherwise.
(2.3)
For fixed y and t the series in (2.2) is a series of orthogonal random variables with mean square limit sum yr(t). Thus integration to the limit with respect to the probability measure is legitimate. Now the process is a Brownian motion with variance parameter j'I2 a2 ; so (Section 1, Example M3) the profess { is a martingale. The validity of the martingale
equality for this process is an identity in y and implies the validity of the martingale equality for each process {Hm and each of these processes is therefore a martingale. The L2 martingale maximal inequality (Theorem III.11) together with (2.2) implies that for b > 0 j ,(1)m
ao
E sup' yr(t) )Sb
Hm, +mN=n m 1 ... MN!
n=k m,+
(ldZa2t)n.
5 4(2na2t)N12 n=k
(N)N
n
mN [H (t)]
632
2.1X. Brownian Motion and Martingale Theory
This inequality implies that almost surely the partial sums in (2.2) converge uniformly for t < b, ft 1 < b, for each pair b, b, .
Harmonic Polynomials
if u is a polynomial on R' and is harmonic, u considered as a function on 9' is a coparabolic (and parabolic) polynomial, and therefore the composition of u with a Brownian motion from a point of W4 is a martingale. This result is generalized in the next section, as is the preceding result for coparabolic polynomials.
3. Superharmonic and Harmonic Functions on R" and Supermartingales and Martingales The following theorem for superharmonic functions on IRN will be extended
in Section 7 to apply to superharmonic functions on Greenian sets but is proved separately because its proof not only is easy but indicates without technical complications why the relation between classical potential theory and martingale theory is so close. Theorem. Let i
_101 (-) } be a Brownian motion in RN from . Let u be a superharmonic_junction on IRN, and suppose that either (a) u is bounded below
or (b) u satisfies for all t > 0 the integrahilit condition
JN i(t,
(3.1)
(27Za21)-N12 nN j exp -r L(Iu+,r,S)rN-' dr < +oo. 2 721
Ju
Then if +oc,. the process is a supermartingale, a martingale if a is harmonic. If u(t) = +oo, the assertion remains true for the parameter interval ]0, + xx F. If a is a harmonic polynomial, (3.1) is satisfied. Thus this theorem includes
the polynomial case noted in Section 2. If u is harmonic, case (a) is trivial because a one-sided bounded harmonic function on RN is identically constant according to Section 1.11.2.
To treat case (a), suppose that u is lower bounded and apply the superharmonic function inequality L(u, r. -) < u(-) to the equality in (3.1) with lul replaced by u to obtain f(t,-,tc)
(3.2)
633
3. Superharmonic and Harmonic Functions on R"
which is precisely the supermartingale inequality, so that is a supermartingale over the set of parameter values t making E{u[w(t)] } finite. If u(C) < + oo, the inequality (3.2) at C yields E{u[w(t)] } = t (t, C, u)
0 so that {u[w(t)],.f(t), t > 0) is a supermartingale:
(3.3)
_ (27ra21)-NIA (u, 1,C) + L(u, 1,C) < +oc.
Here we have used the monotoneity of the function L(u, , C).
To treat case (b), suppose that u is superharmonic and satisfies (3.1), which is simply the statement that Eflu[w(t)] 1} < + oo. Then (3.2) is again valid; so is a supermartingale as stated, with the parameter value 0 omitted when u(C) = +oo. Finally, if u is harmonic and satisfies (3.1), this discussion is applicable to both u and - u ; so is a martingale.
Application to Potentials
If N > 2, a superharmonic potential Gp of a measure µ satisfies the lower boundedness condition (a). If N = 2, a superharmonic potential Gp satisfies (3.1). In fact µ(R2) < + oc because G1t is superharmonic, and the following inequalities (3.4) and (3.5) show that (3.1) is satisfied.
ff
At, C, q)12WO filly-0:5 log I 1 - CI -5 (dC)
,1 R=
I (3.4)
µ(dC) f
I we shall now show that almost no path of a Brownian motion in Q8' hits a specified point q of Q8" at a strictly positive parameter value. It is sufficient to consider Brownian motions from a point of Q8N. The function u = G(q. ) is superharmonic on 118" with value + x at q. This function is positive when N > 2 and satisfies the integrability condition (3.1) for every point (q not
excluded) when N = 2. If
is a Brownian motion from f. the
process ; is therefore a supermartingale except that the parameter value 0 is to be excluded if S = q. This supermartingale is almost surely continuous, so (Section IV. I) almost every sample function is finite valued. Hence almost no path ever hits q except when = q and the parameter
value is 0. It follows that if N > I and if A is a countable subset of 68", almost no path of a Brownian motion ever hits A at a strictly positive parameter value. This result will be extended to polar sets A in Section 5.
635
4. Hitting of an F, Set
4. Hitting of an F. Set Let A be a subset of R", and let A) be the probability (if this probability is defined) that a Brownian motion from hits A at a strictly positive time. A) is defined and Borel measurable if A is an F. set (Section The function VI.6), defined and universally measurable if A is an analytic set according to
the same section. We suppose that A is an F. set in the present section to show how far one can go using relatively elementary methods. See Section 9
for A analytic; such generality will not be needed in the application to be made in Section 5. Lemma. If A is an F. subset of R" for N Z 2, the function harmonic, harmonic on R" - A.
A) is super-
A) is identically I when A is not empty.] If [If N = 1, the function N > 1, let B be a ball in R" with center , and let TT be the hitting time of aB by the Brownian motion from . Since Brownian motion is rotationally symmetric about its initial point, the distribution of ww(T{) is the A) uniform distribution pa(i , ) (harmonic measure) on B. To show that is harmonic on R" - A, suppose that B c R" - A. Apply the strong Markov hits A, property of Brownian motion to find that the probability that which is the same as the probability that
hits A after time TE, is given by
A) = E{u[ww(TS)] } = µ8(i;, u).
(4.1)
Thus A) has the harmonic function average property on R" - A and hence is harmonic there. To show that u is superharmonic on R' observe A - B) is harmonic that if A is an F. set, the set A - B is also an F. set, and on B according to what has just been proved. Since A - B) 5 A) and there is equality in the limit when B shrinks to its center because (Section 3) A) is lower semicontinuous at . Moreover 0, the function u(i;, A) is at least equal to the probability that a path hits A after time T,; so in the present context (4.1) is modified to the superharmonic function inequality A) z E{u[ww(TT)] } =
The function
u).
(4.2)
A) is therefore superharmonic on R", as was to be proved.
Parabolic Context Let A) be the probability, if this probability is defined, that a space-time Brownian motion from hits the set ,4 at a strictly positive time. In view of
the fact (Section VII.12) that parabolic measure on an interval boundary
636
2.1X. Brownian Motion and Martingale Theory
relative to a point of the interval is the hitting distribution on the boundary of a space-time Brownian motion from the reference point, the method of proof of Lemma 4 with the ball B replaced by an interval shows that when A is an F,, subset of A' for N 1, the function ti(-, 4) is superparabolic on AN, parabolic on AN - A. In particular, if A = A x R with A an F, subset of s), A) = RN, the set A is an F, subset of ft N, and A) for all s.
5. The Hitting of a Set by Brownian Motion Theorem. Let
be a Brownian motion on RN.
path meets A at a strictly
(a)
If A is a polar subset of RN, almost no positive parameter value.
(b)
If N > 2, lim,_. w(t) = ao almost surely. Hence almost every
(c)
Brownian path from a point of a Greenian subset D of RN either hits a finite boundary point of D or lies in D and has the boundary point w as a limit. If N = 2, almost every path hits every disk at arbitrarily large parameter values.
Observation (N = 2). According to (a), almost no Brownian path from a point of an open subset D of R2 hits OD if D is not Greenian. Conversely (to be proved in Section 10), almost every Brownian path from a point of D hits OD in a finite boundary point if D is Greenian. Proof of (a). If A is a polar subset of RN and N > 2, there is a positive superharmonic function u on R', identically + co on A (Theorem 1.V.2). Choose
6 > 0. According to Theorem 3, if is a Brownian motion from , the process {u[w{(t)],t z S} is a supermartingale, and according to the submartingale maximal inequality (Section 111.9) applied to the negative of this supermartingale, the restriction to the rationale in [6, 1/6) of almost every sample function is bounded. Hence the same assertion is true if with an arbitrary initial distribuis replaced by a Brownian motion tion. Since u is lower semicontinuous, this function is continuous at each of its infinities, and it follows that almost no path can hit A at a strictly positive parameter value. If N = 2, it will be convenient and sufficient to assume that A is bounded. According to Theorem 1.V.2, there is a measure u whose potential u = Gp is identically + oo on A, and a glance at the proof shows that u can be chosen with compact support if A is bounded. In view of the application to potentials in Section 3 the function u satisfies (3.1), and the proof of (a) now goes through as in the case N > 2.
Proof of (b). It is sufficient to consider a Brownian motion from a I'iJ-1+2 the function u satisfies the hypotheses point . If N > 2 and u(PI) =
6. Superharmonic Functions, Excessive for Brownian Motion
637
is a positive almost surely continuous superof Theorem 3 so that martingale, omitting the parameter value 0 if = 0. Hence (Section II1.15) lim,., u[w4(t)] exists and is finite almost surely; that is, lim,_m Iww(t)I exists almost surely, and this limit is + oo because (Section VII.5(c)) the limit is + oo in measure. B) be the probability that a Brownian Proof of (c). Let B be a disk, and let motion wt(-) from ever hits B. According to Section 4, the positive function B) B) is superharmonic on 982, and it follows (Section 1.I1.13) that is identically constant, necessarily identically 1 because this function is 1 on B. Thus almost every Brownian path from hits B at a strictly positive time. Ifs > 0, the process {wf(s + t), t e 98+ } is a Brownian motion with initial
distribution the distribution of wt(s), and so almost every Brownian path hits B after time s; that is, almost every Brownian path from a point of 982 hits B at arbitrarily large parameter values and therefore hits every disk with rational radius and rational center at arbitrarily large parameter values. Part (c) follows.
Part (c) implies that the cluster set of almost every plane Brownian path as the parameter value becomes infinite is the whole plane. According to Theorem 10 below, (c) is true with "disk" replaced by "analytic nonpolar set." Theorem 5 exhibits the special nature of dimensionality 2 in classical potential theory. Dimensionality I is also special, for example, in that only the empty set of R is polar; so Theorem 5(a) becomes trivial when N = 1. Theorem 5(c) is almost as trivial because almost every Brownian path is unbounded positively and negatively. On the other hand the cases N = 1, 2 are not special for the parabolic version of Theorem 5. Parabolic Context
Theorem 5(a) is true in the parabolic context, that is, almost no space-time Brownian path meets a parabolic-polar set at a strictly positive time. The proof of Theorem 5(a) is applicable, and simpler in the parabolic context in that the proof given above for N > 2 is valid in the parabolic context for N z 1. The parabolic version of Theorem 5(b) is trivial: if is a spacetime Brownian motion lim,_. ord w(t) = - oo. See Section 15 for the hitting of a parabolic-semipolar set by space-time Brownian motion.
6. Superharmonic Functions, Excessive for Brownian Motion Theorem. If u is a positive superharmonic function on the open subset D of RN, then u is excessive jor 6D, and GD(t, , u) < + oo for t > 0.
638
2.1X. Brownian Motion and Martingale Theory
See Section 8 for the converse theorem. The condition that u be excessive is
5 u,
(a)
GD(t, , u)
(b)
lim GD(t, , u) = 11,
(6.1)
t-o
and we show first that (6.1b) and the finiteness of GO, , u) follow from be a Brownian motion in D (6.1a). For each point of D let from , with lifetime S (Section VII.9); for example, this process can be a Brownian motion in U3 killed at the hitting time SS of 8D. We can assume is right continuous. Since u contains the null sets and that that is lower semicontinuous and positive and is almost surely continuous, Fatou's lemma yields
t-o
(6.2)
t-o
and therefore (6.1 a) implies (6.1 b). The finiteness of GD(t, 5, u) follows from co. If (6.1 a) if oe, let B be a ball with center S and closure
in D. The function tBu is positive and superharmonic on D, finite on B, equal to u on D - B. Apply (6.1a) to tBU to find that GD(t, ,
GD(t, , tBU) + GD(t, , u l B)
w,,(i)eBjwjT,J)},
n > 0.
The function T, is optional for because T, is the hitting time by the almost surely continuous process { of a closed set. The function T2 - T, is the hitting time of a closed set by the almost surely continuous process ([ww(Ti + ), w+',,(T1)], -,F,,(T1 + ')}
with state space D x D. Hence T2 - T, is optional for .,F,,(T, + ), and therefore (Section II.2h) T2 is optional for Proceeding by induction we find that T. is optional for for all n. In view of the strong Markov property the discrete parameter process {t',,( T ), .F,t(T ), n e Z+) is a Markov
process with state space D and stationary transition function q, where
6. Superharmonic Functions, Excessive for Brownian Motion
639
q(q, ) is the uniform probability distribution on tB,(q). Since q(q, u) = L(u,q,e A (Iq - aDI/2)), the process {u[w,(T)],.F,(T )} is a supermartingale. Obviously lim,,..m T. = S, almost surely. Choose i > 0 and define {min {n: T.
t}
k--
if S, > t otherwise.
1 +00
and lim,-o Tk = t almost everywhere where S, > t. If u[w,(T+m)] is defined as 0, the supermartingale inequality at times 0 and k (optional sampling theorem) yields
Then k is optional for
u(q) Z E{u[w,(Tk)] }.
(6.4)
When e -+ 0, this inequality yields, in view of the lower semicontinuity of u and Fatou's lemma, u(q)
E{u[w,(t)]; S, > t} = r.D(t, q, u);
(6.5)
so (6.1a) is true, as was to be proved.
Probabilistic Interpretation of Theorem 6 In probability language Theorem 6 states the following. Let u be a positive superharmonic function on an open subset D of R", let be a point of D, be a Brownian motion in R" from , and let SS be the hitting let Then the process time of 8D (Euclidean boundary) by {U[w (t)]I}s4>r) ;
{(t),tER+}
is a supermartingale except that the parameter value 0 is to be omitted if is a Brownian motion in D from u(4) = +oo. Equivalently, if , with lifetime S4, in which case wi(t) has domain of definition {SS > t} when t > 0, we can rephrase the result by stating that {u[w{(t)];
is a supermartingale if defined as 0 for t value 0 is to be omitted if u(t) = +co.
t E R+ }
SS, except that the parameter
Parabolic Context
Theorem 6 and its probabilistic interpretation and proof translate directly into the parabolic context except that balls in the proof of Theorem 6 are to be replaced by intervals.
640
2.IX. Brownian Motion and Martingale Theory
.t be a Brownian motion in 08' from the origin, let So be a point of 08' other than the origin, and define u(s) = I - oI-'. Then EXAMPLE. Let {
u is a positive superharmonic function on 081, and (by Theorem 3 or Theorem 6) the process is an almost surely continuous supermartin-
gale, trivially right closable by 0. This supermartingale has the following properties. (a) is L2 bounded (and therefore uniformly integrable because the function ri-'r2 is a uniform integrability test function on 08+). In fact, if B = B(to, and if e(t, ) = 6(t, 0, ), then
dP = J J (W(fIE R'-S)
f JIw(1)EBI
(b)
e(t, ) I v -B
uz[w(t)]dPS f f(t. 2 - oI JB \
13 (d )= 3I
\
2
/
lim, u[w(t)] = 0 almost surely because [by Theorem 5(b)]
lim,_,,w(t) = co almost surely. and let Td be the hitting time of (c) Choose 6 < Then, under the convention that u[w(+ec)] = 0, the process
{u[w(T, A t)];.F(t),te08+}
S) by w(-).
(6.7)
is a martingale, trivially bounded and therefore uniformly integrable. In fact the process (6.7) is a supermartingale because it can be obtained by stopping at time T6 the supermartingale right closed by 0, and the process (6.7) is a submartingale because 2/6 - u[w(Ta A )] is a supermartingale on the parameter set 08+. The latter process is a supermartingale because if is w(-) killed at T,,, to obtain a Brownian motion in 08' 6/2), then the compoB(o, 6/2) and if u' is the restriction of it to 08' sition 2/6 of the positive harmonic function 2/8 - uwith is a supermartingale which stopped at time Ta yields 216 - u[w(Ta A )]. [Assertion (c) is a consequence of Theorem 7(c) below because u' is a bounded harmonic function on 083 - B(o, 6) and is continuous on the closure of its domain, but it seemed more natural to prove (c) at this stage.] (d) E{u[w(Ta)] } because this is the martingale equality for the process (6.7) at times 0, + oo. (e) {u[w(t)];,f(t), te08+} is not a martingale because if this uniformly integrable process were a martingale it would be right closable by its limit
0 at i = + co (Section Ill. 14) and would therefore almost surely vanish identically.
(f) The process in (e) is a supermartingale potential because it is uniformly integrable; so there is L' convergence to 0 in (b). Ta = + co (g) The process in (e) is a singular potential because almost surely since (Section 3 or Theorem 5) almost no w(-) path hits o,
7. Composition of a Superharmonic Function with Brownian Motion
641
and therefore the process in (e) is a potential which in view of (c) is a local martingale and as such (Theorem V.12) is singular. (h) The process in (e) is not in the class D. In fact, if this process were in D, we could go to the limit (h -+ 0) in (d) to find, since lim P{u[w(TT)] = 0} = 1,
the false evaluation
0.
7. Preliminary Treatment of the Composition of a Superharmonic Function with Brownian Motion; A Probabilistic Fatou Boundary Limit Theorem Theorem 6 makes it possible to analyze by martingale theory the composition of a superharmonic function with Brownian motion. "Preliminary" in the section title refers to the fact that although in Theorem 12 below it will be shown that almost every sample function of such a composition is continuous, only right continuity will be proved in this section. Specifically, the process in (7.1) below is not only almost surely right continuous, as proved in this section, but is in fact almost surely continuous, and the process in (7.1) is almost surely continuous except for a possible jump discon-
tinuity at the parameter value S. Theorem 7 is also preliminary in that (Theorem X.8) it is valid in the more general context of relative superharmonic functions and conditional Brownian motion. In Section 1.XII.19 it was proved that if h is a strictly positive harmonic function on a Greenian subset D of R N and if v is a positive superharmonic function on D, then v/h has a minimal-fine limit at Mh almost every point of the Martin boundary of D. We shall see (Theorem 3.111.4) that an equivalent probabilistic formulation of this theorem is Theorem X.8, which does not involve a boundary and which states that v/h has a limit along certain conditional Brownian paths. When h = 1, this result is part (a) of the following theorem.
Theorem. Let D be a Greenian subset of R", let be a point of D, and let be a Brownian motion from . It is supposed that 9,(0) contains the null sets. Define
S4 =sup{t>0:w{(s)eDfors 0. Then s) is the probability that a space-time Brownian motion in b = R' x ]0, +oo[ from hits A = A x ]0, +oo[ during the process lifetime at a strictly positive time. It follows (Section 9) that ti is superparabolic on b, parabolic on D - A. In particular, if A is closed define Do = 68" - A, Do = Do x ] 0, + oo [. The parabolic version of Theorem 13 applied to the restriction ti of u to Do shows that tto = Do x ] 0, + x[).
14. Probabilistic Evaluation of Reductions Let D be a Greenian subset of R", let be a point of D, let
be a Brownian
motion process from , and denote by T'A [TA] the entry [hitting] time of a set A by
Theorem. If A is an analytic subset of D and if v is a positive superharmonic function on D, then (reductions relative to D)
E{v[w,(T't)]; T'. < T-°} E{v[w,(T )]; TA
T, °, then is an almost surely right continuous supermartingale and O(A) = E{x(T{")}; so the fact that O (B) >_ 4(A) when A c B is the supermartingale inequality for at the pair of times T{', Tc". According to Section 1.VI.5, if 4o(A) is defined as the left side of (14.1), 0o is a Choquet capacity on D, that generated by a topological precapacity, the restriction of 0o to the class F of compact subsets of D. Since 0 is an increasing set function, equal to 0. on r, and since A. T A implies that O (A.) j 4(A), it
follows first that when A is open, ¢o(A) = O(A), second that when A is analytic, ¢o (A) = sup { Oo(B) : B c A, B compact)
=sup {¢(B): B c A, B compact} < O(A), and finally when A is analytic,
¢o(A) = inf {40(B): B
A, B open} = inf {¢(B): B c A, B open} >t 4(A).
Hence 0 = 0o for A analytic, as asserted in (14.1). Equation (14.1sm) is true because (14.1) with A replaced by A becomes (14.1sm) when
r-p0. If (14.1 sm) is applied to v = G°(p, ), the right side of (14.1 sm) becomes G°i.(ry), where ). is the distribution of w,(T,) for T{ < T4°, so that (14.Ism) becomes the equation defining the measure Sp (ry, ). Hence (14.2) is true. Classical Reductions Composed with Brownian Motion
Let D be a Greenian subset of IBN, and let be a Brownian motion in D from , with lifetime S. It is supposed that JF4(0) contains the null sets. Let v be a positive superharmonic function on D. Then
655
14. Probabilistic Evaluation of Reductions
lim u[ww(s)] SfSS
exists almost surely, and we define v [wf(1)] for SS < t < + oo as this limit, fir( )} is an almost surely continuous superso that (Section 7) martingale. Let A be a Borel subset of D, and define ,4 = { (t, a)): wt(t, w) a A}. The set A is nearly predictable because is a nearly predictable process.
In the classical context R' is a positive superharmonic function on D, and we define
RA[w,(t)] = lim RA [w,(s)]
for S4:5 t < + oo. The supermartingale reduction 1 is an almost surely continuous supermartingale, and Theorem 14 combines with Theorem IV.20 to show that this supermartingale is indistinguishable from +vlw
Generalization. Let A and B be analytic subsets of D, and let T{ .B be the hitting time by of B after hitting A but before hitting 8D,
TT B=inf{t: TT 1. Itr(')I2 < 2ca 21 log I log 1
(15.3)
almost surely for all sufficiently small strictly positive values of t, depending
on the Brownian path. but when c = 1, there are almost surely arbitrarily small strictly positive values oft for which there is equality in (15.3). This result is a restatement in probability language of the significance of Section 1.XVIII.6. Examples (d) and (e). To see this, observe that on the one hand according to Example (d) when c = 1, the open set
!)_
0, t > 0, n e D. Hence for fixed the left side of (18.3) increases as t decreases. Define u'(n) as the limit of this left side when t -+ 0. Then u' S u !N almost everywhere on D, and there must be equality IN almost everywhere because the left side of (18.2) has limit u(A) when t - 0. The function u is uniquely defined up to an IN null set, and we now replace u by u'. This change does not affect the left side of (18.3); that is, we now have u' = u, and u is obviously excessive. Conversely, if u is a t -excessive function and is not identically + oc, that is, if u is a positive superharmonic function on D, then u is locally IN integrable, and using again the symmetry of GD(t, , ), we find that
u()GD(t, , 0!N(d) 5 u(n)
(18.5)
with equality in the (monotone) limit when t -- 0. Integrating 1A - IN(dn) yields the fact that the measure p given by (18.1) is GD-excessive.
665
Is. Exmsivc Measures for Browntan Motion
Space-Time (Parabolic) Context. Let D be a nonempty open subset of tfd". We use the notation introduced in Section Y11.2: if A is a subset of G'8",
be the a define A, = {n: (n, 1) e,4) and A, = A, x {t}. Furthermore let algebra of subsets A of b for which A r),6, is a Borel set for all t. A measure t) of Borel subsets of D, µ on the a algebra 2 determines a measure by way of µ(A, t) = µ(A x It)). Conversely, if for each t there is a measure t) of Borel subsets of D a measure 4 on 2 can be defined by setting (18.6)
u(A) = E µ(A,, t)
with the obvious convention for uncountable sums. In the following "measure on !P" refers to a measure defined in this way, but we make no hypotheses
t) on any class of sets. A measure µ on I
on the finiteness of ti or of
is Go-excessive if and only if whenever A e b, Dp(dd, s) J Vin(( s
,
s), (n,
t))lx(dR) 5 µ(A,, t)
(s > t),
(18.7)
A,
and there is equality in the limit when t T .s. A 4-excessive measure is a measure on !' satisfying the obvious dual condition. The counterpart of Theorem 18 in the parabolic context can now be stated : a measure µ on 2 is a 4- [16-1 excessive measure if and only if there is a 6D- [GD-] excessive function ti such that for A e
t)lx0l).
µ(A,, t) = I
(18.8)
A,
In the one direction let ti be a Go-excessive function, so that 0 < ti < + m, u is 2 measurable, and s), (n, t))u(n, t)lv(dn) s
s)
(s > t)
(18.9)
f', with equality in the limit when t T s. Define a measure µ on 2 by (18.8) and (18.6). In view of the duality relation VII(10.6) between GD and e the inequality (18.9) implies that for AE, u(dn, t) J
I
D,
ri((7, t), (S, v))lx(d) 5 µ(A,, s)
(18.10)
A3
with equality in the (monotone) limit when t T s; so u is 4-excessive. Conversely, if 4 is a tD-excessive measure, then following the argument for the classical case as given in the first part of this section it is not difficult to see, but more so than in the classical case, and we leave the details to the reader, that there is a 6,5-excessive function ti satisfying (18.8).
666
2.IX Brownian Motion and Martingale Theory
The dual assertion for G.-excessive functions and lo-excessive measures is derived in the same way or reduced to the one just proved by a reflection of A' in the abscissa hyperplane.
19. Nearly Borel Sets for Brownian Motion In a general context a set A in the (topological) state space of a stochastic transition function p is called nearly Bore! if for every initial distribution it on the state space and an almost surely right continuous Markov process
with transition function p there are Bore] subsets A and A of the state space for which A;, c A c A and for which almost no path hits A',' - A', at any strictly positive time. The subtleties of this definition are needed in more general contexts but are quite unnecessary in the Brownian motion context according to the following theorem. Theorem. A set A is nearly Bore! for Brownian motion in an open subset of R' if and only if A differs from a Bore! set by a polar set. We can suppose that we are dealing with Brownian motions in a connected
Greenian set D. If A is nearly Borel and if p is supported by the singleton the difference A - A. is a Borel set whose hitting time by a Brownian motion from S is almost surely + x so that by Theorem 14 the smoothed reduction of the function I on the difference set is identically 0. Hence (Theorem I.V.4) this difference set is polar; so A differs from A', by a polar set. Conversely, if A differs from a Borel set by a polar set, it can be supposed
that A = Ao u A,, where A is a Borel set and A, is polar. and if so, the sets A, and A can be chosen, respectively, as A and the union of AO with any polar G,, superset of A, .
Near/v Borel Sets for Space-Time Brownian Motion. The parabolic context counterpart of Theorem 19 is a direct translation.
20. Brownian Motion into a Set from an Irregular Boundary Point Let D be a Greenian subset of I8'v, and let aD be the Euclidean boundary. Let 5 be a finite irregular boundary point of D. let be a Brownian motion from C, and let S be the hitting time of aD by is,(-). According to the fine topology criterion of irregularity of a boundary point (Theorem 1.XVIII.7) and the corresponding probabilistic criterion by way of Theorem 15, the random variable S is almost surely strictly positive. Let be the process killed at time S. Then { tr'(t), t > 0} is a Markov process with state space D and transition density (D. Now let v be a positive superharmonic function
667
20. Brownian Motion into a Set from an Irregular Boundary Point
on D. A trivial adaptation of Theorem 8 shows that the process {v[w'(t)], t > 0}, if defined as 0 for t >_ S. is almost surely continuous for t < S and is a positive supermartingale if the expectations of the process random variables are finite. If we replace v by v A c if necessary to ensure finiteness of these expectations and then let c tend to + oc, we find by the backward supermartingale convergence theorem that v has a finite or infinite limit along almost every path back to C, that is, v has a fine limit < + or, at C. We have thereby obtained a probabilistic proof of the limit result in Section I.XI.21; a similar argument yields the corresponding parabolic context theorem in Section I.XVIII.17. For fixed ry in D the function = s) i-+GD(s, , ry) on D x a8 is super-
s > so} for each strictly
parabolic and positive and is bounded on
positive so ; so for t > 0 and rt E D the limit pf limS.tt, 0) f.D(t + s, . q) exists and is finite. Denote this limit by GD(t, C, ?I). Equivalently, there is a deleted parabolic-fine neighborhood A of (Z, 0) such that lira
GD(t + s,, q) = GD(t, C, ry).
(20.1)
. 3h-(4.0)
Here A depends on it = (?I, t), but in view of the parabolic version of Lemma
1.X1.8 we can choose A so that (20.1) is true for , in a countable dense
subset of D x ]0, +oo[. The parabolic context Harnack theorem now implies that (20.1) is true for all 7 in this product set and that the convergence is locally uniform so the function h --+GD(t, (, q) is parabolic on this product
set. Finally we show that GD(t, C") is the distribution density of w'(1). To see this, observe that if f is a continuous function on D with compact support, lim
S-0
GD(t - s, w'(s), D
i) =
I
D
GD(t
a.s. (20.2)
in view of the fine limit result we have just obtained. The integral on the left is a version of E{ f [w'(t)]Jw'(s)}, and in view of the Markov property of Brownian motion this conditional expectation defines a martingale for 0 < s < t, t fixed, and an application of the conditional expectation continuity theorem and the 0-1 law of Brownian motion shows that when s -+ 0
sequentially the almost sure limit of this conditional expectation
is
E{ f [w'(t)] }. This fact combined with (20.2) shows that GD(t, C, ) is the stated density. The corresponding result in the parabolic context is left to the reader.
Chapter X
Conditional Brownian Motion
1. Definition Let D be an open subset of RN, and recall that a Brownian motion in D was defined in Section VI I.9 as an almost surely continuous Markov process with state space D and transition density Go. The probability space on which the process is defined may or may not be rich enough to extend the process to be a Brownian motion in RN.
We now define a Brownian motion in D conditioned by an arbitrary strictly positive superharmonic function h on D. If D is not Greenian, for example, if N = 2 and 982 - D is polar, the definition yields nothing new because then h is necessarily a constant function and conditional Brownian motion reduces to Brownian motion. We therefore suppose below that D is Greenian. Following Section VI.13, define D" = {h < + cc), and define h(i7)
0
if ( , 7) a D" x D
if ( ,7)e(D - D") x D.
(1.1)
Observe that the fact that h is GD-excessive implies that GD(t, , ) = 0 IN almost everywhere on D - D" when e D". An h-Brownian motion in D is a Markov process with state space D, transition density dD, and initial distribution supported by D. After proving in Section 2 that there is always a continuous process satisfying these conditions, we shall add to the definition the condition that the process be almost surely continuous. It will be shown
that then almost no sample path leaves D". Thus a 1-Brownian motion in D is what we have defined as a Brownian motion in D. The Notation P" and E" In previous chapters we have used the notation P and E generically: whenever a probability space was encountered, P and E referred to probabilities
and expectations, respectively, on the space, and if several probability spaces were encountered simultaneously, this notation P and E was used for every space. In the first sections of the present chapter this notation
669
1. Definition
could be confusing because we shall deal with h-Brownian motion and Brownian motion simultaneously. To avoid confusion, we shall use P and E for Brownian motion but Ph and Eh for h-Brownian motion. (If h is identically constant, h-Brownian motion reduces to Brownian motion, and so the superscript will be omitted.) After Section 8 of this chapter, however, and throughout later chapters the special notation Ph and Eh will be used
only when necessary to avoid ambiguity. Ordinarily the notation for the for h-Brownian motion, will warn processes involved, for example, space refer to h-Brownian motion the reader that P and E on the probabilities and expectations. Existence and Lifetime of h-Brownian Motion
According to Markov process theory, if e Dh, the transition density ell, determines an h-Brownian motion from . The distribution of the process lifetime SS is determined by =GD(t'''h)
Ph{SS > t}
h()
Ph{Se = +ao} = lim 1^aC
'
(1.2)
NO
The function ho = lim,-,o GD(t, , h) is an invariant excessive harmonic minorant of h [Section IX.8, Example (a)]. Thus if h is a potential, the function ho vanishes identically, so almost every h-Brownian path has a finite lifetime. If h is a strictly positive superharmonic function on D which is IN integrable over D, for example, if D is bounded and h is a bounded function, then h P h {St= + ao}
0, even when x, implies that then also t (t, S, u) < + oc ; so if u(c) = + oo, the process {u[w' (t)], 0 < t < + oo If is a supermartingale. We shall see in Section 2 that these supermartingales are almost surely continuous aside from a possible jump discontinuity at the parameter value S{, under the condition that is chosen (as we shall see is possible) to be almost surely continuous. if
h-Brownian Motion When h Is Harmonic In particular, if h is a strictly positive harmonic function on D, the h-Brownian
motions play the same role for h-harmonic and h-superharmonic functions that Brownian motions play for harmonic and superharmonic functions. For example, when h is harmonic, a fo-excessive function is, on each connected open component of D, either identically + x or h-superharmonic. from a point The fact that almost every path of a Brownian motion of D [define w{(+ oo) = oc if N > 2] hits the (Euclidean) boundary implies
that almost every path of a Brownian motion in D tends to a boundary point at the path lifetime, but the latter property does not hold for hBrownian motion in D for general harmonic h. We shall show, however, in Section 4 that when h is harmonic almost every h-Brownian path from a
point of D tends to D. although the path cluster set on 8D may not be a singleton.
Canonical Conditional Brownian Motion Let D and h be as above. In Section VII.9 we defined a canonical Brownian motion in D, on a space 6. A canonical h-Brownian motion is defined
similarly, but there is one essential difference: when h a 1, almost every
2. h-Brownian Motion in Terms of Brownian Motion
671
path of an h-Brownian motion in D tends to a point of the Euclidean boundary of D at the path lifetime, but this is not true for all choices of h. Hence we must change the definition of el in that we drop the condition in Section VII.9 that the function oh is to have a left limit in the Euclidean topology of R at the lifetime . (6*h). When canonical h-Brownian motion is discussed with no restriction on h, we shall mean the present definition even though the case h e I is not excluded. In the next section we shall show that h-Brownian motions exist for every choice of h and are almost surely continuous up to their lifetimes. If is such an h-Brownian motion in D, on a space f2, the existence of a canonical h-Brownian motion in D with the same initial distribution (supported by D") follows easily by mapping. In fact the map w F(o) = th takes a point w of fl into a point u, of 6 (we assume that the processes have a common trap point adjoined to D), and the iv(-) measure on 92 thereby induces a measure on 6 under which the coordinate function process is the desired canonical h-Brownian motion. Parabolic Context
The translation of the discussion in this section into the parabolic context is left to the reader. Observe, however, that in the parabolic context an example in Section IX.12 shows that the composition of a superparabolic function with space-time Brownian motion, almost surely right continuous with left limits, may have a dense discontinuity set which is the same for every sample function.
2. h-Brownian Motion in Terms of Brownian Motion In this section we derive relations between h-Brownian motion and Brownian
motion in order to reduce h-Brownian motion properties to corresponding Brownian motion properties. Throughout the section D is a Greenian subset of 68", h is a strictly positive superharmonic function on D, and is a point of D". The process is a Brownian motion in D from with lifetime S defined on a filtered probability space (S2, P), and the process is an h-Brownian motion in D from with lifetime Ph) Observe that S" defined on a filtered probability space (S2", f"
{w(t)eD} = {S> t),
{w"(t)eD} = {S"> t}.
A trap state is supposed adjoined to D to make the transition functions stochastic, and h is defined as 0 at the trap state.
We wish to prove the following, in various contexts. Let T be an fl-) optional optional time on S2, and suppose that A e f(T). Let T" be an time on (2", and suppose that A"e.F"(T"). We wish to show that if T and
672
2.X. Conditional Brownian Motion
A are defined in the same way in terms of sample functions as T" and A" are defined in terms of sample functions, then
P"{A"n{S">T"}}=
h[w(T)] LIS>T)
Although we shall only need very special cases of (2. 1), its intuitive meaning clarifies the subject so we prove (2.1) under very general hypotheses.
Case (a) of (2.1): T - t for some strictly positive constant t. Define a measure Q` on the measurable space (el,.9 (t)) by
Q`(A)=
J.
h[w(t)]h
(2.2)
1s>11
to obtain a measure space on which the process {w(s),s < tis almost surely continuous. Suppose that 0 < i t
t}, M" n {T" > t}) is a linked pair in the same sense because of the simple expression for the probability of hitting an F, set (cf. Section VI.6). Hence Case (a) of (2.1) yields
P"{M" n {S" A T" > t}
h[xt(t) ]
(2.6)
In particular, if Do is an open proper subset of D containing , then (2.6) with B = D - Do shows that w"(-) killed at T" has the same finite-dimensional
distributions as hIDo Brownian motion from , up to process lifetimes, that is w"(-) killed at T " and then provided with a trap state is an hi00 Brownian motion in Do from . Observe that since the almost surely continuous process {w(s),s 5 t}us}
674
2.X. Conditional Brownian Motion
under Q` and the almost surely continuous process {w"(s),s S t}ilsh,t) have the same finite-dimensional distributions, it follows that these processes have the same probabilities of hitting an analytic subset B of D during a specified parameter interval (Section I.12), and more generally the same argument shows that the same assertion is true if the processes are restricted to sets M and M", respectively, linked as described above. That is, if T [T"] is the hitting time of B by then (2.6) remains true. Representation of an Arbitrary h-Brownian Motion in terms of a Brownian
Motion. For t > 0 define a measure Q` on 3F'(1) by Q'"(M") =
h() JMh(Sb>rlh[w(t)]
(2.7)
and note that QV'(M") = 0 if and only if P" { M" n {S" > t)) = 0. The process {w"(s),s 5 1}1(s">,) under Q` has the same finite-dimensional distributions
as the restricted Brownian motion process jw(s),s S t}ils>ti. We can now complete a full circle in the discussion : the integral h["'"(t)]dQ'h
>
JMhSh>r)
whose value is P"{ M" n {S" > t} } when expresses wh(-) process probabilities to time t under the side condition S" > tin terms of a Brownian
motion in D under the corresponding side condition. The important point is that this Brownian motion and the h-Brownian motion involve the same probability space and the same random variables; only the measures are different. In particular, if u is a function on D, then for parameter values < t is not only a function of under P" but also under Q", and we can thereby deduce properties of u on h-Brownian motion from the known properties of u on Brownian motion. The following assertions are immediate consequences of the representation of h-Brownian motion in terms of Brownian motion that we have obtained. (a) If A is a polar subset of D, almost no path of an h-Brownian motion in D hits A at a strictly positive parameter value. In particular, almost no such path hits D - D".
(b) A superharmonic function r on D composed with h-Brownian motion in D yields a process almost surely continuous up to the h-Brownian motion lifetime with finite-valued sample functions except possibly at the parameter value 0.
We have already pointed out in Section 1 that if v is a positive superharmonic function on D and if u = v/h (= 0 where h = +oo), then is a supermartingale if defined as 0 for parameter values z S" and with the oo. understanding that the parameter value 0 is to be omitted if
675
2. h-Brownian Motion in Terms of Brownian Motion
We now see from (a) and (b) that this supermartingale is almost surely continuous except for a possible discontinuity at S". For almost every sample function the supermartingale right limit exists at S" trivially and is 0; the left limit at S' must exist (finite) almost surely because almost surely right continuous supermartingales almost surely have finite left limits
at all parameter values, including the parameter value + oo if the supermartingale is positive. (c)
h-Brownian motion filtrations and the strong Markov property.
Theorems VI.8 and VI.9 are not directly applicable to an h-Brownian motion
unless h is finite valued, in which case D" = D, but the proofs of these theorems are applicable to the h-Brownian motion context. Hence, if ."(t) is is generated by the null sets and .F{w"(s), s 5 t}, it follows that right continuous. Moreover, even without this special choice of filtration, the strong Markov property VI(9.l) holds for These assertions are of course true not only if x,"(0) _ but for an arbitrary initial distribution supported by D". (d) The zero-one law for h-Brownian motion. Let -Ft be the a algebra
generated by nr,o F{tv"(s, s 5 t) and the null sets. Then [assuming as above that tv"(-) is an h-Brownian motion in D from
in D"] .fit is trivial; that is, it consists of the null sets and their complements. Two proofs of the zero-one law for Brownian motion were given in Section VII.6; the second one is applicable to h-Brownian motion for all h. The zero-one law implies that if B is an arbitrary analytic subset of D and if T is the hitting time of B by w"(-), then the probability P{T = 0) must be either 0 or 1. The representation of h-Brownian motion in terms of Brownian motion shows that this probability does not depend on the choice of h, and it follows
(Theorem IX.15) that this probability is if and only if is a fine limit point of A. Roughly, the initial character of h-Brownian motion from a point does not depend on the choice of h. Because of this fact, assertions (c)-(f) of Section VII.6 are valid for h-Brownian motion for all h. 1
h-Brownian Motion at Path Lifetimes [Notation D, , Above]
as
is a Brownian motion in 1V from and if So is the hitting time of the If path killed at Euclidean boundary aD by wo(-), then almost every So tends to wo(S0) at the parameter value So. It follows that almost every path tends to a point of OD at the path lifetime, but it does not follow from the relation between Brownian motion and h-Brownian motion that almost every path tends to a point of aD at the path lifetime. In fact,
when h is a potential, we shall show in Section 4 that almost every h-Brownian
path in D tends to a point of D at the path lifetime, and we shall find the distribution of this limit point. When h is harmonic, it will be shown that almost every h-Brownian path in D tends to aD but not necessarily to a point of aD, at the path lifetime.
676
2.X. Conditional Brownian Motion
3. Contexts for (2.1) As already explained in Section 2, the first problem in finding a proper context for (2.1) is to find appropriate linked pairs (T, T') of optional times and linked pairs (A, A") of sets. The point is that T" should depend on in the same way T depends on and A" should be a set depending in the same way on up to time T" as A depends on up to time T.
In Section 2 the situation was simplified by choosing T" and T to be the same constant function. In this section T" and Twill be nontrivially optional. Case (b) of (2.1): and are canonical, T = T", A = M. Let I be a canonical h-Brownian motion in D from in D, on the
probability function space (&,P, p") When h = 1 , we omit the superscript. Then Define
and the process lifetimes S', .
do(t) _{H(s).s < t} _"(t),
are identical.
A,(+ -0) = Y .o(t) = A011(+ 00). r>0
The basic filtration in this case is A o'(-). Let t bean arbitrary A,'(-) optional
time and let A be an AV) set. We make the obvious choices of fi" and R", namely t" = t and R" = A, and prove (2.1) in this context. Observe that, with the optional time notation defined in Section 11.2, Example (b), and using the fact that A
j2_"
. {9" >
=
[fi"]n} eAo(j2-"),
(2.6) yields
p"{R"
Y
>
j=o
_
P"{A"
{S'" > j2
f
= [fi"]n} }
h[K'(j2_-)]d Pzz
Y_
J=O n"r,{$>j2-n=jf).)
NO (3.1)
=J When n
n "nIJ`>ftt,)
oc, the left side of (3.1) tends to p" {tt" n { 9" > th) }. The right
The sequence ..., h[w([t jt)], ordered as written is a supermartingale, left closable by h[w(t)] and side is
(Theorem 111.17) uniformly integrable; so there is L' as well as almost everywhere convergence to the limit h[w(t)]. Hence we can integrate to the limit (n - oc) in (3.1) to obtain (2.1) in the present context. Application of Case (b). If
[h-Brownian] motion in D from
is a Brownian on the probability space ((),.F, P)
4. Asymptotic Character of h-Brownian Paths at Their Lifetimes
677
[(S2'' " P")] with lifetime S [S"], the problem of finding large classes of pairs (T, T"), (A, A") for which (2.1) is true is partially solved by a reduction to case (b). In fact, if we make the convention that all processes involved have a
common trap, the maps co i- w(-, w) = th, w" H w take 0 and i2" into L = ?", the space of a canonical Brownian motion in D from . In the notation of case (b), an optional time ton 1) has as inverse images under these maps an optional time T on S2 and an optional time T" on 0" if we suppose, as we can, that are right continuous. A set and A in .00'(t) has as inverse images a set A in and a set A" in .F"(T") and (2.1) is true as written if T, A, P. and A" are obtained in this way because (2.1) is true in the context of case (b). For example, if t is the hitting time by of a closed subset of D, then T [T"] is equal almost surely to the hitting time of this set by Generalization of Case (b) for (2.1)
(Not used below.) It is natural to try to extend Case (b) and thereby the application just noted to allow for the P and P" null sets. This generalization is needed, for example, if T is to be the hitting time by of an arbitrary analytic or even merely Borel subset of D. More specifically define .O(t)
[A"(t)] as the a algebra generated by Ao(t) and the P [P"] null sets. According to Section 2, the filtrations and ^-) are right continuous.
If t is an .
optional time and if Ae.*(t), there are [from Section
11.2(j)] an
optional time fio and an Ao(1o) optional time Ao such that,
up to a P null set, to = t and AO = A. Now according to case (b) of (2. 1), o (to) subset of {9 > to} is P null if and only if it is P" null. Hence, up to a P" null set, t = t and Ao = A. Thus if we define T = 11, T" = to, A = A, A" = AO, equation (2.1) is true in the canonical process context, generalizing case (b). We could equally well have started with an optional time t" and an .?"(t") set A". an
4. Asymptotic Character of h-Brownian Paths at Their Lifetimes Theorem. Let D be a Greenian subset of R"', let h be a strictly positive superharmonic function on D, and let be a point of D". (a)
If h is harmonic, almost every h-Brownian path from g tends to 8D at the path lifetime. (b) If h = G0µ is a superharmonic potential, almost every h-Brownian
path from c has a finite lifetime and tends to a point C of D at its lifetime. The distribution dC) _ of C is given by If A is polar, A n D") = 0.
678
2.X. Conditional Brownian Motion
Observation (1). Part (a) of the theorem does not state that when h is harmonic, almost every h-Brownian path from tends to a point of aD, and in fact it will be proved in Section 3.11.2 that if aD is obtained by a metric compactification of D, then almost every h-Brownian path from tends to a point of OD if and only if OD is h-resolutive, and in that case the distribution of the path limit point on aD is OD We can already verify this result in a special case, when h = I and is the Euclidean boundary, resolutive according to Theorem I .VIII.4. In fact, if a Brownian motion in 08" from 6 in D is killed at the hitting time of the Euclidean boundary OD to obtain a Brownian motion in D, almost every path of the Brownian motion in D obtained (without loss of generality for the present purpose) in this way tends to a point of OD at the path lifetime, namely to the point at which the original Brownian path first hits aD. The distribution of this hitting point on aD is according to Theorem IX.13. Observation (2). We shall prove a stronger result than Theorem 4(b): h-Brownian paths from can be obtained in this case by first choosing the asymptotic endpoint for a path according to the probability distribution and then choosing a 0)-Brownian path from (necessarily tending to 5 at the path lifetime). It follows, as stated in (b), that if A is polar. v((', A n D") = 0 because (Theorem I .V.11) h = GDµ = + co p almost every-
where onA;sop(AnD")=0. be an h-Brownian motion in D from 5. with lifetime S", let Do be an open relatively compact subset of D containing , and let Proof of (a). Let
The superscript h is omitted throughout
T" be the hitting time of aD by
when h = 1. It is trivial that P{S > T } = 1. According to (2.1) under case (b) as applied in Section 3, if h is harmonic, dP
P"; S" > T"} = ;s>ri
h[w(T)]h)
E{h w(T)]}
h() (4.1)
I<no( h)
= I.
h()
path hits aDo. To prove (a), we prove that if Cis a of C compact subset of D and if now T" is the first hitting time by after hitting aDo, then P"{S" > T"} is arbitrarily small if D0 is sufficiently Thus almost every
large. With this definition of T" (2.1) yields Csup h}
P"{S"> T"}
T}
C
Now the probability on the right is the probability that a Brownian path in D from f hits C after hitting aD0 and this probability is arbitrarily small
679
4. Asymptotic Character of h-Brownian Paths at Their Lifetimes
for sufficiently large Do since almost every Brownian path in D from tends to the boundary at the path lifetime. Hence (a) is true. Proof of (b). We have already seen in Section 1 that when h is a potential almost every h-Brownian path has a finite lifetime. If Do is an open relatively
compact subset of D containing
and if T' is the hitting time of OD, by
then (2.1) yields
P" {S" > T") =
ItDo(S,h
h(C)
as in the proof of (a). Since his now a potential, the right side of this equality
can be made arbitrarily small by choosing Do sufficiently large (Section I.VIII.I I); so almost every h-Brownian path from has closure a compact C) for some point subset of D. Now consider a particular case, h = path tends C of D other than c. We show that in this case almost every to C at the path lifetime. To see this, define D' = D - {C}, and denote by h' the restriction of h to D'. The function h' is harmonic on D', and h'Brownian motion in D' from can be identified with h-Brownian motion path tends to in D from g; so it follows from (a) that almost every OD' = {C}u aD at the path lifetime. Since we have just proved that almost path has closure a compact subset of D, it follows that almost every every h-Brownian path from tends to C at the path lifetime. More generally, if h = GDµ,
G (t, , 7) = J 1 D(f, , 7) GD(, D
,
C)]
dC),
(4.2)
and if 0 < t, < . . . < t,,, the joint density of v"(t,), ... , Who.) is v(S,dy
f -D(t1+S,S1)GD(t2 -
1,b11S2)...4D(tn-
D
tn-1,Sn-1,
n)GD(SYYM. O
GD(b, 0
C)
(4.3)
This expression shows that an h-Brownian path from can be obtained by first choosing the asymptotic path endpoint C according to the probability C)-Brownian path from , dd) and then choosing a distribution as stated in Observation (2) above. The formal statement of this construction in the language of conditional probabilities is left to the reader. Part (b) of the theorem follows from this construction. h-Brownian Motion for General h
Let h be a strictly positive superharmonic function on D, so that (Riesz decomposition) his the sum of a positive harmonic function h, and a potential
680
2.X. Conditional Brownian Motion
h, . The evaluation (n (t, , tl) =
ht
(t, , n) +
7)
(4.4)
of fD in terms of fo and 60= can be interpreted as follows: to obtain an h-Brownian path from [when x], choose an h;-Brownian path with probability Thus the probability that an h-Brownian path in D from tends to a point of D at the path lifetime is h2()/h(S), and the distribution of the asymptotic endpoint is determined as described in (b), whereas the probability that an h-Brownian path in D from tends to the boundary of D at the path lifetime is h, ()/h(S). In the latter case the path may or may not almost surely tend to a boundary point, depending on h and the choice of boundary, but the probability that an h-Brownian from
path in D from has some asymptotic property at the boundary is the product of times the probability that an h,-Brownian path in D from has this property.
5. h-Brownian Motion from an Infinity of h If D is a Greenian subset of RN, if h is a strictly positive superharmonic function on D, and if oo, an h-Brownian motion in D from o is a process t with the following properties (all densities are relative to IN).
h-BM(l) is an almost surely continuous process from o with state space D, except that if the transition density is not stochastic and if a trap 0 is adjoined to D to obtain a stochastic transition probability, there is a discontinuity at the time of transition to a. h-BM (2) w o(t) has distribution density function Gp(t, o, ) when t > 0. h-BM(3) w0(-). is Markovian with transition density function Go on D; equivalently, under h-BM(2), the reverse transition density (transition from q at time t to at time s. with s < t) is t' 'D(S, o, S)eD(t - S, S, 11)
(5.1)
Suppose now that all probabilities are multipled by This change does not affect the conditional densities in h-BM(3) but replaces h-BM(2) by h-BM(2') wz(t) has distribution density function GD(t, 5o, )h when t > 0. The fact that the measure space on which the process is defined now has measure which may not be I causes no difficulty. From now on
an h-Brownian motion from o will mean as before a process satisfying h-BM(1)-(3) when +x, but satisfying h-BM(l), (2'), (3) when +oo. In the latter case the measure space on which the process is
681
5. h-Brownian Motion from an Infinity of h
defined necessarily has measure + w, and in fact P" {w (0) _ o} = P"{wwa(1) = a} = +oo
when t > 0. We have not yet shown that an h-Brownian motion from an infmity of h exists, and we now proceed to do so. We shall construct the desired process on the space 0 of all functions from R' into D v (a) with value o at the parameter value 0. Observe first that if a process exists,
satisfying either h-BM(l)-(3) or h-BM(l), (2'), (3), then if 'eD and if is finite or not, the process conditioned by s' > 0, and whether wwo(s) has the following properties: the process on the parameter set [0, s'] is independent of the process on the parameter set [s', + oo the process on the first parameter set is Markovian, with reverse transition density (5.1); the process on the second parameter set is an h-Brownian motion from '. Now a process can be constructed on the function space Q with the finite-dimensional distributions of the so-conditioned process. In fact the standard procedure of Kolmogorov can be used, in which a measure P. is assigned to 0 making the family of coordinate functions {x(t), to R+} have the specified finite-dimensional distributions for strictly positive parameter values, with x(0) = o. The class of measurable sets is the smallest class making every coordinate function measurable. The probability assigned to Q itself in this way is 1. If now ' is given the distribution on D with density v'D(s', o, )h, the probability measure P. on t) becomes a measure P. = J 'D(s',
the latter value is finite according to Theorem IX.6. The sequence {P',,' n z I) of measures on 0 is an increasing sequence with P,".(S2)
with the property that if S is a measurable subset of Sl and if S' = S n {x(1/n)eD}, then PiM{S'} = Pig,,,{S'} when n > m. The limit of this increasing sequence of measures is a measure P" on Q for which P"{Q} _ 5 + oc and for which the process is Markovian and satisfies h-BM(2'), (3). The measure P," is the restriction of P" to the class of measur-
able subsets of {x(s')cD}. The measure P" can be expressed in terms of Brownian motion following Section 2 without the measure normalization used there. That is, if {w is a Brownian motion in D from with lifetime S, ifs' > 0, and if A e.F(s'), then the measure
J
h[wro(s')]dP
AnfS>Y
assigns a distribution to {woo(s), s:5 s'} which has the same finite-dimensional
distributions as {x(t), t S s') on the set {x(s') a D} under P. It follows,
682
2.X. Conditional Brownian Motion
has a stochastic as in the corresponding discussion in Section 2, that modification which is continuous except at the time of transition to 0; w .satisfies h-BM(1), (2'), (3) as desired. EXAMPLE. Let h =
Then an h-Brownian motion in D fromo
is a process whose paths have finite lifetimes and tend back to o at these lifetimes.
6. Brownian Motion under Time Reversal Let D be a Greenian subset of RN, and let It be a strictly positive harmonic function on D. Let w4"(-) be an h-Brownian motion in D from , with lifetime S4'. It is tempting to conjecture that this process reversed in time, so that motion. The the paths tend to at their lifetimes, is a following discussion shows that this conjecture is correct if suitably interpreted.
Denote by * the Alexandrov point in the one-point compactification of D, and define wsti (t) =
C
iftS. h
The extended process is a Markov process on the parameter interval R, with nonstationary transition function, and the process reversed in time is therefore also Markovian. This fact is the motivation for the following discussion. It will be useful to reverse the time from a random origin. Let z be a positive random variable, independent of the process w{ with the distribution 1, . More precisely replace the space .0 on which the given random
variables are defined by the product space Q x R, let z be the second coordinate function of this space, and define the measure Q on this product space as the completed product measure of the given 0 measure P" and 1, . The random variable w2(t) becomes a function, also denoted by wf(t), on the product space. Define x(l) for t e R as w{ (z - t), so that x(t) is a function defined on a space of infinite measure. An elementary calculation justified by Fubini's theorem shows that the restriction to D of the distribution of x(s) is absolutely continuous relative to IN with Density at S in D of the x(s) distribution =
S)h(C)
(6.1)
Furthermore for i > 0 the conditional distribution of x(s + t), given x(s) in D can be chosen to be absolutely continuous on D for each value of x(s), with (Density
J.r(s)),x,,
°(t, q,
Q
Gn(S. q) (6.2)
683
6. Brownian Motion under Time Reversal
More generally the conditional density in (6.2) is unaltered if the condition x(s) = n is replaced by the more restrictive condition
x(st) = nt,
..
,
x(s,,) = %,
x(s) = n
for s,
al). m, a) a
w4 (S,Bm + t), t e U8+ } and up to a
null set
{us>a}= U n Am,a+k k=I m=0
so u, is 114, measurable.
Proof of (a2). Suppose first that u is bounded. Then the strong Markov property of conditional Brownian motion yields E{u4l.g2(SfBn)} =
u'[w,h(S B")]
a.s.
(11.5)
Hence (Section 7)
E{u{} = E{u'[ww(Shen)]I _ en( ,u');
(11.6)
so u' is h-harmonic on B. for all n and therefore is h-harmonic on D. If u is
lower bounded but not necessarily upper bounded, this result applied to u A n implies that the function n n} is h-harmonic; so (Section 1.II.3) the function u', the limit of an increasing sequence of bounded h-harmonic functions, is either identically + 00 or h-harmonic and quasi bounded. Proof of (a3). We can suppose that u is positive. Then by (a2) either u' - + 00 or u' is h-harmonic and quasi bounded. In the latter case (11.5) is applicable.
11. The Tail a Algebra of a Conditional Brownian Motion
697
,Ea. and that according to Section VI.7 the a algebra on the right here is Y.e Z. - 4 ( S 4 so when
Note that u, is I,,, measurable, that W, C
114
n
x in (11.5), the conditional expectation continuity theorem yields
Y f"(S4B^)}=limu"[w"(S''B^)] as.
(11.7)
Finally lim,t'I u'[tvw(t)] exists almost surely according to Theorem 8, and this limit is almost surely uS according to (11.7). Thus (a3) is true. Proof of (a4). If u is Borel measurable, the preceding work is applicable with minor changes when u, is defined by (11.2) with the inferior rather than the superior limit. On applying these results appropriately to u v 0 and (- u) v 0
it follows that the function S -- E{ jus } on D is either identically + oo or h-harmonic and quasi bounded, as is then also the function i-' E{u,}, and that in the latter case (11.4) is true. The proof of (a) is now complete. (b)
s{ _". Assertion (al) implies
c
Conversely, fix
in
D and a set A, e 1.R,. We shall now show that then A, 0;V1j'. Define B as in the proof of (a 1) except that now we also suppose that B. c B,,, for all n. According to the Markov property of conditional Brownian motion, P{A4j.f(SeB^)i = P{A{Jw (S,B^)}
a.s.,
(11.8)
and (Section 1.4) one version of the conditional probability on the right has the form 46^[w (SnB^)], where 0^ is a Borel measurable function from aB^ into [0, 1]. We apply the conditional expectation continuity theorem, again using the fact that
yrER',2 (t) = y
1$ B^).
MER,
and find that
lA = lim
a.s.
(11.9)
If u is the function on D defined for all n as 0^ on O B. and defined as 0 elsewhere on D, this function is Borel measurable from D into [0, 1] and
1', = limsup4[w{(t)] a.s. rts{
Thus A, e `o h, as was to be proved.
Observation. According to (a) and (b), it is no restriction on if u in (11.2) and the following definition of c{' is bounded and h-harmonic.
698
(c)
2.X. Conditional Brownian Motion
If h is minimal harmonic, the following assertions are true.
(cl) The tail a algebra Iris trivial for every in D. (c2)
_
If u is an arbitrary Borel measurable function from D into I2, there is a (not necessarily finite) constant c such that P{u, = c} = 1 for every
in D.
In (c3)-(c5) v is a Borel measurable function from D into a Polish space D', and d' is a metric on D' compatible with the D' topology. (c3)
Let n' be a point of D', and define M{(ry') = {q' is a cluster value of v[w{ (t)] when t T S4 h).
Then Mh(q') a ( , and the function P{M,'(q') } is either identically 0 or identically 1. (c4) Let A' be the set of points tI' for which the function P{M"(q')} is identically 1. The set A' is closed and is the cluster set of almost every sample function as t T S2. If D' is compact,
P{ lid'(A', v[ (t)]) = 0} = I ttsh4
for every c.
(c5)
The function
-- P{lim,t5h v[wf (t)] exists a.s.) is either identically
0 or identically 1, and in the latter case there is a point q' of D' such that lim,tsh v[w2 (t)] = PI' almost surely, for every point of D.
Proof of (c 1) and (c2). We shall prove (c2) which implies (c 1). It can be supposed, replacing u by arctan u if necessary, that u is bounded. The function " E{u,} is h-harmonic according to (a2). Since u' is bounded and h is minimal, the function u' is identically constant, u' - c, and (11.7) then yields u, = c almost surely, as was to be proved. Proof of (c3). A point tl' is a cluster value of v[,,' (t, a))] when t T S{ if and only if lim infd'(q', v[tti{(t, co)]) = 0; it S41
so if
-d'(ry',
assertions (a) and (c2) combine to imply (c3).
Proof of (c4). The proof of Section VII.6(e) with obvious changes to the present context yields (c4).
probability space subset for which v{ = Proof of (c5). Let A, be the lim,tshv[nw(t)] exists. Then A,e19{ (Section 11.5); so P{A} is either 0 or 1.
699
12. Conditional Space-Time Brownian Motion
is h-harmonic by the same reasoning [see (11.6)] The function is either making u' an h-harmonic function. Hence the function identically 0 or identically 1. In the second case fix and observe that v, is 94h measurable; so v{ is almost surely a point n of D', that is, lim
s
ltsl
v[w (t)]) = 0
a.s.
According to (c2), it follows that this limit relation is true for all ; so q' does not depend on . The proof of (c5) is complete. 0 The Role of Analytic Set Theory
Although (a)-(c) as treated above involve analytic set theory where, as in (a) and (c2), functions are discussed under very weak hypotheses, the proofs of (b) and (c2) do not involve this theory. Analytic set theory can be avoided
in the other parts of (a) and (c) if appropriate restrictions are imposed on the functions under discussion. For example, in (al) if u is supposed lower semicontinuous, the set {u > c} is open, and therefore (Section VI.6) measurability problems for hitting become trivial.
12. Conditional Space-Time Brownian Motion Let I) be an open nonempty subset of U'B", and let h be a positive superparabolic function on 15. The definition and treatment of h space-time Brownian motion in b follow their counterparts for conditional Brownian motion in a Greenian subset of IB" except that h is allowed to have zeros. We define the transition measure from a zero of h to be identically 0. Define 154 = {0 < Ih < + oo }. We restrict ourselves to the following remarks. An h space-time Brownian motion from a point of 14 is an almost surely continuous process whose sample functions never leave iY' and proceed downward, that is, in the direction of decreasing ordinate values. If ti is a positive superparabolic function on D, the process (v/h)[w is an almost surely right continuous process with left limits at all parameter values and at the path lifetime and is a supermartingale under the same conventions as in the conditional Brownian motion context. If it is parabolic, almost every tv{ path tends to the one-point boundary of b at the path lifetime, and h-parabolic measure µ8(i, ) on the Euclidean boundary of an open relatively
compact subset i of I), containing , is the hitting distribution on this boundary. If h = t) for some point it of 15 and if Oo(4, 0) > 0, then almost every wf path has lifetime ord 4 - ord ri and tends to 1 at the path lifetime. In general if h = d6µ is a superparabolic potential, almost every h space-time Brownian path from a point of I'' has a finite lifetime and tends
at the path lifetime to a point it whose distribution is
i(, q)µ(d1)/h()
700
2.X. Conditional Brownian Motion
The h space-cotime Brownian motion processes are defined in the obvious
dual way. When q) > 0, almost every 66(-, q) space-time Brownian path from tends down to q at the path lifetime and almost every space-cotime Brownian path from q tends up to at the path lifetime. Moreover a reversal of time in one of these processes yields the other process.
Theorem 8 on the composition with conditional Brownian motion of the ratio of two positive superharmonic functions translates directly into the parabolic context, and its statement is left to the reader. Observe, however, that the sample functions of the composed process, up to the conditional Brownian motion lifetimes, are almost all right continuous but not necessarily continuous, as already noted in Section IX.12 for the special case of the
composition with Brownian motion of a superparabolic function. The parabolic counterpart of Theorem 8 can be applied to derive the existence
of a coparabolic-fine limit function for the ratio of two positive superparabolic functions (see Theorem 1 XVIII.14 for an exact statement, and see the classical context argument in Section 8).
13. [Space-Time] Brownian Motion in [W'] R N with Parameter Set II8 All measure densities in this section are relative to IN.
Brownian motion in R' with parameter set R
Let {w(t), tE R} be a Brownian motion in R' with parameter set R. Then (Sections VII.2 and IX.8) there is a positive parabolic function ti on A" such that for s in R the random variable w(s) has distribution density i*, s), with integral + o0 over RN. The process has reverse time transition density (transition from at time s ton at time t < s) u(q, t)G(s - t, t,
s).
In particular, suppose that u is a minimal parabolic function on AN, so that (Section I.XVI.8) ii has the form s) = exp
1 2. If y = 0 and N = 2 [N = 1], almost every w(-) sample path hits every disk [point] at arbitrarily large negative parameter values. If ti is an arbitrary positive parabolic function on RN, then (Section XVI.8) there is a measure A, on RN such that
o expL I
In the following discussion we compare certain results for S.n L° in martingale and potential theory contexts. Suppose that is a uniformly integrable martingale (continuous parameter context), that is, e S. n D. According to the continuous parameter version of Theorem 2.1I1.14 there is an almost sure limit x(+ oo) = lim,... x(t), and x(t) = E{x(+ co)I,F(t) } almost surely, for all t. Moreover according to Theorem
709
6. A Condition for Quasi-Boundedness
is L° bounded, equivalently
2.111.14, E{ Ix(+ oo) IP} < + oo if and only if
a LP. Under the latter condition it
(by Section 2. V.4) if and only if follows easily from Section 2.V.2 that
a.s.
The following are the classical potential theory counterparts of these martingale theory results. Let D be a connected Greenian subset of R', let h be a strictly positive harmonic function on D, and let u = v/h be a positive h-harmonic function on D. According to Theorem 1.X11.10 the Martin boundary is universally resolutive and universally internally resolutive. Hence in the Martin space context the results in the present and preceding sections imply that the class of PWB' solutions is S. n D(µ4_) = Smgb According to Theorem 1.XII.19, if u = Hf then f is the h-harmonic measure
almost sure boundary limit function on approach to 8"D in the minimal fine topology, and u = u' f ). Moreover it follows easily from Section 1.1X.2 (the classical potential theory counterpart of Section 2.V.2) that I f IP) _ J ]P) < + oo if and only if u e LP( 4_ ), and in that event I
LMh uIP.
6. A Condition for Quasi-Boundedness Lemma (Classical Potential Theory Context). If u E Sqb and u e S+, then lim
0
C-M
(Martingale Theory Context). If
q.e.
(6.1)
e SQb,
e
q.e.
(n eV ).
S+, and A, =
{
c l,
then
lim
n-m
0
(6.2)
Observe that (6.1) and (6.2) are identical with the same equations for unsmoothed reductions because {u > c} is an open subset of 1 N and A, is a fine-open subset of Ili3+ x 0. In (6.1) the reduction decreases when c increases. D"d < Oz(-) "c quasi In (6.2) we can only assert that c < d implies that everywhere, and that is why in (6.2) the limit is sequential. The set lL+ in (6.2) can of course be replaced by any other unbounded increasing sequence.
Proof in the classical potential theory context. (The martingale theory proof is a direct translation.) If u = E0 u; with u; in S+ and bounded, then hkrlv>`s
(6.3)
710
3.1. Lattices in Classical Potential Theory and Martingale Theory
and (dominated convergence) it is therefore sufficient to show that t>`i = 0 for all j at the points of finiteness of v. If u; is bounded by aJ, this desired limit relation follows from
lu
'
al
w
c
c
Observation. The relation (6.1) can also be written in the form
lim QuhO" = 0 q.e.
C_W
(6.1')
Parabolic Context
The lemma and its proof need no change in the parabolic potential theory context.
7. Singularity of an Element of S,' Theorem (Classical Potential Theory Context). For a function u in S, to be singular it is sufficient that u A c E Sp for some strictly positive constant c and necessary that u A C E SD for every stric t/v positive constant c. in S,n to be singular (Martingale Theory Context). (a) For a process
it is sufficient that
A cESD for some strictly positive constant c and
A c C SP for every strictly positive constant c. necessary that in Sm is singular if and only ij' lim,_,, z(t) = 0 (b) The process almost surely.
The proof of Theorem 7 in the classical potential theory context was given in Section 1.1X.10, and the martingale proof of (a) is a direct translation of that proof. The martingale assertion (b) was proved in Section 2N. 1I . A classical potential theory counterpart of the martingale theory assertion (b) can be formulated in two ways as follows. (Classical potential theory context) :
(bl) An h-harmonic function u in S, is singular if and only if (Theorem I.XIl.l9) its minimal-fine boundary limit function on the Martin (b2)
boundary of D vanishes Mh, almost everywhere, that is, up to an hharmonic measure null set. An h-harmonic function u in S, is singular if and only if (Theorem is an /t-Brownian motion in D from , with life2.X.8) when
time S,. lim,thu[ t' (t)] = 0 almost surely.
711
8. The Singular Component of an Element of S `
Parabolic Context
Theorem 7 and its proof need no change in the parabolic context. The parabolic criterion corresponding to (b2) does not require a new proof and is of course stated in terms of the limit of a positive h-parabolic function along h space-time Brownian paths.
8. The Singular Component of an Element of S+ Theorem (Classical Potential Theory Context). If u E S+ and has singular component us, then lim hDu0{u'cl = us q.e.
C_00
(8.1)
In particular, u in S+ is singular if and only Y 'u = hOuOD> for every (equivalently some) strictly positive constant c. and has singular component (Martingale Theory Context). If and if Ac = c}, then lim
q.e. (ne7L+).
(8.2)
in S+ is singular if and only if In particular, for every (equivalently some) strictly positive constant c.
quasi everywhere
Observation. Equations (8.1) and (8.2) are equivalent to the same equations with unsmoothed reductions. See the observation in Section 6 on the corresponding point for Lemma 6, and on the explanation of the sequential convergence (n -+ co) in (8.2) rather than the unrestricted approach (c -* oo) in (8.1).
Proof in the classical potential theory context. (The martingale theory proof is a direct translation.) Suppose first that uESs . By Section I.Vl.3(i) as adapted to reductions in the h-superharmonic context, u
hOuO,u>C,
+ hOu0(u5c)
(8.3)
According to the vector lattice decomposition theorem (Theorem 6 of Appendix III), there are members u, and u2 of S+, specific order minorants, respectively, of the first and second terms on the right, with sum u. Since SS is a band the functions u, and u2 must be singular, and since u2 is bounded by c, u2 E Ss n Sqb ; so u2 = 0. Thus u = u, , and u is the first term on the right in (8.3), as was to be proved. For general u in S+ write u = uqb + us,
712
3.1. Lattices in Classical Potential Theory and Martingale Theory
the sum of the quasi-bounded and singular components of u. Then [from Section 1.VI.3(f)] 114(u>c) = IUgb](r>CI + IUAI->cl
The first term on the right has limit 0 quasi everywhere when c - oo according to Lemma 6, and the second term on the right is us for every c because Us =
b1Us0(Us>ci
0 = 0 q.e. PT(d) p vanishes on polar sets. (Martingale Theory Context) The following conditions on a process SP are equivalent:
MT(a) MT(b) MT(c)
in
If A, = (z(-,-)> c}, then lim
0
q.e.
R-M
Observation. PT(c) and MT(c) are equivalent to the same equations with
unsmoothed reductions. See the observation in Section 6 on the corresponding point for Theorem 6 and on the explanation of the sequential convergence (n - cc) in MT(c) rather than the unrestricted approach (c - oo) in PT(c). Proof in the classical potential theory context. In clarification of PT(d) recall that (Theorem 1.V.11) a superharmonic potential G0µ has the value + oo at
p almost every point of a polar set. Hence p vanishes on polar sets if and only if µ {G0µ = + oo} = 0. To simplify the notation in the following proof, we take h =_ 1.
713
9. The Class S,0
PT(a)=:47(b) If u = Eo ujwith each summand in SP and bounded, let be a point of D, and let Bo be an open relatively compact subset of D containing . To show that ueD(µD_), we show that the family of pairs {(u, µB(, )): B - B0, B open and relatively compact in D}
(9.1)
is uniformly integrable. Observe that according to the generalized superharmonic function average property in Section 1.VIII.10, the harmonic u) increases when B decreases, and therefore attains its maxaverage imum value when B = Bo. Now define vn = E', uj, choose s > 0, choose n so large that v,) < e/2, and define c = supD Eo uu. When B is a set in v,) < (9.1), the same superharmonic function average inequality yields E/2, and if A is a Borel subset of 8B, so small that µB(;, A) < e/(2c), then PB(S, UlA) S I1B(S, Vn) + µB (uJlA) < E.
(9.2)
no
Hence the family (9.1) is uniformly integrable. PT(b) PT(d) Suppose that for some it in S+ condition (b) is true but
condition (d) is false. Replacing µ by its projection on a suitable compact set if necessary, it can be supposed in deriving a contradiction that µ is supported by a compact polar set A. Let B be an open relatively compact subset of D containing A, let be a point of B - A, and let A. be a decreasing sequence of compact neighborhoods of A, subsets of B, with intersection A. The boundary of B. = B - A, consists of dB and a subset C. of OA,,. Since u is harmonic on a neighborhood of B,, (9.3)
To show that the second term on the right has limit 0 when n oe, note first that (Section 1.V.4) there is a function v positive and superharmonic on B, with v = +oo on A and v z I on C. for large n; and since for every S > 0 the function Sv so lim sup.- µB.0, C,) 5 µ8n0, 0. Hence satisfies the same conditions as v, it follows that (class D property) limn_.µB.0, ulcn) = 0; so
uG) = li µB.(4, Ul aB) 5 sup U. m
(9.4)
Thus u is bounded on B - A, and therefore (Section 1.V.5) u has a unique superharmonic extension to B, and this superharmonic extension is harmonic, contrary to fact. It follows that PT(b) PT(d). PT(d) PT(a) If u = GDµ is a superharmonic potential for which µ vanishes on polar sets, define A. = in S u < n + 1), and let µ be the projection of µ on An. Then p = Eo µ because y {u = + oo } = 0, and according to
the domination principle, GDµ 5 n + 1. Thus the representation u = Eo GDµ exhibits u as a quasi-bounded potential.
714
3 1. Lattices in Classical Potential Theory and Martingale Theory
PT(a) eo PT(c) See Theorem 8. MT(a) MT(b) If with each summand in SP and bounded, we show that the family {z(T): Toptional} is uniformly integrable.
E. , choose e > 0, choose n so large that E{yy(0)} < e/2, and let c be an upper bound of I" Then if T is optional, the supermartingale inequality yields [Define :(+ x) = 0.] Define
E z(T)} <E{z(0)),
E{y,,(T)} 11
h(5)
t}.
(11.1)
Now by hypothesis uh is a superharmonic potential; so (Section 2.X.1) the probability on the right has limit 0 when t oo, and (c) follows. Observe that E SD implies that x',(+ oo) = 0 almost surely, but that the converse implication is false, although the latter condition is necessary and sufficient that 40 be in So S. The conclusion of (c) would thus be considerably weakened if were replaced by in the conclusion. This weakening is exhibited explicitly in (d) to be proved next. (d) IJ u E then E S, ,for every S. [Recall that under the hypoth-
esis on u. P{x4(+oo) = 0} = I for all .] To prove (d), it is sufficient to prove that if is a bounded supermartingale in S` with parameter set 68+ majorized in the specific order by is a zero process up to a then standard modification, equivalently, E{y(0) } = 0. Now if D is an increasing sequence of open relatively compact subsets of D with union D, containing S. and if SS, is the hitting time of aD, by then the process o :() stopped at time S4 is a martingale. the process stopped at the same time is a supermartingale, and in view of the order relation there is another process which added to yields 0x4() and which when stopped at SS is a positive supermartingale. It follows that stopped at S", is a martingale and therefore that
E{y(0)} = E{y(S )}
E{ r {S -)} < E{ox{(S' -)} = E{xt(+oo)} = 0, (11.2)
as was to be proved.
12. A Decomposition of SS (Potential Theory Context) As in the previous sections h is a strictly positive harmonic function on the connected Greenian subset D of R'. Let S ;. be the cone of functions u in SL for which GD(t, , u) = u for all t > 0, that is, the cone of Gn-invariant
717
13. Continuation of Section 11
excessive positive singular h-harmonic functions on D, and let S , f be the cone of functions u in S.*. for which lim,_m Go(t, , u) = 0. It is easily checked
that if u is in one of these classes, its positive specific order minorants are
in the same class and that a countable sum of functions in one of these classes is also in that class if the sum is in S+. The classes S,,,. = S;,. If u S ;, , and S-f = S+ f - S, , f are therefore orthogonal subbands of is in S,;,, we have seen in Section 2.IX.8, Example (a) (for h =_ I and the 0* (t, -, u) = uo is an h-harmonic general case follows trivially), that minorant of u, in S, . Hence lim,_m GD(t, -, u - uo) = 0, that is, u - uo e S,',f, and therefore S., = S,,,,,, + S,,,,f. The classes S,,,,... and S,,,,f have simple probabilistic characterizations. The is the condition that u is in S,R, and that the process condition that u e ox{(-) in Section I 1 is a martingale for some (equivalently every) , equivdefined in 2.X(8. l) is a alently and more intuitively, that the process martingale for some (equivalently every) . A second probabilistic interpretation is that (Section 2.X.1) uES;,,m [ueS, ,f] if and only if the lifetime of uh-Brownian motion in D from some (equivalently every) 4 is almost surely + oc [almost surely finite]. This characterization inspires the notation.
13. Continuation of Section I 1 Section 11(d) is made more precise by (e) and (f) below. (e) If u e S; ,. , then e S ;,. In fact, as already noted in Section 12, is a martingale, and this process is singular according to the process Section 11(d). (f) If u e S ;, f, then o a S', because the equation
lim Lo(t, , u) = lim E{of(t) } = 0 shows that a Sp , and this process is singular according to Section 11(d). Section I I (c) is made more precise by (g) and (h) below. (g) If ueSpyb, then In view of Section 1 I(c) this assertion is trivial. (h) I f u e Sp then ox a S;, for every for which oo. If u(O < + oo, then by Section l 1(c) the process is in SP +. To prove that
the process is singular, suppose first that the measure associated with the superharmonic function uh has compact polar support A. (According to Section 10, this measure is supported by a polar set.) Define Do = D - A, and let uo be the restriction of u to Do. Since almost no w20 path from the point meets A, the process for uo on Do analogous to x'(-) for u on D can be taken as this same process The fact that may not be in Do will not affect the following reasoning. The function uo on Do is singular as well as h-harmonic because a bounded positive h-harmonic minorant u, of uo has an h-harmonic extension to D by a trivial generaliza-
718
3.1. Lattices in Classical Potential Theory and Martingale Theory
tion of Theorem 1.V.5, and the extension of u, is a bounded li-harmonic minorant of a and therefore vanishes identically. Thus 0.vh()ESs by (e); so as stated. If. however, p does not have compact support. the measure is the sum of a sequence of measures with disjoint compact polar supports, so as was to be proved.
EXAMPLE. Let N = 2. let D be the upper half-plane, and let h = I. The ordinate function u on D is a singular positive harmonic function, singular because any bounded harmonic minor-ant of u has limit 0 at every finite boundary point of D and therefore (Theorem L V.7) vanishes identically. (It
is easy to prove that u is minimal, but this fact will not be needed.) The positive u-harmonic function 1/u has a finite limit along almost every u-Brownian path from a point S to oD so almost every such path tends to oo. Direct calculation shows that u is subsumed under (e) so that if is a Brownian motion in D from with lifetime S,, the process with h = I is a singular martingale. The martingale property amounts to the statement that a one-dimensional Brownian motion from a point a > 0 killed when it hits the origin is a martingale, a fact easily verified by direct computation.
Chapter 11
Brownian Motion and the PWB Method
1. Context of the Problem Let D be an open nonempty subset of R', coupled with a boundary 8D provided by a metric compactification. To avoid trivial complications, we shall assume that D is connected; if D is disconnected, the results are applicable to each open connected component of D. Let h be a strictly positive harmonic function on D. The PWB method of attacking the first boundary value (Dirichlet) problem for h-harmonic functions on D was detailed in Chapter VIII of Part 1. Recall that the a algebra of µ4 measurable boundary subsets is the a algebra of boundary subsets A for which the boundary in-
HA. The class of dicator function 1, is h-resolutive and that Borel boundary subsets for which 1, is h-resolutive is a a algebra, and for each point of D the restriction of µ4(c , ) to this a algebra, on completion. is the measure µ"D( , ) on the a algebra of µhn measurable sets. The class of µ4 measurable boundary functions f which are µ4c, ) integrable does not depend on and is the class of h-resolutive boundary functions, and Hf = This class of boundary functions will also be described as the class of µ4 integrable boundary functions. The PWB method solves the Dirichlet problem when the boundary is h-regular and the given boundary function is finite valued and continuous, but no relation other than the h-resolutivity itself was given in Chapter VIII of Part I to connect an h-resolutive boundary function f on a not necessarily h-resolutive boundary with the solution Hf . In Section 2 it will be shown that in the general case h-harmonic measure has a probabilistic evaluation and
that Hf has f as a boundary limit function along h-Brownian paths. The solution Hf, which (by Lemma 1.4) is necessarily in the class D(µ4_). when composed appropriately with h-Brownian motion defines a uniformly integrable martingale, and the evaluation Hf = becomes the martingale equality for the parameter values 0, +x. Throughout the discussion in Section 2 1, w -Ff (-)) is an h-Brownian motion in D from , with lifetime Si'. With no loss of generality (Section 2.X.2) it can be assumed that .y is right continuous and that F,'(0) contains the null sets. Since h is harmonic, almost every path tends to 8D at the path lifetime (Section 2.X.4). Let I (u)) be the cluster set of ww w) at
720
3.11. Brownian Motion and the PWB Method
the parameter value S,. For almost every to this cluster set is a compact boundary subset. In particular, if h = I and if aD is the Euclidean boundary,
the process iv,'(-) can be identified with a Brownian motion in R' killed at the hitting time of OD, and r!(())) is then almost surely a singleton, { :'[S4 ((o) - ] , . Similarly, if D is bounded, if aD is the Euclidean boundary, and if h has a harmonic extension h' to an open neighborhood D' of D, then can he identified with an h'-Brownian motion in D' killed at the hitting time of (3D. and again r"((,)) is almost surely a singleton, {w2 [S, (o))-];. According to the following theorem, whatever the choice of (D, the distribution of 1 is the h-harmonic measure an h-resolutive function / is for almost every (j) a constant function on the cluster set q(w), and Hf has the prescribed value of./ as limit along almost every path to the boundary. Furthermore aD is h-resolutive if and only if r,' is almost surely a singleton, that is, if and only if the left limit r, (SS -) exists almost surely. Thus the PWB method generalizes the classical concept of solution of the Dirichlet problem but nevertheless leads to solutions which have the prescribed boundary limit function in a reasonable sense.
The Parabolic Context The methods used in this chapter are applicable in the parabolic context, and the statements of the results in that context are left to the reader.
2. Probabilistic Analysis of the PWB Method Theorem. (a)
For every boundary subset A
HA(5)="Ri(S)= R (5)
(=P{T," nA 0 }if A is analytic)
(2.1)
)
and
P{T4 c A; (b)
if aD - A is analytic.
(2.2)
1/ A is pD measurable. then
A) = P;I", c A} =
n A 96 Qf ).
(2.3)
U j is a pll measurable boundary function then, for every S in D, j is identically constant on almost every cluster .set V. Conversely ij f is a Borel measurable boundary Junction and U. for some in D, f is identically constant on almost every cluster .set r4", then J'is µv measurable. (c) A boundaryfunction.f is h resolutive and only if/ is µD measurable and integrable, and in that case
2. Probabilistic Analysis of the PWB Method
721
Hf () = µD( ,f) = E{f(ry) },
(2.4)
lim Hf[wh(t)] = f(r2) a.s.,
(2.5)
.
S141
and the process {x2
x ti (t)
f
defined by
Hf [w" (t)]
if t < Sf
f(ry)
if SS < t < +oU
(2 . 6)
is a un(ormly integrable almost surely continuous martingale. (d) The boundary is h-resolutive if and only if, for each point in D (equivalently, for a single point l; in D), r{ is almost surely a singleton, that is, if and only if the left limit w4 (Sh_ ) almost surely exists in the D u aD metric. The distribution of ww (Sf -) is then p'
Observation. The equality Hf () = E{ f(ry)} is the martingale equality for at the pair of parameter values 0, + co.
'
Proof of (a). The equality H =''R; = RAA /h was pointed out in Section 1.VIII.2. The last equality in (2.1) is a special case of Theorem 2.X.7. An application of (2.1) to OD - A yields (2.2). Proof'of (b). If A is pD measurable, that is, if 1e is an h-resolutive boundary function, then if A is a Borel set, the equality of the terms in (2.1) with those in (2.2) yields (2.3). Since every µD null set is a subset of a Borel uD null set and since the a algebra of 1tD measurable sets is the completion of the restriction of this a algebra to the class of its Borel sets, it follows first that (2.3) is trivial when A is a µD null set and next that (2.3) is true when A is 1c4 mea-
surable. Conversely, suppose that for some point
and Borel boundary
subset A the last equation in (2.3) is true. Then the positive harmonic function Ht -H'', vanishes at and therefore vanishes identically; that is, the set A
is i4 measurable. Thus the assertions in (b) for a boundary function f are true when f is the indicator function of a set, and are therefore true without this restriction.
Proof of (c). The h-resolutivity of a boundary function f implies the µD measurability off and therefore implies that for each point of D the function f is identically constant on almost every cluster set rs ; that is, J (r,) is almost surely uniquely defined. The first equality in (2.4) was obtained in Section 1.VIII.8, and the last equality follows from the probabilistic evaluation of uD in (2.3). The following more direct proof of (2.4) also proves (2.5). If u2 [ut] is in the upper [lower] PWB' class for the h-resolutive boundary function.f, the h-superharmonic functions u2 + c, c - ut, u2 - Hf are posi-
tive if the constant c is sufficiently large. Hence (Theorem 2.10.8) each
722
3.11. Brownian Motion and the PWB Method
function u2, u,, Hf has a finite limit along almost every it`(-) path to the boundary, and u, () 5 E{ lim u, [w (t)] } < E{ lim Hf [w (t)] } < E{ lim u2[W (t)] }
u2( ). (2.7)
Moreover
lim u, [w (t)] < inf f(g) < sup f(C) 5 lira u2 [w (t)] as. fsE
CEr
(2.8)
Erg S*41
Since u, and u2 can be chosen to make u, arbitrarily small, f must be constant on almost every cluster set I'2, and (2.4) and (2.5) are true.
The martingale assertion in (c) follows from Theorem 2.X.8 and Sections 1.4 and 1.5 but will be clearer from the following direct proof. Denote
the 40 process in (2.6) by x2(f; ). All supermartingales and martingales below are relative to the filtration (). If f is an h-resolutive boundary function, Theorem 2.X.8 implies that the following processes are almost surely continuous positive supermartingales:
x(IfI
A n);') (2.9)
The fact that the last two of these processes are supermartingales implies that x( I f l n n: ) is a martingale and therefore (n -* oo) that x{(If 1, ) is a martingale, that is, that (c) is true when f is positive. Since the first two processes
in (2.9) are now known to be almost surely continuous martingales, the process x(./; ) is also, as was to be proved. Proof of (d). If the boundary is h-resolutive, that is, if the Borel boundary subsets are µD measurable, chooses > 0, and let A, , ... , A. be compact boundary subsets of diameter at most r and with union D. Since (2.3) implies that r,' is almost surely a subset of any A, it intersects, the diameter of rh is almost surely at most E. Hence r,' is almost surely a singleton; that is, the left limit tt>2(SS -) almost surely exists. Conversely, if r is almost surely
a singleton for some point , the set function A F-
is an additive function of Borel boundary sets by (2.1); so the boundary is h-resolutive (Section 1.VIII.9). Special Case: It Is Minimal. If h is minimal, then for every boundary subset
A the h-harmonic function ff' = Re/h is either identically 1 or identically 0. In fact by minimality H is a constant function, so that RA = ch, and iterating the reduction operation [Section I.VI.3(h)] yields ch = cRA = c2h; so c = 0 or 1. For every boundary point C if R;; ' - 0, then (since R,; 1 is the infimum of the class of reductions on boundary neighborhoods of C) RA = 0
723
3. PWB" Examples
for A a sufficiently small boundary neighborhood of C. The set B of boundary
points { with Rit} = 0 is therefore open, and Rf,, = 0. Thus the set B' = OD - B is a compact set with the property that IF,' = B' almost surely, for each . The class of µ"n measurable boundary subsets consists of all boundary subsets which are either supersets of if or subsets of B. A boundary function
f is h-resolutive if and only if it is constant (finite) on B' and the PWBh solutions are the constant functions. In particular, a boundary is h-resolutive if and only if if is a singleton {C} which supports every h-harmonic measure 14(x, ), equivalently, if and only if for all in D almost every path tends to C at the path lifetime. See Section 2.X.9 for this situation when D is a ball.
The Role of Capacity Theory
A close examination of the analysis reveals that the theory of analytic sets and Choquet capacities is not needed in the derivation of Theorem 2 if it is supposed that the left limit wf (Sf -) exists almost surely for some (equivalently every) point of D or, going in the other direction, if the boundary is h-resolutive. This hypothesis is satisfied if aD is the Euclidean boundary and if either h - I or D is a relatively compact open subset of an open set on which h has a positive harmonic extension.
3. PWB' Examples EXAMPLE (a). The Alexandrov one-point boundary of a Greenian set D is trivially universally resolutive. From the point of view of Theorem 2 universal
resolutivity corresponds to the fact that for strictly positive harmonic h on D almost every h-Brownian path from a point of D tends to the one-point boundary at the path lifetime. EXAMPLE (b). If D is a ball, its Euclidean boundary is universally resolutive according to Section 1.VIII.9; this universal resolutivity corresponds to the fact deduced in Section 2.X.9 that for strictly positive harmonic h on a ball
D almost every h-Brownian path from a point,of D tends to a point of the Euclidean boundary at the path lifetime. EXAMPLE (c). Let N = 2, let D be a disk, let { be a boundary point, and let h = K(C, ) be the minimal harmonic function corresponding to C (Section 1.II.1). For in D let be the angle between the rays from { to and to
the disk center, - n/2 < 0 < n/2. Then 0 is harmonic on D. Define a distance function
on D x D by
d(4,7)=I4-7I+I4(4)-* (7)I.
(3.1)
724
3.11. Brownian Motion and the PWB Method
If D is provided with a boundary a'D by completion under this distance function, the Euclidean boundary point S is ramified into a compact set A of points corresponding to the directions of approach to S. In this example h is minimal: so the quasi-bounded h-harmonic functions, in particular the PWB' solutions, are the constant functions, and the Euclidean boundary singleton ; y; has h-harmonic measure identically 1. Almost every h-Brownian path from a point c of D has limit r in the Euclidean metric. We now investigate the cluster sets of these paths on a'D, that is. the boundary cluster sets
in the d metric defined by (3.1). Recall from Section 1.XII.12 that if ( is identified with a minimal Martin boundary point of D, no ray from r into D is minimal thin at r. We shall show in Section 111.3 that a Borel subset B of a
Greenian subset of R'5 is not minimal thin at a minimal Martin boundary point if and only if almost every conditional Brownian path from a point of the Greenian set to that boundary point hits B arbitrarily near the boundary point. In the present context, using the notation relative to r7'D. it follows that r is almost surely the set A. According to Theorem 2, a function f on )'D is h-resolutive if and only if f is identically constant (# ± x) on A, and if so, the PWB' solution for f is identically the constant,f(A). The ramified boundary a'D has too many points to be h-resolutive. EXAMPLE (d). Let B be an open subset of a Greenian set D, let D be provided with a boundary OD by a metric compactification, and let aB be the boundary
of B relative to this compactification. Then aB depends on the choice of aD unless B is relatively compact in D. According to Theorem 2. aB is hresolutive if and only if almost every h-Brownian path in D from a point of B either hits D r, aB or tends to a point of aD n aB at the path lifetime. Thus it
is sufficient for h-resolutivity of aB that OD be h-resolutive. This simple resolutivity argument should be compared with the nonprobabilistic argument in Section 1.VIll.8, Example (b). The relation 1.VIII(8.3) connecting Nn and pe is an immediate consequence of the strong Markov property of conditional Brownian motion. EXAMPLE (e). Suppose that the metric compactification D = D u OD of a Greenian set D has the following properties. There is a family ; q5;, i e 1; on D, indexed by some set 1. for which
(e,)
Each function 4,; has a continuous extension (also denoted by q,)
(e2)
to D. If q e aD, a sequence q, in D has limit q if and only if O;(q) for all is that is, the family 0, separates aD.
(e3)
Under these conditions if also for some strictly positive harmonic function h on D lim,1s¢;[x{(t)] exists almost surely simultaneously for all i whenever uh(') is an h-Brownian motion from ( with lifetime S'', it follows that the left limit ich(Sh-) exists almost surely: so aD is h-resolutive. Since one standard technique of defining a boundary
725
4. Tail a Algebras in the PWB` Context
for a set D is to choose a compactification of D which makes each function of a specified family of functions on D have a continuous extension to the compactification, this example is frequently encountered. For example, if D is a Greenian subset of 18" and if o is
a point of D, the Martin compactification of D (Chapter XII of Part 1) can be defined as the space satisfying (ei) and (e2) for the family of functions {04, eD} defined by 0S = c0)Moreover, if convenient, we can restrict to a countable dense subset of D A direct argument in Section 111.5 will show that (e3) is satisfied for every h and therefore that the Martin bound-
ary is universally resolutive, but in fact this universal resolutivity was derived in Section 1.XII.10 from which it follows (Theorem 2) that almost every h-Brownian path from a point converges to a
Martin boundary point and that the distribution of the limit we (Sc -) is
a Algebras in the PWB' Context Let D be a connected Greenian subset of R, coupled with a boundary OD provided by a metric compactification, let h be a strictly positive harmonic function on D, and let w{ be an h-Brownian motion in D from , with lifetime S'. Let I' be the ww process tail o algebra, and let r, (m) be the 4(-, (0) cluster set on aD. Recall that by definition the o algebra I{ contains the 40 null sets. Throughout this section and ww are fixed. Theorem. If OD is internally h-resolutive, IN{ consists [modulo ww of the class of sets of the form
null sets]
{o>: r4'(o)) e A } for Borel po measurable boundary subsets A.
In particular, if OD is also h-resolutive, 5,1 consists [modulo the sets of the form { w : K S (SS - , o)) E A) for Bore! boundary subsets A.
(4.1)
null sets] of
(4.2)
Observation. Since a µD measurable boundary subset differs from some Borel and pD measurable boundary subset by a µD null set, and since the hharmonic measure pD is determined by the distribution of r'2 through (2.3), "Borel" can be omitted in (4.1) and replaced by "µD measurable" in (4.2). Theorem 4 implies that if OD is internally h-resolutive, then a w; process random variable z is Vh measurable if and only if there is a uD and Borel measurable function g on OD such that z = g(r) almost surely, and if aD is also h-resolutive, r 4 can be replaced here by w' (S4 - ).
726
3.11. Brownian Motion and the PWB Method
Special Case: h Is Minimal. If h is minimal, every bounded h-harmonic function is identically constant; so every choice of boundary for D is internally h-resolutive. The a algebra'., and the a algebra of µD measurable sets are both trivial in this case; that is, their sets are all of measure 0 or 1. If A
is a Borel boundary subset of h-harmonic measure I. the a algebra of pt measurable boundary subsets is generated by A and the pt null sets. If A e Sp'.' and if A is not r" null, then T' is the a algebra generated by A and the
null sets. In particular, if OD is also h-resolutive, there is a point S 1. In this case all subsets of 8D are µD mea-
of aD such that surable.
Proof o/'Theorem 4. If A is µD measurable, that is, if the boundary function J = 1,, is h-resolutive, then (by Theorem 2)
lim Hf [w (r)] =l(r) a.s., its
and it follows (Section 2.X.1 I) that f(rs) is
measurable, that is, {w: q(w) c A}esh. Conversely, if Ac-14', it was shown in Section 2.X.1 l that there is a bounded h-harmonic function u such that 1im,tsh almost surely. Since the boundary is internally h-resolutive, u = Hf for some and Bore] measurable function f, and u has almost sure boundary limit function f along tr{ O paths; that is, f(rs) = 1, almost surely. Hence there is a pD and Borel measurable boundary subset A such that f = l,+ up to a µD null set, and A c A} up to a null set. If 3D is also hresolutive, every Borel measurable boundary subset is pD measurable, and ¶
1
a µD measurable boundary subset is a Borel set up to a pD null set ; so the last assertion of the theorem is trivial.
Chapter III
Brownian Motion on the Martin Space
1. The Structure of Brownian Motion on the Martin Space Let D be a connected Greenian subset of 08", let K be a Martin function for D, let h be a strictly positive superharmonic function on D, and let Y,'(-)) be an h-Brownian motion in D from with lifetime SS. For A a subset of D let SS" and Lh", respectively, be the hitting and last hitting times of A by
According to Theorem I.XII.10, if h is harmonic, the Martin
boundary is h-resolutive and ph dC) = K(C, where Mh is the Martin representing measure of h corresponding to K. According to Theorem 11.2, the left limit w!(SS-) exists almost surely and has distribution
supported (Section I.XII.7) by the minimal Martin boundary carD. In particular, if C is a minimal Martin boundary point and if h = K(C, ), then I ; so vt'$(SS-) = C almost surely. With this choice of h we shall sometimes write S44", Le", respectively, for ws SS", L. The analysis in Section 2.X.9 of h-Brownian motion in a ball compactified by its Euclidean boundary is applicable to an arbitrary connected Greenian set D compactified by its Martin boundary, with no change except for allowance for the possible presence of nonminimal Martin boundary points. That is, if h is harmonic, an h-Brownian path in D from S is obtained by first choosing the minimal asymptotic path endpoint C on ??,MD according
to the distribution pD(, ) and then choosing a K(C, )-Brownian motion path from to C. More formally, one version of the conditional distribution of for ws(SS -) = C is the distribution of K(C, )-Brownian motion from
. If h = GDp is a potential and if h(g) < + x, then (from Section 2.X.4) almost surely SS is finite, and the left limit rs(SS-) exists and is in D; wf (SS -) has distribution GD(C, O)p(dC)/h(C). With this choice of h an hBrownian motion path can be obtained by first choosing the asymptotic path endpoint C in D according to the distribution and then C)-Brownian motion path from to C; that is, one version choosing a for r{(SS-) _ C is the distribution of the conditional distribution of
of
C)-Brownian motion from . The case h(C) _ + oo is treated in
Section 2.X.5. If h = h, + h2 is an arbitrary strictly positive superharmonic function on D, with h, positive harmonic and h2 a potential (Riesz decom-
position), h-Brownian motion from , when
+x. is h,-Brownian
728
3.111. Brownian Motion on the Martin Space
motion from S with probability with probability
and is h2-Brownian motion from
Attainable and Unattainable Martin Boundary Points According to Section 2.X.1, if C is a minimal Martin boundary point, the function c - P { SS = + x) is either identically 0 or identically 1. The point s is called attainable in the first case and unattainable in the second. For example, if D is a ball (Section 2.X.9), its Martin (i.e., Euclidean) boundary points are all minimal and attainable. If D is a half-space its Martin (i.e., Euclidean) boundary points are all minimal, and the finite boundary points are attainable, but the infinite boundary point is not attainable. If D = RN with N > 2, the minimal harmonic functions are the positive constant functions, and the point ao is the only Martin boundary point, is minimal, and is unattainable.
2. Brownian Motions from Martin Boundary Points (Notation of Section 1) Brownian Motions from Attainable Boundary Sets Let h be a strictly positive harmonic function on D whose Martin representing measure is supported by the set of attainable minimal Martin boundary
points. Then for each point g of D the h-harmonic measure µDW, dC) = K(C, is also supported by this set. Let wx be an h-Brownian motion in D with initial distribution A and lifetime Sx ; this lifetime is necessarily almost surely finite. We define wx(Sx) as limas" wx(t), a limit which
according to Theorem 11.2 exists almost surely and has distribution According to Section 2.X.6. the process {w,'(Sx - t), to R+} with lifetime S, is a GDA-Brownian motion, and obviously A is the distribution
of path endpoints (t
Sr).
Special Case: Brownian Motion from an Attainable Martin Boundary Point to a Point of D
Let i be an attainable minimal Martin boundary point, and let be a point of D. Then [particular case of the preceding paragraph with h = K(C, ) and with A supported by on the one hand the process {w{(t), to R1 } is a K(C, ) Brownian motion from , and almost every path tends to s at the path lifetime SS. whereas on the other hand the process {ws(SS - t),teR } is a motion from C, and almost every path tends to at the path lifetime S. From now on it will be convenient to write wz (t) instead of t) and denote by t the motion transition density.
2. Brownian Motions from Martin Boundary Points (Notation of Section 1)
729
We now fix and S and discuss the absolute probability distributions of the N4
process: it, A) F--. $w(i) e A} = p(i, A) for A ranging through the Borcl Since
subsets of D -
f1x(di) A
J'(r
- s, q', OAS, dq') = p(t, A).
0 < s < t,
(2.1)
D
it follows that p(t, ) is absolutely continuous relative to /N and that one choice of the Radon-Nikodym derivative dp(t, )/dlN at q is PD(1,C,1)=
1,d(t-s,7',q)P(s,dq').
(2.2)
Observe that the value of the integral on the right does not depend on the choice of s for 0 < s < t because t satisfies the Chapman-Kolmogorov equation; so (2.2) defines C. ) on ] 0, + co[ x (D - {s}). The function p,(t, , ) is the IN density of the distribution of is (t) in D, and the function is parabolic, on that is, ¢ = p, is C, ]0, + oc [ x (D - { }) in view of the following properties of 0 on this product set: 4) is Borel measurable. 4) is locally IA+t integrable.
¢ has the parabolic function average property locally, because fD (- s, q, ) is parabolic on the domain of 0. It is natural to conjecture that when of - C suitably, the transition density (t, q', q) tends to p j(t, S, q), and we now derive one version of such a limit relation : for each strictly positive t,
t
lim4,4(t - s, tsS(s), q) =Pi(t, S, q). To see this, observe that the function (t l', t') ' -,
on (D -
(2.3)
q) is GD(5, )-parabolic
x ] 0, + oo[ and, in fact, is invariant excessive for
space-time Brownian motion on this set. Moreover (2.2) implies that E{GD(t - s, wti (s), q) } = PL(t,
0 < s < t.
(2.4)
Hence the process 14f (t - s, w e(s), q), 0 < s < t} is a martingale. This martingale is almost surely continuous and so (from Section 2.111.16) has an almost sure limit as the parameter tends to 0. This limit is a random variable measurable with respect to the K(C, )-Brownian motion tail a algebra and so (from Section II.4) is almost surely identically constant. Since every martingale is uniformly integrable on a right closed set of parameter values [Section 2.111.3(e)], this constant must be the common expectation in (2.4); that is, (2.3) is true.
730
3.111. Brownian Motion on the Martin Space
Brownian Motion from an Unattainable Minimal Martin Boundary Point to a Point of D If ( is an unattainable minimal Martin boundary point of D, that is (Section 2.X.1), if the function K((, ) is an invariant excessive function for Brownian motion in D, let be a point of D, and let D. be an increasing sequence of open relatively compact subsets of D, containing , with union D. Let L{ be the last hitting time of D. by The function OK((, ) 0°`' (reduction relative to D) is the potential of a measure supported by 8D,,, OK(C, ) Dn = According to Section 2.X.10, the distribution of ,(Lc,) is GD(S, ,1).un(d7)/K(S, L),
and the process {wl'(LC4. - t), to 6F+) with lifetime L4,. is a
motion with initial distribution the distribution of L; = + oc almost surely and that lim,,.
Observe that almost surely.
EXAMPLE. Denote by d{ the Nth coordinate of the point of R', and let D be the half-space {dd > 0}. The Green function GD was evaluated in Section
1.VIII.9, and in Section 1.XII.3 the Martin boundary was found to be the Euclidean boundary. All boundary points are minimal, and the finite bound-
ary points are attainable. By direct calculation, if ( is a finite boundary point and if a2 = 1, lim t (t - s, rl', 7) = lim
G(t - s, 7',,
7)
GD(t, 7')
d°I
dr2ntC12G D
(
if N = 2 ,
,
d,,I(-(IN dd(N - 2) (2n)"r2 t(x+2Y2
'
7)
if N > 2.
According to the results in this section, this limit is p4(t, (,, ). If now
tends
to a second finite boundary point C,, the transition density b, tends to a limit transition density, that of K((,, )-Brownian motion, and p,(t, C, ) tends to a limit density which can be described as the absolute probability density at time t of a Brownian motion from ( to (, .
3. The Zero-One Law at a Minimal Martin Boundary Point and the Probabilistic Formulation of the Minimal-Fine Topology (Notation of Section 1) path tends to If C is a minimal Martin boundary point, almost every at the path lifetime, and (Section 2.X.11) the tail a algebra of this process is trivial. As detailed in Section 2.X. I I for h-Brownian motion with h a minimal
3. Probabilistic Formulation of the Minimal-Fine Topology
731
harmonic function, the asymptotic properties 2.VII.6(b)-(g) relating to set and function properties in the neighborhood of the initial point of a Brownian
motion correspond to set and function properties near the lifetime of the h-Brownian motion, that is, in the present context, near C. For example, Section 2.VII.6(b) corresponds to (b*): if A is an analytic subset of D, the function -. P{ Lf" = S4) is either identically 0 or identically 1. The probabilistic formulation of the fine topology was given in Section 2.IX.15 [see also Section 2.X.2(d)]. The following theorem describes in probabilistic language the minimal-fine topology at a minimal Martin boundary point, defined nonprobabilistically in Section 1.XII.12. Theorem. Let C be a minimal Martin boundary point of D. For C to be a minimal-
fine limit point of the subset A of D it is necessary that for every point C of D,
P{L;R=SC} = 1
(3.1)
whenever B is an analytic superset of A, and it is sufficient that (3.1) be true for a single point C of D whenever B is an open superset of A.
Thus if A is analytic, the point C is a minimal-fine limit point of A if and only if (3.1) is true when B = A. A subset A of D for which D - A is analytic, for example, a Borel subset A of D, is a deleted minimal-fine neighborhood of C if and only if almost every path lies in A during some parameter
interval ] t, S [ with t = t(w) < Si ; the value oft can be taken as the last hitting time of D - A. In both these statements it is necessary that the condition be satisfied for each point C of D and sufficient that the condition be satisfied for a single point C of D. Proof of Theorem 3. An equivalent version of (3.1) is
P{lim sup h[x (t)] = 1) = 1. ttsi
(3.1')
According to Section 2.X.11, this condition is satisfied when A is analytic either for every point of D or for no point C of D. If B is the trace on D of a neighborhood of then (Section 2.X.7) when A is analytic,
K(C, C)
Since the left side is I for all B if and only if C is a minimal-fine limit point of A and the right side is 1 for all B if and only if (3.1) is satisfied, the theorem is true when A is analytic. Since C is a minimal-fine limit point of a subset A of D if and only if C is a minimal-fine limit point of every open superset of A (Section I.XII.12), the theorem is true for every set A.
732
3.111. Brownian Motion on the Martin Space
Application to Function Limits According to Theorem 3 and the discussion at the beginning of this section, a Borel measurable function from D into a Polish space D' has minimal-fine limit [minimal-fine cluster value] tl' at the minimal Martin boundary point C if and only if lim,tsc u[w4(1)] = s has cluster value tl' at 4 along wf(-) paths] almost surely, for each point 4 of D, equivalently, for some point 4 of D. In particular, if D' = R,
limsupu[ww(t)] = mflimsupu(q) a.s. ITSIC
for each point 4 of D, and the corresponding equation for inferior limits is also true.
4. The Probabilistic Fatou Theorem on the Martin Space Let h be a strictly positive harmonic function on a Greenian subset D of R", let v be a positive superharmonic function on D, and define u = v/h. According to Theorem 2.X.8, u4 =
(4.1)
,ts
exists almost surely. In view of the structure of h-Brownian motion discussed in Section 1, the existence of this almost sure limit means that for pD almost every Martin boundary point 4 (which we can suppose minimal) the function u has a limit along almost every K(4, )-Brownian path to 4 from g. According to Section 2.X.11(c), this limit is for each C almost surely a number f(4) which does not depend on 4. In view of Section 3 we have proved that mf lim
rC
f(4)
exists for pD almost every 4, a result we have already proved nonprobabilistically (see Theorem 1.XII.19). Observe that the boundary limit function
was identified in Theorem 1.XII.19 as dM/dM,,, where M,, is the Martin representing measure corresponding to h [to the harmonic component of v in its Riesz decomposition]. The following argument shows how fcan beidentified in the present context, without invoking Theorem 1 .XI 1.19. It is sufficient to identify f for a an h-potential, u a singular h-harmonic func-
tion, and u a quasi-bounded h-harmonic function since (Section 1.IX.11) u is the sum of its components of these types. According to Theorem 2.X.8(c) as now interpreted, the limit function f vanishes pD almost surely if u is an h-potential or is a singular h-harmonic function. According to Theorem
5. Probabilistic Approach to Theorem I.XI.4(c) and Its Boundary Counterparts
733
E{ur} = u°,(g, f) when u is quasi bounded and h-harmonic, in view of the expression we have found in Theorem 11.2 for h-harmonic according to Theorem 1.XII.10. Thus we measure, and f here is have derived Theorem I.XII.19 probabilistically. 2.X.8(d),
5. Probabilistic Approach to Theorem 1.XI.4(c) and Its Boundary Counterparts Throughout this section v is a positive superharmonic function on a connected Greenian subset D of 1 N with N > 1. We have already (Sections
1.XII.13 and 1.XI1.14) proved the Martin boundary counterparts of Theorem I .XI.4(c) which states that if C is a point of D, the function
C)
has a fine limit at C. The first boundary counterpart (Theorem 1.XII.13) of Theorem 1.XI.4(c) asserts that if K is a Martin function for D and if C is a
minimal Martin boundary point of D, then the function v/K(i;, ) has a minimal-fine limit at C. In view of the role of a point s of D as a minimal Martin boundary point of D - {() [Section I.XII.12, Example (a)] Theorem
1.XII.13 includes Theorem I.XI.4(c). As noted in Section I.XII.19 the boundary limit theorem (Theorem 1.XII.13) is a special case of the Fatou boundary limit theorem for the Martin space of D; so in view of the discussion in Section 4 there is no need to discuss this case probabilistically here except to remark that almost every K(C, )-Brownian path from a point of D to the Martin boundary tends to C, and so the probabilistic Fatou boundary limit theorem becomes localized at C in this case. The second Martin bound-
ary counterpart (Theorem 1.XII.14) of Theorem I.XI.4(c) asserts that if v 0 0, the function v/GD(5, ) has a strictly positive perhaps infinite minimalfine limit at each minimal Martin boundary point C, and the probabilistic approach to this result will now be discussed. It is sufficient to prove the existence and strict positivity of this minimalfine limit for v/GD(c, ) bounded, at the expense of replacing v if necessary by vA for a positive constant c. In the following we therefore suppose that 0 < c. Let h be a strictly positive harmonic function on D,
and let w (-) be an h-Brownian motion in D from
with lifetime S,. The
function h will be chosen to be K(C, ) below, but it is instructive and does not complicate the preliminary work to allow the stated generality of h. If Sh is
almost surely finite, that is, if the Martin representing measure of h is supported by the set of attainable minimal Martin boundary points, the process { tv4 (S4 - t), t e O8+ If with lifetime SS is a motion with initial distribution (Section 2). Hence the restriction to D of the function v/GD(C, ) is excessive for this process; so the composition of v/GD(5, ) with this process, on the parameter set ]0, + x[, defined as 0 for parameter values z S,, is a positive bounded almost surely right continuous supermartingale and as such has an almost sure limit x(0+), which we denote by x(0). The process {x(t), to 68+} is a supermartingale, and E{x(0)}
734
3.111. Brownian Motion on the Martin Space
> 0 because v W 0. In particular, if h = K(l, ) for some attainable minimal Martin boundary point C, the initial distribution of wv (SS - -) is supported by "Cl. so has a limit at t along almost every Brownian path from C to ; equivalently (Section 3), has a minimalfine limit at C, equal to E{x(0)} and therefore strictly positive, as was to be
proved. Observe that without the hypothesis that h = K(C, ) we obtain only almost every minimal Martin boundary the weaker result that for point b the function has a minimal-fine limit at C. In view of this
fact we shall assume that h = K(C, ) in discussing limits at unattainable minimal Martin boundary points. Suppose then that S is such a point, and let D. be an increasing sequence of open relatively compact subsets of D with union D, suppose that 5 EDo, let LCn be the last hitting time of D. by w'(-), and define the K(C, )-potential GDin/K(C, -) by Dn
RKu K(C,
Then (Theorem 2.X.7) GD2n/K(C, ) at , is the probability that a K((, -)Brownian path from q ever hits Dn. According to Section 2.X.10, the distribution of w (LC,,) is q)dn(dq)/K((,,), and as discussed in Section 2, the process {ww(LC, - 1), t E R') is a motion with lifetime L and with initial distribution the distribution of The function motion on D is excessive for so the process {xn(t), tE R+ }
= l-
(
Cn 4t1L(j. C
-
)'
tE
R+
(set equal to 0 at times z
is a bounded almost surely right continuous supermartingale. The supermartingale downcrossing inequality is applicable to x and we find that the expected number of downcrossings of an interval
[r,,r2] by
is at most c/(r2 - r,). That is, the expected number of up-
crossings, defined in the obvious way, of [r1, r2] by the process ( 1
v[ws(t)]
tt''(t))'
Lf O 0 and ash for t < 0, the function is parabolic there because this function is already known to be parabolic on b' and has the parabolic function average property on b - D+. In the following we write 6 (t, , h) for dtD(t, c, h)/dt. The function /D is thereby defined, negative, and parabolic on D, identically 0 on D - D. This function vanishes identically if and only if h is /D invariant excessive. h) is 6D invariant excessive on D+ ; that is, (a) The function
-f
S, 1)/v(t, q, h)IN(dn) _ -/D(s + t, , h)
(s > O, t > 0).
(6.1)
D
In proving this it will be convenient to denote by (6.1)5 the relation (6.1)
with "=" replaced by "_ l for which in which case define./'(n,) = S k. Then J'is a continuous map from BN onto A. (Alternatively, it is easy to see that the original domain off is closed and so is itself a Polish space and as such is the continuous image of BN, which makes A the continuous image of BN.) (d) = (a) If (d) is true A = f(BN), where f'is continuous from BN into the Polish space containing A. Let Y ... k be the set of points of BN with first k coordinates n, . . . h 1 . This set is closed and has diameter 11(k + 1). Let X ,, be the closure of /(Y k). If n, a BN, .
.
.
Yn,
nk
1 In.}.
Xn, ..nk 1f(n.)
(k - or).
and therefore A is the nucleus of X. and is analytic, as was to be proved. o
13. Analytic Subsets of Polish Spaces Theorem. Let X and Y he Polish spaces. (a)
Il A [B] is an analytic subset of'X [Y], A x B is an analytic subset of X X Y.
13. Analytic Subsets of Polish Spaces
749
If A is an analytic subset of X x Y and iJ' E X, the set J? J: ry) E'4) is an analytic subset of Y. (c) The projection on Y of an analytic subset of .Y x Y is analytic. (d) The image on Y of an analytic subset of X under a Bore! measurable map from X into Y is analytic. (b)
Parts (a) and (b) are special cases of Theorem 3. To prove (c), observe that the projection map is continuous; so in view of Theorem 12(b) the projection on Y of an analytic subset of X x Y is the continuous image of a continuous image of a Polish space and is therefore analytic. To prove (d), let f be a Borel measurable function from X into Y. The graph off is a Borel
and therefore analytic subset of X x Y. Furthermore, if A is an analytic subset of X, A x Y is an analytic subset of X x Y; so the intersection A of the graph off with A x Y is analytic, and its projection on Y, that is to say J(A), is therefore analytic by part (c), as was to be proved.
Appendix II
Capacity Theory
1. Choquet Capacities If (X, . ') is a paved space for which l is closed under finite unions and inter-
sections, a Choquet capacity on (X,.'), that is, on X relative to X, is a function / from the class of subsets of X into k with the following properties: (a) (b)
/ is increasing; that is, X,
X2 implies that 1(X,) < 1(X,).
X. T X,,, implies that
I(Xj.
X, implies that
I
A subset A of X is called capacitable [relative to (X,'1',1)] if
1(A) = sup {1(B): B c A, BEfa}.
(1.1)
It is trivial that the union of a monotone increasing sequence of capacitable sets is capacitable.
2. Sierpinski Lemma Lemma. Let (X, .:'1') be a paved space, let A he the nucleus of a Suclin scheme X.
over .r', and let h, , h2, ... he a sequence in P. Define Ak =
I
I
X' n X^,nz n .
. .
n :ni56i - oo, choose a < I(A). It will be proved, in the notation of Sierpinski's lemma, that I(B) z a if b, is chosen suitably, and in view of property (c) of a Choquet capacity it is sufficient to choose b,
to make I(B") Z a for all k and therefore sufficient to choose b, to make I(A') > a for all k. To make such a choice, choose b, so large [property (b) of capacity] that I(A') > a, and if b,, . . , bk_, have been chosen so that I(Ak-') > a, choose bk so large that I(Ak) > a, again using property (b). .
4. Lusin's Theorem Theorem. If (X, X, p) is a complete measure space for which p is a countable sum of finite measures, for example, if p it a finite, then .rd(1) = T.
It is sufficient to prove the theorem for p a complete finite measure. Define the outer measure of an arbitrary subset A of X by p*(A) = inf {p(B): A c Be X }. Then p is a Choquet capacity relative to (X, i'). In fact the defining property Section 1(a) is trivial, (b) is a standard property of outer measures defined in this way, and (c) is a standard property of finite measures. According to the Choquet capacity theorem, the sets of .W(X) are capacitable, and in the present context capacitability and measurability are equivalent.
.Capacity Theory Al!.
752
5. A Fundamental Example of a Choquet Capacity Let (D, .F, P) be a complete finite measure space, and define an outer measure P' on the class of subsets of i2 by inf {P(B): B D A, Be .F }.
Let I be the class of finite unions of subsets of R+ x 0 of the form C x A, where C is a compact subset of R+ and A e .F. Then A is a paving of R+ x i2, closed under finite unions and intersections. If A c R+ x i2, let x(A) be the projection of A on i2, and define I(A) = Observe that
.d(A) = s4(9(R+) x .F) z) R(R) x ,F, that sV(.F) = F, and (by Theorem 1.5) that I(A) = P(R(A)) if A e d (F). An important result in capacity theory is that I so defined is a Choquet capacity on R+ x i2 relative to A, that is, that I satisfies Section I (a)-(c). Condition (a) that I is monotone increasing is obviously satisfied. The left continuity condition (b) is satisfied because every outer measure defined like P* in terms of a measure satisfies this condition. To prove that the right continuity condition (c) is satisfied, suppose that S, is a decreasing sequence in A with intersection S. Then the point co is in if and only
if the compact set (t:
is not empty; so x(S,) is a decreasing
sequence in 3F with intersection n(S). Hence lim I(Sn) = lim P(n(S)) =
1(9);
that is, (c) is satisfied.
Application. It is trivial that since I is a Choquet capacity relative to A, this set function is also a Choquet capacity relative to an arbitrary subpaving
of F closed under finite unions and intersections. The Choquet capacity theorem applied to I with various choices of such subpavings of A can be applied (Section 2.11.8) to show that certain subsets of R+ x .0 contain significant parts of the graphs of suitably defined functions from 0 into R.
6. Strongly Subadditive Set Functions Let (X, .f) be a paved space for which :i 0 and a negativity set A- = X - A+ on whose subsets ). < 0. The signed measure A can be expressed as the difference between two measures (Jordan decomposition). For example,
A. = p - v.
p(A) = A(A n A' ),
v(A) = -A(A n A- ).
This particular Jordan decomposition is minimal in the sense that p < p' and v < v' whenever p' - v' is a Jordan decomposition of A. Conversely, if u and v are measures on X, their difference p - v need not be a signed measure, but if both p and v are a finite, there is an increasing sequence B. of sets with union A 'such that both u and v are finite on every set BB ; so the restrictions of p and r to the class of subsets of B, are finite. and we can therefore build a set A' from the positivity sets of these restrictions such that p > v on the subsets of A+ and p < v on the subsets of A- = X - A'. In this way we have found what amounts to a Hahn decomposition of X for p - v and
thereby found a Jordan decomposition of this difference, although the difference may not be defined on all T sets. In Section 7 such differences will be rigorously defined and ordered to obtain a vector lattice.
7. The Vector Lattice #. According to Sections 111.3 to 111.5, the cone ,4Y
can be identified with the
positive cone of a conditionally complete vector lattice ,,4, consisting of pairs (p,. p2) of members of.A//; , with the convention that (p,, p2) = (v,, v2) when when p, + v, = p2 + vi. and with the ordering ( P , p, + v2 t 92 +v , . Each member p of A has a minimal representation
(p ` , p-) in which the component measures are orthogonal. that is. have disjoint supports, and are minimal in the sense that if (p, , p2) is a representa-
tion of p, then p, Z p+ and p, >- p-. If p+ or p- is a finite measure, then p can be identified with the signed measure p+ - p-. A set A is a support for the element p = (p, , p2) of . KQ if there is a representation of p with both components supported by A, equivalently, if A is a support for the measure p, + 42. It follows that two elements of ..* are orthogonal if and only if they have disjoint supports. Charges
Whenever we deal with an extended real-valued countably additive set function on a measurable space (X, 9C), it is to be understood when X has a
773
8. Absolute Continuity and Singularity
topology that . ' is either the class M(.') of Borel subsets of X or the perhaps partial completion of .(9t") relative to some class of measures. Suppose now that X is a locally compact second countable Hausdorff space. (In this book
X will in fact usually be an open subset of IB".) In this context it is to be understood in the absence of a contrary statement that the set functions are finite valued on compact sets. A member y of .,lf, for which 1µ I = µ+ +,uis finite valued on compact sets will be called a charge. If either µ+ or µ- is finite valued, the charge can be identified with the signed measure µ+ -,U-,
in particular with the measure µ+ if µ- = 0. We shall not use the term "Radon measure" in this book, in which "measure" always implies positivity. The class of charges is a sublattice of ..lf, and is itself a conditionally complete vector lattice with the countability property.
In the following sections it is to be understood that the measures and charges discussed are defined on some measurable space, specified further as needed by the context.
8. Absolute Continuity and Singularity If A e.,#,+, an element µ of sV will be called absolutely continuous relative to A if y has a representation (µ,,µZ) for which A null sets are also p, + µ2 null sets; that is, in the usual terminology the measures µ, and P2 are absolutely continuous relative to A. If so, µ, and 92 can be taken as the minimal choices
µ, = µ+, 42 = p. For any acceptable choice (µ,,µ2) there are unique up to A null sets positive finite-valued measurable functions, which we shall denote by dµ,/dt, dµ2/dl, for which µ.(A) =
d2di.
Je
Conversely, if f, andf2 are positive A almost everywhere finite-valued measurable functions and if µ; is defined by
µ.(A) = J JdA,
(8.1)
A
then (p,,µ2) is a A absolutely continuous member of K,. We shall write dµ; = f dal for (8.1). If v = (v,, v2) with dv; = g; di is a second A absolutely continuous member of -#,, then dµ.
µYv=(p',µ')
with dµ', = (ft + g2) v (f2 + 91) dA,
µ A v = (µa, µ')
withdµi =(f, +g2) n (f2+g,)dl.
In particular, if it and v are positive, we can choosef2
= (f2 + 92)dA,
g2 = 0.
774
A.IV. Lattice Theoretic Concepts in Measure Theory
An element p of A'7 will be called singular relative to A if p 1 A, that is, if p and A have a pair of disjoint supports. The Lebesgue decomposition theorem implies that each element of W. has a unique representation as the sum of a A absolutely continuous element and a A-singular element of M,. According to Section III.I 1 the positive elements in the band generated by A are the elements of .,Ko of the form Yo (v A nA), with v in The measure v k nA is majorized by nA and is therefore A absolutely continuous, as is the limit measure y obtained when n oo. Conversely, if p is a A abso-
lutely continuous measure, the measure p A nA has the Radon-Nikodym density (dp/d1) A n; so p = Yo (p A nA), and p is in the band generated by :. Hence the band generated by A is the set of A absolutely continuous members of .,#. and the Lebesgue decomposition is precisely the decomposition of -+Y, into the orthogonal bands of A absolutely continuous and A-singular elements. If (X, X) is a locally compact second countable Hausdorff space
coupled with its class of Borel sets, the lattice .,K, can be replaced in this discussion by the lattice of charges.
9. Lattices of Measurable Functions on a Measure Space Every concept in this section is relative to a specified complete measure A in .,Il; . That is, we are discussing functions on a complete a finite measure space (X. T, A). We conform to the usual custom that in discussing a function from a measurable space into i "measurable" means that the range measurable space is (IB,R(R)). Let .1( be the class of measurable functions from (X, .() into A with the convention that a function need be defined only A almost everywhere and that two functions are identified if they are equal A almost everywhere. Strictly speaking A* is a space of equivalence classes of functions, but we shall adhere to the usual convenient carelessness of analysts in which "function" and "member of .,K * " may be either a function
or an equivalence class, as indicated by the context. The space M* is a lattice under the order 5 to be interpreted as follows: if f and g are measurable functions and if f and g are the respective equivalence classes containing these functions, the inequality f S g means that f 5 g up to a A null set. Similarly f v g [f A g] is the equivalence class containing f v g [f A g]. As already noted, we shall blur the distinction between function and equivalence class.
The subclass KQ of A almost everywhere finite members of #* is a vector sublattice of A' *. The set of positive members of A' * is not a cone. but the set A; of positive members of .# is a cone and .!!; + 0
Theorem. The lattice A' * is complete, and .,# is conditionally complete. Both lattices have the countability property that the supremum [infunum] of a set, iJ it exists, is the supremum [infimum] of a countable subset.
775
10. Order Convergence of Families of Measurable Functions
Since the transformation ft.-. arctan f is order preserving from &* into ..U; , it is sufficient to prove the assertion of the theorem for .,# and therefore for .,/!; ' . Each element f of . K; ' determines an element p f of .,#, by way of du f = f d.1, and the map f " p f is an order isomorphism between . K; and the lattice of A absolutely continuous elements of .11 . The theorem now follows from the conditional completeness of #Q and Theorem 5. Essential Order Terminology
The order of .I(* is called the essential order. If { fa, a E 1) is a subset of ..K*, the elements Vas,fa and ^e,fa are denoted, respectively, by esssup.E,f. and ess inf,E,f.. Thus according to Theorem 9, there is a counttable subset J of I such that supp5 jff zfa for every a neglecting a A null set depending on a. If/is directed, ess lim supat f, is defined (Section III.12) as ess inf [ess sup fa], PEI
a2p
and the essential limit infimum is defined dually. A function f is called the essential limit off if both the essential limit superior and essential limit inferior are equal A almost everywhere to f.
10. Order Convergence of Families of Measurable Functions Theorem. Let I be a linearly ordered set with no last element and with a cofinal increasing sequence. Let be a function from I into the space -&, based
on a complete a finite measure space (X, X, A). Then there is a cojinal increasing sequence I. for which
lim inf x(t) R_.G
lim
It
sup x(t)
R-.0
tt (10.1)
This theorem with I an open interval of R is useful in stochastic process theory. To prove the theorem, assume [if necessary, replacing the given measure A by a finite measure with the same null sets and replacing x(t) by arctan x(t)] that A is finite and that 5 const ( < + oo). Let t; be a cofinal increasing sequence in I, define I.= f1 :1 5 t S and choose t;,o, ... , in I. in such a way that the functions minjsa. x(t.) and maxJSa. x(Q are, respectively, at L' distance at most 2 -" from ess inf,.,^ x(t) and ess sup, E, x(t). Then
lim [minx(t.j) - essinfx(t)] = lim [esssppx(t) - maxx(t;,)] = 0 a.s. jaa t6I, R- W jsa 9eI
,,-.aG
776
A.1 V. Lattice Theoretic Concepts in Measure Theory
because the sums of the integrals of the bracketed A almost everywhere positive functions converge. Hence (10.1) is true with t, the set {1,,j: n >_ 0, j < a j arranged in increasing order.
Generalization Let S be a compact metric space, let S be the space of measurable functions from (X, .1', A) into (S, .4(S)). and let be a function from 1 into S. Theorem
10 can be applied to x(t) =f [y(t)] for J 'a real finite-valued continuous function on S. In fact a slight extension of the proof of the theorem yields the fact that there is a cofinal increasing sequence t, in I for which
limsupj'[ y(t)] n-al
rt
simultaneously for every function f in a countable dense subset of C(S), and
therefore simultaneously for every function in C(S). In particular, if I is itself countable, "ess" can be omitted in (10.2), and it is then easy to conclude that, neglecting a ;! null set, .v(t) has the same cluster values when t increases
through t, as through 1. This result is important in discussing stochastic process separability. If it is known that exists A almost surely whenever t is a cofinal increasing sequence in I, it follows that ess lim, t x(t) exists and is A almost surely this sequential limit, which obviously does not
depend, neglecting null sets, on the choice of t,. If I is countable, it follows that lim,t x(t) exists almost surely. Fatou's Lemma and the Lebesgue Dominated Convergence Theorem
Let ,x(t), te!} be an upper-directed (in the essential order) family of measurable functions from the a finite measure space (X, fit", A) into OB. The ordering of the functions defines an ordering of I, and
esslimx(t) = esssupx(t) = supx(t) a.s.,
it
Icl
tE.I
where (by Theorem 9) J is some countable subset of I and (from Section 111.12) we can choose J to be an increasing sequence t,, cofinal if 1 has a countable cofinal set, so that
esssupx(t) = lim idI
a-m
a.s.
[If 1 does not have a countable cofinal set, this result is trivial because according to Theorem 111.12 there is a point t' of I such that x(t) = x(t') almost surely whenever t >_ t'.] If ess inf,E, x(t) >_ 0, then
777
1. Measures on Polish Spaces
fx ess sup x(t) dA = lim fx x(t) d A. 1cl it Now suppose that is a function from a directed set I into the space of measurable functions from (X, X, A) into R. Then if ess inf c, fp >- 0, the preceding monotone limit result yields Fatou's lemma in the present context: f ess lim inffp dA = lim f ess inf fp dA. < lim inf fX f dA. at x /Za at pt X D rop the positivity hypothesis and apply the usual argument using Fatou's lemma to prove the Lebesgue dominated convergence theorem in the present context: if ess limpt fp =f exists and if ess suppE, fpl is A integrable, then
f fdA=lim f fpdA. X
Pt
x
11. Measures on Polish Spaces Whenever a measure on a Polish space X is discussed in this book, it is to be understood that the domain of the measure is mod" = -V(X) or, if so stated,
the completion of M(X) with respect to a specified measure or family of measures. It is an important property of Polish spaces that a finite measure A on 9t" (and therefore also a a finite measure on T) is inner regular in the sense that for A in i A(A) = sup {A(B): B c A, B compact}.
(11.1)
We sketch the proof for A a finite measure. In the first place (11.1) is true if "compact" is replaced by "closed" because the class of Borel sets A for which (11.1) as so modified is true includes the closed sets and is closed under countable unions and intersections, and is therefore .4(X). Thus it is sufficient to verify (11. I) as stated for closed sets A. Let A be a closed set, and let a be a strictly positive number. In some metric for X let A. be a sequence of nonempty closed subsets of A, of diameter < 1, with union A. Define Bo = A and B, = Ui=o At, where n is so large that A(B,) > A(Bo) - e. Continue by induction: if Bk has been defined as a closed subset of Bk_, with A(Bk) > A(A) - t, we can use the preceding argument to find Bk+,, a
closed subset of Bk, the union of finitely many closed sets of diameter < 1/(k + 1), with A(Bk+,) > A(A) - E. The set B = nk o Bk is a closed set which can be covered by finitely many sets of diameter < 1 /k for every k > 0; so B is compact. Moreover A(B) z A(A) - E. Hence (11.1) is true.
This theorem implies that for any a finite Borel measure p on a Polish
778
A.IV Lattice Theoretic Concepts in Measure Theory
space the class of continuous functions with compact support on the space is a dense subset of L' (µ).
12. Derivates of Measures Let p and v be charges on (E8" with v >_ 0, and lets be a point of RN. If S is a closed convex subset of R' containing i;, let d'(S) be the distance from 5 to OS, and let d"(S) be the diameter of S. A number q will be called the convex derivate at s of p with respect to v if whenever S. is a sequence of closed convex sets containing C for which
d"(Sn) > 0,
lim n-a
lim inf d, ,
0,
> 0.
(12.1)
it follows that v(Sn) > 0 for large n and that lim u(Sn) q n- 7 V(S1) -
(12.2)
In particular, if (12.2) is required only when each set S. is a closed ball with center 5, the derivate is called the symmetric derivate. The convex derivate exists at v almost every point C, and the derivate function is v almost surely the Radon Nikodym derivative dp/dv. The latter notation will also be used for the convex and symmetric derivates, with the intended interpretation always specified. If there is a finite number q such that
d1/`v d
0,
where the derivate is in the convex [symmetric] sense, then (dµ/dv)(i() = q in the same sense, and we shall then say that the derivate is q in the convex [symmetric] variational sense. We outline the proof, standard when N = 1 and v = /, , that dp/dv exists v almost everywhere on U8" in the convex variational sense. If the function dp/dv (convex derivate) is denoted by J'and if q is any real number,
dtp - qvl dv
° 1.I-gj
(12.3)
t' almost surely; so (12.3) is true v almost surely simultaneously for all rational q, and it follows from an easy continuity argument that (12.3) is true almost surely simultaneously for all q. If C is not in the exceptional set for the last statement, set q = J(C) to complete the proof.
Appendix V
Uniform Integrability
This Appendix. consisting of only one section, lists for easy reference what is needed in this book of the uniform integrability concept.
It will be convenient to call a function 1 from R* to 68+ a uniform integrabilityy test function if b is monotone increasing and convex and if lim,_m [O(s)/s] = + x. Let (X,.£',A) be a measure space with i.(X) < +x. All functions on X in the discussion below are measurable from (X, I") into (l . (OB)) unless specified otherwise. Recall that a family , j,, I E I) of such functions is said to be uniformly integrable (relative to i.) if lJ,Idi.=0.
limsup
n-x tel
f1UP"I
The following uniform integrability properties are used in this book. Except in (d), all functions of the family are in L'(;.). (a) The family.f is uniformly integrable if and only if it is L' bounded and if the set function family ; A F-. f A Jj; J(1i., t E i) is uniformly absolutely continuous relative to i. ; more formally J; is uniformly integrable if and only if
sup I 'CI
(b)
,
Jill di._ J5l1n-x
In particular, if limn-. fn = f i. almost surely, if each fn is integrable, and if
780
A.V. Uniform Integrability
lim
f f" dal = fXfdA < + oc,
"-m x
then the sequence f is uniformly integrable. (e) A uniformly integrable sequence f, on X is weakly sequentially compact; that is, there is a subsequence fa. and a A integrable function f on X such that lim fX f,"g di = lx fg dl "-aD
for every bounded function g on X.
Generalization Let (X, ') be a measurable space, and let I be an arbitrary nonempty set. Suppose that for each t in I there is a function f, on X and a finite measure A, on 1'. The family is said to be uniformly integrable relative to A. if
sup,u,(X) < + co, and re/
lim supj
"-'00 iEl
If I dA, = 0. IU4I>")
The obvious translations of (a) and (b) into the present context are valid.
Appendix VI
Kernels and Transition Functions
1. Kernels Let (X, £C') and (Y, 4.1) be measurable spaces. A kernel from the first space into the second is defined as a function from X x Y into G'8+ with the following properties:
(KI)
A) is X measurable when AE*.
(K2) p(l;, ) is a measure on W when e X.
The kernel is called substochastic if I and stochastic if p(-, V) __ 1. If (X, X) = (Y, W), the kernel is said to have state space (X, 1). Kernels Given by Densities
If q is a measurable function from (X x Y, T X'3/) into (98+,.(l)) and if d is a measure on Y, define A) =
q)).(dq)
( e X, A E 'J)
JA
to obtain a kernel p with density q relative to A.
Notation If p is a kernel from (X, f) into (Y, IN) and if f is a measurable function from dry) when this integral is (Y, IN) into (U8,.V(fi)), we write p(, f) for j,. well defined, for example, when f >_ 0 and f is Y measurable. If p has density q relative to a specified measure, we also write f) for p(-, f ).
Extension of a Substochastic Kernel to a Stochastic Kernel Let p be a substochastic kernel with state space (X, 5C'). This kernel can be
extended to be stochastic as follows. Adjoin a trap point (also called a cemetery) 8 to obtain an enlarged space X' = Xu {8}, and define F as the
782
A.V1. Kernels and Transition Functions
a algebra of subsets of X' generated by d and the singleton {e}. A stochastic
kernel p' with state space (X'. X') is then determined by setting p' = p on
Xx.1"and p'
{8}) =
I
if
1
ifg = a.
EX,
This kernel is the desired extension of p.
2. Universally Measurable Extension of a Kernel If p is a kernel from (X..'1.") into ( Y,4) and if the function
X) is bounded, we can extend each measure by partial completion to the a algebra 4'(V) of universally measurable sets over V. Under this extension p becomes a kernel from (X, V(d" )) into (Y, Pl(w1)). To see this, let u be the completion of a finite measure on .f, suppose that A e'W(NJ), define a measure v on .( by
!x p(', )p(dd), and extend v by partial completion to the a algebra *(IN). Since A is v measurable, there are sets A 1, A 2 in ' with A, c A c A2 and v(A 1) = v(A 2). This equality implies that A,) = A) = A2) it
almost everywhere on X; that is, p measurable for every t, as was to be proved. This argument shows that the definition of v(A) for A in i?I remains valid for A in ' /(4'). It is important for the application to probability that we have actually proved more than was asserted. In fact let be the a algebra of those subsets of Y which, for every choice of u, are in the completion of the measure v defined above. This a algebra includes *(ft and its sets will be called the universally measurable sets over ' relative to p. What we have proved is that if is extended by partial completion to this a algebra, then A) is universally measurable, and the definition of v remains valid.
3. Transition Functions Let I be a linearly ordered set, let (X, ') be a measurable space, and suppose
that to each pair (r, s) of points of I with r < s corresponds a stochastic kernel p(r, , s, ) with state space (X, 9C') such that for A in *' and r1 < r2 < r3
in I the Chapman-Kolmogorov equation p(r1, , r2, A) = I p(r2,17, r3, A)p(rt , - r2, dtl)
(3.1)
X
is satisfied. Such a function p is called a Markov transition Junction, or simply a transition Junction. In some contexts it is necessary to allow the kernels to be substochastic or even infinite valued, but any divergence from the above
783
3. Transition Functions
definition will be specified explicitly. The parameter set is usually either the set of strictly positive integers discrete parameter case or the interval ]0, + 00[ (continuous parameter case).
Discrete Parameter Stationary Case
Let (X, _f) be a measurable space, let p(1, , ) be a kernel with this state space, and define inductively a sequence of kernels with this state space by
p(n + 1, , A) = I p(1, it. A)p(n, , dq),
n > 1.
(3.2)
x
Then p(r + s, , A) = J p(r, ry, A)p(s, , dq).
(3.3)
x
Ifp(l, , ) is a stochastic kernel, the function (m, , n, A) -. p(n - m, , A) for 0 < m < n is a transition function with parameter set the set of strictly positive integers, but in this context it should cause no confusion if the function (n, S, A) i-. p(n, S, A) for n > 0 is called a stationary transition function or temporally homogeneous transition function and if (3.3) is identified
as the relevant Chapman-Kolmogorov equation. If the given kernel is substochastic, (3.3) yields a substochastic
transition function p. Let p'(1, , ) be the extension of the substochastic kernel p(1, , ) to be stochastic by way of a trap point. Then the inductive definition (3.2) applied to p'(1, , ) leads to a (stochastic) transition function p', and p'(n, ) is for each strictly positive integer n the extension of p(n, , ) to be stochastic by way of the same trap point.
Continuous Parameter Stationary Case Let (X, ") be a measurable space, and let p be a function from ]0, + cc [ x
X x .£ into R with the following properties: (a)
For each t in ]0, + o t,[ the function p(t, , ) is a kernel with state space (X, X).
(b) The Chapman-Kolmogorov equation (3.3) (with r and s now in
]0, +oo[) is satisfied.
If each kernel p(t, , ) is stochastic, the function (r, , .s, A) r-. p(s - r, , A) for 0 < r < s is a transition function with parameter set ]0, + oo[, but as in the discrete parameter case the function p itself is called a stationary or temporally homogeneous transition function. If the kernels are allowed to be substochastic, in which case p is a substochastic transition function, a
A.VI. Kernels and Transition Functions
784
single trap point can be adjoined to X to make each kernel p(t, ,) stochastic and thereby extend p to be a (stochastic) transition function. Universally Measurable Extensions
In the discrete parameter stationary case it is trivial that if p(l, , X) in (3.2) is bounded and if p (I, , ) is the universally measurable extension ofp(l, , ), that is (Section 2), the extension of this kernel to the state space (X, 'W (52)),
then the nth iterate p (n, , ) of p(1, , )
p
is the extension of
in
is
state
the
the this
if t p (t, , ) is the the kernel for is state space (X, by p,,. To verify (3.3) forp with A in V(52'), fix r, s, and , and choose At and A2 in .t to satisfy be proved.
A, c A c A 2,
p(r + s, , A t) = p(r + s, , A2).
Then
p(r+s..A)=p(r+s,.A1)= f jP(s
,, A1)p(r, , d,),
i= 1, 2;
x (3.4)
so (3.4) is valid when A, is replaced in the integrand by A, as was to be proved.
In this context if we strengthen the hypotheses on p by supposing that for A
in 52' the function p(t, -,A) is R(]0, + oo[) X X measurable, then if f is a Lebesgue measurable function from 18+ into 98+, the function
(,A)-' f(t)p(t,,A)!t(dt) 0
is a kernel with state space (X, i) and so (Section 2) when considered on X x x!(i) is a kernel with that state space, and therefore the above integral defines a W(52) measurable function for fixed A in *(X). Functions defined by integrals of this form are basic in probabilistic potential theory.
Appendix VII
Integral Limit Theorems
1. An Elementary Limit Theorem Let X be a metric space, let A be a completed measure of Borel subsets of X, and let k, be a sequence of A measurable functions from X into A. In many contexts there is a point C of X with the property that A({t}) = 0 and that for f a suitably restricted function from X into 08, continuous at C, lim
(1.1)
'-' Jx The following theorem gives conditions that such a limit relation be true. The conditions are phrased for ease in application rather than for maximal generality. Theorem. Suppose that k, and f satisfy the following conditions: (a) (b)
Each function k is positive, with Ix k dA = 1. The function f is A measurable, and if A is an open neighborhood of Jx_e f I dA = then JX_Ak dA = 0.
Under these conditions lim sup ` kjdA < lim sup
If in addition to (b) the function f is continuous at C, then (1.1) is true.
If the right side of (1.2) is +oo, inequality (1.2) becomes trivial. Thus to prove (1.2), which implies the last assertion of the theorem, it is sufficient to show that if f _< a < + oo on a deleted open neighborhood A of C, then the
left side of (1.2) is at most a. We can suppose, adding a constant to f if necessary, that a Z 0. Then lim sup I k,,,jdA < x+ li m sup x
as was to be proved.
k.I f I dA = a, x_A
786
A.VII. Integral Limit Theorems
2. Ratio Integral Limit Theorems For each positive integer n let k(n, ) be a positive Borel measurable function on QRN. Let H and U be, respectively, a measure and a charge on QRN. Define
a = I'
b = Jkn?l)H(d1l).
k(n, ?l)U(dq),
(2.1)
N
In many contexts k(n, q) is defined in such a way that
lim °" _
Al
a. b dH (0).
(2.2)
We shall derive conditions on k(n, ) for (2.2) to be true. The following remark will be useful. Suppose that H is fixed and that 0 < b < + x for all n. The charge U will be allowed to vary in some specified linear class r of charges, including H and including I UI with U, for which k(n, q)I UI(dq) < + x.
Suppose it has been proved that (2.2) is true under the following special conditions. t1 e F, U > 0. and the derivate in (2.2) is the symmetric derivate and vanishes. It then follows that (2.2) is true for U in f whenever the derivate on the right exists as a symmetric variational derivate (see Section IV.12 for the derivate definitions). In fact if q is the value of this derivate. J'
k(n,rl)IU-gHI(dq).
and by hypothesis I i'% - qH I E F. and by definition of the symmetric variational derivate the symmetric derivate d I U - qHI /dH exists and vanishes at the origin. The preceding remark is of course also valid if the derivate in question is the convex instead of the symmetric derivate. In some contexts ratio integral limit theorems of the type described are easily reduced to rather simple ratio integral limit theorems in one dimen-
sion. Such a one-dimensional ratio integral limit theorem is proved in the next section.
3. A One-Dimensional Ratio Integral Limit Theorem The following ratio integral limit theorem will be used to derive the desired ratio limit theorem for the Poisson integral for harmonic functions on a ball. Let k. be a sequence of positive Bore] measurable functions on R', and let
787
3. A One-Dimensional Ratio Integral Limit Theorem
a, $ be monotone increasing functions on 68+, vanishing at 0, strictly positive on ]0, + oo [, and right continuous except possibly at 0. Define
a"= fo k" da,
b"= f k"dfl.
(3.1)
Jo
The following theorem stating conditions which imply that
lima-"=0 w b"
is phrased for ease in application rather than generality. Theorem. Suppose that a", b" as defined by (3.1) are finite for all n and that the following conditions, in which p is some strictly positive integer, are satisfied.
(a)
k" is positive and monotone decreasing and has a continuous derivative k;,.
If a > 0,
(b)
fa
lim infk"k"(a) ' 1t (dr) = + oo.
o
"°c
There is a strictly positive function f on 68+ such that the function k"/f
(c)
is monotone decreasing for large n and that fo fda < +oo. liro P(r)
(d)
0,
,in
P
soup fi(r) < + oo.
Then (3.2) is true.
To prove the theorem, choose t strictly positive and so small that the limit superior in (d) is strictly less than I/E. If a is so small that cc 0 the function u, with U replaced by I U1, and the function h are locally bounded on RN.
For each s > 0 the function K(s, ) is a positive monotone decreasing function on R+, with a continuous derivative, and K(s, 0) = 1. (c) There is a strictly positive junction on ]0, + oo [ satisfying (b)
lim b(s) = 0,
lim inf K(s, b(s)) > 0.
r0
(d) If a > 0, there is a strictly positive number so = so(a) for which,
uniformly for r z a,
L0
K(s, r)
_ 0.
K(so r)b(s)N -
789
4. A Ratio Integral Limit Theorem Involving Convex Variational Derivates
Then if at a point r; of 08N the derivate dlN/dH exists as a finite convex
derivate and dU/dH exists as a finite convex variational derivate, and if 0 < c < 1, it follows that
u(g,s) h(l:,s) s~o It-Slsca(s) lim
dU (r) = 0. dH
sup
(4.2)
Observe that (Section IV. 12) dU/dH and d1N/dH exist and are finite as convex variational derivates at H almost every point C of RN.
For fixed H the class of charges U satisfying (a) is linear, contains I U1 with U, and contains H. Hence according to the argument in Section 2, it is sufficient to prove that (4.2) is true for C when U is positive and both dlN/dH and dU/dH exist as finite convex derivates with (dU/dH)(C) = 0. To simplify the notation, we assume that C = 0. For r > 0 define r) and r), and define a(l;, 0) _ (3(l;, r), respectively, as the U and H measures of 0) = 0. By definition of the convex derivate
r).
cr = 0,
lim sup r-o W, r)
limsup r~o
rN
r)
I < cr < +00.
To obtain a minorant of h, observe that s) >t LAS)] K(s, r) d
r) z K(s, b(s))
S(s))
(4.4)
so that
K(s r)da(i,r) < fro.6(s))
a(,S(s))
/;(l:, s)
(4.5)
,6(s))*
If a > 0 and if s is so small that b(s) < a, K(s, r) da(l;, r) = K(s,
a) - K(s,
8(s)) (4.6)
K'(s,
r)l, (dr).
)a(s).al
If now e > 0 and if a is so small that a(g, r)
aw).a1
c}
U {up > c} PC I
792
A.VIII. Lower Semicontinuous Functions
simultaneously for every rational c and therefore for all c. Then u' is lower semicontinuous, u' < u', and {u' > c} = {u' > c} for all c. Hence {u' z c} _ {u' >_ c} also, and subtraction yields {u' = c} = {u' = c}; so u' = u'. The second assertion of the theorem is left to the reader.
3. Choquet Topological Lemma Lemma. Let {up, 13 e I } be a family of functions from a second countable Hausdorff space into U8, and if J c I, define u' = infs. j up. Then there is a countable subset J of I such that u' = u'. In particular, if I is directed downward, there is a decreasing sequence up. with limit v such that v = u'.
Let H be the given second countable Hausdorff space, and let H, be a sequence of open subsets of H containing each set of a countable base for the topology of H infinitely often. After replacing up by arctan up if necessary, we can assume that the family u, is uniformly bounded. For n z 1 in H. to satisfy u'(,) choose 1/n. There is a point P. 5 1/n. Set J = f,. Then in I for which inf u'(q) 5 inf u'(ry) + ?
"EH
nen
n
and it follows that u' 5 u'. The reverse inequality is trivial; so there is equality. The second assertion of the lemma is left to the reader.
Historical Notes
The modernization of classical potential theory during the last fifty years has proceeded for the most part in small steps, many of which were more and more rigorous derivations in more and more general contexts of facts stated years earlier. Thus it is certain that the following notes, which give reference to the "originators" of the more important advances, will appear to knowledgeable readers to be inaccurate and unfair. The historical notes to the probability discussion are only slightly more reliable. It is trivial for research mathematicians to discover mistakes in historical accounts of their own subjects, especially when the accounts rely on publication dates and on the references in research papers. Thus no one should have much confidence in a detailed history of the development of a mathematical theory. The difficulty is compounded by the fact that many theorems are proved by several authors in more or less overlapping formulations and that (see the many papers on the Martin boundary) authors are frequently unwilling to read their colleagues' papers carefully enough to state for the record the exact differences between the scopes of their papers. Thus the reader is requested to read the following notes with a grain of salt and a lump of tolerance.
Part 1 See Burkhardt and Meyer [1, 1900] and Lichtenstein [1, 1919] for reviews of nineteenth and early twentieth century potential theory, and see Kellogg [2, 1929] for potential theoryjust before its renaissance. See Brelot [11, 1952]
and [17, 1972] for more modern reviews. The books of Tsuji [1, 1959]. Helms [1, 1969], Brelot [15, 1969], Constantinescu and Cornea [1, 1972], Landkof [l, 1972], and Wermer [1, 1974] on potential theory contain much of the material in Part I but are organized differently with emphasis on different aspects of the subject. The books of Murali Rao [4, 1977], Port and Stone [1, 1978]. and Chung [2, 1981] derive their potential theory as a by-product of probability.
794
Historical Notes
Chapter 1.1
It would be both futile and difficult to give a detailed historical background to the work in Chapter I of Part 1, but note that the development in Section 8 culminating in (8.3) follows [Green I. 1828]. Chapter 1.11 ,Section
/. What was essentially the Poisson integral for N = 2 was
displayed by Poisson in 1820. In [Poisson 1, 1823]. he displayed his integral for N = 3 in its present form and tried to prove the boundary limit part of Theorem I for continuous boundary functions. Schwarz [1, 1872] gave the first rigorous proof of Theorem I (for continuous boundary functions). Sections 2 and 3. See Harnack [ 1, 1887] for his inequality and convergence
theorem, the latter of course only for monotone sequences. The fact that a one-sided bounded harmonic function on R' is identically constant is due to Bocher [I, 1903]. Section 4. F. Riesz [2, 1926; 3, 1930] inaugurated the systematic study of superharmonic and subharmonic functions; the properties and applications
in this chapter are almost all due to him. See [Radii 1, 1937] for early references and further details on these functions. Section 10. Theorem 10 on the concavity of spherical averages of super-
harmonic functions was proved in [Riesz 2, 1926] in the dual form for convexity of spherical averages of subharmonic functions.
Section 12. Lord Kelvin (= W. Thomson) introduced his transformation in [Thomson I, 1847].
Section 14. The theorem that a positive harmonic function on a ball is the Poisson integral of a measure has been called the "Herglotz theorem" with a reference to [Herglotz 1, 1911 ], but this theorem was proved first by F. Riesz [1, 1911]. and in his own paper Herglotz refers to the Riesz paper for this result. Section /5. The essential content of Theorem 15 is that u/h has nontangential limit function almost everywhere (Mb measure) on the ball boundary. Here the derivate dM /c/M,, has different possible definitions, and the exceptional M,, null boundary subset depends on the definition chosen. No distinction is made between these definitions in the following remarks. Fatou [1, 1906] proved Theorem 15 for the case N = 2, h =_ I, and proofs were given later for N > 2 with h = I by other mathematicians. The theorem was proved by Doob [ 12. 1959] in the general case. Section 16. Minimal harmonic functions were introduced into potential theory by Martin [1. 1941].
Pan 1
795
Chapter 1.111
Section 1. F. Riesz [2, 1926] introduced least harmonic majorants of subharmonic functions. Section 3. See these notes to Chapter VI for the history of the Fundamental Convergence Theorem.
Section 4. The reduction concept developed from the dual concept of "extremization." If D is a Greenian subset of l", if u is a negative subharmonic function on D, and if A is a subset of D, the extremization of u on A was defined as the supremum of the class of negative subharmonic functions majorized quasi everywhere on D - A by u. See Brelot [8, 1945] for further details and references to earlier related work by him and A. F. Monna. Brelot [13, 1956] extended work of Martin [1, 1941] by treating reductions of positive superharmonic functions on certain subsets of D. The treatment of reductions in this book extends the standard treatment by allowing reductions on an arbitrary subset of the closure of D in an arbitrary compactification. Section 7. The natural order decomposition is apparently due to Mokobodzki. See Mokobodzki and Sibony [1, 1968] for the relation between the natural order decomposition theorem and additivity of reductions in a very general context. Chapter l .I V
Cartan's paper [2, 1945] on potentials on special sets had an important influence in the modernization of potential theory. See references below to key results in this paper. Sections 7 and 8. F. Riesz [3, 1930] derived the existence of the measure associated with a superharmonic function and proved what is now known as the Riesz Decomposition Theorem.
Chapter IN Section 1. Brelot [5, 1941] introduced polar sets to take the place of sets of inner capacity 0 (called inner polar sets in this book) as the negligible sets of classical potential theory. Cartan proved in [2, 1945], although he announced the result in [1, 1942], that the polar sets are the sets of capacity 0 in the capacity definition of Chapter 1.XIII. Section 2. Choquet [2, 1957] showed, in a trivially different context, that if A is a polar Ga subset of a Greenian subset D of I", then there is a measure p supported by A such that A is the set of infinities of GDu. G. C. Evans had proved the existence of such a potential for A polar and compact.
796
Historical Notes
Section 5. Theorem 5 is usually stated for A a set closed in D. but the formulation in Theorem 5 is sometimes more convenient. The fact that a bounded harmonic function on an open set D less a compact polar subset has a harmonic extension to D was proved by Bouligand [1, 1926]. Vasilesco [2. 1937] proved Theorem 5 for A compact, Brelot [5, 1941], for A closed in D. The fact that a one sided-bounded harmonic function defined on an open
deleted neighborhood of a point must have the form cG(c, ) + h with h harmonic on the full neighborhood goes back at least as far as [Bocher 1, 1903]. The less general fact that a bounded harmonic function defined on an open deleted neighborhood of a point can be extended to be harmonic on the full neighborhood was proved by Schwarz [l, 1872]. Section 6. Szego [1, 1924] proved that in RZ the complement of a compact set A has a Green function if and only if the Robin constant of A (see Section XII1.18 for the Robin constant) is strictly positive. This result is equivalent to Theorem 6.
Section 8. The Evans- Vasilesco theorem references are Evans [2, 1935] and Vasilesco [ I, 1935]. Section 10. The principle of the maximum is sometimes called the MariaFrostman principle of the maximum because Maria [ 1, 1934] and Frostman [1. 1935] proved this special case of the domination principle (with D = UR" and N > 2).
Chapter I
.
V1
Section 1. Szpilrajn [1, 1933] proved the Fundamental Convergence Theorem in the weak version of Theorem 111.3 in the context of a locally L' convergent sequence of superharmonic functions. Radd [1. 1937] remarked that Szpilrajn's result implied Theorem 111.3 in the context of a monotone
decreasing locally lower bounded sequence of superharmonic functions. (Both mathematicians actually phrased their results in the dual context, for sequences of subharmonic functions.) Brelot [2. 1938] strengthened the Szpilrajn Rado result by proving that the exceptional set ;u < u; is inner polar. Cartan [2, 1945], but he announced the result in [1, 1942], proved that this exceptional set is polar. Section 2. In a series of notes in the Paris Comptes Rendus leading up to his definitive "Theory of capacities" [1, 1955] Choquet analyzed set functions satisfying strong subadditivity and related conditions and obtained his capacitability theorem (Theorem 3 of Appendix II), which makes easy the proof that an analytic inner polar set is polar.
Section 3. The reduction properties listed go back to Brelot's early work except for the properties involving sets meeting OD and for property (o) which corresponds to certain crossing inequalities in martingale theory (see Section 2.111.22).
Part 1
797
Chapter 1.VII Writers who describe GD as "the Green's function" should be condemned to differentiate the Lebesgue's measure using the Radon-Nikodym's theorem.
Section 1. The older mathematical literature contains many attempts to show that open sets of various types have Green functions. The modern approach removes the difficulty by generalizing the definition of a Green function. This trick shifts the problem to that of showing that the Green function has certain desired properties, but some of the old smoothness demands have disappeared. For example, it is no longer necessary to define harmonic measure by means of a Green function, as was done in Section 1.8.
Chapter 1.VIII
See Vasilesco [3, 1938] for a survey of progress in the Dirichlet problem just before Brelot's [3, 1939] definitive study of what is now called the PWB method.
Section 1. The fact that relativization of a family of functions having an average property (1.1) yields a family with a corresponding average property (1.3) was pointed out in [Doob 9, 1958]. In a Markov process probabilistic context this relativization corresponds to a modification of transition probabilities (Section 2.VI.13) conditioning paths to go where the conditioning function is large. The analysis of the Dirichlet problem on a Greenian subset of R" provided with its Martin boundary in [Brelot 13, 1956] was essentially, although not explicitly, for relative harmonic functions. The Dirichlet problem for harmonic functions has been attacked in several ways, but it was finally seen that each way involves two distinct problems. (i) The first problem is to find, for a specified suitably restricted boundary
function f, certainly for an arbitrary finite-valued continuous boundary function f, a perhaps generalized "solution" u1. The function u f is to be harmonic, and if the given domain and the given boundary function are sufficiently smooth, the function of is to have boundary limit function f; that is, of is the solution of the Dirichlet problem in the classical sense. (ii) If of exists and if a domain boundary point i; is specified, the second problem is to find conditions on the domain and boundary function ensuring that of has limit f(C) at C. The simple Zaremba [1, 1911] example [Section 2, Example (a)] and the more discouraging Lebesgue [1, 1912] spine example (Section 15) showed that the Dirichlet problem may not be solvable in the
classical sense even when the boundary is the Euclidean boundary and the boundary function is finite valued and continuous. Poincar6 [1, 1890; 2, 1899], however, had shown that for a sufficiently smooth Euclidean boundary the Dirichlet problem has a solution in the classical sense for every finite-valued continuous boundary function. The problem of finding Dirichlet solutions in more general contexts was solved by generalizing the
798
Historical Notes
definition of "Dirichlet solution." The PWB method of finding (generalized) Dirichlet solutions is a development of a method devised by Perron [ 1, 1923]
(independently and almost simultaneously by Remak [I, 1924]) and perfected by Brelot [3, 1939]. Wiener [2, 1924; 3, 1924; 4, 1925] was the first to assign a generalized Dirichlet solution to an arbitrary finite-valued continuous Euclidean boundary function. He used two methods of which one was an elaboration of the Perron method, but he did not obtain Brelot's later definitive results because a miscalculation in [4, 1925] gave him a supposed counterexample which prevented him from proceeding to the class of bounded Borel measurable boundary functions. He showed, however, that for the Euclidean boundary the finite-valued continuous boundary functions are resolutive. Lebesgue was perhaps the first to see clearly the distinction between problems (i) and (ii) above; so it is not surprising that he coined the term "regular boundary point." Kellogg [1, 1928] for N = 2 and Evans [1. 1933] for N >_ 2 showed that every subset of strictly positive capacity of the Euclidean boundary contains a regular boundary point; that is, the set of irregular boundary points is inner polar. This fact implies that the F. set of irregular Euclidean boundary points is polar. Sections 12 to 14. Poincare [1, 1890] used the idea of what is now called
a barrier. Lebesgue [2, 1912] coined the term and used the existence of barriers for a regular region to prove the convergence to the Dirichlet solution (for a finite-valued continuous boundary function) of a certain sequence of
approximants. Lebesgue [3, 1924] found that for the existence of a barrier at a Euclidean boundary point it is necessary and sufficient that the point be regular. Bouligand [1, 1926] showed that it is sufficient for regularity that the barrier be a weak one and proved the related result that a Euclidean of boundary point is regular if and only if the Green function
the given domain D has limit 0 at the boundary point for every pole c (equivalently, a single pole if D is connected).
Section 15. In the years when potential theory was finally conquering the Dirichlet problem the cone regularity condition was ascribed to both Poincare and Zaremba. Only Clio now remembers why Zaremba was later given the sole credit; the condition will be given both names in this book. Section 18. The extension of GD in Section 18 was analyzed by Brelot [6, 1944]. See [Brelot 1, 1938], [M. Riesz 1, 1938], [Frostman 2, 1938], and [Vallee Poussin 4, 1938] for earlier discussions of the relation between GMor (for r superharmonic on a neighborhood of D) and µo(-, c), of the kernel 6', and of related questions.
Section 19. The proof of Theorem 19(b4) follows that in [Murali Rao 3, 1974]. The result was proved by Brelot [6, 1944] in the course of his treatment of potential theory on R' extended by a point at infinity.
Part I
799
Chapter I.IX Everything in this chapter is in the folk lore of potential theory. and much of it has appeared explicitly. Sections 9 and 10. The classes of quasi-bounded and singular harmonic functions were first discussed by Parreau [I, 1951].
Chapter I.X
The concept of sweeping (balayage) goes back at least as far as Gauss [1, 1840]. There are two approaches to the subject. one by operations on functions, now treated by Brelot's reduction technique, the other by orthogonality-minimization techniques originated by Gauss but first made rigorous by Frostman [1, 1935; 2, 1938] and Cartan [2, 1945; 3, 1946]. The first approach is stressed in this book. The following outline of Poincare's
method of solving the Dirichlet problem [1, 1890; 2, 1899] shows why his method is identified with sweeping, but see Vallee Poussin's remark
about Poincare's work. Let D be a bounded open subset of 08^, with a smooth boundary. For it a C'-' function on a neighborhood of D. with Au < 0 (so that u is superharmonic on a neighborhood of D). Poincare evaluated what we now identify as GM Du by the method of proof of Theorem
III.1 and proved that under his smoothness hypotheses on D the function GMDU was the desired Dirichlet solution for the boundary function UIPD. If Au is not supposed strictly negative, Poincare reduced the problem to the case already treated by writing it as the difference between two functions
with the above properties of it. Using this solution and an approximation procedure Poincare solved the Dirichlet problem for an arbitrary continuous boundary function. Thus the basis of Poincare's method was the construction
of GMDU for u superharmonic on D. His construction, in the language of the proof of Theorem 111.1. was to apply operators of the form to with B a ball relatively compact in D. For it superharmonic on D the operation U F - TBu sweeps the Riesz measure for u out of B (observe that tau = R°-"
when u is positive on D. so that reduction notation is applicable), but Poincare did not go on to analyze this aspect of his method. Vallee Poussin in a series of papers including [l. 1932; 2, 1937: 3, 1938. 4. 1938] developed sweeping further and in fact in [3. 1938] he asserted that he had coined the word "balayage" and that (in spite of his own title for [I, 1932]) Poincare's work had nothing to do with sweeping. Vallee Poussin concentrated on the swept measures: so it was he who developed and applied the interpretation of the sweeping kernel as harmonic measure. He applied sweeping to analyze
capacity, the role of sets of inner capacity 0, and many other topics, not always including full details, obtaining among other results one equivalent to Theorem XI.13. The full use of the reduction technique on functions had to wait for the final form of the Fundamental Convergence Theorem, which
Historical Notes
800
was quickly applied to reductions by Brelot [8, 1945]. Most of Chapter X first appeared in the latter paper. Chapter 1.X1
See [Brelot 16, 1971] for further elaboration of the fine topology and its applications. Section 1. Brelot [4, 1940] defined thinness of a set at a point by X1(2.1).
Cartan pointed out in a letter to him that thinness could be interpreted by means of what is now called the fine topology. For some years thereafter, however, the fine topology was merely a tool for phrasing results elegantly; in fact topological language was not commonly used in discussions involving thinness. Cartan [3, 1946] noted that the fine topology is the restriction of a
certain topology of measures to the class of unit measures supported by singletons. According to Constantinescu and Cornea [1, 1972]. the Baire property of the fine topology was noted by Cornea in 1966. Section 4. The existence of the fine limits in Theorem 4(a) was proved by Doob [8. 1957] by probabilistic and [11, 1959] by nonprobabilistic methods.
The present proof is new. The identification of these limits could easily (see the application in Section XII.19) have been made in these earlier papers but was not. Section 5. See [Brelot 6, 1944] for classical potential theory extended to If8" u { oo ; .
Section 6. Theorem 6 is due to Brelot [7. 1944].
Section 9. Theorem 9 is due to Cartan [see Deny 1, 1950, p. 171 ] for fine limits and to Doob [11, 1959] for fine cluster values (in the Martin boundary context). Section 11. Theorem I I is due to Doob [17, 1966].
Section 12. Theorem 12 is due to Brelot [4, 1940]. Sections 14 and 15. Theorems 14 and 15 are due to Brelot [8, 1945].
Section 18. Theorem 18 is due to Brelot [10, 1948], but see also Vallee Poussin 4. 1938].
Section 19. See [Fuglede 1, 1972] for a discussion of classical potential theory based on fine superharmonic and fine harmonic functions. Sections 21 and 22. The material in these sections is taken from [Brelot 9, 1946].
Section 23. See the note to Section XVIII.16 on the parabolic context domination principle.
Pan I
801
Chapter l.Xll Sections 1 to 9. Martin [1, 1941] defined what is now called the Martin boundary and proved what is now called the Martin representation theorem (Theorem 9). By now the Martin boundary of a set D on which harmonic functions in some generalized sense are defined is a rather vague concept. Roughly any set of points (, to each of which corresponds a minimal har-
monic function K(C, ), which is adjoined to D along with possible other points to form a topological space and which then gives as a representation of an arbitrary positive harmonic function u an integral J K(C, )MM(dd) is called a Martin boundary for D. The treatment in Sections 1 to 9 follows Martin as modernized by Brelot [13, 1956] and put into the context of h-harmonic functions. The significance for its Martin representation of quasi boundedness of a harmonic function was pointed out by Parreau [1, 1951 ]. Section 10. Theorem 10 is due to Brelot [ 13, 1956]. Heins [1, 1959] defined
a "generalized harmonic measure" on a Riemann surface D as a positive harmonic function u for which 0:5 u< 1 and GMD[U A (I - u)] = 0. According to the remark "Intrinsic definition of h-harmonic measure," such a function u is actually the harmonic measure of a Martin boundary subset. (The Martin boundary discussion is applicable to Riemann surfaces.)
Sections 11 to 14. Naim [1, 1957] defined the minimal-fine topology on a Martin space (in a more satisfying way than that in Section 12, where her elegant approach is considerably abbreviated) and proved Theorems 11, 13, and 14. Theorems 13(a) and 14 had already been proved in different terminology for D a half-space by Lelong [1, 1949].
Section 16. NaIm [1, 1957] proved the part of Theorem 16 involving minimal-fine limits. The cluster set part of Theorem 16 is taken from [Doob 11, 1959].
Section 18. Theorem 18 is due to Naim [1, 1957]. Section 19. Theorem 19 was proved by Doob [8, 1957] using probabilistic and [11, 1959] nonprobabilistic methods. This theorem has been generalized
to axiomatic potential theory contexts by Gowrisankaran (to Brelot harmonic spaces) and by Sibony (to a context including Bauer harmonic spaces). See [Taylor 1, 1980] for a simple version of Sibony's result with references to previous work.
Section 21. Theorem 21 was proved by Brelot and Doob [1, 1963] along with further theorems on the relations between nontangential and minimalfine boundary limits of functions defined on a half-space. See the references there to work of Constantinescu and Cornea and others on the relations between nontangential and minimal-fine cluster values of functions defined on a ball and see [Brossard 1, 1978] for such relations derived in probabilistic language.
802
Historical Notes
Sections 22 and 23. Theorem 23(a) for n = 2, in the disk rather than the half-plane context, was proved by Littlewood [1, 1928] long before the fine
topology was invented. His result was generalized to the N-dimensional context by Privalov [1, 1938]. The proofs of Lemma 22 and Theorem 23 are taken (corrected) from [Doob 16, 1965], an unreadably garbled paper. Fine topology confusion. Let D be a half-space, let be a finite boundary point of D, and let A be a subset of D. The relations between thinness of A at C, minimal thinness of A at C, and the relations between these two concepts when A is in a cone with vertex S are not obvious and have sometimes been misstated in the literature. The situation has been complicated by the habit
of Doob and some other authors of omitting the qualifier "minimal" in discussing minimal-fine limits and cluster values at Martin boundary points. Jackson [1, 1970] surveyed this situation and made the necessary corrections.
Chapter I.XIII Section 1. In a pioneering paper Gauss [1, 1840] considered measures v supported by a smooth surface A in 083. For f a given function on A he studied the properties of v chosen to minimize JA (Gv - 2f) dv for specified v(A); the measure v was by hypothesis given by a smooth density relative to surface area. He took it for granted that a minimizing measure p existed and showed that then Gp - f is identically constant on the support of p. In particular (f = const), Gauss thereby obtained an equilibrium distribution for A and, if f is the restriction to A of the potential of a measure A, Gauss obtained the sweeping of i onto A. Finally Gauss showed how to manipulate his result to solve the Dirichlet problem for the domain bounded by A. His work was of course not rigorous but offered ideas not fully digested
for a century. Careful formulations suggested by some of his ideas are presented in Section 14. The first rigorous derivation of an equilibrium distribution for an arbitrary compact subset of 1183 was carried through by
Frostman [1, 1935], whose basic tool was a modernization of Gauss's technique of minimization of the energy integral. The capacity of an arbitrary compact subset A of 083 was defined by Wiener [2, 1924] by solving
the (generalized) Dirichlet problem for the unbounded open connected component of 083 - A with boundary function I on 3A and 0 at the point oo. This solution can be identified with the restriction to D of the potential Gp of a measure on A, and Wiener defined the capacity of A as p(A). [Since Gp is in fact the smoothed reduction RA (relative to 083), the potential Gp is the capacitary potential of A.] This capacity definition was extended by Vallee Poussin [1, 1932] who defined the capacity of an arbitrary subset A of 083 as the supremum of v(H3) for measures v supported by compact subsets
of A with potentials Gv 5 1, and he showed that this definition coincided with Wiener's for compact sets. The Vallee Poussin capacity is the inner capacity as defined in Section 11.
Part 1
803
The problem of finding an equilibrium distribution for a set is sometimes called the Robin problem in view of [Robin 1, 1886] in which Robin showed that the problem for a smooth surface in I83 is equivalent to the problem of solving a certain integral equation.
Section 6. Cartan [2, 1945] showed that the space f+ is complete. (He worked with what we have called "special" domains.) See [Brelot 12, 1953-1954] for a discussion of BLD functions.
Sections 7 and 8. Frostman [1, 1935; 2, 1938] and M. Riesz [1, 1938] gave the first rigorous proofs of Theorem 7. Section 10. Choquet was led to his theory of capacities by potential theory problems. He proved Theorem 10 in [1, 1955]. Section 17. The Wiener thinness criterion was derived in [Wiener 3, 1924] in the context of boundary point regularity for the Dirichlet problem. Brelot [4, 1940] put it into the equivalent thinness context.
Section 18. See [Choquet 3, 1958] for an approach to logarithmic capacity
somewhat different from that in Section 18, but observe that he does not use the standard terminology. H. Jackson kindly showed the author an example demonstrating that the logarithmic capacity set function is not subadditive.
Chapter 1.XIV
The material in this chapter is well known and mostly elementary. It is a fine example of material that is easy to develop, useful to have available for reference, and for which it is as difficult as it is pointless to find the original source.
Chapter 1.XV Section 1. Parabolic potential theory has not been treated systematically
in the literature. Petrowsky [2, 1934-1935] is a valuable source for the Dirichlet problem. Tychonoff [1, 1935] has many useful results. Widder's book [2. 1975] is useful for the solutions of the heat equation on a slab. Some work in axiomatic potential theory has included applications to parabolic potential theory. See, for example, [Constantinescu and Cornea 1, 1972], a book which covers contexts far more general than parabolic potential theory with explicit applications to parabolic potential theory. Probabilists have invoked parabolic potential theory as a convenient tool in studying Brownian motion in a potential theoretic context. See, for example. [Doob 6. 1955]. but the reader should be warned that the statement
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Historical Notes
there that for an open set bounded by hyperplanes the Euclidean boundary is regular for the heat equation is false. Section 5. The proof of Theorem 5 is taken from [Tychonoff 1, 1935]. Section 11. See [Moser 1, 1964] for a more precise Harnack-type inequality than that given in Section 11.
Section 16. The Appell transformation dates back to [Appell 1, 1892].
Chapter I.XVI Sections 1 to 6. The essential result in these sections is the equivalence of
Theorem 6 parts (al) and (a2), due to Widder [l, 1944]. The preliminary results leading up to Theorem 6 are mostly taken from Widder's paper and [Tychonoff 1, 1935].
Section 7. Theorem 7 with h = I is old as a special case of a general integral limit theorem; see, for example, [Titchmarsh 1, 1937]. Theorem 7 as stated was first proved in [Doob 13. 1960] along with a general ratio integral limit theorem. See these notes to Appendix VII for references to such ratio integral limit theorems. See [Hartman and Wintner 1, 1950] for a Fatou boundary limit theorem for parabolic functions on an interval. Chapter 1. XVII
Many results in Chapters XVII and XVIII of Part I can be obtained by translating classical results and their proofs into the parabolic context. Such results will not be referenced in these notes. Some of the classical reduction property proofs are no longer valid in the parabolic context, however. Where essentially different proofs are given, they are due to Boboc, Constantinescu or Cornea, separately or together, and have been taken from [Constantinescu and Cornea 1, 1972], which contains proofs valid in a very general context along with detailed references. Section 13. The direct half of Theorem 13 (in a general axiomatic context) is due to Brelot [14, 1962].
Section 18. Theorem 18 is due to Hunt [1, 1956], who proved it in a probabilistic context.
Chapter I.XVIII Section 1. Sternberg [1, 1929] was apparently the first to observe that the Perron method of attacking the Dirichlet problem could be applied in the parabolic context, although he restricted himself to functions on very special domains.
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Section 4. Babuka and Vyborny [1, 1962] showed that a finite Euclidean boundary point of a subset D of RN is regular for Laplace's equation if and only if every Euclidean boundary point s) of D x R is regular for the heat equation.
Section 6. The iterated logarithm criterion of Example (d) is due to Petrowsky [2, 1934-1935] who knew that this criterion corresponded to the Khintchine iterated logarithm law for Brownian motion (Section 2.IX. 15). Sections 11 and 12. Meyer [ 1, 1962] showed that in a very general potential
theoretic context, defined probabilistically, a set is polar [semipolar] if and only if it is copolar [cosemipolar].
Section 14. Theorem 14 and Applications (a) and (b) appear to be new. Applications (c) and (d) in a more general (probabilistic) context are exercises
in Blumenthal and Getoor's [1, 1968] and are proved in their [2, 1970]. Section 15. See [Doob 13, 1960] for a probabilistic version of Theorem 15,
and see Section XIX.15 for a proof of Theorem 15 by Martin boundary technique. It is natural to conjecture that when h - 1, then (15.1) is true as a normal limit (that is, as a limit along the normal line to the boundary point) as well as a coparabolic-fine limit, but in fact Kaufman and Wu [1, 1982] have exhibited a measure v on the upper half-space D of I such that lim sup,_0 GD s) = + oo for 0 < c < 1. According to Theorem 15, a positive parabolic function v on a slab UFN x ] 0, 6[ has coparabolic-fine limit function AMIN at IN almost every point of the abscissa hyperplane. Koranyi and Taylor [1, 1983] have shown
that this result can be used to prove Theorem XVI.7 in the special case h =_ 1, but their method fails for general h. That is, the reasoning used in proving Theorem XII.21 to go from minimal-fine boundary limits to nontangential boundary limits does not seem to have a counterpart in the parabolic context.
Section 16. In a general axiomatic setting including parabolic potential theory Janssen [2, 1975] proved that (notation of Theorem 16) h < ii if suph..4 v(il)/h(,) >_ 1 for Jt, almost every . This result becomes Theorem 16 with the help of Theorem 14. Blumenthal and Getoor state that h 5 6 under condition (c') in a more general context but with ti = const as an exercise in their [1, 1968] and give the proof in their [2, 1970].
Chapter 1.XIX
Various versions of a parabolic context Martin boundary have appeared in the literature but that given here, especially that relative to a Martin point set pair, seems closest to Martin's ideas and the easiest for calculational
purposes. See Janssen [1, 1971], for example, for the Martin boundary obtained by convex set methods in a context including parabolic potential theory.
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Historical Notes
Section 15. In the following discussion of functions on the right half-plane
]0, + oo [ x R we shall write that a function on this set has a [one-sided] parabolic limit a at a boundary point (0, r) if the function has the limit a as for each strictly s - z < fl2S2] r - sl < -.(0,r) with positive choice of Y2 with f, < Y2] P. According to Section 10, if h is a strictly positive parabolic function on the right half-plane, it can be written in the form
where N,; is a uniquely determined measure on Q8 and Nti is a uniquely deter-
mined measure on - 68+. Let a be a positive superparabolic function on the right half-plane with parabolic component (Riesz decomposition) determined by measures N,; and According to Section 15, the function o/h has coparabolic-fine limit dN;IA4 at N,; almost every point of the ordinate axis. This result should be compared with the following results. Kemper
[1, 1972] proved that if it - 1, that is, if N,; = /, and Nh - 0, and if t% is parabolic, then c/h has the stated limit at 1, almost every point of the ordinate
axis in the sense of parabolic approach to the point. On the other hand, Kaufman and Wu [ 1, 1982] have exhibited an example of a positive parabolic function h on the right half-plane specified by a continuous singular relative
0 for all S. to 1, measure Nh and by Nti = 0 such that lim info Thus 1/h does not give the limit d1,/dNh along normal approach to the ordinate boundary, but Kaufman and Wu showed that if r and h are positive and parabolic on the right half-plane, then c/h has the limit dN,;/dNh at Nh; almost every point of the ordinate axis in the sense of one-sided parabolic approach to the point. On the other hand, if c is a superparabolic potential 0 for on the right half-plane, Wu [1, 1979] has shown that
1, almost every s. One-sided parabolic approach does not yield this zero limit, for example, if c is the superparabolic potential of a measure assigning a strictly positive value to each point of a countable dense subset of the right half-plane.
Part 2 The most useful reference books for Part 2 are [Meyer 6, 1966], [Dellacherie and Meyer 1. 1975; 2. 1980]. and [Dellacherie 2. 1972]. For Markov processes see [Dynkin 1. 1960; 2, 1965] and [Blumenthal and Getoor I, 1968]. [Meyer 8, 1968] contains a useful summary of key ideas in the general theory of stochastic processes, including classification of optional times, classification of types of stochastic process measurability, and a discussion of section theorems. The books of K. M. Rao [4, 1977]. Port and Stone [1, 1978], and Chung [2, 1981] are useful references for the relations between probability and classical potential theory.
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Chapter 2.1
Section 1. The concept of a filtered measure space always underlay probabilistic analysis involving the Markov property, and the concept was finally formalized in [Doob 4, 1953] to deal with martingale theory. In the schools of Dynkin and Meyer the concept has become the base of all analysis involving Markov process theory, martingale theory, and stochastic integration. The Dynkin approach to Markov processes in his [2, 1965] leads to a further refinement of a filtered space which is not needed in this book. Section 2. Dynkin defined process measurability, in the context of Markov
processes, in his [1, 1960] essentially as what is now called progressive measurability. Meyer defined and applied progressive measurability in [2, 1962-1963]. This measurability was rediscovered and given its present name in [Chung and Doob 1, 1965], written in ignorance of the earlier work.
Section 5. Let ..., xo, x,, ... be mutually independent random variables, each uniformly distributed on the interval [0, 1], and define 9(n) = 9f. In what amounts to this context Theorem 5 was proved by Jessen [1, 1934] and for n-. +co by Levy [1, 1935], who supposed x in Theorem 5 to be the indicator function of a set. It is clear in Levy's discussion, however, that he does not use the special independence context, and in fact this special context is not presupposed in his treatment of the same convergence result in his [2, 1937]. Theorem 5 as stated is in [Doob 1, 1940].
Section 9. The first rigorous treatment in a general continuous parameter
context of the hitting of a set by a process was given by Hunt [2, 1957]. Meyer [2, 1962-1963] was the first to discuss the hitting of progressively measurable sets.
Section 10. Kolmogorov [1, 1933] gave the measure theoretic basis for probability, including the definitions of expectation and conditional expectation and a construction of measures on infinite-dimensional product spaces. The latter construction is what is needed for canonical processes. Section 13. For variations of Theorem 13 see [Chung and Doob 1, 1965], [Meyer 6, 1966], and [Dellacherie and Meyer 1, 1975]. Section 14. See [Dellacherie 2, 1972] for a full discussion of the important a algebras of subsets of 68' x S2, first defined by Meyer. See [Meyer 6. 1966; 8, 1968] for earlier discussions. Chapter 2.11
Section 1. Like the Moliere character who spoke in prose without realizing it, mathematicians had dealt with optional times for centuries without finding it necessary to define them precisely. Precise definitions were finally intro-
duced, along with a algebra filtrations, to answer the needs of Markov
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Historical Notes
process and martingale theories. Meyer [6, 1966; 8, 1968] first classified optional times.
Section 4. Hunt [2, 1957] was the first to show that in an appropriate context the hitting time of an analytic set is optional. Doob [5, 1954] had shown that the hitting time by Brownian motion of a capacitable set is optional, but in 1954 it was not known that all Borel sets are capacitable in his context. Meyer [2, 1962-1963] proved that the hitting time of a progressively measurable set is optional. Section 7. Observe that predictable optional times are defined slightly differently in [Dellacherie and Meyer 1, 1975]. Section 8. Section theorems were introduced into probability theory by Meyer [2, 1962-1963], more successfully in [6, 1966]. See Dellacherie [1, 1972], from which the proof of Theorem 8 is taken, for an elegant unified approach to them. Section 9. The proof of Theorem 9 is taken from the correction sheet in [Dellacherie and Meyer 1, 1975].
Section 10. Dellacherie [1, 1969] proved that, in the notation of this section, a predictable subset A of If8' x n is semipolar if for almost every w the set {t: (t,w)EA} is countable. Chapter 2.III Before martingales had been formally christened, Levy [1, 1935; 2, 1937], Bernstein [1, 1937], and other mathematicians had analyzed some of their properties in special contexts; usually the martingales in question arose as partial sums n -. Eo y) of a sequence y, of random variables under the condi-
,, . y'1-i } = 0 so that the sums arose as generalizations of sums of independent random variables with zero means. Ville [1, 1939] defined a martingale very nearly as a positive martingale is now defined but tied it to a sequence of independent random variables under analysis. His fundamental tool, a fact he proved, was that almost every sample sequence tion E{ y,I
ofa positive martingale sequence is bounded (see Theorem 9). Doob [1, 1940] discussed martingales and proved the basic martingale convergence theorems
under the name "family of random variables with the property E." ("E" was chosen not as the initial letter of "expectation" but as the first letter in the alphabet following D.) Under the respective names "semimartingale" and "lower semimartingale," submartingales and supermartingales were introduced in [Snell 1, 1952] and [Doob 4, 1953]. This obviously inappropriate nomenclature was chosen under the malign influence of the noise level of radio's SUPERman program, a favorite supper-time program of Doob's son during the writing of [Doob 4, 1953]. For further work in martingale theory see [Neveu 1, 1972] (discrete parameter context) and [Dellacherie and Meyer 2, 1980] (continuous parameter context).
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Section 4. The trivial but useful map in this section appears to be new. Section 6. Theorem 6 and its generalizations to various forms of optional sampling in Chapters III and IV are adapted from [boob 1, 1940; 4, 1953] with improvements from [Meyer 6, 1966].
Section 9. Theorem 9 is taken from [Doob 4, 1953]. In the martingale case or the Imartingalel case Theorem 9(a) was proved by Levy [2. 1937]. Bernstein [l, 1937], and Ville [1, 1939]. Section /1. Theorem I I is taken from [Doob 4, 1953]. Section 12. Inequality (12.2) is new, as are (12.3) and (12.4) except when 1. Crossings (implicit in [Doob 1. 1940]) were formally introduced
into martingale theory in Doob [3, 1951], where a variant of (12.3) was obtained for 1. Snell in [I, 1952] extended a martingale and crossing inequalities to submartingales and supermartingales. Dubins [1. 1966] obtained a variant of (12.4) with
1.
Sections 13 to 17. These results are taken from [Doob 4, 1953]. In the martingale case Theorems 13 and 17 were in [Doob I. 1940].
Section 19. This may be the first time that the natural decomposition theorem appears in print in the martingale theory context, but the result is merely a special application of the general theory Section 22. The First application of reduction theory to martingale theory was made by Snell [I, 1952]. See Neveu [I, 1972] for further discussion of related problems. The infimum of the class of supermartingales majorizing a given process under appropriate side conditions is called the Snell envelope of the given process.
Chapter 2.I V
Section 1. The study of the continuity properties of continuous parameter martingale sample functions was initiated in [Doob 1, 1940]. completed in
[Doob 3, 1951], and extended to supermartingales in [Doob 4. 1953]. Perhaps the first sophisticated use of optional times appeared in the first of these papers to obtain martingale right continuity properties (without crossing inequalities, unknown at that time).
Sections 2 and 3. Theorems 2 and 3 are adapted from [Doob 4, 1953] with improvements in formulation from [Meyer 6, 1966).
Section 4. Theorem 4, the counterpart of, but much deeper than, the classical potential theory rather elementary fact (Section 1.11.4) that the limit of an increasing sequence of superharmonic functions on a connected open subset of R" is either superharmonic or identically + 00, is due to Meyer [6, 1966]. The elementary right continuity proof is new.
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Historical Notes
Section 5. This theorem appears to be new, but see related work of Mertens [1, 1972]. As Mertens' work indicates, the choice of standard continuous parameter supermartingales need not be the almost surely right continuous ones. For some purposes the right continuity hypothesis can be advantageously replaced by a combination of an appropriate type of measurability together with invariance of the supermartingale inequality under optional sampling. Sections 6 to 12. Doob [4, 1953] proved the rather trivial discrete parameter decomposition Theorem 8 and after searching for several years found a mathematician who could prove the continuous parameter version Theorem I 1 [Meyer 4, 1962; 5, 1963]. See the appendix to Meyer [6, 1966] for earlier
decomposition results due to Volkonski, Sur, and Meyer which are for supermartingales associated with Markov processes. Perhaps as tribute to Doob's persistent search for this holy grail, the Meyer decomposition is sometimes inappropriately called the Doob-Meyer or even the Doob decomposition. Meyer proved Theorem 11 with the monotone increasing process natural in the sense of Section 7. The equivalence between natural and
nearly predictable [Theorem 7(a3)] was proved by Doleans [1, 1967]. Rao's [1, 1969] proof of Theorem 11 is followed in the text. See [Dellacherie and Meyer 2, 1980] for extensive applications of the Meyer decomposition.
Section 13. The supermartingale domination principle appears to be new.
This example of the potential theory of supermartingales indicates how closely such a theory parallels classical potential theory. Unfortunately, it was impractical to include the elegant Airault-Follmer [1, 1974] results or the Azema [1, 1972] and Azema-Jeulin [1, 1976] results in this direction. Among many other things these papers contain a counterpart for supermartingales of conditional Markov processes. The most glaring probabilistic
potential theory omission from this book, however, is that of additive functionals of Markov processes (see, for example, [Dynkin 2, 1963] and [Blumenthal and Getoor 1, 1968]). The author has three convincing excuses for this omission, of increasing cogency: (1) the existence of these books, (2) lack of space, (3) ignorance of the subject. Sections 17 to 20. Reduction theory in this form seems to be new, but see also related work involving Snell envelopes by Mertens [1, 1972], Azema
[1, 1972], and Azema and Jeulin [1, 1976]. Results in the latter paper together with Theorem 13 suggest that in some contexts R i.) in Section 17 should be defined as the infimum of the class of almost surely right continuous positive supermartingales whose left limit processes majorize that of on A. This definition is also suggested by the awkward extra hypotheses in Section 18(m). Section 21. This energy treatment is due to Meyer [3, 1962-1963].
Part 2
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Chapter 2.V
Sections 2 and 4. Krickeberg [1, 1956] introduced lattice ideas into martingale theory. In particular, he discussed the LM operator and proved Theorem 4(c) for p = 1, the "Krickeberg decomposition" of a martingale. See [Dellacherie and Meyer 2, 1980] for further elaboration of this decomposition. Section 12. Local martingales were introduced into martingale theory by Ito and S. Watanabe [1, 1965]. Section 13. Quasimartingales were introduced by Fisk [1, 1965]. Theorem 13 is due to K. M. Rao [2, 1969]. See his paper and that of Dellacherie and Meyer [2, 1980] for further discussion of the decomposition of a quasimar-
tingale. See also [Follmer 1, 1973] for a correspondence, unfortunately omitted from this book, between quasimartingales relative to a filtered probability space and measures on the a algebra of predictable sets determined by the filtration. Chapter 2.VI
This chapter treats only those aspects of Markov process theory needed in this book. Readers interested in going further into probabilistic potential theory, which is based on Markov processes, may consult [Dynkin 2, 1965], [Blumenthal and Getoor 1, 1968], and [Chung 2, 1981]. Knowledgeable readers will observe that translation operators are not mentioned in this chapter. They are not needed in this book.
Section 1. A. A. Markov introduced what are now called Markov processes into probability theory in [1, 1906]. Section 3. The strong Markov property in the discrete parameter case is
trivial to prove, but the realization that such a property is relevant and requires proof had to await the sophistication not present before the property was needed in the technically more difficult continuous parameter case. Even now rather than explicitly recalling and applying this property, it is frequently easier, when the discrete parameter strong Markov property is relevant in a
computation, to make the computation detailed enough to avoid explicit statement and use of the property. Section 4. Kakutani [3, 1945] defined random walks on domains of R2 as in the example in this section. He proved, omitting some details, what amounts to the fact that the distribution of x(n) in the example tends to harmonic measure relative to the initial point.
Section 6. Hunt [2, 1957] was the first to consider in depth the hitting probabilities of sets by Markov process trajectories.
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Historical Notes
Section 8. See Meyer [7, 1967] for a more complete analysis of the filtrations generated by Markov processes. The 0-1 law in this section is known as the Blumenthal 0-1 law in view of [Blumenthal 1, 1957]. Blumenthal's paper is probably the first paper to treat a Markov process explicitly as a process adapted to a general filtration and having the Markov property relative to the filtration. Section 9. The first version of the strong Markov property in a continuous parameter context was proved by Doob [2, 1945] in the context of a countable state space. Hunt [1, 1956] proved a version for processes with independent increments. The first generally applicable versions were proved independently by Dynkin and Yushkevich [1, 1956] and Blumenthal [1, 1957].
Sections 10 to 12. Hunt [2, 1957] introduced excessive functions and measures into Markov process theory as part of his fundamental development [1, 1957; 2, 1957; 3, 1958] of probabilistic potential theory based on Markov processes. Section 13. Conditioned Markov processes were introduced in [Doob 8, 1957] in the context of Brownian motion. See [Dynkin 2, 1963] for conditioned Markov processes in the context of multiplicative linear functionals. Section 15. See [Meyer, Smythe, and Walsh 1, 1972] for a deep discussion of Markov processes killed at various kinds of random times.
Chapter 2.VII The Brownian motion process is the link between classical potential theory and martingale theory, and only those Brownian motion properties relevant to this linkage are discussed in this book. For more information on Brownian motion see, for example, [Levy 4, 1948], [Ciesielski 1, 1966], [Freedman 1, 1971], [Ito and McKean 1, 1974], [Knight 1, 1981], and, especially for the physical significance of Brownian motion, [Nelson 1, 1967]. The process
takes its name from the English botanist Brown who in 1827 observed irregular motion of pollen particles in a liquid. Einstein in 1905 obtained from physical considerations the fact that the mean square displacement of a Brownian particle is a multiple of the displacement time and evaluated this multiple. Bachelier [1, 1900; 2, 1901] and later papers derived many distributions involving Brownian motion considered as a limit of random walks and saw the connection with the heat equation. The first rigorous construction of a Brownian motion process was given by Wiener [1, 1923], but his work remained unknown to or at least ignored by other probabilists
for about 15 years. Thus probabilists were at a disadvantage in treating sample functions of Brownian motion; although they realized that in some
sense these sample functions were continuous, it was not clear how to
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formulate this property without Wiener's work. It was intuitively clear, however, that many distributions involving Brownian motion from a point of a domain involved a process whose paths were ordinary Brownian motion paths until they hit the boundary of the domain, where they were absorbed or reflected in some way so that the transition probabilities involved, like the transition densities of Brownian motion in 08", were governed by the
heat equation in the domain. The conduct of the paths at the boundary then determined corresponding heat equation boundary conditions. Sections 2 and 3. The definition of a Brownian motion process used here implies that one example is the canonical example: a measure is assigned to the space of continuous functions from R into I8" making the coordinate functions random variables with the properties BM I -BM4. This measure is known as Wiener measure, and Brownian motion is sometimes defined as this canonical example. Under this definition a process satisfying BM I -BM4 is not necessarily a Brownian motion.
Section 4. Theorem 4 is a special case of a theorem of Meyer [7, 1967]. The proof in the text follows that of Chung and Walsh [2, 1974]. Section 6. The zero-one law for Brownian motion can be considered as a special case of a general zero-one law of Kolmogorov [1, 1933].
Section 8. The Andre reflection principle goes back to [Andre 1, 1887], where a reflection idea is used in a ballot counting problem. The distributions in Section 8 were derived by Bachelier (see the above notes to this chapter), but his work was ignored and rediscovered by many others.
Section 9. Hunt [1, 1956] derived the basic properties of the transition density for Brownian motion in an open set.
Section 11. The transition density for Brownian motion in an interval was found by Bachelier (see the above notes to this chapter). Chapter 2.VIII a Brownian The integral J, f dw with f a function from I into U8 and motion was first discussed by Wiener [1, 1923]. Ito [1, 1944] allowed the integrand to depend on the Brownian paths, thereby inaugurating a farreaching further development. See McKean [1, 1969] for many applications of Ito's integral, and see [Dellacherie and Meyer 2, 1980] for stochastic integrals with differentials more general than Brownian motion differentials. Section 3. The problem of finding conditions that Y. as defined by (3.10) be a martingale is a special case of a problem proposed by Girsanov in 1960 is a local martingale. Various sufficient conditions have been in which found; for example, Novikov [1, 1972] found a condition reducing to (k) in our context.
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Historical Notes
Section 8. Theorem 8 is a consequence of Ito's [3, 1951] construction of a complete orthonormal system on a Brownian motion measure space. See also [Kunita and S. Watanabe 1, 1967] for a systematic study of general stochastic integrals in L2 contexts. The proof in Section 8 is taken from [Parthasarathy 1, 1978]. Dudley [1, 1977] proved that every finite-valued .F w(t), sE IlB+; measurable function can be expressed as an Ito integral with integrand in r*. Section 10. Levy [3, 1940] proved that if t, is a dense sequence in [0, b] and if t4"', ... , are t,,, ... , t" in increasing order, then lim i [w(ti"') - w(tj"'1)]2 = a2 b a.s. See [Doob 10. 1953] for a martingale proof of this limit theorem. Section 12. It6's lemma, also called Ito's formula, was proved in [2,195 1].
Section 14. The fact that the composition of an analytic function with plane Brownian motion is Brownian motion with a new time scale is due to Levy [4, 1948]. Chapter 2.IX The martingale theory properties of the composition of superharmonic and harmonic functions with Brownian motion were treated in [Doob 5, 1954],
and the corresponding discussion in the parabolic context was given in [Doob 6, 1955]. Nonroutine results in Chapter IX, aside from results specifically attributed otherwise, are taken from these papers, although most of the proofs are different and some of the results are slightly refined.
Section 5. Levy [3, 1940] proved Theorem 5(c) and stated Theorem 5(a) for A a singleton. Kakutani [1, 1944] proved Theorem 5(b) and [2, 1944] sketched a proof of Theorem 5(a) for N = 2.
Section 10. Theorem 10(a) was indicated without proof details in [Kakutani 2, 1944]. Section 11. Theorem I 1 in its classical and parabolic versions [Doob 5, 1954; 6, 1955] was generalized by Hunt [2, 1957], Dynkin [2, 1963], and Meyer [7, 1967] to yield various continuity properties of the composition of excessive functions with their corresponding Markov processes.
Section 13. The idea that Theorem 13(a) and its parabolic counterpart must be true is old, but a rigorous proof had to await the rigorous development of harmonic measure and of Brownian motion. Courant, Friedrichs,
and Lewy [1, 1928] obtained harmonic function Dirichlet solutions for smooth domains by solving a corresponding problem for difference equa-
Part 3
815
tions and going to the obvious limit. They refer to the probabilistic interpretation of their difference equation. Petrowsky [1, 1933-1934] gave a similar discussion, and Khintchine [1, 1933] in a corresponding discussion for both harmonic and parabolic functions mentions the Brownian motion interpretation of the limiting case. Such ideas were common at the time, but Wiener's [1, 1923] treatment of Brownian motion was unknown or unappreciated, and, for example, Khintchine's Brownian motion interpretation was therefore somewhat artificial. Kakutani [2, 1944; 3, 1945] stated Theorem 3(a) with indications of a proof. Section 14. Theorem 14 (in a much more general context) is due to Hunt [2, 1957].
Section 15. The dichotomy in Theorem 5 in the classical and parabolic contexts [Doob 5, 1954; 6, 1955] has been generalized to define fine and cofine topologies in the probabilistic potential theory of Markov processes. Khintchine [ 1, 1933] proved the iterated logarithm law for Brownian motion, aside from the fact that he could not quite define the probabilities involved without a rigorous definition of Brownian motion.
Section 17. Theorem 17 is part of the folk lore of the subject but is new as stated. Chapter 2.X
Section 6. See [Chung and Walsh 1, 1969] for a general discussion of reversing a Markov process. Section 8. The anplication of martingale theory to derive a probability version of Theorem 1.XI.4 is taken from [Doob 8, 1957]. See [Weil 1, 1969] and [Airault 1, 1973] for such work in a more general context. Airault shows that the specification of the limit in Theorem I.XI.4(b) is possible because if the Riesz measure associated with a positive superharmonic function h is
supported by a polar set, the lifetime of an h Brownian motion process is predictable. Unfortunately the stated hypotheses underlying most probabilistic analysis of this sort, including the above references, are too strong to cover the parabolic context. Section 10. The evaluation of the capacitary distribution in terms of last hitting distributions (in a more general context) is due to Chung [1, 1973].
Part 3 Chapter 3.1
The lattice theoretic results in this chapter are mostly routine or in the folklore; so few references to their origin will be given.
816
Historical Notes
Section 4. Lemma 4, the key to the connection between uniform integrability and the PWB method, is taken from [Doob 7, 1956], where the potential theory version is proved in an axiomatic potential theory context.
Section 8. See [Arsove and Leutwiler 1, 1974] for a general approach to Theorem 8.
Section 12. Lamb [1, 1971] pointed out the decomposition S., = +Smsf Chapter 3.11
Section 2. The delicate parts of Theorem 2 not treated in previous chapters are taken from [Doob 7, 1956; 9, 1958]. Chapter 3.1I1
Section 1. See [Kunita and T. Watanabe 1, 1965. 2. 1966], [Meyer 9, 1968], and [Dynkin 3, 1969; 4, 1971] for probability-potential theory approaches to the Martin space and process paths on it, carried out in Markov process contexts at various levels of generality. The reader will observe that this chapter is rather skimpy and that a parabolic context counterpart is missing. The point is that the author is a firm believer in finite-valued stopping times.
Section 4. Observe that, as pointed out in the notes to Section XI.4, the probabilistic proof of the Fatou boundary limit theorem for the Martin space preceded the nonprobabilistic proof.
Section 6. The example is taken from [Doob 10, 1958], which contains several examples of conditional Brownian motions in a half-space, involving paths to and from boundary points.
Appendixes Appendixes I and II The operation constructing a nucleus from a Suslin scheme was devised by
Suslin in [1, 1917]. Apparently Lusin coined the name "analytic sets." Choquet created his capacity theory in [1, 1955; 4, 1959]. Theorem 11.5, which makes analytic set theory conveniently applicable in many probabilistic contexts, is taken from [Meyer 6, 1966]. See [Dellacherie and Meyer 1, 1975; 2, 1980] for more on analytic sets and capacities and their application to probabilistic potential theory.
Appendixes
817
Appendixes III to VI The material in these appendixes, except possibly for Section 12 of Appendix
IV for which see [Besicovitch 1, 1946], is traditional and routine and is assembled for the reader's convenience. [Peressini 1, 1967] is a useful source for a reader who wants enough but not too much vector lattice theory.
Appendix VII
See [Doob 13, 1960; 14, 1960-1961] for ratio integral limit theorems; those in Appendix VII are adaptations to fit the needs of this book.
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[4] "Points irregulier. Determination des masses par les potentiels." Acad. Beiq. Bull. Cl. Sri. (5) 24 (1938), 368-384. Florin Vasilesco: [I] "Sur la continuite du potentiel i1 travers les masses, et Ia demonstration d'un femme de Kellogg." C. R. Acad. Sri. Paris 200 (1935), 1173 -1174. [2] "Sur une application des families normales de distributions de masse " C. R. Acad. Sci. Paris 205 (1937), 212-215. [3] La notion de point irrbgulier dans Ic probleme de Dirichlet. Act. Sci. Ind. 660 (1938), Paris, Masson. Jean Ville: [I] Etude Critique de la Notion de Collectij. Pans, Gauthier-Villars, 1939. Michel Weil: [I] "Proprietes de continuite fine des fonctions coexcessives." Ztschr. Wahrsrheinlichkeitstheorie verve. Geb. 12 (1969), 75- 86. John Wermer [1] Potential Theory. Lecture Notes in Mathematics 408. Berlin, Springer-Verlag, 1974.
D. V. Widder: [1] "Positive temperatures on an infinite rod." Trans. Amer. Math. Soc. 55 (1944), 85- 95. [2] The Heat Equation. New York, Academic, 1975. Norbert Wiener: [1] "Differential space." J. Math. Phys. Mass. Inst. Tech 2(1923),131-174. [2] "Certain notions in potential theory." J. Math. Phys. Mass. Inst. Tech. 3(1924),24-51. [3] "The Dirichlet problem." J. Math. Phys. Mass. Inst. Tech. 3 (1924),127-146. [4] "Note on a paper by O. Perron." J. Math. Phys. Mass. Inst. Tech. 4 (1925), 21-32. Jang-Mei G. W u : [ 1 ] "On parabolic measures and subparabolic functions." Trans. Amer. Math. Soc. 25 (1979), 171-185. S. Zaremba: [I] "Sur Ie principe de Dirichlet." Acta Math. 34 (1911), 293 316.
Notation Index
b n Hf, Hf, Hrk
101
7 L , Z , 7L
xxiv xxiv
AD
114
00
xxiv
GD
132
1 - Al
xxiv xxiv
WD D(µ4_)
136
Xxiv
xxiv
LP(4_) St, Si
144 145
xxiv
08,
68+,
68,
6) IN
A-B L(k'(D) nN A D.
L(u, A(u,
6) , 6)
142
S
146
3
Sm
148
3
SP
149
3
Sqb
150 150
4 4 4
Smqb, Spqb Ss, Sms, Sps SD
4
f lim
6
Af
167
7
u*
172
hD(S, A) GB
13
u#
'IN
G
a
14
151
156
166
180
R
188
fAD
191
fGD
193
K(q,
14
Pi
15
TB
16
L`(uB-)
27
DM, aMD
197
D(JB-)
27
a;MD
200
[u, v], 1IN11
227 227
K
196
LM, GM
35
u
37
R,
38
9+ 9
0v04
39
-9 (u, v), -9 (u)
231
TB
BLD
233
GM", LMh
99 99
hRA
99
C*, C*, C r(A)
252
99
Al A
262
ftN
262
IDu0A
PWB, PWBh
100
229
240
Notation Index
82_8
Hm1
m5
Hm l
m,v
264
E{xj.F}
265
QS, TP
266
G
266 271
TB
A(u, q, c)) Aa(ti, rj) PI
LI(NB
)
GM. LM
274 275 276
276 285 290 290 295 296 296
GD,
6b
298
pflim
308
APt. At'
309 309
p*flim fU
320 326
UN
326
GMh, GMh
329
rI N
[S].
391 414 415
O*, O**
421
Dn[f;g,h]
445 456
rsr, Ti
GM, LM R^1.,
Tsr, LM, GM
st
'S S 'Sm, Sm, 'Sp
530
'Sqb, 'Smgb, 'Spgb Sqb, 'ST, Ss
531
Sms
GX
t(t,
r
r, rZ, ro
r', r, r2
329
1'a
FIj, Hf
329
Sms00
D
332
Smsf
r,, (w)
u5(t)
351
359
LP(µp_)
362 363
K DM, b
364 366 371
387
E{ x}, El x;A
390
fD G,
532 533 561
, ?)
hTB htA
Pf1r, PfpD Pf(,D
458 459 460 500 525 526 528 529
'S', S±, S+
329
343
.
Rz1.,
(t, . h)
572 590
592 599 600 602 604 716 717 719 728 728 735
.a1(4)
741
4+/(X)
756 769
41+ ,#.+
W.
771
772
Index
References are to sections, say 1.11.3 or, for the Appendixes, A.1.2. The index covers neither the Historical Notes nor the Bibliography.
Above: a point of (8", relative to a set, I.XV.I Abscissa hyperplane: of ft", 1.XV.1 Absolute probability function (entrance law): 2.Vl. I; of Brownian motion, 2.X.13 Adapted: family of functions, 2.I.1; stochastic process, 2.1.8 Analytic function: generalizations of Liouville's theorem, 1.11.2, 1.11.13; -J° is subharmonic, Hadamard three circle theorem, 1.I1.11; generalization of Cauchy's removable singularity theorem, 1.V.5; generalization of
the maximum theorem, I.V.7; composition with Brownian motion, 2.Vlll.l4 Analytic set: over a paving, A.1.2; over a product paving, A.1.3; projection characterization of sJI(Y), A.I.5, A.I.7; d(sl), A.1.6; inverse image under a measurable transformation, A.1.8; in a metric space, A.1.12, A.1.13; hitting of by a progressively measurable process, 2.11.4 Andre reflection principle: 2.VII.8 Appell transformation: 1 .XV.16 Approximation: of a superharmonic function by infinitely differentiable superharmonic functions, 1.11.8, I.IV.10; of a potential by continuous
potentials, 1.V.9; of a continuous function by potential differences, 1.V11.9; of a superparabolic function by infinitely differentiable superparabolic functions, 1.XV.14, I.XVII.7(e); of a random variable by its conditional expectations, 2.1.5, 2.111.14
definition, in particular for a half-space or interval, 2.VII.9, 2.VII.11; as a Green function, 2.Vl1.9, 2.1X.17 "D, lD: definition, 2.VII.10; i"D = Go, 2.IX.17 Backward parabolic equation : 2.1 X.17
Baire property of the fine topology: classical context, I.XI.I ; parabolic context, I.XVII.9 Baire null space: A.1.11 Balayage: see Sweeping
830
Index
Ball
Classical context: the Green function and the Poisson integral, 1.11.1; Riesz-Herglotz theorem, 1.11.14: L'(µ8_) and D(µB_) classes of harmonic functions, 1.11.14, 1.1X.12; Fatou boundary limit theorem, 1.11.15; minimal harmonic functions, 1.11.16; reductions. 1.111.4; potential of a uniform boundary distribution, I.IV.2. PWB solutions, I.V1II.2: universal and universal internal resolutivity of the boundary, I.V111.9, lattices of relative harmonic functions. I.IX.12; Martin boundary. 1.X11.3: classical versus minimal fine boundary limit theorems for relative harmonic functions, 1.X11.20: ball capacity. 1.X111.13; conditional Brownian motion in, 2.X.9
Parabolic context: regularity of the boundary, parabolic ball and the irregularity of its highest point, I.XVlll.6 Band: of a vector lattice, A.I11.8; projection on, A.111.9; generated by a singleton. A.111.I I
Barrier Classical context: definition, local nature, versus weak-, at oo, I .V11l.12, relation with boundary point regularity, I.V111.13, 1.VI1I.14, 1.VIll.16, Poincare- Zaremba examples, 1.VIll.12, 1.Vlll.15
Parabolic context: definition, local nature, versus weak-, relation with boundary point regularity, I.XVII1.3 Below: a point of 1W relative to a set, I .XV. I BLD function: 1.X111.6 Boundary limit theorems
Classical context: for the Poisson integral, semicontinuous boundary function, 1.11.1; ratio, for harmonic functions on a ball, 1.11.15, 2.X.8: for superharmonic functions at an irregular boundary point, 1.X1.21: ratio, for superharmonic functions on a Martin space, 1.X11.13, 1.X11.14, 1.X11.18, 1.X11.19; classical versus fine topology approach to a ball boundary, 1.X11.20: for potentials on a half-space, 1.X11.22, 1.X11.23: ratio, for superharmonic functions, along stochastic process paths. 2.Vl.4, 2.1X.7, 2.1X.13, 2.X.8, 3.111.4, 3.111.5: for Dirichlet solutions, 2.1 X.13, 3.11.2
Parabolic context: for the Poisson integral on an interval or slab, semicontinuous boundary function, I.XV.9, I.XVI.1 ; ratio, for superparabolic functions on a slab, 1.XVI.7, I.XV111.I5: for superparabolic functions at an irregular boundary point. 1.XVIII.17. ratio, for superparabolic functions on a Martin space, with applications to a slab and
to the fourth quadrant of 082, l.XIX.13-l.XIX.15; ratio for superparabolic functions, along stochastic process paths. 2.1X.7, 2.1X.13, 2.X.12, 3.11
Brownian motion (see also Conditional Brownian motion and Space-time Brownian motion): definition, 2.VlI.2; path continuity, 2.VII.3; right
continuity and predictability of-filtrations, predictability of-optional times, continuity properties of supermartingales relative to-filtrations,
Index
831
2.VI1.4, 2.VI1.6, 2.V111.8, 2.X.2(c); in an open set, relation between transition densities and Green functions, 2.V11.5, 2.VI1.l1, 2.1X.9, 2.IX.17; law of large numbers for and path length of, 2.VI1.5; zero-one law for, 2.VI1.6; tied down, the Brownian bridge, 2.VII.7; in an interval and the evaluation of parabolic measure, 2.VIl.11, 2.V11.12; under change
of parameter, 2.V111.9: composition of a function with, 2.V1I1.13, 2.VII1.14, 2.1X.11, and, if the function is superharmonic, 2.IX.2, 2.1X.3, 2.IX.6-2.IX.8, 2.1X.11-2.IX.13, 2.IX.16, 2.X.2; excessive functions and measures for, 2.IX.6, 2.IX.8, 2.!X.18; for large parameter values, 2.IX.5;
hitting of sets by, 2.IX.4, 2.1X.5, 2.1X.9, 2.!X.10; from an irregular boundary point, 2.IX.20; killed, 2.X.2, 2.X.6; to and from Martin boundary points, 3.111.1, 3.111.2
Canonical stochastic process: definition and construction, 2.1.10; Markov case, 2.VI.3, 2.V1.5, 2.V1.6, and Brownian motion case, 2.VI1.3, 2.VI1.9 Capacitary measure and capacitary potential: see Equilibrium measure and equilibrium potential
Capacity: Choquet-definition, A.I1.1; Choquet capacitability theorem, A.ll.3: projection example, A.11.5; generated by a strongly subadditive set function, in particular by a topological precapacity, A.II.7, A.11.8; generated by the reduction operation, I.V1.2(e), 1.VI.3(k), I.VI.4(k), I.VI.5, LXVII.16(o), I.XVII.17(o); generated by upper PWB solutions, I.V1I1.6(h); classical, I.XIII.I, I.XIII.10-I.XIII.13, logarithmic, 1.X111.18
Chapman-Kolmogorov equation: A.VI.3, 2.V1.1, 2.VI.5, 2.VII.1
Charge: definition, A.IV.7; energy, l.Xlll.l, 1.X111.3; energy is positive, 1.X111.7, I.XIII.8 Choquet capacity: see Capacity Choquet topological lemma : A. V I 1 I.3
Closability: in martingale theory, 2.111.1-2.111.3
Cluster set: of a function along Brownian and space-time Brownian paths, 2.VII.6, 2.1X.5, 2.1X.15, 2.X.2, 3.11.1-3.11.4 Condenser: definition, l.XII1.1 Conditional Brownian motion (see also Brownian motion): definition, transition density, lifetime, excessive functions for, 2.X.1; in terms of Brownian
motion, 2.X.2, 2.X.3; at path lifetimes, 2.X.4; from an infinity of the conditioning function, 2.X.5: time reversal of, 2.X.6; and the PWB method, 2.X.7, 3.11.1-3.11.4: last hitting distribution, 2.X.10; tail a alge-
bra, 2.X.11: parameter set R, 2.X.13; application to boundary limit theorems, see Boundary limit theorems; in a Martin space, see Martin boundary
Conditional expectation: properties. 2.1.4; continuity theorem for, 21.5. Fatou lemma for, 2.1.6; dominated convergence theorem for, 2.1.7; iteration of at optional times, 2.111.2(a); sample functions of the process of conditional expectations relative to the a algebras of a filtration, 2.IV. I
832
Index
Conditional Markov process: definition, 2.V1.13; see Conditional Brownian motion for the application to Brownian motion Conditional inequalities: in martingale theory, 2.I11.10 Cone: definition. A.I11.3: specific order generated by, A.1I1.4 Continuity properties: of a progressively measurable process sample func-
tion, 2 11.5. 2.11.6: of a supermartingale sample function, 2.IV.l: of a Brownian motion sample function, 2.V11.3; of the composition of a function with a Brownian motion sample function, 2.1X.11, 2.1X.12, 2.1X.16, 2.X.2 Convergence (see also Fundamental Convergence Theorem): of a directed set
of harmonic functions, Harnack's convergence theorem, 1.11.3; of an upward directed set of superharmonic functions, 1.I1.4, 1.IV.4; of families
of superparabolic functions, I.XV.12; of families of supermartingales, 2 111 5. 2 IV.4: forward, of a supermartingale, 2.111.13-2.111.15, 2.IV.3, backward, of a supermartingale, 2.111.16, 2.111.17
Coparabolic function (see also Parabolic function): definition. I.XV.2: coparaholic polynomials, I.XV.3: Coparaholic polynomials and martingale theory. 2.IX.2 Copotential (see also Potential): definition in the parabolic context. I.XVI1.5 Cylinder: superparabolic function on, I.XV.15, parabolic Dirichlet problem on, I XV1lI.4 D class
Classical context (D(pn )): of h-harmonic functions on a ball. 1.11.14. I IX 12. of functions on an open set, in particular h-harmonic and positive h-suhharmonic functions, I.IX.3 Parabolic context (D(µn_ )): of parabolic functions on a slab, I.XVI.6; of functions on an open set, 1.XVIII.19 Probability context: of stochastic processes, 2.11 11, includes right closed positive submartingales, 2.111.6, 2.V.3: includes potentials generated by increasing processes, 2.IV.6
Combined context: notation, 3.1.3: the PWB method. 3.1.4: S. n D = Smgh. 3 1 5. SPyg, = D n S'. 3.1.9 Decomposition See Krickeberg decomposition, LP. Rao decomposition, and Riesz-Herglotz
representation for decompositions in various contexts of a vector lattice element into a difference between positive elements, see A.1I1.5 for the abstract version of such a decomposition, and see A.IV for such decompositions in measure theory. See Lattice for decompositions of a vector lattice into bands. Riesz: of a superharmonic function. 1.1.8, I.IV.8, I.IX.I 1 ; of a superparabolic function, I.XVI1.7: of a supermartingale, 2.111.21, 2.V.8
Natural order: for superharmonic functions, 1.111.7; for superparabolic functions, I.XVII.2: for supermartingales, 2.111.19, 2.V.5
833
Index
-: of optional times, 2.11.12 -: of a supermartingale into continuous and jump components, 2.IV.22, 2.I V.23
-: of conditional Brownian motion, 2.X.4 Meyer: of a supermartingale, 2.IV.11; of a submartingale, 2.IV.12 d, A: definitions, 1.XV.2 Derivate: in the convex and symmetric senses, and in the convex and symmetric variational senses, A.IV.12 Dirichlet integral: 1.XIII.6 Dirichlet problem: see PWB method Dominated convergence theorem: A.IV.10, for conditional expectations, 2.1.7 Domination principle: classical context, I.V.10. 1.XI.23, I.XII1.2: parabolic context, 1.XVII.5, l.XVII1.16. martingale theory context, 2.IV.13 Downcrossing inequalities: 2.111.12, 2.111.23, 2.IV.18(o), 2.IV.19(o)
Energy: electrical, l.X1II.l ; energies and mutual energies of charges, I.XI11.2, 1.X1I1.3; dependence of energy on the containing set, 1.XII1.5; the Dirichlet functional.9 and Hilbert space methods, I .XII1.6; positivity of, 1.X111.7, 1.XIII.8, 2.IX.17; Gauss minimum problems, I.XII1.14; relative to 182, I.XIIl.16; of a supermartingale potential, 2.IV.21 Entrance law: see Absolute probability function
Entry time (see also Hitting time): definition, 2.11.4; of a predictable set, 2.11.9
Equilibrium measure and equilibrium potential: I.XIII.1, I.XIII.10. I.XII1.12; relative to I82, 1.XI11.18
Essential order: of measurable functions, A.IV.9, 2.1.3; convergence, A.IV.10; of stochastic processes, 2.1.8, 2.V.1, 2.V.2, 2.V.5; convergence of supermartingale families, 2.111.5 Euclidean boundary: definition, Notation and Conventions; resolutivity and set of regular points in the classical context, 1.VII1.4; resolutivity and set of regular points in the parabolic context, I.XVIII.I Evanescent set: definition, 2.1.8 Evans-Vasilesco theorem : 1 . V .8 Excessive functions and measures (see also Invariant-): definitions, 2.VI.10; excessive functions as generators of supermartingales, 2.VI.I I ; excessive functions defined by hitting probabilities, 2.VI.12; excessive functions for [space-time] Brownian motion versus positive [superparabolic] superharmonic functions, 2.IX.6, 2.IX.8; a excessive functions for Brownian
and space-time Brownian motion, 2.IX.16; excessive measures for Brownian and space-time Brownian motion, 2.IX.18
Extension: of a superharmonic function through a polar set, 1.V.5; of a Green function GD to GD, 1.VII.4, I.VIII.18, I.VIII.19; of a parabolic function defined on a truncated cylinder, 1.XV.17; and contraction of OD, I.XVII.4; of a superparabolic function through a polar set, I.XVII.8; of a Green function GD to GD, 1.XVIII.9
834
Index
Fatou boundary limit theorem: see Boundary limit theorems Fatou's lemma: A.IV.10, for conditional expectations, 2.1.6 Filtration of a measurable space, 2.1.1; composition .F(T) with an
optional time T, 2.11.1 ; predictable-and F(T- ), 2.11.7; choice of for a supermartingale. 2.111.1, 2.IV.1; choice of for a Markov process, 2 VI.2: right continuity of. for a Markov process, 2.V1.8; right continuity and predictability of Brownian motion filtrations, predictability of Brownian motion optional times, continuity properties of supermartingales relative to Brownian motion filtrations. 2.V11.4. 2.VI1.6, 2.VIII.8, 2.X.2(c). for conditional Brownian motion, 2.X.2(c), (d) Fine topology (see also Thinness)
Classical context: definition and basic properties, I .X1.I-1.X1.3, I.XI.5, and, in probabilistic formulation, 2.IX.15. character of derived set, boundary, and interior, I.XI.I, 1.XI.6:-limits versus Euclidean topology limits. 1.X1.9; identification of a derived set by a special function u''. I X1.10: quasi Lindelof property. 1.X1.11; relation to boundary point regularity, 1.X1.12; -open sets as domains for superharmonic functions, I.X1.19: minimal-fine topology, 1.X11.1 I, 1.X11.12, 1.X11.17, and, in probabilistic formulation, 3.111.3; minimal-fine topology limits
versus Martin topology limits at the Martin boundary, I.XII.15, I XII 16: normal boundary limit points of a half-space subset versus minimal-fine boundary limit points of the set, 1.X11.22
Parabolic context: definition and basic properties, I.XVII.9, I.XVII.12. I XVIII 10. and, in probabilistic formulation, 2.IX.15; character of a derived set and its identification by a special function ta', I.XVII.15, I XVII 16(r). I.XVII.17(r): quasi Lindelof property. I.XVII.19: minimal-fine coparaholic limits versus Martin topology limits at the Martin boundary. l .XIX. l2 Probability context: in Q8' x 0, 2.IV.16 Finite dimensional distributions (see also Stochastic process): of a stochastic process. 2 1 10 First boundary value problem: see PWB method Forward parabolic equation: for Brownian motion, 2.1X.17 Fundamental Convergence Theorem: classical context. 1.111.3, 1 VI I. I AI 7,
1 XIV 3. parabolic context. I.XVII.2. I.XVII.l3- I.XVII.l5. martingale theormy context. 2.IV 5 Gh sets: homeomorphs of, A.1.9 Gauss Integral Theorem: 1.1.6, 1.1.7 Gauss minimum problems: 1. X 111.14
GM (see also LM)
Classical context: definition. 1.111.1. 1111.2. I.XIV.3: of a potential, 1111 1, I IV 3. 1 IV 8(a), relation with the reduction operator, 1.1114. 111 6: unaffected by deletion of polar sets, 1.V.5; relation with Dinchlet solutions. I VIII 3. I Vill.I8(c): relation with the r operator, 1
I VIII
I
I
835
Index
Parabolic context (GM, OM): definition, 1.XVII.1; of a potential, I.XVII.5; relation with Dirichlet solutions, I.XVIII.I
Martingale theory context: definition and basic properties, 2111.20, 2.111.21, 2.IV.14; of a potential, 2.111.21
Graph: of an optional time, 2.11.1;-predictability, 2.11.9 Green function Classical context (GD): preliminary definition, 1.1.5, 1.1.8; of a ball, 1.1I.1,
definition, I.VII.!; in terms of a Dirichlet solution, 1.VII.1, I.VI11.3, I.VII1.18, I.VHII.19; extremal property, I.V11.2; boundedness property, 1.V11.3; symmetry property, 1.V11.4; extension to GD, I.VII.4,
1.VIII.l8, 1.VIII.19; vanishing of C) at 8D, 1.V11.4, l.X1I.14, 1.X11.18; minimality of G,(-, C), 1.VII.10; (N=2) with pole oo, the Robin constant, I.VI11.20, 1.X111.18; the case N = 1, I.XIV.6; in terms of GD, I.XVII.18; probabilistic expression for, 2.1X.17 Parabolic context (GD, GD): preliminary definition, 1.XV.4, 1.XV.7; of an interval, 1.XV.8; definition, examples, properties, GD = GD, I.XVII.4; minimality of I.XV11.8; vanishing of i;) at 815, 1.XVI1.14; in terms of a Dirichlet solution, 1.XVIII.1; extension to GD , I.XVIII.9; probabilistic expression for, 2.IX.17 Greenian set: definition, trivial in l when N > 2, but I82 is not one, 1.1I.13; conditions that an open subset of I8Z be Greenian, 1.V.6, 1.VII.7 in nonprobabilistic terms, and 2.IX.10 in probabilistic terms Hadamard three circle theorem: 1.11.11 Hahn decomposition: A.IV.6
Half-space
Classical context (-of I8"): Green function, harmonic measure, Poisson integral and Riesz-Herglotz representation, I.V1II.9; Martin boundary, I .XII.3 ; minimal thinness, minimal-fine versus nontangential limits at a boundary point, I XII.12, ratio boundary limit theorems for harmonic functions, 1.X11.21 ; normal versus minimal-fine boundary limit func-
tions, 1.XII.22; boundary limit function of a potential, 1.X11.23; Brownian motion, 2.VII.9 Parabolic context (-of C'8", see also Slab): upper and lower half-space of
ftN, 1.XV.1; Green function, l.XV1.l, I.XVI1.4; Martin boundary, 1.XIX.8, l.XIX.9 Harmonic function (see also Subharmonic function and Superharmonic function): definition, average properties, 1.1.3; maximum-minimum theorem, 1.I.4; Poisson integral, 1.1I.1; Harnack inequality, 1.11.2; Harnack convergence theorem, 1.11.3; classes L", D, and the Riesz-Herglotz theorem, 1.11.14, 1.IX.3, 1.1X.4; ratio boundary limit theorem in a ball, 1.11.15; minimal, 1.11.16, 1.X11.5; removable singularity set, 1.V.5; relative,
l.VIIl.l ; boundary poles, 1.X.7, I.XII.5; Martin representation, 1.XII.9, 3.111.6; vector lattice Sand its band decompositions, 1.1X.7, 1.IX.9I.IX.12, 3.1.12; first boundary value (Dirichlet) problem, see PWB method; composition with Brownian motion, see Superharmonic function
836
Index
Harmonic measure (p'): for domains with smooth boundaries, 1.1.8; null sets. I.Vl1I.5; of fool, LVIII.5; definition, l.V1I1.8; for ball and halfspace, I.V111.9; dependence on D, 1.VIII.17; relation with the sweeping kernel. 1.X.2; on a polar set, 1.X.8; relative to an irregular boundary point, I .X1.22. for the Martin space, 1.X11.10; as a conditional Brownian motion hitting distribution, 2.1X.13, 2.X.7, 3.11.2 Harnack convergence theorem: classical context, 1.11.3, parabolic context. I.XV.I I Harnack inequality: classical context, 1.11.2; parabolic context, 1.XV.11. 1.XVI11.17
Hermite polynomials: 1.XV.3; composition of space-time-with spacecotime Brownian motion, 2.IX.2 Hitting: by progressively measurable processes, and the dependence on the choice of process with prescribed finite dimensional distributions, 2.1.92.1.12; hitting, entry, and last hitting times, 2.11.4: the hitting probability
for a Markov process of an analytic set, in particular of an F. set, as a function of the process initial point, 2.V1.6, 2.1X.4, and the excessive function so defined, 2.VI.12; by Brownian motion, 2.VI1.6, 2.V11.8, 2.1X.1, 2.1X.4, 2.1X.9, 2.1X.15; of a parabolic semipolar set by space-time Brownian motion, 2.1X.15: harmonic measure as a conditional Brownian
motion hitting distribution. 2.1X.13, 2.X.7, 3.11.2; capacitary distributions in terms of last hitting distributions, 2.X.10 Hunt potential theory: 2. V1.10
Increasing process: definition, support of measure generated by, 2.IV.6: potential generated by. 2.1V.6, 2.IV.8: natural versus predictable, 2.IV.6, 2.IV.7: as the generator of the Riesz measure for a supermartingale. 2.IV.13 Independent increment process: 2.111.2(c). 2.VII.I Indistinguishable processes: 2.1.8 Integral limit theorems: A. V I I Internal potential theory limit theorems: classical context, I.XI.4, 1.X11.19;
parabolic context, I.XVIII.14 definition, upper, lower, lateral boundaries, I.XV.I : paraInterval of
bolic Green function, I.XV.8: parabolic measure. I.XV.9, 2.VI1.12; Dirichlet problem, I.XVI11.1, I.XVIII.6, Brownian motion in, 2.VII.1 I Invariant excessive functions and measures: definitions, 2. VI.10 ; for Brownian
and space-time Brownian motions, 2.VII.10, 2.1X.8 Inversion in a sphere: the Kelvin transform, 1.11.12, 1. VII I.1: image of a polar set, IN. 1: and the PWB method, I.VI11.2: and h-harmonic measure null sets. I.V111.5: and boundary point regularity, I.V111.14 Irregular boundary point (see also Regularity of a boundary point) Classical context: limit of a superharmonic function, of harmonic measure, and of a Green function at one, 1.X1.21, 1.X1.22; Brownian motion from one, 2.IX.20
837
Index
Parabolic context: limit of a superparabolic function, of parabolic measure, and of a Green function at one, I.XVIII.17, 1.XVIII.18 Iterated logarithm: regularity criterion and Brownian motion law, I.XVIII.6, 2.IX.15
Ito integral: integrand classes, 2.VII1.1, 2.VIlI.2; properties, 2.VIII.3, 2.VIII.6; definition, 2.VIII.5, 2.VIII.7; integration by parts, 2.VIII.1I Ito's lemma : 2. V I I1.12
Jensen inequality: applied to subharmonic and harmonic functions, 1.11.9,
applied to subparabolic and parabolic functions, 1.XV.12; applied to conditional expectations, 2.1.4(h): applied to submartingales and martingales, 2.111.3(c)
Jordan decomposition: A.IV.6 Kelvin transform: see Inversion in a sphere
Kernel: definition, extension from substochastic to stochastic, A.VI.1; universally measurable extension, A.VI.2 Kolmogorov construction of canonical processes: 2.1.10 Krickeberg decomposition: of a martingale, 2.V.4(c)
L° bounded: stochastic process, 2.1.8; submartingale, 2.V.4; Krickeberg decomposition of an-martingale, 2.V.4(c) L° class
Classical context
of h-harmonic functions, 1.11.14, I.IX.4,
3.1.3
Parabolic context (L°(4_)): of h-parabolic functions, I.XVIII.19, 3.1.3
1.XVI.6,
Last hitting time: definition, 2.11.4: Markov process killed at a, 2.VI.15; Brownian motion killed at a, 2.X.6: capacitary distribution and-, 2.X.10 Lattice Abstract theory: definitions, countability property. A.III.2; cone, vector lattice, specific order, A.I11.3-A.I11.5: decomposition property of a vector lattice, A.111.6; orthogonality, bands, projections, A.111.7A.M. 11; order convergence, A.II1.12, A.I11.13, A.IV.10 Application to measure theory: lattice of set algebras, A.IV.1; lattices of measures, A.IV.4-A.IV.7; absolute continuity and singularity, A.IV.8;
lattices of measurable functions, A.IV.9; essential order and the corresponding convergence, A.IV.10
Application to classical potential theory: St, S', I.IX.5; S, I.IX.6; Sm, I.IX.7; SD, I.IX.8; Sgb, Spgb, Smgb, the relation between Smgb and the
PWB method, 1.IX.9; S.,, Sps, S.,, l.IX.10; refinement of the Riesz decomposition of a positive superharmonic function, I.IX.11, I.IX.12; S,,,
, Smsf, 3.1.12
Application to parabolic potential theory: translation from classical to parabolic context, 1.XVI I I.19
838
Index
Application to martingale theory: 'S', 'S', S. S+, 2.V.5; 'S, S, 2.V.6: 'Sm, Sm, 2.V.7; 'Sp, Sp, 2.V.8; Se,, Spgb, Smgb in the specific order, and
their primed counterparts, 2.V.9; Ss, S,,,,, Sps in the specific order, and their primed counterparts, with orthogonal decompositions of 'S and S, 2.V.10, 2.V.I I Combined application to nonprobabilistic potential theory and to martingale
theory: background, 3.1.1; band relations, 3.1.2; L° and D, 3.1.3; D and the PWB method, 3.1.4, 3.1.5; characterizations of Sqb, Ss, Sms, Spgb, Smgb, Sps, 3.1.6-3.1.10; band identification of the composition of an h-superharmonic function with h-Brownian motion, 3.1.11, 3.1.13
Law of large numbers: for Brownian motion, 2.VI1.5 Lebesgue decomposition: A.IV.8 Lebesgue spine : I . V 111.15 Levy Brownian motion square increment theorem: 2.V111.10
Lifetime: of a Markov process, 2.V1.3, 2.VI.7; of conditional Brownian motion, 2.X.1. 2.X.4, 2.X.9, 3.1.12 Liouville's theorem: 1.11.2, 1.11.13, 1.V.5
LM (see also GM)
Classical potential theory: definition and existence. 1.111.1, 1.111.2: for subharmonic functions in various classes, 1.1X.2-I.IX.4 Parabolic potential theory: definition and existence, I.XVII.I Martingale theory: definition and existence, 2.111.20, 2.IV.14; for submartingales in various classes, 2.V.2-2.V.4 Logarithmic potential on l : defined, I .IV.I . Riesz type decomposition of a superharmonic function, I.IV.9: domination principle. I.V.l0; infinity set. I.V I I Lower semicontinuous functions: smoothing of a function, A.VIII.I ; supremum of a family of, A.VlIl.2; Choquet topological lemma, A.VI11.3 Lusin's measurability theorem: A.11.4
Markov process: defined, Markov property, initial distribution, transition
function, absolute probability function, 2.V1.1; choice of filtration. 2.VI.2: strong Markov property, 2.V1.3, 2.V1.9: stationary stochastic and substochastic transitions. 2.V1.3. 2.V1.5, 2.VI.6; application of martingale theory, 2.V1.4: lifetimes and trap points, 2.V1.3, 2.VI.7; right continuity
of filtrations, 2.VI.8. conditional-, 2.V1.13; tied down-. 2.V1.14: killed . 2.V1.15: Brownian motion as a--, 2.VII.2 Markov time: see Optional time Martin boundary
Classical context (aMD): Martin space D'M, 1.X11.1, 1.X11.3; Martin function K, 1.X11.2; Martin representation, I.Xll.4, 1.X11.6, 1.X11.9: minimal harmonic functions and the set arD of their poles, the minimal boundary, 1.X11.5, 1.X11.7, I.XII.8; resolutivity, I.XI1.10; minimal thinness and the minimal-fine topology, I.Xll.11, 1.X11.12, I.XIl.15
839
Index
1.XII.17; attainable-points, Brownian motion to and from-points, 3.111.1, 3.111.2
Parabolic context: Martin point set pairs, flat point set pairs, measure set pairs, 1.XVIII.17, 1.XV111.18, I.XIX.2; exit and entrance boundaries, I.XIX.1; Martin function k and admissible superparabolic functions, I.XIX.2; Martin space and boundary, I.XIX.3; Martin representation, 1.XIX.4, I.XIX.7, 3.II1.6; minimal boundary, 1.XIX.5, I.XIX.6; slab, I.XIX.8, I.XIX.9; right half-slab, I.XIX.10; resolutivity, 1.XIX.11; minimal thinness and the minimal-fine topology, 1.XIX.12 Probabilistic context: attainable -points, Brownian motion to and frompoints, 3.111.1, 3.11I.2; zero-one law at a minimal-point, 3.111.3
Martingale (see also Submartingale and Supermartingale): definition, examples, elementary properties, closability, 2.111.1-2.111.3; optional sampling of a uniformly integrable-, 2.111.2(a), 2.IV.2; subsets of l8 as general parameter sets, 2.111.4; convergence if L' bounded, 2.111.13; convergence if uniformly integrable, 2.111.14; backward convergence, 2.111.16: Krickeberg decomposition if L° bounded, 2.V.4(c); 'Sm, S. and their decompositions, 2.V.7, 2.V.11, 3.1.2, 3.1.4-3.1.6, 3.1.12, 3.1.13; local-, 2.V.12, 2.VIII.3(e), 2.VII1.3(j), 2.VIII.6(e), 2.VIII.6(j); relative
to a Brownian motion filtration, 2.VIII.8; generated by composing functions with Brownian or space-time Brownian motion, 2.IX.1-2.IX.3, 2.IX.7(c), 3.11.2(c) Maximal inequalities: in martingale theory, 2.111.9-2.111.11 Maximum: -minimum theorem for harmonic functions, 1.1.4, for parabolic
functions, I.XV.5; theorem for analytic functions, I.V.7; principle of and complete principle of, 1.V.10 Measurable family of functions: 2.1.2 Measurable space: A.IV.2
Meyer decomposition: of a supermartingale and of a submartingale, 2.IV.11, 2.IV.12 Minimal function
Classical context: harmonic, 1.I1.16, relative harmonic, 1.VI1I.1; on D - Jr }, I.VIl.10; Martin boundary pole of, 1.X11.5
Parabolic context: parabolic, 1.XV.2; on the upper half-space of OB', extension of, when defined on a cylinder, 1.XV.17; on a slab, 1.XVI.8; 1.XVII.8 e) on D Modification: see Standard modification
Nearly: progressively measurable, predictable, and so on, 2.1.8; Borel sets for Brownian motion, 2.1
Nontangential: deleted neighborhood filter for a ball boundary point, 1.11.15: limits versus minimal-fine limits at ball and half-space boundary points, I.XII.12, I.XII.20, I.XII.21 Normal: boundary limit points of a half-space subset versus minimal-fine limit points of the set, I.XII.22; boundary limit function of a classical
840
Index
context potential, 1.X11.23; boundary limit function on a slab boundary determines a bounded parabolic function, I.XVI.2 Nucleus: of a Suslin scheme, A.1.2 Optional sampling and stopping theorems: 2.111.6-2.I1I.8, 2.IV.3 Optional time: definition and properties, 2.11.1, 2.11.2; predictable, 2.11.7, 2.11.9; decomposition, totally inaccessible and accessible-'s, 2. 11. 12 Ordinate: of a point of U8^'. I.XV. I Orthogonality: in a vector lattice, A.111.7
Parabolic function (see also Subparabolic function and Superparabolic function): definition. 1.XV.2; maximum-minimum theorem, I.XV.5; average
properties. I.XV.I0. Harnack convergence and inequality theorems, LXV.I I. extension of, when defined on a cylinder, 1.XV.17, Poisson integral and the L' and Dclasses. for functions on a slab, 1.XVI.1, 1.XVI.5,
I XVI.6: ratio boundary limit theorem on a slab, I.XVI.7, I.XVIII.IS: removable singularity set, 1.XVII.8: relative, I.XVIII. I ; minimal, I.XIX.5. Martin representation. I.XIX.7-I.XIX.10; vector lattice Sm and its band decompositions. 3.1.1: first boundary value (Dirichlet) problem. see PWB method; composition with space-time Brownian motion, see Superparabolic function Parabolic limit: at a slab boundary point, 1.XVI.7 Parabolic measure (p h)): for domains with smooth boundaries, I.XV.7; for an interval, I.XV.9. I.XVIII.2; definition, 1.XVIII.2: relation with the
sweeping kernel, I.XVIII.8; relative to an irregular boundary point, I.XVIf1.17: for the Martin space, I.XIX.I 1; as a conditional space-time Brownian motion hitting distribution, 2.V11.12, 2.IX.13. 2.X.7, parabolic version of 3.11.2 Paving: A.1.1
Poincare-Zaremba barrier: I .VII1.12, I .VIII.15 Poisson equation: classical context, 1.1.7; parabolic context, I.XVII.6 Poisson integral: classical context, for ball and half-space, 1.11.1, I.VIIl.9. parabolic context, for a slab, I.XVI.1, I .XVI.5, I .XVI.6. 1.XVII.5 Polar set Classical context (in R'): definition and properties, inner-, l.V. I -1.V.4, I .XIV.2: removable singularity set of a lower bounded superharmonic
function, I.V.5: role in characterization of Greenian subsets of OZ, 1.V.6: infinities of a potential on a-, I.V.I l: exceptional set in the
Fundamental Convergence Theorem, I.V1.I: analytic inner-is-, I.V1.2: h-harmonic measure on. 1.VIII.5. 1.X.8; vanishing of sweeping kernel on. 1.X.6: characterized by absence of fine limit points, I.X1.6;
null for a measure of finite energy, I.X1l1.2; not hit by Brownian motion, 2.1X.5, 2.X.2. but analytic non polar sets are surely hit if N = 2, hit with probability > 0 if N > 2, 2.1X. 10
Parabolic context (in AN): definition, counterparts of classical context
841
Index
properties, 1.XVII.8; criteria for, parabolic- = coparabolic-, 1.XVII1.1 I ; not hit by space-time Brownian motion, 2.1X.5 Probability context (in R' x f2): definition, 2.11.10
Pole (see also Martin boundary): boundary pole of a minimal function, L X.7, 1.X11.5, 1.XIX.5
Polish space: definition, A.1.10; measure on, A.IV.1I Potential
Classical context: kernel G, 1.1.5, I.IV.I, I.IV.9; of a smooth density, 1.1.7; on a special open set, 1.IV.1; vanishing of GM(-), 1.IV.3, I.IV.8; smoothing of, by averaging, I.IV.5; generating measure, 1.IV.6, 1.IV.7; condition that a superharmonic function be a potential, 1.IV.8, 1.VI11.1 I, 1.X11.17 example; infinities of, on a polar set, I .V.11; boundary limit function, 1.V11.5, 1.X11.18, 1.XI1.23, 2.1X.7; lattice S,,, 1.IX.8; the case N = 1, I.XIV.7
Parabolic context: kernel GD, 1.XV.4, I.XVII.5; of a smooth density, I.XVI1.6; boundary limit function, 1.XIX.14, 2.IX.7; lattice, I .XVIII.19
Martingale theory context: definition, 2.111.21; generation of, by an increasing process, 2.IV.6-2.IV.11: associated measure, domination principle, 2.IV.13; potential theory on R` x S2, 2.IV.15; energy, 2.IV.21
Precapacity: topological, and the generated Choquet capacity, A.II.8 Predictable:-a algebra, and--family of functions, 2.1.14; predictability of
a function family at an announced time, 2.11.3;-optional times andfiltrations, 2.1I.7;-process defined by an accessible optional time, 2.11.7,
2.11.12; section theorem for a-set, 2.11.8; graph of a-optional time, 2.11.9;-increasing process = natural-process, 2.IV.7; continuity properties of--supermartingales and martingales, 2.IV.23 Progressively measurable: definition, right and left continuous processes
are-, 2.1.2; hitting of sets by-processes, 2.1.9, 2.11.4, condition for existence of a-standard modification, 2.1.13;-process at an optional time, 2.11.3; continuity properties of-processes, 2.11.5, 2.11.6 PWB method (see also Barrier, Regularity of a boundary point, Resolutivity) Classical context (PWB''): upper and lower classes and solutions, I.VI11.2, I.V111.6, 1.V111.7; solutions as reductions, 1.VI11.2, 1.VII1.10; solu-
tions are quasi bounded, I.IX.9 Parabolic context (PWB''): upper and lower classes and solutions, relation with reductions, 1.XVI I1.1. classical context as a special case of parabolic context, l.XV11I.4 Probabilistic solution: 2.1X.1, 2.1X.13, 2.X.7, 3.1I.1-3.11.4
Quasi-bounded Classical context: the classes Sqb, Smgb, Sp'gb, 1.IX.9; significance for the
PWB method of quasi-boundedness, 1.IX.9; significance of quasiboundedness of a harmonic function in relation to the class D property and the Riesz-Herglotz and Martin representations, 1.IX.12, 1.X11.9
842
Index
Parabolic context :
I . X V I I I.19
Martingale theory context:-positive supermartingales and their generation by increasing processes, 2.IV.6-2.IV.I l ; classes 'Sqb, Sqb and their subclasses 'Smgb, 'S pqb, S; qb, Spgb, 2.V.9 Combined context: see Lattice
Quasi-everywhere: classical context, inner-, IN. I, probabilistic context, 2.1.8
Quasi-Lindelof property: classical context, I.XI.11; parabolic context, I.XVII.19 Quasi-martingale: definition, Rao representation as an element of S, 2.V.13 Reciprocity law: l .X111.2 Reduction (see also Sweeping) Classical context: definition and properties, 1.1I1.4, 1.111.5, I .VI.2-l .VI.4;
GM(-), 1.111.6; relation to capacity, I.VI.5; generalized, 1.X1.20; probabilistic evaluation, 2.IX.14, 2.X.7
Parabolic context: definition and properties, I.XVII.3, I.XVII.I1, I.XVII.14, 1.XVII.16, I.XVII.17; probabilistic evaluation, 2.X.7 Martingale theory context: discrete parameter case, 2.111.22; application to crossing inequalities, 2.111.23, 2.IV.18(o), 2.IV.19(o); continuous parameter case, 2.I V.17-2.I V.20 Regularity of a boundary point (see also Barrier)
Classical context: definition, I.VIII.2; Euclidean boundary, I.VIII.4; Poincare and Poincare-Zaremba criteria, Lebesgue spine, I.VI1l.12, I.VIII.15; in terms of harmonic measure. I.VIlI.8; in terms of the fine topology, 1.X1.12
Parabolic context: Euclidean boundary, I.XVlll.l, 1.XVIlI.6; for an interval, 1.XVIJI.2; classical context regularity in terms of parabolic context regularity. I.XVIII.4; for a parabolic ball, I.XVIII.6: iterated logarithm criterion, I .XVIII.6; in terms of the fine topology, I .XVIII.7 Resolutivity
Classical context: definition, universal and internal, 1.VII1.2; Euclidean boundary. I.VIII.4: condition for-of a boundary function, 1.V111.8,
I.V111.9, 3.11.2; condition for-of a boundary, I.VI1I.9, 3.11.3(e); universal and universal internal- of a ball boundary. I .VIII.9. I .IX.12: of a Martin boundary, I.XII.10
Parabolic context: adaptation of the classical context definitions and results, I.XVIII.I - I.XVIII.3, 3.11.1; universal and universal internalof a Martin boundary, 1.XIX.I I Reversal: of Brownian motion. 2.X.6 Riesz decomposition: of a superharmonic function, 1.1.8, 1.IV.8, 1.IV.9. 1.1X.1 I. I.XIV.9; of a superparabolic function, I.XV.7, I.XV1I.7: of a supermartingale, 2.111.21 Riesz measure: associated with a superharmonic function. I.IV.7; associated
with a superparabolic function. I.XVII.7; associated with a supermartingale. 2.1 V.13
843
Index
Riesz-Herglotz representation: of a harmonic function on a ball, 1.11. 14 Robin constant: I .XII1.18 Sample function: definition, integrability, 2.1.8; continuity properties, 2.11.5,
2.11.6; for properties of sample functions of specific process types, see those types Section theorem: 2.11.8 Semipolar set
Parabolic context, in A": definition, I.XVII.10; exceptional set of the Fundamental Convergence Theorem, I.XVII.13; criteria for, parabolic semipolar = coparabolic semipolar, 1.XVII.15, 1.XVIII.12; hitting of by space time Brownian motion, 2.1X.15 Probabilistic context, in I x 0: definition, 2.11.10; role in supermartingale smoothing, 2.IV.I; exceptional set of the Fundamental Convergence Theorem, 2.IV.5 Sierpinski lemma: on Suslin schemes, A.II.2 Singular Classical context: classes SS , SS, Sps of positive singular functions, respectively superharmonic, harmonic, and potentials of measures, 1.IX.10; conditions that a function in S be singular, 3.1.7; evaluation of the singular component of a function; in S+, 3.1.8; conditions that a function in SD be singular, 3.1.10 Parabolic context: 1.XVIII.19 and the parabolic context part of Sections 3.1.7, 3.1.8, 3.1.10
Martingale theory context: classes 'Ss , S, of positive supermartingales
with their subclasses 'S,
,
'S', Sms, S ,, 2.V.10; condition that a
process in SP be singular, 2.V.12; martingale theory part of Sections 3.1.7, 3.1.8, 3.1.10
Slab in ft': definition, I.XV.13; Green function and the Poisson integral, 1.XVI.1, 1.XVI.5, I.XV1.6, 1.XVII.4; generalized superparabolic average
inequality, 1.XVI.2; L` and D classes of parabolic functions, 1,XVI.6; ratio parabolic function boundary limit theorem, 1.XVI.7, 1.XVIII.15, I.XIX.15; minimal parabolic function, 1.XV1.8; Martin boundary, I.XIX.8, I.XIX.9; coparabolic minimal thinness and minimal-fine limits at a boundary point, l.XIX.12 Space-time Brownian motion (see also Brownian motion): definition, 2.VI1.2;
properties induced by Brownian motion properties, 2.VI1.4; transition
function for-in an open set, 2.VII.10; transition function for-in an interval, parabolic measure as a-boundary hitting distribution, 2.VII.12,
2.VII.13; composition of a superparabolic function with-, excessive functions and measures for-, 2.1X.2, 2.IX.3, 2.IX.6-2.IX.8, 2.IX.12, 2.IX.13, 2.IX.16, 2.IX.18; probability of hitting sets by-, 2.IX.4, 2.1X.5;
from a parabolic irregular boundary point (parabolic counterpart of) 2.1X.20; conditional, 2.X.12 Special open set: definition, I.IV.1; transition from, to Greenian sets, 1.VII.8 Specific order: definition, A.II1.4
844
Index
Standard modification: of a stochastic process, 2.1.8: measurable-, 2.1.13 State space: of a random variable, 2.1.3: of space-time Brownian motion, 2.V11.2
Stochastic interval: definition, 2.1I.1; fine open, closed, 2.IV.16 Stochastic process (see individual types under their own names): definition, adaptedness, indistinguishability, almost sure properties, 2.1.8; canonical, 2.1.10: dependence of certain probabilities on the choice of probability space, 2.1.11, 2.1.12, 2.11.4 Stolz domain: definition, 1.11.15 Stopping time: see Optional time Strongly subadditive set function: definition, A.11.6, generation of a Choquet I.VI.2(b), I.VI.3(j), I.VI.4(j); capacity by. A.I1.7. A -+ R,,,. and A R", +11
classical capacity function. I.XIII.10; negative of the Robin con-
I.XVII.11(d), I.XVII.16(k), stant, I.XII1.18; 4- R", and A l.XV11.17(k); A , and A - k , 2.1V.18(i), 2.IV.19(i) Subharmonic function (see also Harmonic function and Superharmonic function): definition. 1.11.4, 1.XIV.2: Jensen's inequality applied to, 1.11.9; LMU operator on, I.IX.2-1.1X.4 R,_,_",
Submartingale (see also Martingale and Supermartingale): definition, 2.111.1 ; Jensen's inequality applied to, 2.I11.3(c); is right closable if uniformly integrable and is uniformly integrable if positive and right closable, 2.111.3(e), 2.111.6, 2.V.3, 2.V.4; LM(--), 2.V.2 Subparabolic function (see also Parabolic function and Superparabolic function): definition, Jensen's inequality applied to, I.XV.12; maximum theorem for a slab domain, I.XVI.3 Superharmonic function (see also Harmonic function and Subharmonic function): definition, 1.11.4, 1.XIV.2: minimum theorem, 1.11.5, I.V.7; characterization in terms of harmonic functions. 1.11.7; differentiable, 1.11.8; approximation of by differentiable ones, 1.11.8, I.IV.5, 1.IV.10; on an
annulus, 1.11.10; Riesz measure and decomposition. condition that a positive- be a potential, 1.1.8, I.IV.7 1.IV.9. 1.XIV.9: relative-, I.VIII.I: smoothness properties when N = I, 1.XIV.4; is a superparaholic I.XV.15; composition with function of the space coordinate of Brownian motion, 2 IX.3, 2.1X.6-2.1X.8, 2.IX.12: composition of ratio of 's with conditional Brownian motion, 2.VI11.13. 2.X.1, 2.X.8, 3.1.11, 3.1.13, 3.11.2
Supermartingale (see also Martingale and Submartingale): definition, closability. 2.111.1 2.111.3; first treatment of limits of monotone sequences
of. 2.111.5: at optional times. 2.111.6-2.111.8. 2.IV.2, 2.IV.3: subset of R as general parameter set. 2.111.4: maximal inequalities, 2.111.9, 2.111.10; downcrossing inequalities, 2.111.12, 2.111.23; convergence. 2.111.132.111.17, 2.IV.3; potential. 2.111.21, generated by an increasing process,
2.IV.6 2.IV.11: sample function continuity properties, 2.IV.1, in particularas influenced by the reference filtration, 2.IV.23. 2.VII.4. 2.V111.8; limit of an increasing sequence of- 's, 2.IV.4: Fundamental Convergence
845
Index
Theorem, 2.IV.5; measure associated with a-, domination principle, 2.IV.13; decomposition into jump and continuous components, 2.IV.22, 2.IV.23; lattices of-'s, 2.V.5-2.V.1I Superparabolic function (see also Parabolic function and Subparabolic function): smooth context Riesz decomposition, 1.XV.7; definition, 1.XV.12; minimum theorem, I.XV.13; approximation of, by differentiable ones, I.XV.14; on a cylinder, 1.XV.15, I.XV.17; generalized-inequality on a slab, 1.XVI.2; Riesz measure and decomposition, I.XVII.7; relative-, 1.XVIII. I ; admissible-, I.XIX.2; composition with space-time Brownian motion, 2.IX.3, 2.IX.6-2.IX.8, 2.[X.12, 2.IX.13 Suslin scheme: A.I.2 Sweeping of a measure Classical context: definition, the sweeping kernel, 1.X.1, 1.X.4-1.X.6; relation of the sweeping kernel to harmonic measure, 1.X.2; sweeping symmetry, 1.X.3; support of a swept measure, I.XI.14, 1.X1.18, 2.IX. 15
Parabolic context: definition, the sweeping kernel and parabolic measure, dualization of sweeping symmetry, I.XVIII.8; support of a swept measure, 1. X V I I I.13, 2.I X.15
Tail a algebra: of a conditional Brownian motion, 2.X.11, 3.11.4 T operator Classical context (rhq): definition, 1.11.1, 1.XIV.3; application to superharmonic functions, 1.11.6; generalized, application to GM"" operator, l.V1II.11; variant.of, I.VIII.21 Parabolic context (ihj): definition, I.XV.9; application to superparabolic functions, 1.XV.14, I.XVIII.2 Martingale theory context (rST): definition, 2.111. 18, 2.IV.14 Thinness (see also Fine topology)
Classical context: definition, 1.XI.1; criterion, 1.XI.2, 1.XI.3; at oo, 1.X[.5; minimal, 1.XII.11; a ray from a half-space boundary point is not minimal thin at the point, I.XII.12; Wiener criterion, I.XIII.17; probabilistic criterion, 2.IX.15, 3.111.3 Parabolic context: definition, criterion, 1.XVII.5 Example (c), 1.XVII.9, I.XVII.12; minimal thinness at Martin boundary points of a slab and of the fourth quadrant of 12, I.XIX.12; probabilistic criterion, 2.IX.15 Probability theory context: at an optional time, 2.IV.16 Transition function: definition and universally measurable extension, A.VI.2, A.VI.3 Trap: definition, A.VI.1, 2.VI.3, 2.VI.7 Tulcea construction: of a canonical Markov process, 2.VI.3
Uniform integrability: definition, test function, implications, A.V; of the class of conditional expectations of a random variable, 2.1.4(i); of a process, 2.1.8; of a right closed positive submartingale, and the right
846
Index
closability of a uniformly integrable submartingale, 2.111.3(e), notation
UI, quasi-bounded martingales are the uniformly integrable ones, 2.V.9: example of a uniformly integrable but not class D supermartingale, 2.IX.6 Universal measurability: definition, A.11.9; of hitting probability as a function of a Markov process initial point, 2.V1.6 Upcrossing inequalities: see Downcrossing inequalities Version: of an equivalence class of random variables, 2.1.3 Wiener measure: defined. 2.V11.3
Wiener thinness criterion: I.Xl1l.l7
Zaremba barrier: see Poincare-Zaremba barrier Zero-one law: for the classical and parabolic sweeping kernels, I.X.6(c),
I.XVIII.IO; at the initial point of a Markov process, in particular for Brownian motion, 2.VI.8, 2.VI1.6, 2.X.2(d); at a minimal Martin boundary point, 3.111.3 Zero set of a positive superparabolic function : l. X V I 11.14, l. X VI 11.17
CLASSICS IN MATHEMATICS
Springer-Verlag began publishing books in higher mathematics in 1920, when the series Grundlehren der mathematischen Wissenschaften, initially conceived as a series of advanced textbooks, was founded by Richard Courant. A few years later, a new series Ergebnisse der Mathematik and ihrer Grenzgebiete, survey reports of recent mathematical research, was added. Of over 400 books published in these series, many have become recognized classics and remain standard references for their subject. Springer is reissuing a selected few of these highly successful books in a new, inexpensive softcover edition, to make them easily accessible to younger generations of students and researchers. Classical Potential Theory and Its Probabilistic Counterpart From the reviews: "This huge book written in several years by one of the few mathe-
maticians able to do it, appears as a precise and impressive study of this bothsided question that replaces, in a coherent way, without being encyclopaedic, a large library of books and papers scattered without a uniform language. Instead of summarizing the author gives his own way of exposition with original complements. This requires no preliminary knowledge....The purpose which the author explains in his introduction, i.e. a deep probabilistic interpretation of potential theory and a link between two great theories, appears fulfilled in a masterly manner". M. Brelot in Metrika, 1986
"In the early 1920's, Norbert Wiener wrote significant papers on the Dirichlet problem and on Brownian motion. Since then there has been enormous activity in potential theory and
stochastic processes, in which both subjects have reached a high degree of polish and their close relation has been discovered. Here is a monumental work by Doob, one of the masters, in which Part i develops the potential theory associated with Laplace's equation and the heat equation, and Part 2 develops those parts (martingales and Brownian motion) of stochastic process theory which are closely related to Part C. G. E. H. Reuter in Short Book Reviews, 1985
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