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0, W'o" is the Banach space formed of all continuous functions w: [0, T] -+ R" such that w(0) = 0 with the norm 11 wI(= max,,,., Iw(t)I. Let H = {h = (h*((t))G,E W;; each of h, is absolutely continuous with square integrable derivative}. H is a separable Hilbert space with inner product (hl, h2)H =
2 J: I;i(s)h:(s)ds
h, E H , i = I , 2?
where
If P is the Wiener measure on W;, then it is easy to see that { W;, H, P"} is an abstract Wiener space which we call the n-dimensional Wiener space. For 0 < oc < 1/2, set W;,a= {w E W ; ; \\wl\,, < m}, where
Then W;,, is a Banach space with the norm 11 . I), and it is easy to see that { W;,e,H , P w Iw0" } is also an abstract Wiener space. 1U
Example 3.2. (n-dimensional pinned Wiener space). For a fixed T > 0, W ; and H are as in Example 3.1 and consider their subspace @, = {w E W;; w(T) = 0) and ~, = H. Let PrTbe the ndimensional pinned Wiener measure on W;,T, that is, the Gaussian measure with the mean 0 and the covariance
n
1
w'(t)wj(s)PzT(dw)= &p(T - t ) ,
*0".T
0
<s 5 t 5 T ,
f i T , PET} is an abstract Wiener (cf. [5]). It is easy to see that space which we call the n-dimensional pinned Wiener space. From now on, we fix an abstract Wiener space { W, H, p} and discuss a differential calculus for p-measurable functions on W. Our first aim is to introduce a similar class of functions as D" in Section 2. Before proceeding further, however, we prepare several notions and notations. In the following, E, El, . . . are separable Hilbert spaces with norms and and ( ., * ) E , ( . , * ) E l , . .. inner products denoted by I . IE, I . IEl , ' A(El, E,), denoted also by El 0E,, is the Hilbert space of all linear mappings A : E, -+ E, such that
-
11
An Introduction to Malliavin's Calculus
for some (= any) ONB {e,} in El, which is endowed with the HilbertSchmidt inner product m
( 4 ,A P ) H S : =
C (Al(ed, i=l
Az(e,))E2
'
As in Section 2, for k = 0, 1, . . we define Ak(El,E,) successively by Ed = A(& Ak-SEi, Ed). If Ez = R, AdEi, Ez) -'!,(El, Ez) = Ez, is denoted simply by A,(E,). A,(E,) is also denoted by El 0E, 0 - . . 0El \
7-
/
k
and called the n-fold tensor product of El. Note that Al(El) = E f = El and the norms coincide under the identification. For x E El and y E E,, x 0y E El 0E, (= A(El, E,)) is defined by ( x 0y)[z] = (x, z),, .y, z E El. In the following, E Dc E, E! c El, . . . are some dense subspaces of Hilbert spaces E, El, . . . . A typical example is the case E = H and Eo= W*. It is convenient to introduce a dense subspace AO(E!,E;) of 44, Ed by
Ao(E;,E;) =
(.E A(El,Ez);
A is of the form A =
5e, 0fi
i=l
for some n, e, E Et and f i E E:, i = 1,2, . . ., Note that, in the case El = H and E; = W * , A
E
do(W*,E;) C A(H, E,) n
extends to a continuous linear mapping A : W 3 E: by A[x] = C e,(x). fi. i=l
For k = 0, 1, . . ., A",E;, E:) is defined successively by AXE!, E:) = E,O and Al(E!, Ei) = do(E&A;-](E;, E,O)). If Ei = R, we write simply .4i(E!, E;) = Ai(E;). If A E A:(E!, E:), then A is of the form of a finite
sum A
n
=
C
e, 0[el O h l ]with e, E E! andfi, E E ; and the trace of A
i,j=l
is an element in E: defined by
where { C k } is an ONB in the linear span of {el, e,, . . ., en}. We denote Lp(W 3 E, p) := L,(E), 1 p < 03, the usual L,-space of p-measurable functions f: W-+ E with
12
N. IKEDA AND S. WATANABE
L,(W-+ R,p) is denoted simply by Lp(W, p):= L,. Finally we shall introduce a convenient class of functions on W called polynomials on W. We set 9 = {F;F W + R which is expressed in the form F(w) = p(l,(w), Iz(w), . . ln(w)) for some It, I,, l,, - - I, E W* and a real polynomial p(tl, t2, - ., t,)} and B(Eo)= {F; F: W+ E o which is expressed in the form F(w) = C;=lF,(w)e, for some n, F, E B and e, E E" c E } . a ,
+,
Proposition 3.1. l I p < m .
B ( E o ) is a dense subspace of L,( W -+ E, p) for any
Let 8, be the set of E-valued polynomials on W of degree ) ~ ( w ) p ( d=w ) ~ ( w ) ~ ~ ( w ) p ( d w ) W
J
= -
W
( D F ( ~ )~, ~ ( w ) ) , p ( d w )
+
for every F E Di and G E Di such that l/p I/q 1, (see [78], [89] and [94]). Futhermore we note that there exists a stationary Gaussian diffusion on W with the semigroup {T,},,, given by (3.13) and the invariant measure p, called the Ornstein- Uhlenbeck process. As stated in Introduction, in [58] Malliavin defined the notion of derivatives for Wiener functionals in terms of Ornstein-Uhlenbeck processes over Wiener spaces, (also see [82]).
0 4.
The pull-back of Schwartz distributions and the regularity of image measures
Let { W, H , p} be an abstract Wiener space as in the previous section and F(w) = (F’(w),F2(w), . . ., Fd(w)) be a mapping W - t Rd. If F is “smooth” and “non-degenerate”, we can expect as in the finite dimensional case that F has such nice properties as ( i ) every smooth vector field X on Rd can be lifted to a “smooth” vector field W +. H such that D F [ f w ] = X F ( w ) , (ii) the image measure F*(p) on Rd of p has a smooth density. We shall study these properties of the mapping F by defining the pullback or lifting of Schwartz distributions on Rd as elements in D-” of the previous section. This approach has some advantage: If we can give a correct mathematical sense to 6,(F), the pull-back of the Dirac &function 6, at x E Rd by the mapping F, then we can justify a formal expression
s:
18
N.IKEDAAND S. WATANABE
like EP[G,(P)Gl for the density Ep[G(w): P(w) E dx]/dx, often used intuitively, and the smoothness of the density is a by-product of this justification. Here EP[ -1 denotes the expectation with respect to the probability p. Furthermore, in this framework, the various properties of Schwartz distributions can be lifted to those of generalized Wiener functionals. First, the above smoothness of F: W + Rd is defined in the sense of Sobolev spaces: We assume that (4.1)
F E D"(R')), equivalently, Ft E D", i = 1,2, .
a ,
d.
Then a(w) = (oij(w)) is defined by (4.2)
= (DFi(w), DF'(W))H, i,.f = 1, 2,
8 ( W )
* * *,
d
and u'f E D". The matrix o is called the Malliavin covariance of F. As in Section 2, we consider the following non-degeneracy assumption (A. 2), for F: We say that I; satisfies (A. 2), ( p > 1) if a(w) is strictly positive definite for almost all w ( p ) (this assumption is refered to as (A. 1)) and if
co
det (rtj(w)) E LJW,
(4.3)
where a-'(w) = (j' 1. Using the identity d
DTfj = - C Y;J;ZDO:',
(4.4)
where
k,l=l
+
= uL1 edLzand (?;')
Proposition 4.1. 1. fiedforp > k
+
Ti,
E
E
> 0,
= (utl)-', it is easy to verify the following:
09 for all 1 I r
O
9"(Rd):=the Schwartz space of tempered distributions on Rd.
Then we have the following (see [93] and [94]):
Theorem 4.2. Let k be a positive integer. If F: W--t Rd satisfies (A. 2), for some p > 4k and if we take q > 1 satisfying (4.12)
l/q
+ 4k/P < 1
Y
then the mapping Y ( R d )3 4 -+ q5 0 F E D" is continuous with respect to the norm 11. ll-2k on Y ( R d )and the norm 11 .IIpn, - 2 L on D" where
(4.13)
I/Po
+ l/q
=
1.
More precisely, we have
i
(4.14) where
~ / ~ ° F ~ ~ P n , - Z kK k , q l I $ I I - Z k ~
Kk,*
is given by (4.10).
Proof. We have
$ EY(Rd),
An Introduction to Malliavin’s Calculus
21
and by (4.9)
I lI(1 =
+ 1x12 - ~)-k$lI~IIi71~(~~8>111
ll$ll-2k
ll~2k(+v;g)ll*.
This, combined with (4.10), concludes the proof. Corollary 4.1. 4
Under the same assumption as Theorem 4.2, Y ( R d )3 9
9 F E D” extends to a unique continuous linear mapping qwzk 3 T 0
---f
T ( F ) E D;:k.
Since we can choose any q such that (4.12) is satisfied, we can choose any p , such that
1 1. In particular, Y ’ ( R d )3 T -+ T ( F ) E D-” is dejned and
This T ( F ) E D-” is called the composite of the Schwartz distribution T and the Wiener functional F or the pull-back of the distribution T on Rd under the mapping F: W-+ Rd. T ( F ) coincides with T o F if T is given by a bounded continuous function on Rd. Now the above result can be applied to the existence of smooth First, we note the following facts easily verified by density of F&). Fourier analysis: If 6, E Y ’ ( R d )is the Dirac &function at y, then 6, E %?-2, if and only if m > d/2. Furthermore, Rd E y --+ 6, E V-,,-, is 2ktimes continuously differentiable for k = 0, 1,2, . . .,where (4.15)
WZ =,
[d/2]
+ 1.
Therefore, we can conclude the following (see Watanabe [93] and [94]): Theorem 4.3. Let k = 0, 1,2, . . . and suppose that Fsatisjes (A.2), for p > 4(m, k). Then for every 1 < po < p/4(mo k), 6,(F) E D;tmo
+
+
22
N.IKEDA AND S. WATANABE
for every y E Rd and Rd 3 y +-6,(F) t D;:mo-2kis 2k-times continuously diferentialbe. In particular, y 4(6,(F), g ) E R is 2k-times continuously diferentiable for any g t DiY+2k,l/p, 114, = 1, where (., *) is the x D:Y+2k. canonical bilinear form on D;a2mo-2k
+
--
Corollary 4.3. IfFsatisjies (A. 2)-, then fbr any k = 0, 1,2, . and po > 1, Rd 3 y -+ 6,(F) E D;tmo-2kis 2k-times continuously direrentiable and hence Rd 3 y -+ (S,(F), g ) is 2k-times continuously direrentiable for every
In particular, it is infinitely differentiable i f m
In Theorem 4.3, if g E DiY then y -+ (6,(F), g } is continuous and it is easy to show by the Riemann sum approximation that
J
(4.16)
Rd
f(y)(G,(F), g)dY = ( f o F, g> = J w f o F(w)g(w),u(dw)
for every continuous f: Rd -+ R with a compact support. This shows that (6,(F), g) coincides with the density F,(,ug)(dx)/dx of the measure F*(,ug)(A) =
[ g(w)l,{F(w))p(dw), W
for Bore1 A c Rd ,
with respect to the Lebesgue measure dx on Rd. In particular (6,(F), 1) = F,(,u)(dx)/dx where F&) is the image measure of ,u by the mapping F and
( w 9 9
g> = Er(g I F
= Y)1)
by using the well-known notion of conditional expectation. Thus we have
Corollary 4.4. Let k = 0, 1, 2, . . . and F satisfy (A. 2), for p > 4(m0 k). Then Y = F*(,u) has a C2*-density Y(X) with respect to the Lebesgue measure dx and Er[g I F = X ] Y ( X ) has a C2k-versionif g E DiY+” with l/q, 4(m0 k)/p < 1. In particular, i f F satisjies (A.2)m, then v = F,@) has a C”-density Y(X) with respect to the Lebesgue measure h and for any
+
+
+
23
A n Introdirction to Malliavin's Calcidirs
Ep(g IF = X)Y(X) has a CZk-versionwith respect to the Lebesgue measure dx for every k = 0, 1 , 2 , - ... Example 4.1. Let { W ; , H, P"} be the d-dimensional Wiener space (Example 3.1) and for fixed t > 0 and x E Rd, let F: W -+ Rd be defined by F(w) = x w ( t ) . Then F E D"(Rd)with utf(w) = &,t. Hence Y,, = &,/t E L, for all p 2 1 and therefore (A. 2)- is satisfied. Let k(x) E P ( R d )such that I k(x) I 5 K ( l [ x I) for some positive constant K and derivatives are all polynomial growth order. Then it is easy to verify that
+
+
g(w) = exp
for all p have JRd
> 1.
Hence y
-+
[Jl
1
+ w(s))ds
k(x
E D : ~
(6,(F), g ( w ) ) is C Z kif k = m - m, > 0. We
0 and x E Rd be fixed. Then ( i ) X i = F ( t , x, w) E Dm,i = 1,2, . d, (ii) The Malliavin covariance ad' = ( D X ; , O X ; ) , is given by
-
(5.4)
a'? =
I ,
i , j = 1, 2, . . ., d. 2 s' (Y,Y,'Vo(X,))i(Y,Y;'Vo(Xs))'ds, 0
n=l
Here (Ax)' = C;=lA:xj $ A = (A:.) is a d X d-matrix and x = (x') E Rd.
A rough idea of the proof is as follows: Since
a DX:[h] = -Xi(t, x, w a&
+~ h ) ,
h E H,
IL0
we obtain from (5.2) DXf[h]= (5.5)
2 5 s' 3, I/,i(X,)DX:[h] dw"(s) + 2 f V:(X,)k(s)ds + 5 s' 3,V6(Xs)DX:[h]ds, 0
o=l k=l
0
a=l
0
k=l
0
where h = (hn(s)). By the method of variation of constants, we obtain
DXI[h] =
(5.6)
2 s' [r,Y;' V,(X,)]"(S)dS,
o=l
0
or, if one likes, it is easy to verify by the It8 formula that the right-hand side of (5.6) actually satisfies (5.5). Similarly, we obtain the equations for DLXt[h,,h,, .,h,] successively and we see easily D k X E Lp(W;+ Ak(H),P") for all p 2 1. These formal arguments can be made rigorous by the standard Cauchy polygonal approximations, cf. [34] for details. For a vector field V(x) on Rd identified with the Rd-valued function V ( x ) = (V'(x)), define a function&: RdX GL(d, R)+Rd by
--
(5.7)
f T ( r ) = Y - I V ( x ),
r = (x, Y ) E Rd x GL(& R) ,
(see [34]). Now (5.4) can be written also as
Set (5.9)
6'' =
2 rfGa(rs).fJa(r,)ds, o=l
i , j = 1,2, . . .,d .
0
Our next aim is to study the conditions for "non-degeneracy" of the Malliavin covariance a = (a"). Noting the well-known fact (cf. [34])that
30
N.IKEDA AND S. WATANABE
E(ll Y#) < co and E(lI Y;'Il*) < co for allp 2 1, we can replace u = (at') by 2 = (P)in this study, that is, (A. 1) is satisfied if and only if det 2 > 0 a.e. (P") and further (A. 2)m is satisfied if and only if
(5.10)
E((det 6)-")
6). Lemma 5.6. (cf. [34], Lemma V-8.6). Brownian motion on s E [O, 61. Then (5.17)
P"[V,o,al(B)< el
Let B ( s ) be a one-dimensional
I 2 exp [-6/(2'~')1
for every positive E and 6. Lemma 5.7. Let B ( s ) be a one-dimensional Brownian motion on [0,a] where a is a positive constant. Then for every 0 < Y < 112, there exist positive constants a, and a, such that
This is a consequence of well-known Fernique's theorem (cf. Kuo [47], Theorem 111-3.1). In Example 3.1, we remarked that for 0 < 'i < 1/2, {Wi,r,H, P"} is an abstract Wiener space and Fernique's theorem
35
A n Introduction to Malliavin's Calculus
implies that E[exp [ E l [ wli:]]< 03 for some E > 0. It is easy to conclude (5.18) from this. For details, see [86]and [94]. Let ~ ( s )be an It6 process with bounded characteristics vo(s). As is well-known, there exists a one-dimensional Brownian motion B ( t ) such that
where A ( s ) = C J Tn(u)2du. n=l
0
Hence, by Lemma 5.7, we have the following
Lemma 5.8. Let ~ ( s )be an Ztd process with bounded characteristics. Then for any 0 < Y < 1/2, ci,i = 0, 1, 2 exist such that (5.20)
Pw[ sup O<s ~ * 2 - ~( ba ) - ] 2 48-'2(b
(S
- tl),ds
- a)-'lZ13
which completes the proof. Proof of the key lemma. Suppose t = 1 for simplicity. In proving (5.15), we may clearly assume n 2 2. In the following a,, i = 1,2, . . . and d,, i = 1,2, are positive constants independent of n. First we note that a one-dimensional Brownian motion B ( t ) with B(0) = 0 exists such that
-
where
Set
and
Here c, is given and c, will be determined later.
Set
and Wl,z= [A(&) 2 l/nal].
Then (5.23)
Wl C Wl,l U W1.:
if a, 2 c,
+ 1,
An Introduction to Malliavin's Calculus
since then l/nal 5
l/nC4+l
l/(2nC4)for n 2 2. Set
w,= [& - & = l/n] w,= [ sup -Ieo(u) - €o@)I opz3 >
We say that x E Rd satisfies the assumption (A. 3) i f there exist A4 0 and A , , A,, . . ., A , E ,f" such that A,(x), A,(x), . A d ( x ) are linearly Clearly x satisfies (A. 3) if and only if there exists A4 0 independent. such that
-
a ,
(5.38) If (5.38) is satisfied, then we can find > 0 and bounded neighbourhoods U ( x ) in Rd and U(I) in GL(d, R ) of x and I respectively such that (5.39)
C
inf 2ESd-I
(f.fA)'(r)
2 E,
if r
E
U ( x )x U ( I ) .
ACfx
Set u = inf {s;re 4 U(x) x U(1))
where re = (Xe, YJ is the solution of (5.2) and (5.3) put together with ro = (x,I ) . Then, by Lemma 5.5, for some c, > 0, P"[u
Set a: = 0 and (5.40)
C$
< t/n]
exp [-c,n],
= u A (t/n),n = 1,2,
n = 1, 2, . . . .
. -. Then
41
A n Introduction to Malliavin’s Calculus
and P”[W;] I exp [ -cln]. Let # (JM)= N, say. Then for every 1 E Sd-’ we can find ao,al, ., a,,0 I k I M such that 1 I a. I r, 0 I a‘,a,, ...,aI,< rand
-
(5.41) where
Noting (5.35), we can apply the key lemma successively to conclude the following: for each j = 0, 1, . ., k, we can find positive constants c{, i = 1 , 2 , 3 , 4 , independent of n and Z, such that
-
In particular, we can conclude that sup P w [ k Z€Sd-l
a=l
J’[l~fva(rs)]2ds < l/ncl] I c, exp [-cgnc4] 0:
for some ci,i = 1,2, 3 , 4 . By Lemma 5.3, we can conclude that (5.42)
E[(det & ) - P I
0, F: W:
Let 9” = {x E R d ;x does not satisfy (A. 3)). By a slight modification of the proof, we can show that the conclusion of Theorem 5.2 still holds if x E 3’ but the set 3 is “thin” at x in a certain probabilistic sense, (cf. Malliavin [59] and [34]). For a further study on this line, especially applications to hypoellipticity and global hypo-ellipticity of the operator
c va, + vo
1 ‘ A =2 a=l
and to an ergodic theorem of A-diffusion, cf. Kusuoka and Stroock [56]. For some examples to which Malliavin’s method can be applied, see Stroock [82]. [34] and [86]. In [63], Malliavin also discussed an application to estimate the resolvent.
N. IKEDAAND S. WATANABE
42
9 6.
Several topics related to the Malliavin calculus
(A) Stochastic oscillatory integrals In Example 5.1, LCvy's stochastic area was introduced: It is a stochastic line integral (cf. [33]) S ( t , x, w) = with respect to the 2-dimensional Wiener process X,(t) starting at x and a is a differential 1-form on R2given by (Y
It is easy to see that for R
=
E
1 -(x'~x'
2
-
x'dx').
(0, co)
and by the Feynman-Kac formula and a formula on Hermite polynomials, we obtain
m x , y ; 4=
Rt 2 sinh 2
(6.3)
(cf. 1211 and [34]). Then there exists a positive a(t; 2) =
and hence
TRt ~
Rt sinh 2
to such
for t 2
to,
that
An Introduction to Malliavin’s Calculus
1
- lim - log a(t; 1) =
(6.4)
cr-
t
R
-
2
43
> 0.
In [62], Malliavin used the stochastic calculus of variation to study similar asymptotic properties of stochastic oscillatory integrals. Let, for a fixed t > 0, { 6 ‘& f i t , pTt} be the d-dimensional pinned Wiener space (Example 3.2) and set X ( s ) = X ( s , x, y , w) = x
+ ( y - x)s/t + w ( s ) ,
s E [O, t l .
Let
c d
cr =
crY,(X)dXZ
i=1
be a smooth I-form on Rd where q ( x ) are smooth with bounded derivatives of all orders 2 1. We set
and
K,(x, y ; R, a) = E[exp
(6.6)
{mW t , w ;x , Y , or>}]
where E [ . ] denotes the expectation with respect to the probability Then it is easy t o see that ( i ) for x , y E R d a n d O I u , s I t ,
(6.7)
~d
K J x , z ; 2, cr)K,(z, y ; 1, a)(&)d’2exp
[--Id2 2s
We now assume that
Malliavin’s result is as follows: Take a 1e (0, m) and set
Prt.
44
N. IKEDAAND S. WATANABE
Then there exists a positive constant a(,?)such that - lim
(6.9)
tt-
1 log a(t; I , a) = a(I) t
He used the integration by parts on the Wiener space to obtain necessary estimates. In fact, by using (4.6) and (4.7), we can rewrite K,(x, y ; 2, or) in the following form : m x ,y ;I,@)
where K is a positive constant independent o f t , x,y and I . Hence the proof of (6.9) can be reduced to estimate the right hand side of (6.10). For details, see Malliavin [62]. Asymptotic results of this type are related to some problems in the theory of quantum physics and stochastic holonomy in the representation theory of semisimple Lie groups, cf Gaveau [22], Gaveau and Vauthier [27], Malliavin [%I, [60] and [61], M. P. Malliavin and P. Malliavin [67]. (B) Capacities In his paper [64] in these proceedings, Malliavin introduced a family of capacities Cp3,,1 < p < 03, r > 0 on the Wiener space and discussed applications of this notion: Using the same notations as in Section 3, let, for an open set 0 c W, Cp,,(8)= inf{llullp,r;u E D;, u 2 0 and u 2 1 on O } ’
and call it C,,,-capaciry of 8. For any A c W, its (outer) capacity is defined as usual by
crJm=
inf
A C B : open
CP,,(O)
-
A n Introduction to Malliavin’s Calculus
45
A set A C W is called slim if C,,,(A) = 0 for all 1 < p < co and r > 0. C,,,-capacity (corresponding to the equivalent norm ~ ~= IIu11, u ~ /I IDu Ix11J coincides with that in Fukushima’s sense of the Dirichlet space associated with the Ornstein-Uhlenbeck process on W ([19]) and a set of C,,,-capacity 0 coincides with a polar set of the Ornstein-Uhlenbeck process on w,(cf. [20]). A slight different but equivalent and more manageable notion of C,,,-capacities was introduced by Takeda [90]. The notion of C,,,-capacities has been used to obtain an implicit function theorem on W by Malliavin. For various applications of this notion, see [64] and [65]. Also see Kusuoka [51] for some topics related to the notion of C,,,-capacity on W. Recently many attempts have been made to measure by the capacity a set of p-measure 0: In the case of the ordinary Wiener space, some classical theorems like the law of iterated logarithm or LCvy’s Holder continuity are shown to hold not only almost everywhere but also everywhere except a set of C,,.-capacity 0 for all co >p > 1 and r > 0, while some typical subsets of W with the Wiener measure 0, related to the recurrence of paths or multiple points of paths are shown to have positive C,,,-capacity for some p > 1 and r > 0. For details, see [69], [20], [40], [41], [42], [90] and [go].
+
(C) Applications to filtering theory and the partial Malliavin calculus The Malliavin calculus has been applied to obtain regularity results in filtering theory by, e.g., Michel [72], Bismut-Michel [14], Kunita [44] and Kusuoka and Stroock [57]. The problem is to show the regularity of certain conditional distributions and therefore, in the study of laws of Wiener functionals some components are supposed to be fixed. Such a kind of calculus is formulated and called the partial Malliavin calculus in [57]. Also see Michel’s paper [73] in these proceedings. (D) Applications to infinite dimensional diJEusions A theorem in Stroock [82] enables us to deal with the finite dimensional marginal distributions of some infinite dimensional interacting diffusion. For related topics, see also Holly and Stroock [31].
(E) Bismut’s approach to the Malliavin calculus Bismut [6] obtained the following formula of integration by patrs. Under the same situation as in Section 5, let g($) be a real, bounded, FrCchet differentiable function on the Banach space C([O, TI -+ Rd) with the supremum norm and its FrCchet derivative
~
~
,
46
N. IKEDAAND S. WATANABE
be given by
with signed measures dgi(. ; $) on [0, TI. Let W ; 3 w + X ( w ) E C([O, TI + R d ) be defined by the solution X,(w) = X ( t , x,w) to SDE (5.2). Let u = (ul(t),uz(t), . . ., u , ( t ) ) be a bounded and {g,}-adapted process on W;. Then
where E [ - ]denotes the expectation with respect to Pw. We shall comment this Bismut formula in our context: Since
by (5.6), the right hand side of (6.11) coincides with E [ ( D ( g ( X ) ) ,u),] where u, identified with
E
H , is an element in D:(H). Thus
(6.1 1) amounts to (6.12)
-c
6u=
This is seen most simply by the Girsanov theorem: For any F E D:,
c
E [ F ( w + E L I ) ] = E exp
C 1: K1 E:
u,(s)dw"(s) -
-E'
2
'St
a-1
u,(s)'ds
I I F(w)
and by differentiating in E , we have
proving (6.12). The formula (6.12) suggests a generalization of stochastic integrals t o a class of anticipative integrands, cf. Skorohod 1811, Shigekawa [77], Gaveau and Trauber [26], Ramer [75] and Taniguchi [92]. In a dual argument, we see also that for any F E Di, its predictable representation
A n Zntroduction to Malliavin's Calculus
47
is given by
and this is closely related to the results of Haussmann [30]. The method based on the Girsanov theorem can be applied to more general cases than Wiener measures, cf. Bismut [l I] and [I21 and Bichteler and Jacod [4]. Also see Bismut's paper [13] in these proceedings.
(F) The Malliavin calculus and boundary value problems Bismut [I21 discussed the regularity of boundary semigroups by obtaining a very elaborate formula of integration by parts. Cf. also his paper ([13]) in these proceedings. Arous, Kusuoka and Stroock [l] discussed the regularity of harmonic measures for operators of the Hormander type as an application of the partial Malliavin calculus. (G) de Rham complex of Wiener functionals The following results are due to Shigekawa [79]. Using the same notations as in Section 3, let the alternation A : A,(H) + A , ( H ) be defined as usual by
and let d ; ( H ) = { T Ed,(H); A(T) = T}. In particular, &( H) = R and & ( H ) = H. Then the Sobolev space D;(d;(H)) are defined and we set
Define
d,: A"(T*(W)) by setting d, = A o D and = (n 1)s. Then
+
a,+,
0 -+ R
2:
o
A" +' (T* (W))
d"+'(T*(W)) --+ d"(T*(W)) by s.tting d, = 0, 6,o 8,+, = 0 and both do
i
{constant functionals} +k'(T*( W ) )-+ A1(T*(W ) ) da 5 AZ(T*(W))+ .
*
.
48
N.
0cR
IKEDA AND
li"AO(T*(W ) )
s. WATANABE
A*(T*(W ) )
are exact. Here i is the injection and i*(F(w)) =
A,; A"(T*(W))
-+
-
A2(T*(W ) )
83
F(w)p(dw).
... Let
A"(T*(W)), be defined by
Then A , = L - n I where L is the Ornstein-Uhlenbeck operator of Section 3. From these results, we obtain the following de Rham-Kodaira decomposition: A"(T*(W))= 8% 0CJ~+~(A"+'(T*(W))) 0&-I(A"-'(T*(W)))
where 2'"= { F E A"(T*(W));A,F = 0) and it holds that Zn = {constant functionals} if n = 0 and 2, = (0) if n 2 1.
(H) The law of a system of multiple Wiener intergrals In case of a one-dimensional Wiener space, it is well-known that the Wiener-It8 decomposition is realized by the multiple Wiener integrals (Ita [38]). We consider an Rd-valued Wiener functional F = (F', F2, . -,F d ) has a finite order expansion by multiple Wiener integrals. By where using the Malliavin calculus, Shigekawa [78] obtained a sufficient condition for the absolute continuity of the law of F. Kusuoka [52] gave a necessary and sufficient condition by using an algebraic method. In concluding this note, we would refer the reader to the following interesting and important expository papers on the Mallianvin calculus : Williams [95], Bismut [9], Kusuoka [54], Malliavin [65] and Stroock [87]: The last three papers will give more complete and up-to-date information on the topics in the Malliavin calculus and its applications.
References G . B. Arous, S. Kusuoka and D. W. Stroock, The Poisson kernel for certain degenerate elliptic operators, preprint. [ 2 ] P. Baxendale, Stochastic flows and Malliavin calculus, to appear in Proceedings of 20th IEEE conference on Decision and Contl., 1981, 127152. [ 3 1 Ya. I. Belopol'skaya and Yu. L. Daleckii, It6 equations and differential geometry, Russian Math. Surveys, 37 (1982), 109-163. 141 K. Bichteler and J. Jacod, Calcul de Malliavin pour les diffusions avec sauts: Existence d'une densitt dans le cas unidimensionnel, Lecture [ 1]
An Zntroduction to Malliavin’s Calculus
49
Notes in Math, 986 (1983), 132-157, SCminaire de Probabilitts XVII, ed. par J. Azima et M. Yor, Springer-Verlag, Berlin. 1 5 1 P. Billingsley, Convergence of Probability measures, John Wiley and Sons, New York, 1968. 161 J. M. Bismut, Martingale, the Malliavin calculus and Hypoellipticity under general Hormander’s conditions, Z. Wahr. verw. Geb., 56 (19811, 469505.
171 181 191
-, Martingales,
the Malliavin calculus and Hormander’s theorem, in “Stochastic Integrals” ed. by D. Williams, Lect. Notes in Math., 851 (1981), 85-109, Springer-Verlag, Berlin. -, MBcanique Altatoire, Lect. Notes in Math., 866 (1981), SpringerBerlag, Berlin. , An introduction to the stochastic calculus of variations, in Stochastic Differential Systems, ed. by M. Kohlmann and N. Christopeit, Lect. Notes in Control and Inform. Sci., 43 (1982), 33-72, Springer-Verlag, Berlin. -, MCcnique albatoire, Ecole dCtB de Probabilites de Saint Flour, Lect. Notes in Math., 929 (1982), 1-100, Springer-Verlag, Berlin. -, Calculus des variations stochastique et processus de sauts, Z. Wahr. verw. Geb., 63 (1983), 147-235. -, The calculus of boundary processes, to appear in Ann. Ec. Norm. sup.. -, Jump processes and boundary processzs, in these Proceedings, 63-104. J. M. Bismut and D. Michel, Diffusions conditionnelles, I, 11, J. Fuc. Anal. 44 (1981), 174-211, 44 (1982), 274-292. R. H. Cameron and W. T. Martin, Transformation of Wiener integrals under translations, Ann. Math., 45 (1944), 386-396. Ju. L. Daleckii and S. N. Paramonova, Integration by parts with respect to measures in a functions space, I, 11, Theor. Prob. Math. Statist. 17 (1979), 55-56, 18 (1979), 39-46. R. M. Dudley, J. Feldman and L. Le Cam. On seminorms and probabilities and abstract Wiener spaces, Ann. Math., 93 (1971), 3 9 0 4 0 8 . K. D. Elworthy, Stochastic differential equations on manifolds, London Math. SOC.,Lecture Notes Series 70, Cambridge Univ. Press, Cambridge, 1982. M. Fukushima, Dirichlet forms and Markov processes, Kodansha/NorthHolland, Tokyo/Amsterdam, 1980. , Basic properties of Brownian motion and a capacity 03 the Wiener space, J. Math. SOC.,Japan, 36 (1984), 161-176. B. Gaveau, Principle de moindre action, propagation de la chaleur et estim k s sous elliptiques sur certain group nilpotents, Acta Math., 136 (1977), 95-153. -, Systtmes dynamiques associes a certain opkrateur hypoelliptiques, Bull. SOC.Math., 102 (1978), 203-229. -, An example of a stochastic quantum process: interaction of a quantum particle with a boson field, in these Proceedings, 135-147. B. Gaveau and J. Moulinier, GBomCtrie differentielle stochastique et inttgrales stochastiques non anticipantes de bruits blancs B plusieurs paramkires, to appear in C. R. Acad. Sci. Paris. -, IntCgrals oscillantes stochastiques: Estimation asymptotique de fonctionnelles caracteristiques, J. Func. Anal., 54 (1983), 161-176. B. Gaveau and P. Trauber, L‘intkgrale stochastique comme optrateur de divergence dans l’espace functionnel, J. Func. Anal., 46 (1982), 230-238. B. Gaveau and J. Vauthier, Intkgrales oscillantes stochastiques I’iquation de Pauli, J. Func. Anal., 44 (1981), 388-400. I. M. Gelfand and G. E. Shilov, Generalized functions, Academic Press, New York, 1964, Vol. 2. L. Gross, Abstract Wiener spaces, Proc. Fifth Berkeley Symp. Math. Statist.
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N. IKEDAASD S. WATANABE
Prob. 11, Part 1, 31-41, Univ. Calif. Press, Berkeley, 1965. U. Haussmann, On the integral representation of functionals of It8 processes, Stochastics, 3 (1979), 17-27. R. Holley and D. W. Stroock, Diffusions on an infinite dimensional torus, J. Func. Anal., 42 (1981), 29-63. L. Hormander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. N. Ikeda and S. Manabe, Integral of differential forms along the path of diffusion processes, Publ. R.I.M.S. Kyoto Univ., 15 (1979), 827-852. N. Ikeda and S . Watanabe, Stochastic differential equations and diffusion processes, Kodansha/North-Holland, Tokyo/Amsterdam, 1981. -, Stochastic flows of diffeomorphisrns, to appear in Adv. in Probability, Vol. 7, ed. by M. Pinsky, Marcel Dekker, Inc. New York, 1984. K. It6, Differential equations determining Markov processes, Zenkoku Shij6 Siigaku Danwakai, 244 (1942), No. 1077, 1352-1400, (in Japanese). -, On stochastic differential equations, Mem. Amer. Math. SOC.,4 (1951). -, Multiple Wiener integral, J . Math. SOC.Japan, 3 (1951), 157-169. A. N. Kolmogoroff, Zufallige Bewegungen, Ann. Math., 11, 35 (1934), 116117. T. Komatsu and K. Takashima, Hausdorff dimension of quasi all Brownian path in the d-dimensional Euclidean space, to appear in Osaka J . Math., 21 (1984). N. Kono, Proprittts quasi-partout de fonctions altatoires Gaussiennes, ExposCs du 29 Avril au 20 Mai 1983, Stminaire d’Analyse des Fonctions Altatoires, preprint. N. Kono, 4-dimensional Brownian motion is recurrent with positive capacity, preprint. H. Kumano-go, Pseudo-Differential Operators, The MIT Press, Cambridge, 1982. H. Kunita, Densities of a measure-valued process governed by a stochastic partial differential equation, Sys. Cont. letters, 1 (1981 ), 100-104. -, Stochastic partial differential equations connected with non-linear filtering, Proc. C.I.M.E. Session on “non-linear filtering and stochastic control”, ed. by K. Mitter and A. Moro, L x t . Notes in Math., 972 (1982), 100-168, Springer-Verlag, Berlin. H. H. Kuo, Integration by parts for abstract Wiener measures, Duke Math. J., 41 (1974), 373-379. -, Gaussian measures in Banach spaces, Lect. Notes in Math., 463, Springer-Verlag, Berlin, 1975. --, Uhlenbeck-Ornstein process on a Riemann-Wiener manifold, Proc. Intern. Symp. SDE, Kyoto, 1976, ed. by It6, 187-193, Kinokuniya, Tokyo, 1978. -, Donsker’s delta function as a generalized Brownian functional and its application, Theory and application of random fields, Proc. IFIP-WG 7/1 Working conf. at Bangalore, ed. by G. Kallianpur, Lect. Notes in Cont. and Inform. Sci., 49 (1982), 167-178, Springer-Verlag, Berlin. S . Kusuoka, Dirichlet forms and diffusion processes on Banach spaces, J. Fact. Sci. Univ. Tokyo, 29 (1982), 79-95. -, Analytic functionals of Wiener process and absolute continuity, in Functional analysis in Markov process ed. by M. Fukushima, Lect. Notes in Math., 923 (1982), 1-46, Springer-Verlag, Berlin. -, On the absolute continuity of the law of a system of multiple Wiener integral, J. Fact. Sci. Univ. Tokyo, 30 (1983), 191-197. -, The Malliavin calculus and hypoellipticity of second order degenerate elliptic differential operators, Probab. theory and Math. Statist. Proc. Fourth USSR-Japan Symp., ed. by K. It8 and J. V. Prohorov, Lect. Notes
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51
in Math., 1021 (1983), 365-371, Springer-Verlag, Berlin. The Malliavin calculus and its applications, to appear in Sfigaku, 1984, (in Japanese). S. Kusuoka and D. W. Stroock, Applications of the Malliavin calculus, Part I, in theses Proceedings, 271-306. -, Apllications of the Malliavin calculus, Part I1 and 111, in preparation. -, The partial Malliavin calculus and its application to non-linear filtering, to appear in Stochastic Processes. P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proc. Intern. Symp. SDE, Kyoto, 1976, ed. by K. ItB, Kinokuniya, 1978. -, Ck-hypo-llipticity with degeneracy, Stochastic Analysis, ed. by A. Friedman and M. Pinsky, 199-214, 327-340, Academic Press, New York,
-,
1978.
-,
GBometrie diffkrentielle stochastique, Les Presses de I’UniversitC de MontrBal, Montrkal, 1978. -, Stochastic Jacobi fields, Partial differential equations and geometry, Proceedings of the Park City Conference, 203-235, Lect. Notes in pure and applied Math., 48, Marcel Dekker, Inc., New York, 1979. -, Sur certaines inttgrales stochastiques oscillantes, C. R. Acad. Sci. Paris, 295 ( 1982), 295-300.
-,
Calcul des variations stochastiques subordonnt au procesm d: la chaleur, C. R. Acad. Sci. Paris, 295 (1982), 167-172. -, Implicit functions in finite corank on the Wiener space, ia thes- Proceedings, 369-386. -, Analyse diffkrentielle sur l’espace de Wiener, Proc. ICM, 1983. -, Diffusion on the loops, to appear. M. P. Malliavin and P. Malliavin, Factorisations et lois limites de la diffusion horizontale au dessus d’un espace riernannien symetrique, Lect. Notes in Math., 404 (1974), 166-217, Springer-Verlag, Berlin. P. A. Meyer, Demonstration probabilite de certaines inkgalitis de Littlewood-Paley, SCminaire de Prob. X, ed. par P. A. Meyer, Lect. Notes in Math., 511 (1976), 125-141, Springer-Verlag, Berlin. -, Notes sur les processus d’ornstein-Uhlenbeck, Siminaire de Prob., XVI, ed. par J. Az6ma et M. Yor, Lect. Notes in Math., 920 (1982), 95133, Springer-Verlag, Berlin. -, Quelques resultats analytiques sur le semigroupe d’ornstein-Uhlenbeck en dimension infinie, Theory and application of random fields, Proc. IFIP-WG 7 / 1 Working conf. at Bangalore, ed. by G. Kallianpur. Lect. Notes in Cont. and Inform. Sci., 49 (1983), 201-214, Springer-Verlag, Berlin. A hand-written manuscript of Meyer distributed in the seminars at r711 -, Paris and Kyoto. [721 D. Michel, Regularit6 de lois conditionnelles en thkorie du filtrage nonlin6aire et calcul des variations stochastique, J. Func. Anal., 41 (1981),
-,
8-36.
Conditional laws a i d Hormander’s condiiion, in t h x e Procezding, 387408.
M. A. Piech, A fundamental solution of the parabolic equation on Hilbert spaces, J. Func. Anal., 3 (1969), 85-1 14. R. Ramer, On nonlinear transformations of Gaussian mearures, J. Func. Anal., 15 (1974), 166-187. I. Shigekawa, Absolute continuity of probability laws of Wiener functionals, Proc. Japan Acad., 54 (1978), 230-233. -, Analysis on abstract Wiener spaces, Master t h s i s of Kyoto Univ., 1979, (in Japanese). -, Derivatives of Wiener functionals and absolute continuity of induced
52
N. IKEDA AND S. WATANABE measure, J. Math. Kyoto Univ., 20 (1980), 263-289. de Rham-Hodge-Kodaira’s decomposition on abstract Wiener spaces and its application to infinite dimensional diffusion processes, preprint. -, On a quasi everywhere existence of the local time of the 1-dimensional Brownian motion, to appear in Osaka J. Math. V. Skorohod, On a generalization of a stochastic integral, Theor. Prob. Appl. 20 (1975), 219-233. D. W. Stroock. The Malliavin calculus and its auulications to second order parabolic differential operators, I, 11, Math. System Theory, 14 (19811, 25-65, 141-171. -, The Malliavin calculus, a functional analytic approach, J. Func. Anal., -U - ( 1 9 8 1 ) , 217-257. The Malliavin calculus and its auulications. in “Stochastic Integrals” ed: by D. Williams, Lect. Notes in Math., 851 (1981), 394-432, SpringerVerlag, Berlin. -, Lectures on topics in stochastic differential equations, noted by S. Karmaker, Tata Inst. Fund. Res., 1982. -, Some applications of stochastic calculus to partial differential equations, Ecole d’6t6 de ProbabilitCs de Saint Flour, ed. par P. L. Hennequin, Lect. Notes in Math., 976 (1983), 268-382, Springer-Verlag, Berlin. -, Stochastic analysis and regularity propxties of certain partial differential operators, Proc. ICM, 1983. D. W. Stroock and S . R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proc. Sixth Berkeley Symp. Math. Statist. Prob., 111, 333-359, Univ. Calif. Press, Berkeley, 1972. H. Sugita, Sobolev spaces of Wiener functionals and Malliavin’s calculus, to appear in J. Math. Kyoto Univ. M. Takeda, (r, p)-capacity on the Wiener space and properties of Brownian motion, preprint. S. Taniguchi, Malliavin’s stochastic calculus of variations for manifoldvalued Wiener functionals and is applications, Z. Wahr. verw. Geb., 65 (1983), 260-290. -, The adjoint operator of weak derivative as It6 integrals, preprint. S . Watanabe, Malliavin’s calculus in terms of generalized Wiener functionals, Theory and application of random fields, Proc. IFIP-WG 7/ 1 Working conf. at Bangalore, ed. by G. Kallianpur, Lect. Notes in Cont. and Inform. Sci., 49 (1983), 284-290, Springer-Verlag, Berlin. -, Lectures on stochastic differential equations and Malliavin calculus, noted by M. Gopalan Nair and B. Rajeev, Tata Inst. Fund. Res., 1984. D. Williams, “To begin at the beginning. .”, in “Stochastic Integrals”, ed. by D. Williams, Lect. Notes in Math., 851 (1981), 1-55, Springer-Verlag, Berlin. N. Wiener, Collected Works, Vol. 1, ed. by P. Masani, MIT Press, Cambridge.
-,
[831 I841 [851 1861 1871 1881
1891 1901 1911
~
-.
.
IKEDA NOBUYUKI DEPARTMENT OF MATHEMATICS OSAKA UNIVERSITY TOYONAKA, OSAKA560 JAPAN
SHINZOWATANABE OF MATHEMATICS DEPARTMENT KYOTO606 JAPAN
Taniguchi Symp. SA Kyoto 1982, pp. 53-104
Jump Processes and Boundary Processes Jean-Michel BISMUT
9 0. Introduction In this paper, we will present the results which we have recently obtained in our two papers [4] and [6]. Since the starting point of these two papers has been the development of the Malliavin calculus by Malliavin, we start by giving a brief history of this method. Consider the stochastic differential equation in Stratonovitch form m
dx
=
X,(x)dt
+ C X,(X) dwt , *
1
x(0) = x,
where X,, XI, . . . ,X , are smooth vector fields, and w = (w’, . . . , w m ) is a Brownian motion. (0.1) defines a Markov continuous diffusion whose generator 9 is given by
The smoothness of the transition probabilities for the diffusion (0.1)-which define the semi-group etY-is usually studied via Hormander’s theorem on hypoelliptic second order differential operators [lo]. In particular we know from Hormander’s theorem that under conditions on X,, Xi, . -,X , and their Lie brackets, the operator a/at 64 is hypoelliptic, and that the transition probabilities are smooth. In [20] and [21], Malliavin developed a purely probabilistic method of proof for the existence of smooth transition probabilities. The idea in [20] and [21] was to use the stochastic differential equation (0.1) itself to get a direct proof for smoothness. To do this, Malliavin showed that it was possible to integrate by parts on the Wiener space, and that a wide class of functionals of x.(o) given by (0.1) could be submitted to such a calculus of variations. To prove that integration by parts is indeed possible, Malliavin used the Ornstein-Uhlenbeck operator d which is an unbounded self-adjoint operator acting on the L, space (for the Wiener measure), and the corresponding infinite dimensional Ornstein-Uhlenbeck
-
+
54
J.-M. BISMUT
process whose generator is d . Still using the Ornstein-Uhlenbeck operator, Shigekawa [24], Stroock [26], [27] and Ikeda-Watanabe [12] simplified and extended Malliavin’s original approach. In particular the estimates which prove the smoothness of the transition probabilities were proved in Malliavin [21], Ikeda-Watanabe [121 and improved in Kusuoka-Stroock [19], [28] where the full Hormander’s theorem was in fact obtained. In [3], we suggested a different approach to the Malliavin calculus, based on the quasi-invariance of the Wiener measure, which is expressed by the well-known Girsanov transformation. A formula of integration by parts was then obtained in [3], which was in fact deeply related to a result of Haussmann [9] on the representation of certain FrCchet differentiable functionals of the trajectory x.(o) as stochastic integrals with respect to the Brownian motion w. In [4], we developed a calculus of variations on jump processes. Our motivation was : a) To try to exploit the resources of the stochastic calculus on jump processes, and in particular the existence of a Girsanov transformation on jump processes (see Jacod [14]) in a framework where no clear-cut extension of the Ornstein-Uhlenbeck process exists. b) To understand better how the calculus of variations on diffusions works, in particular in its relation to the classical It6 calculus and martingale theory. c) To obtain specific analytic results on a class of jump processes. In [4], the computations seem to be difficult. One of the key reasons (which may appear in Section 1) is that in spite of all the randomness of Brownian motion, all the a-variations of the Brownian motion are deterministic processes, which is not the case for jump processes. The calculus of variations developed in [4] consists in doing an “elementary” integration by parts at the level of each jump, so that in the end an infinite number of classical integration by parts has to be done. In the limiting Brownian motion case, many complications are smoothed out due to continuous stochastic integration, which makes the unpleasant variation terms disappear. In Section 1 of this paper, we present another approach to the calculus of variations on jump processes, which is based on more elementary arguments than in [4], and does not rely on the Girsanov transformation on jump processes. Section 1 should make the reading of [4] easier although much of the technical work is done in [4]. The estimation techniques are briefly indicated. In our later paper [6], we still focused on a special sort of jump processes, which are the boundary processes of continuous diffusions. Our motivations were:
Jump Processes and Boundary Processes
55
a) To understand better the relation between certain pseudodifferential operators [30] and the stochastic calculus. b) To exhibit the interplay between the continuous diffusion (and its continuous martingales) and the discontinuous boundary process (and its discontinuous martingales). In particular, we felt that the It6 theory of excursions (It6-McKean [13], Ikeda-Watanabe [12]) could be a powerful analytic tool to study the boundary semi-groups. c) To understand the possible interplay between the calculus of variations on the continuous diffusion and the calculus of variations on the jump boundary process. d) To try to exhibit some “Hormander-like” interaction between the “drift” and the LCvy kernel of the boundary processes or between two Ltvy kernels. e) To find degeneracy conditions on the continuous diffusion so that the boundary process would exhibit a slowly regularizing behavior which is typical of some jump processes. Section 2 gives a simplified account of our results in [6]. Proofs are in general briefly indicated. The proofs exhibiting the interaction described in d) are given in detail. Relations with the techniques of enlargement of filtrations [15], [16], [17], and [38] are exhibited.
0 1. The calculus of variations on jump processes The purpose of this section is to present some of the methods and results which we obtained in [4] on the calculus of variations for jump processes (these results were announced in [5]). Recall that in [3], we had given an approach to the Malliavin calculus using the quasi-invariance of the Brownian measure which is expressed through the Girsanov exponential formula. In [4], our idea was to explore if the Girsanov transformation on jump processes (see Jacod [14]) could be the starting point for the development of another calculus of variations whose purpose would be to study the transition probability laws for Markov jumps processes. An integration by parts formula was proved in [4] using such arguments. In particular it appeared that such a formula could be obtained as the consequence of an infinite number of integration by parts in the LCvy kernel of the considered jump processes. In [4], we applied this technique to study the transition probabilities of a special class of pure jumps processes whose construction was elementary using auxiliary independent increment jump processes. The Malliavin calculus of variations on diffusions is based on certain stochastic differential equations. The Brownian motion model is still important for two reasons:
56
J.-M. BISMUT
a) It is essential to build explicitly the solutions of stochastic differential equations, which can then be submitted to the calculus of variations on the Brownian motion space. b) The necessary estimates are obtained by reference to the Brownian motion model, [12], [19], [21], [28]. Of course, if the Malliavin calculus of variations is applied on the Brownian motion itself, it gives-not unexpectedly-essentially trivial results (for an illuminating discussion of this case, see Williams [36]). As it appears in [4], this is not the case on jump processes, even when the calculus is applied to independent increment jump processes. Moreover although computations are elementary in their principle, the resulting formulas are extremely heavy to manipulate. In this section, we will try to present the calculus of variations on independent increment jump processes in an elementary way, i.e. based on the most elementary aspects of their structure. Once this is done, the reader can at least have an intuition for how to study more complex Markov jump processes, constructed by means of such independent increment processes, as in [4]. Note that we will only briefly address the question of knowing what is the "right" formulation for expressing a Markov jump processes in terms of independent increment jump processes. In a) an integration by parts formula is proved on the probability space of an independent increment jump process. In b) this formula is used to obtain an integration by parts formula on the semi-group of the considered process. In c) the estimates which are necessary to make such a formula valid are derived. In d), the application of such methods to more general processes is briefly considered along the lines of [4]. Finally e) is devoted to some geometrical considerations.
a) Integration by parts on independent increment jump processes Let g(z) be a function defined on R"/{O}with values in R' which has the following properties. a) g is differentiable with a continuous derivative g,. b) g is such that
Note that since g is 2 0, if g(z) = 0, then also gz(z) = 0. For t 2 0, a E R", set
57
Jump Processes and Boundary Processes
D(R") denotes the space of functions defined on R' with values in denotes R" which are right-continuous with left-hand limits. {Fl}ttO ro the canonical filtration of D(R") (here F,= 9(z,Is 5 t)). { S t } lwill eventually be made right-continuous and complete as in Dellacherie-Meyer [8] without further mention. B is the predictable a-field on R' xD(R") [8], [14]. Let z, be the independent increment right-continuous process whose characteristic function is given by (1.2). Let I7 be the probability law of z. on D(R"). If &(o,z) is a function defined on R' X SZ X R"/{O} with values in R such that a) 17, is B 0B"(R"/{O})-measurableand b) for any t 2 0 (1.3) (resp. (1,3')
00
),
then we denote by SsSt2, S:
A> Is]
NT
=
C E"C(dzf(ZT),A$"(SI, . .
A ; ~ ( S , ., Sn-1, AZ,,,
' *
' 3
Sn-1, A Z S , , . . ' 7 Az~n-,, AZ,S,))JSI
., ~ z , , - , z, ) ) g ( z ) d z ~ s .]
Since A(z) has compact support, it is feasible to integrate by parts in the variable z in each of the terms of the sum in the right hand side of (1.1 I), so that (1.11) is equal to
Observe that (1.12) is well defined even if g may be 0 at some points since g(z)dz gives measure 0 to these points. Now clearly (1.12) is equal to (1.13)
-
This equality also holds if NT = 0. Using (1.1 I), (1.13) and integrating in the variables S,, . . ., S,, . . ., we get
Now clearly since A(z) has compact support (1.15)
so that
(1.16) (1.5) is proved when (1.6) is verified. Let p be a C” function defined on R with values in [0, I] which is equal to 1 for I t ] > 1, and to 0 for It]< 1/2. Let z:, z:I be the independent increment processes associated to the LBvy measures p(lz)/c)g(z)dz and (I - p(izI/E))g(z)dz, and let 17” be their corresponding probability laws. On (D(R”)xD(R”),17’X17”), the law of the process z, given by
n’, 2,
= 2;
+
2;
is exactly 17. Now by (l.l), we have (1.17) Moreover 17/63li”’ as., z’ and z” do not have the same jump times. Using (1.17), and reasoning on z’ as previously for each fixed z”, it is not hard to obtain EU‘@lI‘’
[(dzf(zT), SsO,(1.5) still holds. Proof. (1.5) still makes sense. In fact, if (1.19) holds, then, if < 2(z), and so using (1.1) and the boundedness of A, we
1z1;I 1, [ Ag(o,z) I
1
see that IA8(w,z)lg(z)dz is uniformly bounded. Similarly if (1.19f) holds, by using (1.1) and Schwarz's inequality, [ A8(o,z)]g(z)dz is still bounded, and so SSsTAexists and is integrable. Similarly, if (1.21) (resp. (1.21') holds, S:,T(div,g(z)A(z)/g(z)) is a martingale (resp. a square-integrable martingale). We now prove (1.5). Assume M > 0 exists such that if IzI 2 My A(z) = 0. Take p as in the proof of Theorem 1.1. For 0 < E < My set
J
(1.22)
(1.5) applies for As. Now we make E + 0. Clearly
J.-M. BISMUT
62
and so
S,,,A"
(1.24)
+ SsSTA
L,(n).
in
Assume that (1.19) and (1.21) hold. Then E"
S
J:
ds Idiv,gAc - div,gA/dz
(1.25)
I E " r 0d s S Izl M, A(z) = 0.
Remark 2. In general, if (1.19') and (1.21') hold, the equality (1.28) does not make sense any more, since the sum S,,,(div,gd/g) is not well A(z) exists. defined. Also observe that since A , is bounded, A(0) = limE40 If (1.19) or (1.19') hold,
J
IA(z)\g(z)dz is bounded.
J g(z)dz = +
CQ
So if
,
it is clear that A(0)=0. Moreover if grad,gA/g is bounded, (1.21) implies (1.219. Observe that by (1. l), we may take in (1.19') 1(z) = z
In this case the constraint (1.20) reads (1.32) This is no accident, since (1.32) shows that the perturbation S,vlTAcan at most be of the order of the quadratic variation of the process z,.
Remark 3. If n = 1, it is easy to prove that (1.33)
A = lim (2 - 7)p.p.
2
rTt2
~
dz IzI'+r
in
g'(~)
Let u, be a bounded predictable process. Let z E R -+ Y(Z)be a nonnegative C" function with compact support such that for Iz] 2 1, Y(Z) = z2. For 0 < Y < 2, set g,(z) =
-7 ~2(zI'+r - - - .
J.-M. BISMUT
64
Let 1 7 r be the probability law of the process z. associated to gr. Set
As(a,z ) = usY ( Z ) . If 2(z) = C[zl, (1.19') holds, and moreover for C large enough (1.20) holds. It is trivial to check that (1.21') also holds. For a n y f e C:(R), we have
Now make Y t t 2 in (1.34). Using (1.33), it may be proved that {IP} is tight on 9 ( R ) (endowed with the Skorokhod topology) and moreover that as r +. 2, li'r converges to the Brownian measure P on %'(R). We will take a formal limit in (1.34) without too many justifications. Observe that if for 0 2 s 2 T, I Az,]
1, then SssTu,~ ( d z , = )
JOr
u, d[z, z],.
Now
as Y + 2, the jumps of IIr become "smaller and smaller". The quadratic variation of z, for P being equal to t, as Y t t 2 , the first
[
term in (1.5) "tends" to E P f ' ( z J have
(&)'(z)zI+r
sr
u ds].
= (1 -
Moreover for Iz/ I 1, we
Y)z ,
so that S;sTu , [ ~ Y / z ~ + ~ ] ' (isz )the z ~ +compensated ~ sum of jumps, which, when I LIZ, I 5 1, are exactly usLIZ,( 1 - 7). It is then reasonable to expect that as l'tf2, the limit of Sg,,u,[~/z'+r]'(z)z'+rwill be the It6 integral -
uSz. So as Yf72, (1.34) becomes formally
It is gratifying that (1.35) holds, and this is shown by a rigorous argument given in [3],[9]. (1.5) then appears to be the natural extension of integration by parts on elementary functionals of Brownian motion to jump processes.
b) Integration by parts on the semi-group of an independent increment process We will now extend Theorem 1.2 in order to obtain a formula of integration by parts on the semi-group of z,, i.e. we will prove that for adequately chosen T > 0 we have that for f E C;(R"),and 1 5 i I n
65
Jump Processes and Boundary Processcs
(1.36)
;:[
E" - ( z T )
1+
E U [ f ( z T ) D $= ] 0
As previously pointed out, formula (1.36) itself is not very interesting, since in this case, we know the characteristic function +T(ac) explicitly. However the method to obtain (1.36) is closely related to our work [4], where the calculus of variations is still performed on independent increment jump processes, but the considered functionals are much more complicated. R(z) is a measurable function R"/{O} R + such that ---f
(1.37)
J
R2(z)g(z)dz< 12151
+
03
.
v(z) is a function defined on R"/{O}with values in R+ having the following
properties. a) Y ( Z ) is bounded and differentiable, and has a bounded differential VZ
.
b) C > 0 exists such that if Iz/ 5 1
(1.38)
YlZl
I CR(z)lzl.
c) The following inequality holds (1.39) Choose p as in the proof of Theorem 1.1, and for (1.40)
p,(t> = p ( t M
7jr
> 0 set
'
We now have
Theorem 1.3. For any T > 0, 7jr > 0 , f ~C;(Rn),and i, (1 5 i 5 n),
Proof. (1.41) makes sense. First note that p,(S,,,v) and ~ ' ( S , , , V ) so that p q ( s 8 5 T v ) / s 8 < T Y, p , ( s 8 < T Y ) / ( s S < T ')', p : ( s s 5 T v ) / S , < T are ~ bounded. Since av/az, is bounded, S,,,(av/az,)v exists and is integrable for the same reason as SsSTuin the proof of Theorem 1.2.
are if S s < T Y 5
66
J.-M. BISMUT
To prove (1.41), the easiest way is to go back to the assumptions in the proof of Theorem 1.1. Namely, we assume that E, M, 0 < E < M, exist such that if Iz/ 5 E , or Iz] 2 M , Y ( Z ) = 0. We also temporarily assume that (1.6) holds. Using the notations in the proof of Theorem 1.1, we have if NT # 0,
Set for n (1.43)
< NT NT
K, =
C Y(Az~,,,),
Z, =
zT - Az,,
I
n'#n n'=1
(1.42) is then equal to
(1.44)
If N , = 0, pll(S,,,v) = 0, so that equality between (1.42) and (1.44) still holds (if NT = 0, Cy.. . is taken to be 0). By integrating (1.44) in all variables, we find easily that (1.41) holds. Assumption (1.6) is released by using the same argument as we used in the proof of Theorem 1.1. The support condition on Y is released as in the proofs of Theorems 1.1 and 1.2, using in particular (1.37), (1.38) and (1.39). We now make 7 -+ 0 in (1.41). To obtain (1.36), we need that (1.45)
P,(S,,T
Y)
+
and so we need that SssTv
> 0 as.
(1.46)
J
as.
1
A necessary condition is that
g(z)dz =
+ . 00
Jiinip
67
Processes and Bouridary Processes
(for ( I .36) to hold (1.46) is needed since otherwise a Dirac mass is left in the law of z.!). From Theorem 1.3, we get Theorem 1.4. If T > 0 and u are such that I/S,,,u is in a given L,(I7) with p > 2, then for any f ' C;(R"), ~ (1.41) still holds with pq replaced by 1. For any k E R', t 2 kT, Ictlk$,(ct) is a bounded function. For any l? E N , t > (l? n)T, the probability law of z , is given by ql(y)dy, where q,( E Ci(R"). In particular, if for any T > 0, l/SIIsTu is in a given L,(n) with p > 2, then for any t > 0, the law of z , is given by qt( .)dy, where q,( .) E C;(R").
+
a )
Proof. We make 7 -+ 0 in (1.41). Clearly (I .46) holds so that the first term in the left hand side of (1.41) is taken care of. It is not hard to prove, using (1.38) and the boundedness of (au/az,)u that S,sT(av/az,)~is in all the L,(I7) (1 < q < co). S,s.(au/az,)u/(S,..u)2 is then in L,(Z7). Moreover
+
For t 2 7, pi(t) = 0, so that if t
> 0, p:(t) + 0 as 71 +0.
Clearly
(1.47)
Finally S,C,T(a(gu)/az,)/gis in L2(17). Using (1.47) we see that the dominated convergence theorem applies in the three last terms of the left hand side of (1.41). Using (1.41) with f ( z ) = e-i(".z), we see that for any i ( l 2 i I n), Ia'I+.(a) is a bounded function. Since +,(a)= [+T(c~)]"~, the result on function $, follows. The results on the law of z , are then standard results on Fourier transform. Remark 4.
s
+
A result of Tucker [31] states that if g(z)dz = 03, for any t > 0, the law of z, has a density with respect to the Lebesgue measure. Although the proof in [31] is probabilistic, this result admits an easy analytic proof by differentiating qt(cx). Remark 5. We use the same assumptions and notations as in Remark In particular, assume temporarily that n = 1. As we shall see later (in Theorem 1.6) if u is C" with compact support and is such that u(z) = z2 for IzI 1, then for every 'i < 2, the conditions of Theorem 1.4 are 3.
J.-M. BISMUT
68
verified. Now when all the jumps of z are in size 5 1, then S s a T u = [z, zIT. As TT72, it is reasonable to replace SsaTu by T. Moreover (av/az)u = 2z3. The 3-variation of z for the Brownian measure P is 0. so for P, we may formaZZy cancel SZsT*u. aZ
"
For Iz]< 1, -(g,u)
g, = (I - T)z.
aZ
to become - z,. Taking the
As 7772, we may expect S:,,
formal limit in (1.41) (with ps = l ) , we get that for T > 0, f~ C;(R),
[;
(1.48)
EP
~
(z,)
1
[
- EP f(zT)-
ITTI=O
(1.48) is of course trivially true.
Remark. 6 . As should be expected, formula (1.41) with ps replaced by 1 can be directly obtained by non trivial manipulations on the characteristic functions (see [4]). c) Some estimates on independent increment processes We now are left to find sufficient condition under which the assumptions of Theorem 1.4 are verified. As we shall see, the effect of the calculus of variations is to transfer an estimate on a Fourier transform to an estimate on a Laplace transform. Let 7, be the non-negative measure on 10, w[ which is the image measure of g(z)dz by the mapping z -+ ~ ( z ) .Since [ vg(z)dz < 03, we
+
s
J
+ +
+
have xdp.(x) < co. More generally, let m be a non-negative o-finite measure on 10, w[, such that x A l dm(x) < 03. For /3 2 0, (Y E
R, set (1.49)
ir,(B)
=
s
s
- I)dm(x) , M,((Y)
(e-j'
+
=
s
(cos ax - l)dm(x),
Clearly, for p 2 1 1 (1.51)
E"[l-sz
For (1.51) to be finite for at least one p
> 2, it suffices that E > 0,
69
Jump Processes and Boundary Processes
C 2 0 exist such that for ,~9E R’
(1.52) or equivalently, that for j3 large enough
(1.53)
Tr,LP) 5 - (2
+ €1Log P . +
We will now sufficient give conditions on a measure m on 10, m[ so that T,@) behaves adequately as P -+ + co. The following is proved in [4]. Theorem 1.5. The following two conditions are equivalent a) A s x + O
m(]x, b) A S P + +
1 + co[) - C Log . X
m
r,(P)
-
-
c Log P .
A suficient condition for a)-b) to hold is that
+
- - CLogjaj.
C) AS -+ 00, Mm(a) Either of the two conditions d) l hmlx 2t-4 5-0 log l/x
=
C, or
implies f) lim
P-+-
~
T,(P)
LogP
2 - c.
Proof. The proof relies on standard Abelian and Tauberian techniques. For the full proof, see [4]. Corollary. Assume that D (0
< D 2 + m), exists such that
(1.54)
Then for T > 210, l/S,,,u belongs to one given L,(I7) with p
> 2.
Proof. This is obvious using (1.54) and the implication d) Theorem 1.5.
+ f) in
J.-M. BISMUT
70
Remark 7. In general the function R appearing in Section 1 b) is a bounded function, so that for C > 0 v(z) 5 Clzl. If (1.54) holds, then also
If n = 1, it is crucial to remark in Theorem 1.5 that in general a) or b) do not imply c), and d) or f) do not imply e). Otherwise using (1.551, we would find that
I d
(1.56)
-
j (cos ax
20
L o g LI
I.l-+m
and so for E
- l)g(x)dx
> 0, as a -+ + co
(1.57)
which would make the conclusions of Theorem 1.4 trivially true without any calculus of variations! The counter example is as follows. Let m be given by
'Ix.
A s x + O , m ] x , +co[--
Log 2
Moreover for k E N
c(1 cos 211.2X-n)= C (I cos 2n2"-") c(1 - cos 2n2-") Mm(27C), m
- Mm(2n2k)=
n
-
-
n=k+1
+m
=
= -
1
+
and so lM,(a)I does not tend to +co as la1 + 00. In this case, the probability law whose Fourier transform is
is singular with respect to the Lebesgue measure. So if n = 1, at the critical logarithmic concentration, the calculus of variations gives non trivial results for independent increment jump processes. In a private communication, Prof. H. Delange has shown us how to construct a function g 2 0 and C" on 10, co[ such that as x + 0
+
Jump Processes and Boundary Processes
71
(1.58) and that if
M(a) =
(1.59)
so +"
(cos a x - l)g(x)dx ,
IM(a)I has an arbitrarily slow growth at infinity. It is then clear that even when g is C", (1.58) is not enough to imply a logarithmic behavior of (1.59) as la1 --f 00. Note also the following result of [4].
+
Theorem 1.6. Let Y be such that 0 < i' < 1. Then the following conditions are equivalent: a) As x -+ 0+, mflx, m[) (3x7. - CT(1 - Y)(sinlr/2(1 - Y ) ) l a l r . b) As IayI + + m , M,(a) C) AS p -+ m, ~,(p) - c r ( i - r)p.
+
-
+
-
-
If the equivalent conditions of Theorem 1.6 are verified, then for
T>O
(s
+=(a)= exp T (e-"""
-
I)dm(x))
is such that for any n E N , lal"+r(a)is bounded. So qT(a)is the Fourier transform of a probability measure which has C" density with respect to the Lebesgue measure. In this case, it is the concentration of m which determines the regularity of the corresponding probability. d) The calculus of variations on general jump processes As we have already said, the previous method is not devised to be applied to independent increment jumps processes, but can be used on processes which are constructed by using independent increment jump processes as an instrument, in the same way as continuous diffusions are constructed using Brownian motion, which is plugged into a stochastic differential equation. I n [4],we have treated the case where the process x, with values in R" is given by the solution of
(1.60)
s, = x,
+ +!
X,(x,)ds
+ vt + . . + Y!
where y', . . .,yq are (mutually independent) independent increment jump processes, where the probability law is modified by using the Girsanov transformation on jump processes [14]. To treat this case, the previously developed calculus of variations
72
J.-M. BISMUT
must be applied to functionals of y’, . . ., y* which are much more complex than those which we previously considered. Namely, functionals of process x, which involve jump martingales constructed by means of x , must be submitted to the calculus of variations. In the same way as in the calculus of variations on diffusions, a random flow vt : x, -+ x , must be considered and lifted to various bundles, and these lifts must also be submitted to the calculus. In [4], we have treated the case where the jump process is “elliptic”, i.e. the support of the jump measures for arbitrarily small jumps spans the whole space R”. Let us note that in the proof of Theorem 1.4, as soon as we are able to control the differential of order 1 of the law of z and so prove that lal+T(a)is bounded, the boundedness of [a]”+,(a) as long as t 2 nT is obvious. For the case of the process x given by (1.60), no such argument exists a priori. However, it is possible to mimic the previous argument by using a step by step integration by parts procedure, i.e. to make a variation of the processes y’, . . -,y 4 first on [0, TI, then on [T, 2T], [2T, 327, . . ., [(n - 1)T, nT], so that at each step, a control is obtained for the differentials of higher and higher order of the law of x n T . This procedure is fully developped in [4], and avoids the iteration of the calculus on the same interval [0, TI which would require: more differentiability on the Ltvy kernels. frightening computations. Note that this procedure can also be applied to ordinary diffusions whose generator is everywhere hypoelliptic, and can be localized using the localization procedure of Stroock [26]. e) Some geometrical considerations The structure of equation (1.60) is not completely satisfactory. In fact it makes full use of the vector space structure of R” since the various jumps are “added” to each other. In particular, there is no interplay between the jumps of y l , . .,y 4 which would be similar to the interaction of vector fields by the bracketing in ordinary continuous diffusions. A natural idea would be to replace (1.60) by more general stochastic differential equations with jumps (see Jacod [ 141). However technical difficulties do arise, essentially because contrary to (O.l), such general equations do not define flows of diffeomorphisms i.e. trajectories starting from different points may collide. As will appear on an example later, it seems that for general jump processes, the Ltvy kernel gives an analytically useless description of the process (except in the case of independent increment processes). Namely, it is very hard to describe explicitly how does a LCvy kernel M(x, dy) vary with x.
73
Jump Processes and Boundary Processes
Although we now know how to describe in a geometrically invariant way a much larger class of jump processes, we will concentrate on the boundary processes of certain continuous diffusions, where, hopefully, our point will clearly appear.
8 2.
The calculus of boundary processes
In this section we report on some results which we have obtained in our forthcoming paper [6] on the calculus of boundary processes. Assume that z is a reflecting Brownian motion on [0, m[ ([13], p. 40, [12], p. 119), L its standard local time at 0, w = (w’, -,w m ) a Brownian motion independent of z. Consider the stochastic differential equation in Stratonovitch form
+
+ cXi(X, z) .dwt + D(x)dL m
dx (2.1)
= Xo(x, 2)dt
1
x(0) = x,
where X,, . . ., X,, D are smooth vector fields. A drift b(x, z) is introduced on z using a Girsanov transformation. If A , is the right-continuous inverse of L, we study in [6] the transition probabilities of the Markov process ( A , , x A J . Of course ( A t , xAt)is a jump process, whose jumps correspond to the excursions of z out of 0. In [6], we use the fact that the LCvy kernel of the jump process is itself the image of the excursion measure of (z, w ) corresponding to the excursions of z out of 0 through the solutions of a stochastic differential equation. We will essentially focus on some aspects of our work [6], and insist on some possible connections with other recent developments in probability. In a) the main notations and assumptions are given. In b) a stochastic flow is associated to the considered stochastic differential equation. In c) the Girsanov transformation is briefly introduced. In d) the boundary process is defined. In e) a partial calculus of variations on w is presented along the lines of Bismut-Michel [7]. The key quadratic form C y is introduced as in [20], [21]. In f) some simple considerations relating the calculus of variations to the method of enlargement of filtrations (see Jeulin [15], Jeulin-Yor [16], [17], Yor [38]) are developed. In particular the “non differentiability” of local time L with respect to any natural differential structure on the space C(R+;R’)associated to z forces us in [4] to use the calculus of variations on jump processes to study the component A , in ( A t , XJ. This is briefly done in g ) . In h), the key problem of the a s . invertibility of C z is studied. As in [20], we know a priori from g) that if this is the case, the law of ( A t , x A Chas ) a density
J.-M. BISMUT
74
with respect to the Lebesgue measure. Sufficient conditions under which this is the case are proved. The consequences are interesting since they ) the vector field D show that the LCvy kernel of the process ( A t , x A C and may interact through some sort of Lie bracketing which is precisely expressed through true Lie brackets of (Xo,X,, . . ., X,, d/az, 0). In i), j ) the so called ''localizable'' and "non localizable" cases for regularity of the transitions probabilities for the boundary semi-group are considered. In particular, in the non localizable case, conditions are given under which the boundary semi-group is slowly regularizing. In k), z is now a standard Brownian motion, and x is driven by the vector fields (Xo,XI,. ., X,) when z > 0, by (XA, X : , ., X;) when z < 0, and D for z = 0. Existence of densities for the transitions probabilities of the boundary process is proved under conditions which still exhibit interactions between all the considered vector fields. In l), regularity results for the two-sided case are briefly presented.
-
Assumptions and notations m is a positive integer. Q(resp. Q') is the space C ( R + ;R") (resp. C ( R +; R + ) ) . The trajectory of o E 8 (resp. o' E Q') is written as w, = (wt,. . -,wp) (resp. z). The o-field 9, (resp. 9;) in Q (resp. 8') is de= 9?(z,Is < t ) ) . is the space fined by .F,= 9?(w,Is I t ) (resp. 9; D x 8',endowed with the filtration {.F,}trO, where $, = .Ft09;. All filtrations will be made right-continuous and complete as in Dellacherie-Meyer [8], without further mention. P is the Brownian measure on 8, such that P[wo= 01 = 1. For z E R + , PI is the probability measure on Q' associated to the reflecting Brownian motion on [O, co[, starting at z, i.e. P;[zo= z] = 1. For notational convenience we will write P' instead of PL. On (Q', PL),L, denotes the local time at 0 of z. By [12], p. 120, we know that B, = z, - z - L, is a Brownian martingale with Bo = 0. We also know that if zo = 0 a)
+
A, is the right continuous inverse of L, i.e. A,
(2.3)
=
inf{A 2 0; L,
>t } .
d is a positive integer. Y = ( x , z) is the standard element in Rd+',with x E Rd and z E R. Rd will be identified to Rd x (0). Xo(x,z), . -, X,(x, z ) are m 1 vectors fields defined on Rd+' with values in Rd, whose components are in C;(R'+'). D(x) is a vector field defined on Rd with values in Rd whose components are in C:(Rd). b(x, z) is a function
+
-
75
Jump Processes and Boundary Processes
defined on Rd+',with values in R, which is in C;(Rd+'). If X , is a continuous semi-martingale, dX denotes its differential in the sense of Stratonovitch, and 6X its differential in the sence of It6 [22]. If h is a C" diffeomorphism of Rd onto R d , and if K ( x ) is a tensor field on Rd, (h*-'K)(x) denotes the pull-back of K(h(x)) to x through the differential ah/ax(x) (see [l]). If Y(x) is a vector field, we see that
.
( h * - ' Y ) ( x ) = [?]-'Y(h(x))
b) The reflecting process and its flow We now build a reflected process as in Ikeda-Watanabe [12], p. 203. Take (x,, z,). E Rd X R'. On P 0Pi,) consider the stochastic differential equation
(a,
dx = X,(X, z)dt x(0) = x,
(2.4)
(the summation sign form of (2.4) is
Cy=lis systematically omitted).
X,(X, Z )
(2.5)
+ D(x).d L + Xi(x, 2).dwt
I ax. + --'X 0, and any p 2 1, I A t S T B ris in Lp(P(8 P'). b) for any f E C;(R X Rd),i f x , is the process y8(u,x,), then
+
(2.20) Proof. For the complete proof, see [6]. The partial calculus of variations is very close to Bismut-Michel [7]. Remark 6. Because of Remark 5, it is essential that f has compact support. However, by using (2.18), this assumption may be released, so that smoothness of the law of xAcwill follow (the component A , is no longer needed). Theorem 2.9. Assume that t' > 0 is such that a) for every x E Rd,C;t, is P 0P' a.s. invertible. b) for every T 2 0, there is q > 2 such that for any x E Rd,
is in L,(D, P 0P') with a norm which is bounded independently of x E Rd. Then for any x, E Rd,any multiindex m, any t 2 [mit', on (B,P (8 P') there exists a random variable 0 7 having the following properties: a) For any T 2 0, IA,t
By the zero-one law, we know that P 0P' as., V,t is a fixed space, not depending on a. Let us assume that V; # T,,(Rd). If S is the {gc}tro stopping time (2.38)
S
=
inf{t
> 0; V, f
V,t}.
then S is > 0 a.s.. Let f be a non-zero element in T,*,(Rd)orthogonal to V;. Then
(2.39)
(f, (~~-*X~)(xo)) =0 for
Now from [11-Theorem IV. 1.1, we know that
s
<S.
J.-M. BISMUT
86
or equivalently
Now (2.41) gives the Its-Meyer decomposition of the { g t } t r semi-martino gale (f, p,*-'X,), which is 0 for s 4 S. By canceling the martingale terms, we find that for s 5 S
J: (f, p:-'[XJ, Xt1)6w' (2.42)
so'(f,
1
=09
< i, j < m ,
X*])6B = 0 .
p:-1[$
An elementary reasoning on the quadratic variation of the local martingales (2.42) and the continuity of the processes p:-'[X,, X,], p:-'[a/az, Xi] (see [3]-Theorem 5.2) show that P 0P t a.s., for s 5 S 1
(f, p:-lPj, X i ] ) = 0 , (f, v:-'[T$ a Xi])= 0 .
(2.43)
< i, j < m
Reapplying (2.41) on (2.42), we find that for s (f, p,*-"X,, [X,, Xill)
(2.44)
(f, P:-"az,
a
=
0
9
S
9
a [a ,, Xi]])= 0 .
We now cancel the bounded variation process in the Meyer decomposition of (f, p:-'Xi) (s < S), i.e. using (2.41)-(2.43), we get for s 4 S (2.45) Since P
[
(f,p:-l[X0, xild.>
+ ( L j:p:-'[D,
0P' a s., the support of the measure d L is exactly the closed set
Jump Processes and Boundary Processes
87
(z, = 0) which is negligible for the Lebesgue measure ([13], p. 44), from (2.45) we deduce that for s S
By a result of Yamada (see Ikeda-Watanabe [12],p. 168), we know that (2.69) has a unique strong solution. Using (2.68) and the comparison theorem of Yamada [37](Ikeda-Watanabe [12],p. 352), we find that p: 2 r:
(2.70)
for
t
0, the Hausdorff-Besicovitch dimension of (r, = 0, 0 < t E ) is the constant 1 - d/2. Using this result with d = 1, we find that a.s. the dimension of (z, = 0, 0 5 t < S ) is 1/2. Because of (2.72), it is clear that a.s. the dimension of (zdt= 0, t < 7,) is 1/2. Now (2.69) shows that for 0 5 t I Y,, r: is the square of a Bes (1 C) process and moreover 0 < C < 1. We then know that as.,
+
Jump Processes and Boundary Processes
91
the dimension of ( r : = 0,O 5 t < 7,) is (1 - C)/2 < 1/2. This is a contradiction to (2.71). So we find that M, = 0. Assume that the second line in (2.51) does not hold. By using the stopping time optional selection Theorem [8]-IV-84, we can find a {.F;t}t20 T such that (P6 I")(T< co) > 0, and moreover if T < 03, then T < S , Z, = 0, MT f 0. Now using the strong Markov property of (w,z), the whole reasoning can be restarted after time T (instead than after time 0); we still arrive at a contradiction. The second line in (2.51) holds. By iterating the whole procedure on (2.51) we find in particular that f is orthogonal to U:"(E,UFJ(x,, O)(by taking s = 0 in (2.51)). By the assumption in the Theoremfis 0. This is a contradiction to S > 0.
+
+
Remark 9. Instead of using the dimension properties of the set of zeros of a Bes2(d) process rt (0 < d < 2) starting at 0, we could as well use the fact (It6-McKean [13], p. 226) that if Ld is the local time of r at 0, and Ad is its inverse, then Ad is a stable process with exponent 1 - d/2, so that its characteristic measure will be proportional to I,to(dx/x2-d/2)). By proceeding as in It6-McKean [13], p. 43, this shows that if N f ( t )is the total length of the intervals in (r > 0) fl[0, t ] whose length is 5 L , then a.s., for any t > 0, as E J 0 (2.75)
Nf(t)
- Cd&d'2Ld(t)
where Cd is > 0. Using (2.75), a contradiction is easily obtained from the assumption M, # 0 in the proof of Theorem 2.14.
Remark 10. Theorem 2.14 exhibits clearly that the excursions of the process (x, z) out of z = 0 can interact with the process when it stays on z = 0 through D, so that the probability law of ( A t , xAt)has densities, although the LCvy kernel of this process may be degenerate. i) Regularity of the boundary semi-group: the localizable case W e h o w give sufficient conditions under which the assumptions of Theorem 2.10 are verified. Definition 2.15. For d
(2.76)
k'(x, z) =
E
N , the function ke(x,z ) is defined by
C (f, Y(x,2))'.
inf
f€R~,llfll=lj = 1 T E E /
We then have Theorem 2.16. I f x ,
(2.77)
E
Rd is such that for a given 8 E N , 0 > 0
lim zLog inf kC(x,z) 2>O,z-O
15--101
0, T
2 0, IAtsTI[C2;]-'} is in all the L,(PO P') (1 < p
0, T: is the stopping time
T: = inf {t > 0; z,
(2.78)
=
7')
then (2.79) j) Regularity of the boundary semi-group :the non-localizable case We now give conditions under which the assumptions of Theorem 2.1 1 are verified. For 4 E N , define me = 6 x 20E-'.
Theorem 2.17. Assume that for a given 4 such that
E
N, there exists C
>0
lim z Log inf ke(x,z ) = - C .
(2.80)
z>o,r-o
xERd
For any t > 1 6 n m , C , T 2 0, there exists q > 2 such that for any xoE Rd,IAILTIIC$]-'lis in L,(P P') with a norm bounded independently of xo. For any t 0, the law of ( A , , xAt)is given by p,(a, y)dady andp,(a, y ) is such that a) it is C" on 10, w [ X Rd, b) if t > ( k d 2) 1 6 f l m , C , p,(a, y ) E C * ( R X Rd).
>
+ + +
Proof. See [6]. Remark 11. The condition (2.77) is a local one, while (2.80) is a global condition, which justifies the terminology which we have used. Moreover, in [6],we show how instead of assuming that (2.77) is verified on a neighborhood of the starting point xo, it may be verified on a neighborhood of the final point y as well (for such a problem on standard diffusions see Stroock [26]). Remark 12. The conditions of Theorem 2.17 give exactly the analytical conditions under which the boundary semi-group is slowly regularizing in the sense of [4](also see Theorem 1.4). Moreover it is shown in [6] that under conditions like (2.80), the generalized symbol of the
Jump Processes and Boundary Processes
93
generator of the boundary process exhibits a logarithmic behavior (see [6], Section 6, Remark 5). The conditions of Theorem 2.17 are minimal. In fact consider the stochastic differential equation
dx
=
exp{
(2.81)
- -Id$. 1 22,
x(0) = 0 .
Conditionally on z, the law of x A tis clearly a centered Gaussian whose variance is (2.82) Now, if n+ and u are defined as in Section 2, g),
(2.83)
sup 2, 2 ___ I c + n+[O<s<s Logl l / a l =
c + Log l / a
(the last equality in (2.83) is classical [12]). From (2.83), we see that
and so
Since (2.86) we see from (2.85) that for t small enough, (2.86) is of xdcunder P 0P" is h,(x)dx, where (2.87)
+ co.
Now the law
J.-M. BISMUT
91
+
We then find that for t small enough, h,(O) = 03. h, is not even continuous at 0. It should be pointed out that this result has nothing to do with the fact that we are considering xAt instead of ( A t , x A J , since differentiation in x as in (2.7) is irrelevant here.
k) Two sided boundary processes: existence of a density Let Xi(x, z), . . ., X’,(x, z) be another family of vector fields having the same properties as X,,. X,. z now denotes a standard (i.e. non reflecting) Brownian motion. The corresponding probability space is still written as Q’ and the probability law of z’ is written as P:,. L,is the local time at 0 of IzI (i.e. L is twice the standard local time at 0 of z), so that a ,
(2.88) z:,
lz,l
=
lzol
+ Stsgnz,az, + L , .
z; are defined by z;
= 2,
v0,
z; = z, A 0 .
A, is still the inverse of L as in (2.3). Consider the stochastic differential equation dx = I,>,[X,(X, z)dt
(2.89)
+ X,(x, z) dwi] *
+ I,,,[XXx, z)dt +
x(0) = x,
XKX,
z).dwil
.
+ D(x)dL
3
A flow vt(a,x,) can be associated to (2.89) in the same way as in Theorem 2.1. If b’(x, z) is a function having the same properties as b(x7z), the Girsanov exponential is changed into n/r, = exp[Jl (2.90)
IZ*>,(NXY,
I z,)6z, - 2 bZ(X,,ZJdS
11
1 + J: I*s~((v~-1~~)(x~, zY), $. J: ‘3,
(xo,z
~)(vb*-l~t)
r2 0, C:50 is invertible.
Proof. U, is the vector space spanned by Iz,,o(y~~’Xi)(xo) (I < i < rn) (1 5 i < m). V, is the vector space spanned by and Is,~,,(y~-lX~(x,,)) UsstU, and V,+is defined by
K+
(2.93)
=
n v,.
s>t
We then proceed as in the proof of Theorem 2.14. Namely assume that Vo+(which is a non random vector space) is # T,,(Rd). Then if S is the stopping time
S = inf{t
(2.94)
> 0; Vt f
Vo+}.
S is positive a s . Let f be a non-zero element of Tz0(Rd)orthogonal to Vl. Then (2.95)
(A
=
0
(L ( v : - ~ ~ x ~=D0
on (zU> 0 )n 10, SI, on (zEl < 0) n [o, SI .
Using the optional selection Theorem [8]-IV-84, it is easily proved that > 0) and (z < 0). (2.95) can be replaced by (z = 0) is included in both closures of (z
(2.96)
(L ( v : - ~ ~ x ~=D0 ( A (P,*-~X:)(X~)> =0
From (2.95), we find that
w7
on (zU 2 0) n S [ , on (z, I 0 ) 10, s[
n
.
J.-M. BISMUT
96
From It6-Tanaka's formula, we know that
(2.98)
(there is no integral
s:
. . - d L because the support of d L is ( z = 0), and of
(2.96)). From (2.97), we find easily that for 1 < j 5 m
( A P:-i[x,7 (2.99)
xi])= 0
on
4)
( L c p ~-I[$, *
0
=
> o m 10, SI , on (z, > 0) n LO, SI . (2,
By iteration, using (2.99) again as in (2.44), and reasoning as in (2.45), we find that for 0 j m.
0, the sum is (f, &lH) .
if t
If z, = 0, and if S is a left cluster point of ( z = 0), the sum is still 0, and moreover by (2.103), (f, y$-'H) = 0. If z, = 0 and z is positive on a left neighborhood of S, the sum is (f, & - l H ) , and G, = 1. If z, = 0 and if z is negative on a left neighborhood of S, both sides of (2.107) are
J.-M. BISMUT
98
0. Finally if z, < 0, the left hand side of (2.107) is 0 and G, = 0. Let E,rand E: be the processes Et =
1:
Iz>o((f,P:-'(Wo,
HI
+ $[4,[XI3H11)))d.
respectively. By using the first line in (2.101) as well as (2.107), we find that for any t 2 0, a s . (2.109)
EL,,
=
G,,s(f, P ? / . H ) *
Now the process G,,,(f, cpfiiH) is continuous. This is clear if t < S, by using (2.103). If S is a left cluster point of (z = 0), (f, pZ-lX,) = 0 and continuity at S still holds, while if S is isolated on the left from (z = 0), G will be continuous at S. From (2.109), we find that a.s.
Et
(2.1 10)
=
G , ( f , p?-'H)
on [0,S ] .
Similarly (2.111)
E: = G : ( f ; pf-'H)
We claim that for t (2.1 12)
on
[0, S ] .
S
(G,
+ G X f , lo?-'H) = (f; &'W
*
We only need to prove (2.1 12) if z, = 0. If t < S, (f, y$-lH) = 0 and (2.112) is true. If t = S, and S is a cluster point on the left of (z = 0), the same reasoning applies. If S is not a cluster point on the GL = 1, and (2.112) still holds. From (2.110)left of (z = 0), G, (2.112), we see that
+
(2.113)
(f, g ~ ? - l H )= E, + E:
on [0, S ]
.
Comparing with (2.109, we find that (2.1 14)
(f, Jl p Y [ D , H ] d L
=0
on [0,S]
Jump Processes and Boundary Processes
99
so that
(2.1 15)
on (z=O)n[o,s[.
(f,y;-'[D,H])=O
The first line in (2.104) has been proved. Now from (2.103)-(2.110) it is clear that (2.1 16)
E,
=
0
on (z, = 0) fl[0, S [
.
Set ps = 2
fils =
+ E:,
2 (f,y:-"x,, j=1
Hl(x0
o))2.
From (2.1 16), we see that (2.1 18) c,,
lit
on
p, = 0
(z, = 0)
n [0, S[.
have been defined in (2.102). Set
We know that z is a reflecting Brownian motion, and that (w', . ., wm,B ) is a {g.,}t20 Brownian martingale. Moreover cs is a {,F.t},20 stopping time. Using (2.118), we have
Pt
on
=0
(2, =
n
0) [O, cs[ .
Moreover using standard results on semi-martingales we know that E,, and hence p t is a continuous process. Using (2.108), we find that (2.120)
E,
=
Jl ES
ds
+ J: (f, yz-yx,, H ] ) 6 w j .
Assume that Mo # 0. By renormalizing f, we may assume that Mo = 1. Let g be a positive constant such that g < 1/4. Since Eo = 0, we may suppose that S > 0 has been chosen to be small enough so that ifs<S
J.-M.
100
so that if t (2.122)
BiSMUT
< cs 1-T71 0. Remark 13. Since under the assumptions of Theorem 2.19, the boundary semi-group has densities, we see that unexpected interaction may occur at the boundary between the two sides (z > 0 and z < 0) of the process (x,, z,) as well as with the vector field D. This is a clear example that Levy kernels do interact. The two sided boundary process: regularity of the semi-group The assumptions to get regularity for the boundary semi-group are in general much stronger than for one-sided processes. The following counter example is developed in [6]. Consider first the stochastic differential equations 1)
(2.124)
dx
=
I,,,dwi ,
~ ( 0=) x ,
which can be put in the equivalent form (2.125)
dx = Z,,,dwt, dh = dt ,
~ ( 0 )= 0
h(0) = 0 .
The calculus of variations applies to the component x A t in (2.125). In fact conditionally on z, the law of x g e is a centered Gaussian, whose variance is CAIhTwhere C , = s. By [13], p. 26, the law of A , is
Jump Processes and Boundary Processes
101
One finds that l / d G is in all L,, and the law of xAcis proved to be smooth. Consider now the system (2.126)
dx = Iz>O I h < T dwt dh = I,>,dt I,,,dt, 9
+
x(0) = x, h(0) = 0 .
(2.126) appears as a two-sided perturbation of (2.125). We claim that now the law of xAt is not smooth. In fact conditionally on z, the law of xAtis a centered Gaussian whose variance is C(PtAT, where (2.127) Clearly, C;,,,
C: =
< C&.
(2.128)
s:
I,,,dS
By LCvy’s Arcsine law ([I31 p. 57), the law of C$ is IO<s 0). In the two-sided case there is in general no “localizable” result, except when both sides are equally regularizing. We prove however a regularity result in [6]. Recall that the families of vector fields E,, E: have been defined in Definition 2.18. Definition 2.20.
For 4 E N , X‘(x) is the function defined by
We then have Theorem 2.21. Assume that 4 E N , 7 > 0 exist so that for any x (2.130)
X“x) 2 ’I
E
Rd,
J.-M. BISMUT
102
Then for any xoE Rd, any T 2 0, Z A t S T ] [ C ~is] ~in' all 1 the L p ( P @P'). Proof. Note that (2.130) is a global assumption on Rd. (2.130) indicates in particular that UI(EnU E@( x , 0) spans Rd for each x . The techniques of estimations of Theorem 2.16 based on classical stochastic calculus do not work any more. Apparently, no single side ( z > 0) or (z < 0) is enough by itself to get the desired result. Assume for instance that C:=l - Pa(x,*
'Psl$we get llve+P611; 2 E-2k"llP,ll:+41d
(2.5)
with some positive constant k" depending only on the dimension. (Note that the best p so it is within the integration limits at least for E sufficientlysmall). Using (2.5) we get
-
2 2~11~''P.llaZ2 C,~-eIlP,1122+"d for c, > 0 depending only on the dimension and on the ellipticity bound for the u,'s. From (2.6)
d
-(llp,/l~)-~/d
dt
2 cg2
or
(llp611;)-2/d2 C,t €1
+1
114
Rodolfo FIGARI, Enza ORLANDI AND George PAPANICOLAOU
which implies
Finally the Chapman-Kolmogorov equation and (2.7) provides us with the pointwise bound P i x , x', t ) =
cP
k Y , t/2)P,(Y, x', t/2)
Y€ZP
The estimate a) now follows easily
Proof of b). Because min, [a log a
+ la] = -e-2-1
C,(P,(x, t ) lOgp,(x, t ) + 4xIP,(x,
t)
XEZ,
for any real 2, we have
+ bp.(x, t ) ) 2 --e-"-'
C
XEZf
or
Choosing b such that e-b-l
=
+
(1 1 - e-ar
and a = l / ( M e ( t ) we ) have -Q,(t>
+ 1 +b 2 -1.
Hence
Because 2/a
+ 1 2 (1 + e-.)/( 1 - e-.)
for any positive a, we have
e-'lzl
DiffusiveBehavior of a Random Walk
115
which is what we wanted to prove.
0 3. Outline of the convergence proof Let ~ ' ( x 0 , ) be the solution of the elliptic problem: (3.1)
(L,u')(x,0 )
+
(YU'(X, 0 ) = f ( x )
x E Zt,
f E 1'(Z;).
A heuristic argument based on a multiple scales [4]suggests that the continuum limit u(y), y E Rd, of uais the solution of the equation
Where qiJ is defined by
and the XJ are formally solutions of d
(3.4)
-
.z[V;(u,(x,
J=1
O)r:)]xk(x,0 ) = V;(u,k(x,0))
x E Z? .
The precise definition of the X's will be given below. Let us indicate briefly the physical interpretation of the quantities which appear in (3.3). (3.4) identifies Xk(x) as the steady temperature distribution due to a uniform heat flux that is generated by an applied unit temperature gradient in the k-direction. 6k,J V;Xk is then the temperature gradient in the J-direction induced by the applied unit temperature gradient in the kdirection. (3.3) identifies then qrJ as the average heat flux for unit gradient which is, by definition, the average conductivity of the medium. If the solution U' of (3.1) (for a! = 0) is going to become smooth in the microscopic scale we will have around each point a real temperature gradient {au@/ax,}. In this scheme we expect that C JqtJ (aue/axJ)is the effectiveheat flux and (3.2) the effective heat equation. To make the argument above rigorous one must first prove the existence of a unique solution for the heat flux induced by a unit temperature gradient. We state this, which is entirely analogous to the one in [4],as follows. There exists a unique set of functions @(o) belonging to L*(sd)satisfying
+
Rodolfo FIGARI, Enza ORLANDI AND George PAPANICOLAOU
116
5
(3.5)
E{GJ(J(0)(BkJ
J=l
+
fi)(Tej
- I)!&U)}
=
(3.6)
E($:(o)) = 0
(3.7)
E(+2CJ(Tei - I)$) = E($Z(TeJ - I>&
4
for any E L2(Q),J,k = 1, . . ., d. We want now to define the non-stationary potential P ( x , o) corresponding to the field (-J."J. Putting P(0, o) = 0, Vo, we can define Xk(x,o)by just adding the corresponding to a path of bonds having 0 as the starting point and x as the end point. Condition (3.7) just means that this definition is path-independent.
(4:)
Let us for example fix a specific order of the coordinates and choose a path reaching x going parallel to each axis i until reaching x1
with this definition
so that there is a constant B such that
TeJbeing unitary the von Neumann ergodic theorem implies that
converges strongly in L2(Q)and because of (3.7) it is Te,invariant for any k = 1, . - .,d and has average value equal to 0. We have then (3.9)
1 lim -E{(Xk(mx, 0))2}'/~ m
=0
.
m-m
Choosing for the path going to mx the m-times iterate of the path chosen from 0 to x, we have also m
Xk(mx,o) =
C TkXk(x,
0).
1=0
DiffusiveBehavior of a Random Walk
117
Because X*(x, o)is in L'(Q), the individual ergodic theorem tells us that X'"(mx, o)/m converges for a.e. o to an L'(sZ)function. Because we have L*(Q)-convergenceto 0 we get the result that for almost each o,Xk(mx,o)/m converges to 0. From here on the proof goes exactly as in [4] or [2]. It is first shown that the following is true. Let u'(x, o)be the unique solution of the variational problem
) of the equation Let ?i be the unique f 2 ( Z t solution
for all 6 E / * ( Z : ) where , the qrJ are defined by (3.3). Then lim
(3.12)
C
E{Iu'(x, w ) - ti'(x)r} = 0
r I O XEZ!
where +:(x, o) = (TX$9(o). Denote by Re(,?, o) E (LI")- A)-' and R,(R) = (La- A)-' the resolvents of L: and Et, where R $ R + . We can then restate the previous result as
For each also have (3.15)
E
and o,with the notation TJt,o) = etL: and T,(t) = etLiwe lim EII(T,O, 0 ) - w)fll:*cz.d) = 0 *lo
uniformly for t over compact sets. Let Q:( -,o) be the measure on D([O, m), Rd) corresponding to the Markov process x,(t) defined by the transition probabilities p,(x,y , t, a),x,y E Zf and by the initial condition
118
Rodolfo FIGARI, Enza ORLANDI
AND
George PAPANICOLAOU
P{x,(O) = x} = 1. Let us define in the same way Qf(-) as the measure on D([O, m), R d )corresponding to the Markov process whose infinitesimal generator is L, and subject to the same initial condition. One can rewrite (3.15) in the form
We have then shown that the difference of the one point distributions of the two processes goes to 0 when E goes to 0. This result extends immediately to all finite point distributions of the two processes. We now prove that both the Q:(. ,0)and the Q;( .) lay in a compact set of measures on D([O, m), Rd). (3.16) will then imply that they have the same limit points in the weak topology of measures, and if one of them has a unique limit, so does the other. A sufficient condition for compactness is (see [l], Chapter VI, Section V, Chapter 4) that lim lim sup {
(3.17)
h10
8 1 0 O 0, a, r E [O, 11) is the fundamental solution of the operator (1;/2)4 with the boundary conditions :
[
$(t,
i
0, r ) = -aP (t,
aa
p ( t , 0 , r ) = -(t, aP ao
1, r ) = 0
1, r ) = 0
1 p ( t , 0, r ) = p ( t , 1, r ) = 0
in the case (I) , in the case (11) , in the case (111) ,
and the second term of the right hand side of (2.7) is defined by a stochastic integral with respect to the cylindrical Brownian motion B,(a). We see that the equation (1.2) has a unique solution X,=X,(-) which is a %'-
[f
valued continuous process satisfying ~ u p ~ E ~ ~ ,iX,(a)Ydo] , ~ , each T
0 ) of independent %?-valued random variables such that each Y , has the same distribution as Y. Theorem 3. For each 0 < t, < t, < . . < t,, the joint distribution of { Y J - ;K ) } ; = ~ on the space V" tends weakZy to that of { Y t $ (. )} :=as, K-+
00.
Remark 2. In the case (11), the random variable Y( .) has a realization: Y(a) = a(Ao)w, with a d-dimensional Brownian motion w, starting from the origin with the time parameter E [O, 11.
0 4. Brownian strings in a potential field In this section we discuss the equation of a Brownian string in a potential field U ( x ) ( x E Rd),that is, the equation (1.2) with a(x) = 1, and b(x) = - (1/2)FU(x):
We give a stationary measure of the equation (4.1) explicitly and also investigate the equation in the case when the function U ( x ) diverges on some region in R d . (1) The stationary measure. The Hamiltonian of an elastic string X E %? in the potential field U with modulus x of elasticity is given by
Random Motion of Strings
127
With this H ( X ) , the equation (4.1) is rewritten in the following form:
where 6H/6X denotes the functional derivative of H ( X ) . This is an infinite-dimensional analogue of Einstein-Smoluchowski equation and is sometimes called Ginzburg-Landau equation in physical literatures (see [31). T o give the stationary measure of the equation (4.1), we set p the standard Wiener measure on the space C([O, K-'I, R d ) which satisfies the following conditions according to the cases (1)-(111). ( I ) p(X(0) E dx) = dx (.= Lebesgue measure on Rd), (11) p ( X ( 0 ) = A,) = 1 (i.e., the probability law of the Brownian motion starting from A,), (111) p(X(0) = A,, X(c-') = A,) = 1 (i.e., the probability law of the pinned Brownian motion). We note that p is an infinite measure in the case (I). Let p. be a measure on the space %? defined by p,(B) = p ( X ( r - ' .) E B ) for every topological Bore1 subset B of V. We introduce a non-negative measure v G us on the space %? by (4.3)
{
d v ( X ) = exp -
s'
U(X(o))do}dp.(X).
To show that v is a stationary measure of the equation (4.1), we assume the following.
Assumption 1. The function U is bounded from below and belongs to C1(Rd). The function V U is Lipschitz continuous. We denote by P,, the probability distribution on C([O,co), %?) of the the expectation solution X,of (4.1) with initial state X,E %? and by Ex*[-] with respcet to Pxo. Let Cb(%?)be the space of real valued bounded continuous functions on %? and let Co(%?) be the family of all f E C,(%?)which IX(o)l 2 M with some M = M ( f ) < 00. satisfy that f ( X ) = 0 if sup,,E.co,ll Theorem 4. Under Assumption 1, in each of three cases (I)-(III), the measure v is a stationary reversible measure of the solution X,of (4.1), that is, we have
T. FUNAKI
128
where ?hefunction space @(V)
Co(V)in the case (I) and C,(@ in the cases
=
(11) and (111).
This theorem is shown by using Theorem 1.
(2) The case with a divergent potential. The discussions here are limited to the case (111) which is interesting from the topological view point. We can define the measure v by (4.3) for a divergent potential U too, namely, U is a continuous function from Rd into (- 03, co]. We = 1. We set denote by Y again normalizing the measure to be ~(9)
D = {X E R d ; U ( X )< m} , D, = {X E R d ; U ( X )< N } ,
N
2
1,2,
* * *
,
and assume the following. Assumption 2. The function U is bounded from below and belongs to Cl(0). The function U is Lipschitz continuous on D, for each N = 1,2, . . .. Two points A , and A , belong to the region D and can be connected with a continuous curve in D. We also assume the following technical condition. Assumption 3. There exists a sequence { U(N'};=,of functions on Rd which satisfy the following four conditions. U ( N )satisfies Assumption 1 for each N = 1, 2, . . . . U")(X) = U ( x ) for x E D,. U(")converges non-decreasingly to V ( x )as N-+m for each x E Rd.
sup Je N
f j L'V(")(X(a))
(lo
dadP"(X)
0. U(x) = U(r) = with some 0
03
{(r - ro)-c
for r
ro ,
r
> ro ,
for
< r, < co and c > 2 where r = 1x1.
Random M o t i o n of Strings
129
Let D be the closure of the domain D and let 4 = CAo,A1([O, I], D) be the space of all X ( E % satifying X ( 0 ) = A,,, X(1) = A , and X ( a ) c D for every (i E [0, 11. Since the support of the measure Y is included in the space 8,the process X , introduced by Theorem 5 is a @-valued continuous process. The space @ has a decomposition according to the relative fundamental group ?cl of the connected component of B which include A,, and A , : qi. a )
Q=u zExl
This gives the following decomposition of the probsbility distribution P on C((- 00, w), 8)of the stationary process X,. Theorem 6.
We have
and P , is a conditional probability measure P(. I
where a$ = Y(%J
a(-
03, 0O>,
%,)I.
Since P , depends on the parameter
K,
we denote it by Pi,+.
( 3 ) ,The limit of P i , =as li -+ 00. The results of S. A. Molchanov enable us to investigate the limit of P 6 , Kas li -+ 00. We assume that d = 2 and D is a non-convex connected open polygon in R2. For each i E z,, let Ti( -)be a shortest continuous path in 9, which has a representation in terms of length. We also assume that the function U and the region D satisfy the following condition. U ( x ) 2 C (dis ( x , aD))-., x
Assumption 4. ( i )
E
D, with some C
> 0 and 0 < c < 1 + f l ,where aD is the boundary of D. (ii) For each i E r,, {u E (0, 1); T,((i)
Theorem 7.
8 5.
aD} is a finite set.
Under Assumptions 2, 3 and 4, we have that
Pi,+ -+ 6,;pszi where X$)((i)
E
= T,(a),
t
E
as
K -+ 03
,
R', (i E [0, I] and 6 denotes the Dirac's 8-measure.
Two dimensional Brownian strings
In this section we discuss the recurrent property of 2-dimensional strings with free edges. We assume that d = 2, K = 1 , a(x) = 12,b(x) = 0 and discuss the case (I) so that the equation is
dX,((i) = dBt((i)
+ 'dX,((i)dt . 2
T.
130
FUNAKI
This is a sort of Langevin equation and we can easily show that the solution X,is recurrent as an L2([0,I], R2)-valuedstochastic process. We get stronger results on recurrence stated as follows. Theorem 8. (i) The process X, is recurrent as a %-valued process, that is, for each X,E $fand non-empty open subset 0 of V, we have
P,,(X,,
E
0 for some t , f
03,
n
=
1,2, . . .) = 1 .
(ii) The string X , sweeps away allpoints in R', that is,for every x and X,E %, we have Px,(Xt,(u,) = x for some u,
E
[0, I] and t ,
co, n =
1,2,
-.
a )
E
R2
=1
.
To prove the theorem, we use a decomposition of the process X , : X,(O)
where
+ y , + Z,(fJ)
= X,t(fJ)
7
X,(u)is a non-random function introduced in
§ 2 and Y, =
J : ~ , ( u ) d u .We see that two processes Y , and Z , ( . ) are mutually inde-
pendent, Y , is a 2-dimensional Brownian motion and that Z , ( + )has the following property. Lemma. Let { v , ; t > 0) be a family of probability measures on the space %' = C([O, 11, RZ)induced by 2, and let v be that induced by
where w(u) is a 2-dimensional Brownian motion with a time parameter t sE [0, I]. Then v , tends weakly to Y as t + 03.
We can prove the assertion (i) by using the strong Markov property of the proeess X,and the recurrent property of Y , as an R2-valuedprocess. The assertion (ii) follows by noting the assertion (i). Remark 4. When d 2 3, since the process Y , on Ra is not recurrent, we can show that the solution X , of (5.1) is also not recurrent as a %'valued process.
0 6. Ornstein-Uhlenbeck theory for the string We consider the following equation:
13 1
Random Motion of Strings
I
UE[O,l],
tzo,
a>O,
where B,(a) is the cylindrical Brownian motion on L2([0,11, R d )and b(x) is assumed to be an Rd-valued Lipschitz continuous function defined on Rd. This equation represents the Ornstein-Uhlenbeck theory which describes the dynamics of the string with friction intensity a forced by white noise. We discuss only the cases (11) and (111) assuming A, = A , = 0. The operator - A with boundary condition (11) or (111) defined on the space L2([0,11, Rd)has pure point spectrum repeated according to multiplicity. Let $, = #,(a) be the normalized eigenfunction corresponding to 2,. Setting x , ( t ) = ( X t ( a ) ,$,(a)) and u,(t) = (V,(u), $i(u)) formally, the equation (6.1) turns into the following system, where, ( ., ) stands for the inner product in the space L2([0,11, Re).
jdx,(r) = u,(t)dt (6.1)’ {du,(t)
=
adw,(t)
+ ab,(x(t))dt - ER,x,(t)dt - avi(t)dt, 2
= 2 R;x: < -},
x = {xi};=l;x, E R’, Ilxll:
i=l
n = 0 , -1
we can show the following existence and uniqueness theorem of the solution of the equation (6.1)’. Theorem 9. For every initial state {x(O), ~ ( 0 )E)H, X H-’, there exists a unique solution { x ( t ) , u ( t ) } of the equation (6.1)’ such that SUP
tEC0,TI
E[IIx(t)IIi
and { x ( t ) , v(t)} E C([O,
+ Ilv(t>lI~ll
0) .
The parameter Q, > 0 represents the inverse temperature of the system. Physicists [3] are studying this type of equation with 3-dimensional para~ and with d replaced by the Laplacian ~ ~ = 1 2 / i 3 u ~ meter o = { u ~ } ; E= R3 in order to investigate the kinetic drumhead model of interface. The problem is to look for the limit of the solution of (7.1) as ,k?+ 03 tie., low temperature limit). The first step to answer this problem is to study the asymptotic behavior of the invariant measure Y()) of (7.1) as p ---f 03. By the result of 0 4, Y()) is given by d v ( b ) ( ~= ) 2;' exp
{ - B, 1;~ ( ~ ( o ) ) d o } d , u ( x ) x E v , )
Random Motion of Strings
133
where 2) is a normalizing constant and ,u is a probability distribution on %? = C([O, 11, R’) of the pinned Brownian motion X( .) satisfying X(0) = --A and X(1) = 2. Introducing a class { X e ( . ) ;f E [0, I]} of functions on 10, 11 by
we get the following partial result on the asymptotic behavior of dp).
Theorem 11. For every 6 lim P--
P( J 1
~ ( a)
> 0, we have
~ ( ( 0rda 1 > 6 f o r every f
E [0, I]
)=0 .
It seems to be an interesting but difficult problem to give the limit process of X , as p-+ 03. Remark 5 . The asymptotic behavior as E -+ 0 of the following equation was discussed by Faris and Jona-Lasinio [I].
i
dX,(O) = €dB,(O) - U’(X,(a))dt
+ dX,(O)dt ,
XJO) = X,(l) = 0 ,
where U is the function given by (7.2).
References W. G. Faris and G. Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A: Math. Gen. 15 (1982), 3025-3055. [ 2 ] T. Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math. J., 89 (1983), 129-193. [ 3 ] K. Kawasaki and T. Ohta, Kinetic drumhead model of interface I, Prog. Theoret. Phys., 67 (1982), 147-163. [ 1]
OF MATHEMATICS DEPARTMENT NAGOYA UNIVERSTY NAGOYA 464, JAPAN
Taniguchi Symp. SA Katata 1982, pp. 135-147
An Example of a Stochastic Quantum Process: Interaction of a Quantum Particle with a Boson Field Bernard GAVEAU
5 1. The study of field theories leads immediately to the study of infinite dimensional partial differential equations of Schrodinger type (the so-called Schwinger equations). These kinds of equations appear also in quantum statistical mechanics for systems which are not at thermodynamical equilibrium; in this context, a new class of problems has been introduced under the name of “quantum stochastic processes” but unfortunately, there are very few rigorously treated examples and models of such processes. I should like to develop, in this lecture, an example of a quantum stochastic process, which can be also considered as an elementary field theory. The general idea of a stochastic model is the following: we start with a purely deterministic model 2 with an infinite number of degrees of freedom, but we are interested only in a small part X I of this system. The evolution of I,, is controlled by the other part 2, of the system, about which we have only partial knowledge: thus, in a vague sense, X 2can be considered as “random” and 1,has some kind of random behavior, it exchanges energy with 2, (it can receive energy, and also loses energy, and in this last case, we speak of “friction”) and ZZcan be thought as a heat bath. In classical statistical mechanics, the situation is more or less clear (at least near equilibrium) and, in general it is useless to come back to an exact mechanical model of a heat bath; for example, in the Langevin equation, we have a heavy particle interacting with very light particles; in time At (which is short time for the observer, but a long time for the particle), the heavy particle loses momentum by friction at a rate -kpAt where k is a constant and p its momentum, and also the momentum varies by all the collisions with the light particles; if, in time At, the number of such collisions is very big, we can assume that the variation of momentum is given by d B ( t ) , (i. e. by a white noise) and in this way, we get a Langevin equation (1)
dp
=
-kpdt
+ d B ( t ).
Here we have not described the “heat bath” (i.e. the light particles) by
136
B. GAVEAU
classical mechanics but a rigorous theory should do so. Also, we have not considered the precise dynamics of the collision.
0 2. In quantum statistical mechanics, in non equilibrium situation, these questions are far from being clear and in fact not at all understood, in fact, if we try to adapt (1) in quantum mechanics, p becomes an operator and dB(r) should also be a kind of stochastic operator, but it is not clear what statistics we should impose on d B ( t ) ;the major problem is that (1) is already a macroscopic description, but quantum mechanics is, by its very construction, a microscopic description, even though the quantum statistics has finally macroscopic effects (for example, in conductivity problems, in laser physics and non linear optics, in some “chemical” models. . .). So, the idea is to try to come back to an exact mechanical system 2 and to look at the behavior of a small part by 8,forgetting the other bigger part Zz that we want to treat as a “heat bath”, or as a random system. The model that I should like to discuss here, is the following: El is a point quantum particle and we are interested in its behavwe suppose that 2, is in a ior; Z2is a boson field which interacts with Z,; state of equilibrium; here, to avoid difficulties, we will not consider thermodynamical equilibrium of 2, (so our model is totally caricatural); 2, has some kind of evolution under the influence of 8, and we are only interested in the behavior of 2, so we shall integrate on the degrees of freedom of XZ,or, in probabilistic language, take a conditional expectation with respect to 2,; in the case of a boson field these two operations are equivalent, but it is not, if we take for Zza fermion field or even a boson field in representation different from Fock representation. What is the interpretation of this model? From the point of view of statistical mechanics, we can think of it, as a motion of a particle in a crystal, the particle interacting with the phonon field which describes the quantization of lattice vibrations; in same sense, it is a polaron model [l]. From the point of view of field theory, we could think of a non relativitic particle with an electric charge, interacting with a quantized electromagnetic field (so the number of photons can vary because the particle reacts with the field, but the particle can never disappear as in true quantum electrodynamics) [21. Moreover we shall not be interested in renormalization problems. 0 3. Let us first recall how one can describe the interaction of a point charge with an exterior electromagnetic field. The Schrodinger equation is
137
An Example of a Stochastic Quantum Process
Here, m is the mass of the particle, e is its electric charge, his Planck’s constant, t is time, B is (J/ax, a/ay, a/az), is the wave function (I take a scalar particle), A is the external vector potential and V is the scalar potential. We can assume that we are in Lorentz gauge, so that
+
(3)
div A = 0
and if we write Fi = 1, m = 1, we get for the Hamiltonian
We shall use Kac’s idea to study this Hamiltonian; namely we shall study the heat equation
It is well known that we can write a probabilistic formula for this equation: call Ptq0the heat semi-group solution of (5); we have for V 0
=
In this formula, b,(t) is the 3-dimensional Brownian motion, Ep, is the mathematical expectation under the condition that b,(O) = p o ; in the Feynman-Kac exponential we see that there is an It6-stochastic integral (not a Stratonovich one); the fact that we use It6 intergral takes into account the term - (e2/2)I A 1’ of (4); moreover from the quantum electrodynamical point of view it is more natural to use It6 integral (because of a causality condition). It is rather difficult to get sensible estimate of (6); such estimates were given in [4] in the case of a particle with spin 1/2 and some symmetry condition on A ; in this conference, Professor Malliavin has given estimates on (6) for a scalar particle which are based on his stochastic calculus of variation; in fact, we can think of (6) as a Fourier stochastic integral operator, and the idea is to get an estimate of (6) just as in usual theory of Fourier integral, by integrating by part in Wiener space, using the Ornstein-Uhlenbeck operator for which the Gaussian measure Epois an invariant measure (see also [7], [S]). In this model we have a particle, and an external field acting on the particle, but the particle does not react on the field. We want now to consider the possibility of the reaction of the particle on the field. To
B. GAVEAU
138
do that we shall have to quantize the electromagnetic field.
9 4. a). Classically the electromagnetic field can be expanded in Fourier integrals (or Fourier series if the system is in a parallelepiped periodic boundary condition). We shall write
s
+
A(x) = p(k)-1/2(a(k)e-'k'x a*(k)e"k'x)dk
(1)
where the sign
s
denotes either a Fourier integral or a Fourier series, p(k)
in a given function of k , related to dispersion relations; moreover
where the e(')(k),i = 1,2 are orthogonal, length 1 and orthogonal to k and the a(Q(k)are the polarisation components. We shall group the terms in A(x) in the following way: let us note to
J
s*
an integral analogous
but where we identify k and - k and we write 4 w =
1 ~
p(k)l/z
(a@)
+ a*(k) + 4 - k ) + a*(-k))
p(k) = -iF(k)'/'(a(k) - a*(k) P(k) = p(k)'''(a*(k)
+ a(-k)
- a*(-k))
+ ~ ( k-) ~ * ( - k )- ~ ( - k ) )
(see Landau-Lifschitz [6]). With these notations we obtain (2)
[q(k)cos (kax)
Moreover we also obtain
and here, a*(k).a(k)is a scalar
+ Q(k) sin (kex)]dk
A n Example of a Stochastic Quantum Process
139
b). Let us now pass to the quantization of electromagnetic field; in are the annihilation (resp. creation) operator this case u("(k)(resp. a(*)*(k)) of a photon (or a phonon, or more generally a boson) of momentum k and polarisation e(')(k)(transverse vector to k) and the quantity given by the right hand side of (3) is the Hamiltonian of the free field; in fact for each i = 1, 2 and each k, (q(*)(k), p(*)(k))(resp. (Qct)(k),P(t)(k)))are set of two canonically conjugate operators satisfying
[q'"(k),q'J'(k')]= 0 [p")(k),p("(k')] = 0 [q@)(k),p(j)(k')] = isij8(k - k') and the same for Q(*)(k), P(j)(K). If we replace formally in the right hand side of (3) thep, q and P, Q by these operators we get 1 F(k)(a*(k).a(k) -
2
+ a(k).a*(k))dk
(because u and u* do not commute). Unfortunately this operator has a lowest eigenstate which has infinite energy and which is
J p(k)dk = + m
(in general) .
The idea is that we subtract this "quantity" from the right hand side of (3) and doing so, we get only
This is a trivial renormalization, even though the point 0 energy that we have subtracted is really useful from other points of view. Let us now fix i E {1,2} and k and introduce (5)
+
p[*'(k)* p(k)2q'"(k)2 - p(k)
9
P ( ~and ) qci)satisfying the commutation relations given above. The von
Neumann theorem about representation of commutation relation tells us that we can represent this operator as an operator of Ornstein-Uhlenbeck type
Here we have an infinite number of such operators ( 5 ) and the von Neu-
140
B. GAVEAU
mann theorem is false for such objects. But, nevertheless, we shall assume that the Hamiltonian (4) can be represented by a formal functional differential operator: namely we introduce for each k, variables q ( ( ) ( k )and Q(*)(&) and we define
This is the free field Halmiltonian and it acts on functionals T(g(k),Q(k)).
0 5. a). We consider now the interaction of a non relativistic quantum particle in R" with the boson field described above. The wave function of the total system is a functional Y(x, g(&), Q(&),t ) depending on x E R3, and of the coordinates of the field. Its temporal evolution is given by (7)
A is given by formula (2) and H F p by formula (6). In the case where x is particle in a crystal and A is a phonon field describing the quantization of lattice vibration, (7) is justified because the generalized momentum of the particle is p - eA instead of p . b). If we want to study the stationary states of (7), we have just to study the Hamiltonian on the right hand side of (7). Using the ideas of Kac [5] it is then easier to study the heat semi-group of (7) (or to go to imaginary time or to pass to so called Euclidean version); in this case we can write a probabilistic formula for this heat semi-group. More precisely let us introduce: (1) The classical Brownian motion b,(t) in R3 (generator 4 2 ) (2) For each k and each i let us introduce an Ornstein-Uhlenbeck process q:)(k, s) the generator of which is the Ornstein-Uhlenbeck operator
and also we suppose that q?)(k,s) are independent for i f j and & # k' and also independent of Brownian motion. Let us also introduce a second set of Ornstein-Uhlenbeck processes
An Example of a Stochastic Quantum Process
141
independent of the preceding processes and independent between themselves for i # j and k # k'. Let us finally denote
(we suppress the sample a). (3) Let us denote by E the mathematical expectation on the Brownian motion, E" the mathematical expectation on the degrees of freedom of the field, with the invariant Gaussian measure, namely the integration over 4(k, s), Q(k, s), these Ornstein-Uhlenbeck processes being distributed at time s = 0 according to the invariant Gaussian measures of their generators (and then at all time s). (4) Let us suppose that at time t = 0, the system is in the following state : the Ornstein-Uhlenbeck processes are distributed according to their invariant distribution and the particle according to some wave function T(x). Then at time t the particle has a wave function F(t, x) which is given by
I
(
F(t, x) = E,E exp -ie
.
J:
A(b,(s)).db,(s)
because of 5 3, but here, A contains in its expression the Ornstein-Uhlenbeck processes (see formula (2)). We can rewrite this as
+
Q 6. In formula (9), E" is a Gaussian expectation, do not depend on the degrees of freedom of E" and then we have an expectation E" of an exponential of a linear functional of a Gaussian process; we then have the general well-known rule
But the q(t)(k,s), Q@)(k,s) are independent
142
B. GAVEAU
E((J:S*. . .)’)
=
J* E((J~.. .)‘) .
In our case this quantity is exactly
where we denote
Moreover it is well known that
and the same with Q instead of q ; then the last computation gives
This is an evolution equation on the wave function p for the quantum particle in R3which we are studying. But this evolution contains a double Kac functional [3], so that this evolution is non Markovian. U p to now the computations are exact.
5 6. a). Let us first remark that e(‘)(k)depends only on the direction of k, and that in general we can suppose y(k) depends only on the modulus 1 k I.
Let us replace S*dk by
A n Example of a Stochastic Quantum Process
143
b). Let us also suppose that .\G. is bounded; we have then to study the quantity
X exp (- p(1 k \)Is - s")db?)(k, s)db?)(k,s'))]
.
If p is only in L2(R3)by Cauchy-Schwarz inequality we have to study the same quantity J ( t ) but with e2/4instead of e2/8. Because exp is a convex function we can put out of it the
[
du(k) and interchange this sign with
s
s2
the sign E and we can write that J ( t ) < do(ko)J(ko,t ) where J(ko,t ) depends only on k, in S2and is
(here bc)(k,s) = e@)(k) .b,(s) depends only on the direction of k). (c) But in the preceding formula for J(ko,t ) , b2)(ko,s), b?(k0, s) and ko.b,(s) are the three orthogonal projections of b&) on the three orthogonal directions eCt)(ko),e(')(k0),k,; these three projections are then three independent Brownian motions in R, so that the stochastic integrals in (15) are stochastic integrals of certain functions knowing ko.b,(s). By changing the frame of orthogonal coordinate, we can then suppose that ko is in direction 3, e(')(ko)and e(')(k0)in directions 1 and 2. Let us define the stochastic kernel
Then (15) becomes, using the independence of dbf)(s)and dbf)(s)
(17)
J(ko,t ) = E(exp ( - 1
where R = -(e2/4).
f f R,(s, s')db~')(s)dbt')(s'))) 0
0
B. GAVEAU
144
d). To obtain an estimate we shall compute (16) by discretizing the time, we shall also denote
So (oo +m
Ro =
p2dp
assuming that the classical dispersion relation p of the boson field is such that R, < m. We have then to study in the discretized version
+
Because this is a quadratic exponential in bj:) - b:iLl and because R,(st, s,) is independent of the first component of the Brownian path, we can partially integrate the first component of the Brownian path. Denote by 92 the n x n matrix the elements of which are R(s,, s,) for i # j and R, on the diagonal and denote As = si - si-l. The integral (18) is
where Ec3) denotes the expectation of the third component of Brownian path. The result is then (19)
J(k,, t ) = F3)((det(I
+ 2 1 A ~ 9 ? ) ) - ’ /.~ )
Let us now pass to the limit for As-+ 0; this means the following: we consider on [0, t ] the following integral equation depending on the random o ;
This integral equation has a symmetric, Hilbert-Schmidt kernel, so that its eigenvalues are discrete 1 p,(t) and
+
It is also clear that this kernel R is positive by its construction; in fact, we have
An Example of a Stochastic Quantum Process
=
145
+ sin pb(s) sin pb(s'))f (s)f(s')dsds'
1;"
P'dP."((J:aPY s) cos (Pb(s))f(s)ds)z
+
(s:
>
!XP, s) sin (Pb(s))f(s)ds)2) 09
where Q(p, s) is an Ornstein-Uhlenbeck process which is independent of b(s) and E" is its expectation. e). From this last fact we see immediately that J(k0Y
t)
< E((1
Now +==
C pt(t) = R i=l so that (22)
s:
+ EPa(t))-li2). i=l
R(s, s)ds = RtRo ,
J(k,, t ) < (1
+ RtRo)-1/2
from which we also deduce (23)
J(t)
< (1 + RtR0)-'/*
which means the mapping +(x) -+ +(t, x) given by ( 9 ) is a linear mapping from L" to L" or from L2 to L" and that its norm is decreasing like t - ' I 2 when t + + m .
5 7. Let us now describe some ameliorations of the preceding results: a). The mapping +(x) -+ +(t, x) is not a semi-group and the functional (9) is not a Markov functional; its memory is due to the degrees of freedom of the boson fields that we have integrated by saying that at time 0, there are no bosons around (vacuum states or integration with respect to the invariant Gaussian measure of the Ornstein-Uhlenbeck processes). b). Let us now give a slightly better estimate of the norm of the mapping (9) in a particular case. This particular case is the situation in
B. GAVEAU
146
which the quantized boson field has only degrees of freedom in the direction ko (and polarisation orthogonal to ko). This means that the sign
1*
in
(9) has to be replaced by
(so that we have a Dirac measure on the unit spahere instead of the uniform measure). We can suppose that ko is the z-axis and we are studying the scattering of a particle by a boson field having a given polarization. We replace +(x) by a Fourier integral +(x> =
J exp i(l,x)$(l)cil.
In (9) we can integrate first on the components x and y of the Brownian path and we obtain: $(l)E,E"(exp
[-4[ds
x exp (il,z&>) where z,(s) denotes the z component of the Brownian path and I = (Iz, I,, l,); finally q ( " ) ( p s) , . . - denote the degree of freedom of the boson
field for x-polarization and momentum p , E" is the expectation of the field and E, the expectation on z,(s) starting at time 0 at z. In this situation we can perform the E" expectation, in fact at that point we have reduced the problem to an ordinary Kac exponential in one variable but in a random potential depending on time given by the following formula:
Using this formalism, it is possible to obtain an exponential decrease in t of the norm of the mapping +(x) -P +(t, x) but the details are complicate and will be given elsewhere.
A n Example of a Stochastic
Qiraritiirii
Process
147
0 8. The preceding text is a detailed version of the second part of my talk at Katata, the first part was concerned with fluctuation for a chemical reaction and has already appeared in Journal of Statistical physics, 1982. Finally, the lecture at the Kyoto symposium was concerned with various problems on some points of field theory and has already appeared in other journals, one part of this lecture was concerned with the quantization of Yang Mills field with cut off in a non perturbative way (J. Funct. Analysis, 38, 1980, 324 and 42, 1981, 356); another part explained a new way of doing a renormalization by changing the Lagrangian frame in infinite dimension and is directly inspired by the work of Professor Leray and Professor Maslov (J. Funct. Analysis, 1982, and C. R. Acad. Sci. Paris, 295, 1982, 189 and C . R. Acad. Sci. Paris, 293, 1981, 469). I should like to express my deepest gratitude to M. Taniguchi and to Professor It6 for their kind invitation to attend this symposium. References [11 [2] [3] [4] [5 ] [6] [7 ]
181
R. P. Feynman, Statistical mechnics, Benjamin, 1972. Quantum electrodynamics, Benjamin, 1961. B. Gaveau, Estimation de fonctionnelles de Kac et de certaines fonclionnelles non markoviennes singulikres, C.R. Acad. Sc. Paris, 292 (198 1 ),
-,
577-580. B. Gaveau and J. Vauthier, Intkgrales stochastiques oscillantes: 1'6quation de Pauli, J. Funct. Analysis, 44 (1981), 3 8 8 4 0 0 . M. Kac, On some connections between differential and integral equations and theory of probability, Proc. Second Berkeley Symp. Math. Statist. Prob., 189-215, Univ. California Press, Berkeley, 1951. L. D. Landau and E. M. Lifschitz, Theorie quantique relativiste, Mir, Moscou, 1963. P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proc. Intern. Symp. SDE Kyoto 1976, ed by K. It8, 195-263, Kinokuniya, Tokyo, 1978. -, Sur les intkgrales stochastiques oscillantes, C.R. Acad. Sc. Paris, 295 (1982), 295-300.
D~PARTMENT DE MATH~MATIQUES TOUR45-46,s ETAGE U N I V E R S PARIS I~ VI 4 PLACE JUSSIEU,75230, PARISCEDEX 05 FRANCE
Taniguchi Symp. SA Katata 1982, pp. 149-167
Convergence in L2 of Stochastic Ising Models: Jump Processes and Diffusions Richard HOLLEY')
0 0.
Introduction
The purpose of this paper is to investigate the convergence of the semigroups of stochastic Ising models in the L2 spaces of their Gibbs (equilibrium) states. In particular we are interested in finding conditions which guarantee that the convergence takes place exponentially fast. Exponentially fast Lz convergence has several interesting consequences. In the first place it means that the system returns to equilibrium very quickly if it is displaced from equilibrium by a small amount. Secondly it gives quite a bit of information on the space time correlations when the system is in equilibrium. This information can in turn be used to study renormalization and the resulting limiting process. Finally, if the interaction is attractive (ferromagnetic) then it is sometimes possible to draw conclusions about the rate of convergence in the uniform norm from information about the rate of convergence in L2. Section two is devoted to a more detailed explanation of these consequences of exponentially fast convergence in L2. We are primarily interested in stochastic Ising models in which changes take place via continuous diffusion with reflection at the ends of a bounded interval, and we choose the diffusion and drift coefficients to minimize the technical difficulties in the construction of these processes. Section one contains the description of these processes and the corresponding Gibbs states. The semigroup of a stochastic Ising model converges exponentially fast in L2if and only if the generator has 0 as a simple eigenvalue and 0 is an isolated point of the spectrum. The former condition is equivalent to the tail field of the Gibbs state being trivial, so we concentrate on the isolation of 0 in the spectrum. Our tool for attacking this problem is a relationship between the spectrum of the diffusing stochastic Ising model and the spectrum of a stochastic Ising model in which all changes are made by jumps. The latter seems to be more tractable. Section three ')
Research supported in part by N.S.F. Grant MCS 80-07300.
R. HOLLEY
150
contains a description of the jump process and explains the relationship between the spectrum of the diffusion and jump generators. Finally in sections four and five we obtain sufficient conditions for 0 to be an isolated point of the spectrum of the generator of the jump process. The method in section four involves a study of the resolvent using the explicit form of the generator, and in section five we take an indirect route by finding sufficient conditions for the semigroup to converge exponentially fast in the uniform norm uniformly on a dense set of functions. The conditions we obtain involve considerable notation, so we postpone giving them precisely; however, basically what they say is that at high temperatures there is exponentially fast convergence in L2for both the diffusion and jump processes. $j1.
The diffusion stochastic Ising model
The stochastic Ising models with which we will be dealing are Markov processes on the state spaces E = l-IkEZdI,, where = [a,b], a k e d closed bounded interval for all k. We denote elements of E by x and xk E I, denotes the value of the configuration x at site k. The specific stochastic Ising model is determined by an interaction or potential which we will take to be a pair potential (@(s,j)} of finite range, R, (i.e. @ ( k , , ) ( xxk j, ) = 0 if Ik - j l > R). In addition we assume is translation invariant (i.e. @ ( k , J ) ( x k , x j ) = @ ( , , , , - k ) ( x , , , X j - k ) that {CD(~,~)} if x , = x, and x , = x , - k ) and that
This last assumption is made to minimize technical difficulties. An example of such an interaction is (1.2)
@(k,j)(xk>
=
-sin
(xk)
sin ( x j ) X ( l ) ( l
- j I),
where X, is the indicator function of A and we take each Ik = [ -(n/2), n/2]. The Gibbs states determined by the interaction {@(k,j)} at temperatures 1//3 are probability measures on the Bore1 sets of E ( E is given the product topology) which satisfy the following conditions: For A c Z d define a-algebras
F A= u { y , : k
E
A} and g A= u { x k :k $ A } .
Let PA(Xk:
I j l x ) = exp [-p(
@(k,j)(xk, k,j€A
xj)
+
kEA
ieA
@(k,j)(xk>
xj))1/z(n9
151
Convergence in L3 of Stochastic lsing Models
where Z(A, x) is the normalizing constant needed to make ,on(.Ix) a probability density in the variable . with respect to the normalized product Ik. Then ,u is a Gibbs state for the interacof Lebesgue measures on tion 0 at temperatures I/! (we write p E Yo,@) if ,u(. I g,,)(x) has density ,on(. 1 x)
(1 -3)
.
As will be seen below the elements of Y o , are @ stationary measures for the stochastic king model determined by { @ ( k , j ) } and 8. In order to define the diffusion stochastic Ising model we first let
{
9 = f : f E Cm(E),f depends on only finitely many coordinates,
and
af- (x) = 0
-
1.
if x, is an endpoint of Ik
In order to simplify the notation we set Uk(xk,x) = C jO ( * , J x Lx, j ) . For f E 59 define
In our example (1.2) this becomes (1.5)
9 f ( x )=
x (-((x)ax; azf + ,8 cos (x,) x
sin ( x j )
lj-kl=l
kEZd
~
af
-
(x))
ax,
.
Given x E E it can be shown that there is a unique measure Px on C([O,a), E ) such that for all f
E 9,f ( x ( t ) ) -
s:
9f(x(s))ds is a P, mar-
tingale (with respect to the natural a-algebras) and such that
P, (Lebesgue measure It: x k ( t ) = a or b} = 0)
= 1
for all k
and P,(x(O) = x) = 1. (See Quinn [lo]). The Markov family {Px:x E E } is the diffusion stochastic Ising model (see [12]). We let Tt be the semigroup of this Markov process, and it is clear from the definition that the generator of the semigroup, when restricted to 9 is given by 9. Let ,u E Yo,, and define P, =
1
P,,u(dx).
We conclude this section
E
by showing that P, is reversible, i.e. that (1.6)
P,W
E
r,,x(s) E r,)= ~ , ( x ( t )E rz,x(s) E r,)
R.HOLLEY
152
for all 0 5 s, t
0 such that forfE L2(p) (2.1)
IlTtf -
< I l f - P.(f)llLzip) e - ‘ ,
P(f)llLY,81
7
An immediate consequence of this is that if the system is perturbed slightly away from p then it returns to p very quickly. To see this suppose that Y is a probability measure with Y 0 (depending on {@,,,,)} and E ) such that for all A, c A, c Z d and a l l y e B which are 2FAomeasurable and g E
Convergence in L2of Stochastic Ising Models
153
Lz@)which are PAlmeasurable there is a constant A , such that (2.2)
IEJ"(~(t>lg(x(o)>l - , 4 f ) A g ) I 0 such that for all n
Convergence in L2of Stochastic lsing Models
< Ilf - pCln+(f)llLn(p:)
f 2 - c ( t - 1 l)g;
llLZ~t$)
155
7
where g,' is the Radon-Nikodym derivative of (T(")+)*6, with respect to p,'. It can be shown (see [lo]) that there is a R < 03 such that llg; 1 J L 2 ( r ~ ) I elna. Hence the first term on the right side of (2.4) is bounded by 2 sup,If(x)l e 8 e - ( 6 t - 2 n. d )Taking n = ( ( ~ / 2 2 ) t ) 'we / ~ see that the first term (2.4) will go to zero exponentially as fast as t + 00. on the right side of That leaves the other two terms. The second term involves only the equilibrium states, and whether or not it goes to zero and if so how fast has nothing to do with the stochatic Ising model. We postpone our discussion of it for a moment. The third term on the right side of (2.4) causes us the most trouble, and we have been unable to handle it in general. The problem is that even though we know that T?)+f(x) -+ T,f(x) as n -+ 03 for fixed t and x , we have already decided to let t go to infinity with n. The general theorem which we can prove in this situation requires that n goes to infinity at least as fast as a constant (>e) times t . This is clearly not the case for our choice of n. We get around the difficulty by adding another hypothesis which will allow us to forget about the third term. j , } is attractive, The additional hypothesis is that the interaction {a(,, or ferromagnetic. Specifically we assume that
This has the following consequences (see [9] Chapter 9). For all f e 9, which are increasing i.e. satisfy (2.8)
f ( x ) 2 f ( y ) if x , 2 y ,
for all k
E
(here a, = a and b, = b for all k E Z d ) and for all p
E
Zd
we have, for all t 2 0 and x E E,
(2.10) The latter consequence is a standard result about ferromagnetic systems (see [9]). The former, (2.9), is proved in [lo]; however it is intuitively
R. HOLLEY
156
plausible since (2.7) implies that if x, 2 y r , k E Z d and x ( t ) = y and d R ) + ( t= ) x then the drift at every site in A(n) is larger in the x ( ~ ) + ( - ) process than it is in the x( .) process. Thus since the diffusion coefficients are always constant, the two processes can be coupled in such a way that if xp)+(O) 2 x,(O) for all k then the same inequality holds for all time. We now take an f E 9 satisfying (2.8) and argue as in (2.4). (2.9) implies that
As in (2.6) the first term on the right side of (2.12) is bounded by ( f ( b ) - f(a))eae-('t-.'"d), and we denote the second term by F,(n). Then taking n = ((c/2R)t)'ld we see that for all increasingfc 9 there is a constant A , such that
In some cases (see Remark (5.16) below) it is possible to show that there is a constant ?' > 0 such that for all f~ .9 there is a constant B, such that (2.14)
I p m - p,(f)l I B F r n*
In this case we see from (2.13) that there is a number 8 all increasingfe 9 there is a constant C, such that (2.15)
sup I T t f ( x )- p ( f > I < C, exp.(-6t l / d ) X
> 0 such that for .
It is well known that (2.14) holds if d = 1. Thus in one dimension the semigroup converges exponentially fast in the uniform norm on the increasing elements of 93 if it converges exponentially fast in L2. § 3. Jump stochastic Ising models
In the jump stochastic Ising model each site jumps at the times of a Poisson process with intensity one. The Poisson processes used by different sites are assumed to be independent, and when a jump occurs at . where site k the distribution of the new position has density P ( ~ , (Ix(t)),
Convergence in L2of Stochastic Ising Models
157
~ ( tis) the configuration at the time of the jump. To describe the generator, 8, of this process we let 9 be as before and define 8 on 9 by (3.1)
=
cs1 k
(f(ZkY
a
u) - f(x))pk(Y I x)dy
*
Here (Zk, JJ) is the element of E which is equal to x at every site except
k, where it is equal to y. Let { S t :t 2 0) be the unique semigroup whose generators, when restricted to 9,is given by (3.1) (see [7]). By a computation very similar to (1.7) we see that if p E Y,,,, then forf, g E 9
x
k("3
v) - g(x))pk(Y 1 x ) d y d d x )
Hence again using the results of [ l l ] we see that Also
SZP
*
is selfadjoint in L2(p).
But
pk(zIx)-Vz. Since Uk(z,x) is bounded, c > 0. From (3.4) we obtain
R. HOLLEY
158
Integrating both sides of (3.5) with respect to p, summing over k and using (1.7) and (3.3) we obtain
Note that the inequality in (3.6) still holds for processes which have only finitely many sites such as those in the previous section. Our goal is to prove, under certain conditions, that for some E > 0 and for all f E L2(p) (3.7) This is equivalent to
In view of this equivalence and (3.6) it seems plausible that to prove (3.7) it should be enough to show that there is a constant E~ > 0 such that
J
-J f ~ ~ d2 pgo (f - p ( f ) ) ' d p
(3.9)
for all f e 9 .
That proving (3.6) is enough to prove (3.7) has been proved by Quinn [lo]. In turn (3.9) is equivalent to
and the last two sections are devoted to proofs of (3.10) at high temperatures.
3 4.
L2 convergence for jump processes
be fixed. In this section L2 always means L'@) and all Let p E expectations are computed with respect to p. Our first observation is that in L2, S is just a sum of projectives. To see this note that
is a projection, and that if we set Qk = I - Pg, then (4.2)
Qf(.>=
-C Qef(*) -
Convergence in L2of Stochastic lsing Models
159
(4.3) Remark. Throughout this section we will work with infinite sums even in some situations where it is not clear that they make any sense. To make all of this rigorous one should work with finite sums, get bounds which are uniform in the number of terms, and then pass to the limit. See [7], which is the basis for the ideas in this section, where this type of argument is done. We leave it to the reader to supply these steps. FOrfe Lz we define lllSlll= Clr II Q n f l l m Note that {f:lllflll < m) is a subset of the domain of W) and that
We begin by finding sufficient conditions to guarantee that (4.5)
IIlSJIII I e - t lllflll ’
In order to state these conditions we first define
and set
x Pa,’(% (4.8) Lemma.
U
v)dudv}
.
160
R. HOLLEY
where
). We now choose ha(?) and h,(v) to minimize M h o , h l ( ~(i.e.
and hI(v) = 1 - ha(7).) Call the resulting minimum M(7). Note that the factor not involving Mho,hl(ljl) on the right side of (4.10) is Jw,g - PkgY 1 5=Wl(.I). Thus
Convergence in Lp of Stochastic Ising Models
161
Therefore applying Lemma (4.8) we have (4.16)
(2
+ 1)II Q&ll
< II
Qgf
I1 + jC A,,j(Il Q4l+ /I Qjgll) +k
Summing (4.16) over k and recalling that g = R2f yields (4.17)
(2
+ 1 - 4llIRAfllI Il l l f 111 -
Finally the theorem is proved by using the formula
st
--
lim &A(AEA-C
(see [I] Chapter 13, Section 10).
2-m
(4.18) Corollary. If A < 1 and ,u has trivial tailjield (i.e. is an extremal point of .40m,s),then for all f E L2 (4.19) Proof.
IIStf
- P(flIIL.1I I l f - P(flll e-cl-A)t
Let 0 5 t, < tz. Then forfE 9,
*
R. HOLLEY
1 62
Hence SJconverges to something, say
4, in L2 and
(4.20) IlIS,flll = 0. i.e. Pk# = 4 for all Also 11141I1 = ll[limt4mStfill 2 ht-m k , and hence q5 is tail measurable. By hypothesis 4 must be constant and the constant is clearly p(f>. Thus we have (4.19) for all f e 9. Since 9 is dense in L2 the result follows (see [lo]).
(4.21) Example. In example (1.2) if d = 1 one can improve the above results to conclude that (4.19) holds if A = 3 4 , < 1. In example (1.2) with d = 1, 3 4 , < 1 if ,Q < 0.47. We don’t pursue this here though because the results in the next section allow us to conclude that (4.19) holds for Q, < 1 in this example. (4.22) Remark. Note that the results in this section make no assumption of attractiveness even though our principle application is to that case. In the next section we take another approach which relies heavily on attractiveness.
Q 5. The attraction case: Jump processes In this section we take an approach which has no hope of working if there is more than one Gibbs state. Nevertheless in the attractive case it seems to yield better results than those obtained in the last section. The general method that we use here would yield something for arbitrary finite range interactions ;however, we add the assumption of attractiveness in order to interchange an integral and an absolute value and thus obtain better results. The idea is to couple two jump processes in such a way that at any given site all jumps in both processes occur simultaneously and to show that no matter what the initial configurations are, the difference between the two processes at any given site goes to zero exponentially fast. We begin by constructing the coupled process. First note that if
and
163
Convergence in L' of Stochastic lsing Models
The coupled process is now constructed as follows. Let E 2 = E X E be the state space. We denote elements of E 2by (x('),x(')). Now define an operator b on the functions on E 2 which depend on only finitely many coordinates by (5.5)
Of(X('),
x(2)) =
cs' (f(PX,I x(')),Pn, I $b,(U
$b&4
k
0
- f ( x ( ' ) X'2'))dU ,
I f f really only depends on one variable, say x('), then bf(x('),d2)) = Q f ( x ( l ) )and hence the semigroup gt generated by b satisfies gtf ( x ( ' ) , = S,f ( x ( ' ) )for such functions f. (see [8] Section 2.1). We now assume that the interaction {@(*,,)} is attractive (or ferromagnetic). i.e. that (2.7) holds. It is easily checked that !.7) implies that if xp) 2 xy) for a l l j then for all y and z
From [9] we know that (5.6) implies that + k ( y \ x ( ' ) ) +*(y x@))for all y , hence (5.7) if xy) 2 xy) for allj, then for all u E [0, 11, $bk(uI x(')) 2 #&(u I x@)).
It follows that if xY'(0) 2 xY'(0) for all j , then xY)(t) 2 xiz)(t)for all j and all t 2 0 (see [3]). For a general review of coupling arguments see Liggett [8]. (5.8) Lemma.
If the interaction is attractive and
then for all f E 9 and all x('),x(') E E (5.10)
;:1
[ S t f ( x ( ' )) S,f(x")')[5 (b - a ) C sup -(x) k
x
I
e-('-a)t.
Proof. By the monotonicity mentioned above we see that
R. HCLLEY
164
To complete the proof it suffices to show that
+ (1 - R)xfZ)(t)for some R E [0, 11.
where z is of the form Rx(')(t)
(5.12)
X E,,,[xY)(t) = -(1
Thus
- xY'(t)] - E,,.[~f)(t) - xh"'(t)]
- CY)E,,,[XA''(~) - ~:'(t)],
since Eb,,[xY)(t)- xp)(t)] = E,,,[xk)(t) - x:)(t)] for all j by the translation invariance of the interaction. Finally (5.1 1) follows immediately from (5.12). (5.13) Theorem. rfthe interaction is attractive and (5.9) holds then there is only one p E Yo,) and for all f E L2(p), (5.14)
IIStf-
P(f)IILa(p'
I Ilf-
pU(f)IILl(p)e-(l-"'t
*
Convergence in L2of Stochastic Zsing Models
165
Proof. Since all elements of Y,,, are stationary for S, the uniqueness of ,u follows from Lemma (5.8). Since 9 is dense in L2(,u)(5.14) follows from (5.10) and the spectral theorem (see Quinn [lo]).
(5.15) Example. Again we consider the example @W(Xk,x,) = --in (x*>sin (x,)ql)(lk -ill with I* = [-(n/2), n/2]. In this case
where I, is the modified Bessel function of order 0. In order to get functions which we recognize we note that if in Lemma (5.8) we replace
and differentiate with respect to sin (xj) instead of x,, then the conclusion (5.10) still holds except the constant depending on f is different. The result is that if
then (5.14) holds. Thus we need to find ,8 so that sup, (d/da)(Il(~a)/Io(~a)) < 1/2d. = /Isup, (d/du)(Zl(u)/Io(a)). First note that sup, (d/da)(Il(pu)/Io(pa)) Using the relations (d/dz)I,(z) Thus we need to find sup, (d/da)(Il(a)/Io(a)). = Il(z), and 2(d/dz)I,(z) = Iv-l(z) Iv+l(z),and (d/dz)I,(z) = Iv-l(z) (v/z)Iv(z),we proceed as follows.
+
_d_ _ Il(4 _ --
4(Io(4
da lo@)
+ Iz(a>>Io(a)- I1(aIZ Z0(a>Z
=-(11 2
2Ida)Z - Io(4Iz(4 10(4z
Note that since 11(0)= 12(0)= 0 and Io(0)= 1, we have sup, (d/du)(Il(u)/Io(a)) 2 $. To obtain the opposite inequality it suffices to show that 211(a)22 Io(a)12(a)for all a. Consider 2Z,(a)z - Io(a)12(u). At a = 0 it is equal to 0 and its derivative is
R. HOLLEY
166
From the power series expansion of I , it is easily seen that this has the same sign as a, and hence
Finally we conclude that in this example if /3
< l/d then (5.14)holds.
(5.16) Remark. What lemma (5.8) shows is that if its hypotheses are , A , is satisfied then for all f e 9 sup,( S,f(x)- ,u(f)l 2 A , e - ( l - a ) twhere a constant depending onf. Just as in [5]it can be shown that this implies that there is a 7' > 0 such that (2.14) holds. Thus when the interaction is attractive and (5.9) holds we may conclude that (2.15) holds for the diffusion stochastic Ising model.
References 1 1 1 W. Feller, An introduction to probability theory and its applications vol. 2 John Wiley and Sons, New York, 1971. 121 D. Forster, Hydrodynamics fluctuations, broken symmetry and correlation functions, Benjamin Reading, Mass, 1975. t 3 1 R. Holley, An Ergodic theorem for interacting systems with attractive interactions, Z. Wahrscheinlichkeitstheorie verw. Geb., 24 (1972), 325-334. 1 4 1 R. Holley and D. Stroock, L, Theory for the stochastic Ising model, Z. Wahrscheinlichkeitstheorie verw. Geb., 35 (1976), 87-101. Applications of the stochastic Ising model to the Gibbs states, Commun. 1 5 1 -, Math. Phys., 48 (1976), 249-265. Rescaling short range interacting stochastic processes in higher dimen 161 -, sions, Colloquia Mathematica Societatis Janos Bolya, 27 Random Fields, Esztergom (Hungary), 1979. T. M. Liggett, Existence theorems for infinite particle systems, Trans. Amer. Math. SOC.,165 (1972), 471-481. -, The stochastic evolution of infinite systems of interacting particles, Lecture Notes in Mathematics, 598 (1977), 187-248, Springer-Verlag, Berlin-Heidelberg-New York. 191 C. Preston, Random fields, Lecture Notes in Mathematics, 534 (1976), Springer-Verlag, New York. T. Quinn, Ph. D. Thesis, University of Colorado, 1983. D. Stroock and M. Fukushima, Reversibility of solutions to martingale problems, Adv. in Math., to appear.
Convergence in L2 of Stochastic Ising Models
167
[12] D. Stroock and S. R. S. Varadhan Multidimensional diffusion processes, Springer-Verlag,New York, 1979. [13] W. G. Sullivan, A unified existence and Ergodic theorem for Markov evolution of random fields, Z. Wahrscheinlichkeitstheorie verw. Geb., 31 (1974), 47-56. DEPARTMENT OF MATHEMATICS COROLADO, BOULDER BOULDER,COROLADO 80309 U.S.A.
UNIVERSITY OF
Taniguchi Symp. SA Katata 1982, pp. 169-195
On the Asymptotic Behavior of the Fundamental Solution of the Heat Equation on Certain Manifolds Nobuyuki IKEDA') Dedicated to the memory of Dr. Hitoshi Kumano-go.
0 1. Introduction Let M + be an open d-dimensional manifold with (d- 1)-dimensionaI smooth boundary N endowed with a smooth Riemannian metric g +. We consider the case where there exists a symmetric double M = M +U N U M - of the Riemannian manifold ( M + ,g + ) . Let g be the Riemannian metric of M and A be the associated Laplace-Beltrami operator, i.e., in local coordinates (2,x2,. . .,xd)
where g&) = g,((a/ax,),, @/ax,),), G = det k i j ) and (g") = (gtj)V'. certain cases, the coefficients G-'/2a(gtjG'/2)/ax~ occurring in A jump as x crosses the submanifold N, but the heat equation
au ~
at
=
1 -Au, 2
( t , x) E (0, w ) x M ,
still has the continuous minimal fundamental solution p of (1. l), (see [25] and Section 3). In this paper, we are going to study short time asymptotics of the minimal fundamental solution p of (1.1). Following Varadhan [25],we have the following asymptotic formula: if the Riemannian manifold ( M , g) is complete, then for fixed x,y E M ,
x # Y,
where p(x, y ) is the distance between x and y induced by the Riemannian metric g . In [3], Buslaev took up the matter of finding corrections to (1.2) related to the "short-wave asymptotic behavior of diffraction" on This work was supported in part by the Grant-in Aid for Scientific Research.
170
N. IKEDA
smooth convex surfaces.') The problem is to find how the asymptotic behavior of the minimal fundamental solution p of (1.1) reflects the shape of the hypersurface N of M (also, see [17]). To illustrate the situation, we first introduce some geometric notions. We first impose the following : Assumption (i). N endowed with the induced Riemannian structure is a totally geodesic hypersurface of M , i.e., every geodesic of N is also a geodesic of M. If the metric tensor is smooth, it is well known that N is a totally geodesic hypersurface of M if and only if its second fundamental form a vanishes identically (e.g., see [l] and [16]). However, in case where the Riemannian metric g is not smooth, although N is a totally geodesic hypersurface of M , we may assume the following: Assumption (ii). For every V E T,(N) such that g,(V, V) # 0, a,( V , V ) < 0 for every x E N. As usual, a geodesic joining x and y is called an enveloping ray if it is a curve having more than one point in common with boundary N of M ' , (see Buslaev [3]). Under the assumptions (i) and (ii), Buslaev [3] was led to the following conjecture: for a wide class of manifolds it holds that if the geodesic #,,*(t), 0 5 t 5 1, joining x and y , (x,y E M' U N ) , is an enveloping ray, then as t 4 0 , where C is a non-negative constant depending on the shape of the hypersurface N , the arc-length of #z,u([O,11) n N and p(x, y). Although the intuitive background of his considerations is transparent, it is heuristic. Buslaev [3] used the concept of continuum product integral to show (nonrigorously) the asymptotic formula (1.3). Our main aim is to give a rigorous proof of the asymptotic formula (1.3) for a special class of manifolds, (see Theorem 1 in Section 3). The constant C in the right hand side of (1.3) will also be calculated explicitly. We restrict ourselves to the case where x, y E N and x # y . Under the assumptions mentioned in the following sections, by using skew products of diffusion processes, we will reduce the problem to the Laplace asymptotic formula for Wiener functionals similar to Schilder's one in [21] (also see Donsker-Varadhan [5], [6] and Dubrovskii [7], [S]). For details, see Lemma 4.5. As a result, we finally arrive at the calculation of the Wiener *) For a deep connection between the various types of second order partial differenal equations, see [IS].
On the Asymptotic Behavior of the Fundamental Solution
171
integral E"[exp
{- z j :
Iw(s)lds)/w(l)
for
= 01
z
>0 ,
where E W [ . / w ( l )= 01 denotes the expectation with respect to the Wiener measure with the fixed endpoint 0. As proved by Kac [13], the explicit expression for this integral could be obtained in terms of Bessel functions of order 1/3 (or the Airy function), (see also [14], [I51 and [23]). The organization of the paper is as follows. In Section 2 we prepare some geometric notions which will be needed later. The assumptions and main results are stated in Section 3. In Section 4, by using skew products of diffusion processes and the Feynman-Kac formula, we give a sketch of the proof of main results. Section 5 contains the details of the proof. In Section 6, we will give some comments.
3 2.
Preliminaries
First we will give some notations and notions which will be needed later. Throughout this paper we assume that manifolds are connected and a-compact. In general, for every mapping we denote by the Let $ be a curve in a manifold. Then we denote by 4 differential of the tangent vector field of the curve $, i.e., 4 = &(d/ds) where d/ds denotes the standard vector field on the real numbers. Let a : R' -+ R' be a positive continuous function satisfying the conditions :
+
+.
+*
(A.1): ( i ) a([) = a(lE]) for s' E R'. of the function a to [0, co) is a non(ii) The restriction increasing C"-function. (iii) a(0) = 1 and a+(O)< 0 where a + ( [ )is the right derivative of a at [. Let us fix a positive integer d 2 2. Let S be a ( d - 1)-dimensional smooth complete Riemannian manifold with metric g. Consider the product differentiable manifold M = R' X S with its projections rl : R' X s-+ R' and z2:R' x S -+ S. From now on we use the following notations x' = n,(x) and
X = z,(x)
for x E M .
We define a Riemannian metric g on M by gzv, Y)=
(2.1)
szl((d*(x), (rJ*(Y)) + &'>-'&((d*(m,(x*)*(Y)) for X , Y E T z ( M ) and
x = (x', X) E M
172
N. IKEDA
where g is the standard flat Riemannian metric on the 1-dimensional Euclidean space R'. The Riemannian manifold (M, g) is called the warped product of R' and S by the function a-' (cf. [2]). Then it is easy to see that the Laplace-Beltrami operator A on M has the form
A =L
(2.2)
+ a(x')iI
where
L
=
a ax'
A(xl)-l-
(A(x')-a
3
,
and 0 is the Laplace-Beltrami operator on S. Let us consider the submanifolds M " and N given by
M + = {x;x = ( X I , x) E M , x' > O}, M - = {x;x = (XI, x) E M , x' < 0 ) and N = {x;x = ( X I , x) E M , x' = 0 ) . Let g + and g - be the restrictions of g to M + and M - respectively. It is easy to see that ( M , g ) is a symmetric double of the Riemannian manifold ( M + g, + ) . Although the Riemannian metric g is not C', we can still define the notions of the arc-length Z(q5) of a piecewise smooth3) curve q5: [a, b] + M and the global Riemannian distance p(x, y ) between two points x and y in M as in case of smooth Riemannian manifolds. As well known it is easy to see that the arc-length I(#) is independent of the parametrization of the curve. For details, see Varadhan [25], [26] and Milnor [18].
Lemma 2.1. Take any two points x,y E N . Then, for every piecewise smooth curve T,!P : [0, 11 + A4 joining x and y , there exists a piecewise smooth curve q5: [0, 11 -+ N joining these points such that
I(+) L I(#) where the equality can hold only if+([O, 11) c N. Furthermore
p(x, y ) = p(X, 7 )
for
x = (0, x)
and y
=
(0,p)
where p is the global Riemannian distance on (S, g). By piecewise smooth we mean 4 is continuous and piecewise Cm.
On the Asymptotic Behavior of the Fundamental Solution
173
This is an immediate consequence of (2.1) and the assumption (A.l) and so the proof is omitted. Take two points x and y in M. As usual, we can also define the notion of geodesics joining x and y. Take a local coordinate (x2,x3,. . .,x") in 5'. Let Ttj be the coefficients of the Riemannian connection P in (S, g) with respect to the local coordinate (2,xs,. . ., x"). Now, keeping Lemma 2.1 in mind, we can show the following.
-
Lemma 2.2. Let us consider a local coordinate (2,x2, , x") such that x' E R' and X = (x2,x3, . .,xd) E S. Let # ( I ) = ( @ ( t ) ,$ ( t ) ) , 0 5 t 1, be a geodesic. Then (i) for every t such that # ( t ) E M\N,
for i
=
2, 3, . . ., d ,
where $ ( t ) = (#'(t), #'(t), . ., #"(t)) E S and a'(t) = da(E))/dCfort # 0, (ii) for every t E (a, b) such that #((a, b)) c N , #'(t) = 0
(2.5)
3 % dt2
-
Now let # ( t )
=
2
for i = 2, 3, . ., d
f:,($(t))&t)$X(t)
+
j,x=2
(#'(t), $ (t)), 0
5 t 5 1, be a geodesic of M and set
where k ( t ) is the inverse function of h ( t ) defined by h(t) = c
j' a(#'(s))ds ,
c =
([
a(#'(s))ds)-I
.
Then the curve $ ( t ) , 0 t 5 1, is a geodesic of S such that $(O) = $(O) and $(1) = $(1). This is an easy consequence of Lemma 2.2 and the proof is omitted. Before closing this section we give some remarks on assumptions (i) and (ii) in Section 1. We note that the Riemannian connection V + on M' can be naturally extended to the Riemannian connection V; on p.
N. IKEDA
174
Let a:T ( N ) x T ( N ) + T ( N ) I be the second fundamental form of the hypersurface N in ( Fg,) , i.e., for every X , Y ET J N ) and x E N , a,(X, Y ) is the normal component of (P:Y), (see [16]). Since, under the assumptions mentioned above, it is possible to choose a field of unit normal vectors globally on N , we can regard a as a mapping from T ( N )x T ( N ) to the space of C"-functions on N ([16]). Combining the considerations mentioned above we have the following: Proposition 2.1. 1) The hypersurface N in ( M , g ) is totally geodesic, i.e., every geodesic of N is a geodesic of M . 2) For every X , Y ET J N ) and x E N ,
where K
=
-a+(O).
The proof is straightforward and is omitted. Throughout this paper we denote by K the positive constant -a+(O).
0 3.
Main results
We consider the heat equation
with the initial condition (3.2)
u(O+,x)=f(x),
XEM.
Let p : (0, m) x M XM -+ (0, m) be the minimal fundamental solution of (3.1) with respect to the Riemannian volume m(dx), i.e., p is the fundamental solution satisfying the condition: Let f be a non-negative continuous function with compact support and u be a solution of (3.1) with (3.2). Then setting
we have
On the Asymptotic Behavior of the Frrndamental Solution
I75
From now on, throughout this paper, we assume the following: (A.2): There exists a bounded non-negative function b defind on [0, w) such that
(3.3)
1 - a(S) = KS - b(E)Ez
for
E E [O,
m)
I
Roughly speaking, the use of the stationary phase method combines the contributions to the asymptotic behavior of p from the various parts of the geodesic in a inultiplicative manner. Then, using the considerations of Varadhan [25], [26] and Molchanov [20], we can restrict ourselves to the calculations in a neighbourhood of the hypersurface N. Hence the assumption (A.2) plays a similar role to the following assumption: the function aIco,.,,is convex in a neighbourhood of the origin. Let 2, be the first eigenvalue of the eigenvalue problem (3.4) with the boundary condition (3.5)
d -u(A, dx
Of)
= 0.
It is well known that 1, > 0 (see 1121, 1141 and [23]). Let Q(3,p) be the set of all minimal geodesics of S joining X and 7,(n, 7 E S ) and set n(x, p) = #Sa(x,y ) , i.e., the number of elements of Q(X, 7). We are now ready to state the main results of this paper. Theorem 1. Let us assume (A.l) and (A.2). Take two points x = (0, X) and y = ( 0 , ~ )in N . V X and 7 are non-conjugate points along each element o f s Z ( ~7) , and I 2 n(X. 7)< CO, then
as t 5.0. Theorem 2. Let us assume (A.l) and (A.2). For every compact subset D of S, there exist positive numbers pt, i = 1,2, 0 < p, < pa < 00 such that
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176
uniformly in X, J
E
D with pl 5 p(x, J )
p2.
Remark 3.1. If M is a smooth Riemannian manifold, the second term in the right hand side of (3.6) does not appear in the asymptotic behavior of p . For details, see Molchanov [18] and MinakshisundaramPleijel [19]. Therefore the second term is a correction depending on how smooth the Riemannian metric g is. Remark 3.2. Let us consider the heat equation with the Neumann boundary condition
where ajan denotes the differention in the direction of the normal of N . Let p + be the minimal fundamental solution of (3.8). Since ( M , g ) is a symmetric double of the Riemannian manifold ( M + ,g'), we have
__
where x* = (-XI, X) for x = (XI, x) E M + . Combining this with Theorem 1, we can obtain the asymptotic formula o f p + ( t x, , y ) as t J 0. Before turning to the proof of Theorem 1 we give a typical example. Example 3.1. Let us consider the Euclidean space Rd endowed the standard flat Riemannian metric g and we denote by Sd-' the (d - 1)sphere of radius 1 endowed the induced Riemannian metric g from Rd. Let M + be the exterior domain of Sd-' in R d . Let (r, 01,02,. . ., t 9 - l ) be the standard polar coordinate in Rd and we define a local coordinate (XI,x2, . ., xd) in M' by
Then, in the local coordinate (x',x2, . . -,xd),the metric tensor g + = (g;) of M is expressed in the following form : +
On the Asymptotic Behavior of the Fundamental Solution
177
g:, = 1, g&(x) = u(x1)-1g&)
for x
=
(XI,X) E M' , and
for j
g&(x) = 0
where u(E) = (1
=
i, j = 2, 3,
. . ., d
2, 3, . . . , d
+ lE1)-z and
g&)
=
g3(a/axf,a/axj) ,
i,j = 2, 3,
. . -,d .
It is easy to see that the function u satisfies (A.l) and (A.2) with K = 2 and b(E) = (3 + 2E)/(1 + E)z. Let p' be the fundamental solution of (3.8). Now, by (3.9), we arrive at the following asymptotic formula: if X and J are not antipodal points on the Sd-' and x = (0, X), y = (0,J ) ,
as
t40.
We will again return to this example in the final section.
9 4.
Skew products of diffusion processes and the Feynman-Kac formula
Let W' be the space of all real continuous functions defined on [0, w). There exists a minimal one-dimensional diffusion measure { Q E ;E E R'} generated by L/2 (see [lo] and [ll]). We denote by Ef the expectation with respect to the measure Q , on W'. We also denote by E f [ - / w ( t )= 71 the conditional expectation with respect to the measure Q , on W' under the condition w ( t ) = 7 . Roughly speaking, this is defined by the usual formula
In Molchanov [20], a rigorous justification of various definitions of conditional processes with fixed endpoints was discussed in detail (cf. [22]). Let p be the minimal fundamental solution of
_a _u _- 1- - 2-4 , at
2
(tyX)€(O,w)XS
with respect to the Riemannian volume FFZ on S. We set
N. IKEDA
178
where $;'(w) is the inverse function of q 5 r ( ~ ) . Then, letting + ; I ( W ) inverse function of +,(w), we have
be the
Combining this with the formula (2.2) and using skew products of diffusion processes, we have
where k(t, E, 7 ) is the minimal fundamental solution of
(4.3)
-au - - L-u ,I at 2
(t,X)E(O,W)XR1
with respect to the measure A(C))df, (see [lo] and [ll]). Combining (4.2) with Molchanov's results (Theorem 2.1 and Theorem 3.1 in [20]) we have the following lemmas.
Lemma 4.1. Assume (A. 1) and (A.2). For every compact set D there exists a positive constant po such that for every p, with 0 pl as t -1 0,
c S,
< < po,
where Hl(f):[0, w) -+ R' is a positive continuous function with Hl(0) = 1.
0
for
t 2 to.
To avoid non-essential complexities, from now on, we assume (A.3). For the simplicity of notations, we set
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180
We now consider the l-dimensional Wiener measure P y on W' starting at E and denote by EY the expectation with respect to P y . Then, setting
for p
>0,
K(d - l)Jt .t(l, w;O)}/w(I)
=
X k(t, 0, 0)
we obtain the following Lemma 4.3.
For every p
> 0,
(4.9)
+
4
01
where t(t, w;z) is the local time at z of the l-dimensional Brownian motion { w ( t ) , t 2 0 ) and E r [./w(l) = 01 denotes the conditional expectation with respect to P r under the condition w(1) = 0. Proof. By using the transformation of drift by C and (2.3) it holds that for every bounded continuous function j
Setting u ( t ) = (log A(E))/2 and using Tanaka's formula (see III-(4.1) in [lo]), we can rewrite (4.10) in the following form
On the Asymptotic Behavior of the Fundamental Solution
181
qs'
- 2t
0
(1 - a(w(ts)))ds)z(s' a(w(ts))ds)-l}
Using the scaling property of the Brownian motion, we can again rewrite this in the equivalent form
(f)
(d-l)/Z
exp { - p ' / 2 t } E , W [ ( A 1 ( J f w ) ) - ( d - 1 ) / i
{
x exp - $ ~ ( 1 / tw)
+ a ( ~ , ( ~ ~ I) t . ) ) ) j
which completes the proof. The Feynman-Kac formula allows one to obtain the asymptotic formula related to the Wiener integral in the right hand side of (4.9). Let {An}, 0 < 2, < A, < . . . and {q5n} be the eigenvalues and the normalized eigenfunctions of the eigenvalue problem (3.4) with (3.5) respectively. Then we have
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182
For details, see [9], [12], [13], [14] and [23]. The following is an easy consequence of (4.1 1).
Lemma 4.4. As t
4 0 ,for every p > 0,
Furthermore, for every 0 in p with C, 2 p 2 C,.
< C, < C,
0,
Since u and C are bounded functions, it is easy to show that for every p > 0, as t .10, Er[exp { -p2A,(d tw)/2t}Gj1)(,lt w ; p)/w(l) (4.14)
=
0]
= E,W[exp { -pzA2(d 7 w)/2t}Gt2)(47w;p)/w(l) = 0](l
+ O(t))
, uniformly in p. In the following section, by using direct asymptotic evaluations of (4.15)
E,W[exp { -p2A,(dJt w)/2t}GjZ)(z/fw ;p)/w(l) = 01 ,
we will prove the following:
Lemma 4.5. As t J 0,fur everyp
> 0,
(4.16)
E,WMP {-pZA,(J7w)/2t}Gt2’(J -~ t w ; P)/W(I) = 01 E,W[exp { -P‘A,(Jt w ) / 2 t } ~ t ~ ywd;tp)/w(l) = 01
(4.17)
Er[exp { - p z A ~ ( ~ t ~ w ) / 2 t } G ~ 3p)/w(l) ~ ( ~ / t= w ;01 +I, E,W[exp { - p 2 A , ( d 7 w ) / 2 t } / w ( l ) = 01
~
~
1,
On the Asymptotic Behavior of the Fundamental Solution
183
and
(4.18)
where
(4.19) Furthermore, for every 0 < C, < C,< 00, (4.16), (4.17) and (4.18) hold uniformly in p with C, p 5 C,.
On the other hand, it follows from Lemmas 4.2 and 4.3 that under the assumptions of Theorem 1
(4.20)
x EY[exp (-p(x, y)2A2(Jfw)/2t}GI’)(J~i w ;p(x, r))/w(l) = 01 x Hz(P(4JMl + o(1)) *
The proof of the theorems. Combining (4.14), ‘(4.20) and Lemmas 4.4 and 4.5 we can easily complete the proof of Theorem 1. Furthermore, by using Lemmas 4.1, 4.4 and 4.5 and (4.14) we arrive at the conclusion of Theorem 2.
It should be noted that it is sufficient for the proof of (3.6) to show
0.
By the Markovian property of the Brownian motion, we have Pow"; Iw(s)lds 5 Azt1/6,€t-'/6 5 M,(w) 5 A,t-'/"lw(l)
5 Pow[aaZ(w)- a,(w) (= A2P, a,(w) < l / w ( l ) = 01 =
(5.14)
J-; PF[a,(w)
E
=
1
0
d s ] P L [ o , ( w )I (A#) A (1 - s)/w(l - s)
=
01
On the Asymptotic Behavior of the Fundamental Solution
x exp
= Jz(t, a),
I89
[- 2(1 -(t")2 ]dud% s - u)
(say).
E;
Since 3Aztd
< ( ~ t - "-~ t'>z
for every
t, 0
< t < tr ,
we have
Combining this with the inequality (t"Y
4 2 4 s - u) exp
[- 2 ( 1 - s - T ) I % z F for u
1
1
< (Aptd)A (1
- s),
we have
Hence, for every t with 0 equal to the following
< =
< t < t,,
8A2 d2nA,E
t3d/2
the last term of (5.14) is less than or
exp [ -A,/t 'I3]
5 exp [ -A s / t 1 / 3.]
This completes the proof. We now turn to the proof of Lemma 4.5. Take two any positive numbers 0 < C, < C, < 00 and fix an any positive number E . Take sufficient large constants A,, i = 1 , 2 , 3. Then, by combining Lemma
N. IKEDA
190
5.3 with Lemma 5.4, we can conclude that there exists a positive number t, independent of p with C, p 5 C, such that for every t, 0 < t < t,,
+ exp [ -A 3 / t 1 q By combining this with Lemma 4.4, we get
Since e is any positive number, this implies (4.18). Similarly, using Lemmas 4.4, 5.1, 5.2, 5.3 and 5.4, we obtain (4.16) and (4.17).
0 6.
Remarks
In this section, we give some remarks on the asymptotic behavior of p ( t , x,y) in the case where x = (0, X) E N a n d y = (y’, p) 4 N . For the sake of simplicity, we assume that there exists a unique minimal geodesic #(t) = ($(t), &t)), 0 5 t 5 1 , joining x and y. We also assume that the geodesic 4 is an enveloping ray. Then, as stated in the section 2, there exists tl, 0 t , 1 such that
<
:Xt(f) =
*
This is clearly an OU 9(g)-process with characteristic norm pa = p. Since X , : (g,p") + (L,11 llL) is isometric, it can be uniquely extended to It is easy to check that B = an isometric map Xt: (E,p) 4(L, /I \IL). { X t , t E T } is an OU 9(E)-process with charactristic norm p H = p. Since ( E , p ) is isomorphic to (Xt(E),11 [IL), our separability assumption of (L, 11 / I L ) implies that (E,p) is a separable Hilbert space. Using the natural injection 8 : E -+E we obtain X from X as follows:
XU>= mY)*
(2.7)
Definition 2.5. X is called the completion of X . If X = X, then X is called complete. Theorem 2.3.
X is a complete OU L?(E)-process, then
Ornstein-Uhlenbeck Processes
(2.8)
X,(E> = M 4 8
20 1
*
Proof. Since X = X and so ( E , p ) = (E,p), the argument above implies that ( E , p ) is a separable Hilbert space. Being isomorphic to (E,p), (X,(E), 11 llL) is also a separable Hilbert space. Hence X,(E) is a closed linear subspace of (L, (1 \IL). Thus we obtain
M,Q
= CLS(X,(E))= X,(E)
.
In the subsequent sections except in section 9 we assume that X is a complete OU P(E)-process, so that (E, p ) is a separable Hilbert space and X , ( a = M,(X).
9 3.
The characteristic operator
Let X = { X , } be a complete OU 9(E)-process, and p its characteristic norm. Then (X,(E), 11 IIL) is isomorphic to (E,p) and X,(E) = M,(X), as we mentioned in the previous section. Since Xis centered Gaussian and Markov, we obtain
m,(f 1I M,(X))
= -&X,(f
1I Mo(X)) E Mo(X) = Xo(E)
9
for t 2 0, so we can find a unique g = g ( t ,f ) E E R X d f 1I MIAX)) = Xo(g(t,f
Y
which implies that (3.1)
QXt
+B(f
1I M ; ( X ) ) = Xs(g(t9 f 1)
Y
Yt s 2 0
3
by the stationarity of X . Defining an operator S , : E -+ E by S,f = g ( t , f ) for each t 2 0, we can rewrite (3.1) as follows: (3.2)
~‘(X,+,(f)IM,(X))=Xs(S,f),
t 2 0 ,S
E T .
Theorem 3.1. { S t , t 2 0) is a strongly continuous semigroup of linear contractions on the separable Hilbert space (E, p ) . Proof: Sincef- ,!?(X,(f)lM;(X)) is linear inf, (3.2) (s=O) implies that S, is linear. Since P ( S , f ) = llXO(~tf)IlL= l
5 IIX(f)llL
l ~ ( x ( f ) l ~ m ) l l L
=P ( f )
Y
S, is a linear contraction in ( E , p ) . To prove the semigroup property, observe
K. IT6
202
( M i = M8 (r n ) (by M ; 2 Mi) = &jw,+,(f)lM;)IMo) = &~,(Stf)IMi) (by (3.2)) = XO(S8Stf) (by 3.2)) *
XO(S8.J) = &K+t(f)IM,)
So = I is obvious. Hence {S,, I 2 0) is a semigroup of linear contractions in (E, p ) , which is strongly continuous, because
P ( S , f - f ) = llXo(Stf) - XO(f)llL =
I1m 3 f ) - Xo(f>I MdX)) IIL
5 IIXt(f>- XO(f)llL
+
0
( t -1 0)
-
Now we will apply the Hille-Yosida theory [5] to {St}. Let A be the infinitesional generator of {St}. Since (E,p ) is a separable Hilbert space, A is characterized by the following three conditions : (A.l) A is a closed linear operator in E and D = 9 ( A ) is dense in E. (A.2) a(1- A ) = E. p ( A f , f)5 0, f e D. (A.3) (dissipativity) Also fED===+S,fED,
d
-Slf dt
=
AS,f = S,Af,
where the differentiation is taken in the sense of norm convergence in ( E ,p ) for each f. Definition 3.1. A is called the characteristic operator of the OU 2’(E)process X . Since X = { X t } is centered Gaussian, the finite joint distributions { X t ( f ) } t , , are determined by its covariance function, which is given as follows : Theorem 3.2.
203
Omstein-Uhlenbeck Processes
Similarly for the case s 2 t.
5 4.
Three Hilbertian seminorms on D
We define two Hilbertian norms q and r and a Hilbertian seminorm b on D = 9 ( A ) as follows:
(4.1) (4.2) (4.3)
+
q ( f , g) = P ( A s) P V f , A d . r ( f , g) = P ( ( 1 - 4 A (1 -
*
b ( A g) = - P M A g) - P ( A A g ) *
It is obvious that q is a Hilbertian norm and r and b are Hilbertian seminorms by the dissipativity of A . We can see that r is a Hilbertian norm by observing
Thus we obtain the following: Theorem 4.1.
(4.4) Theorem 4.2.
(D, q ) and (D, r ) are separable Hilbert spaces.
Proof. Let G be the graph of A , i.e.
(4.5)
G:
=
{(A A f ) : f ~D}.
Since A is a closed linear operator, G is a closed linear subspace of the separable Hilbert space ( E , p ) @ ( E , p ) . Hence ( G , p O p ) is also a separable Hilbert space. Being isomorphic to this space by the projection (f,Af) H A (D, q ) is a separable Hilbert space and so is (D,r ) by (4.4).
0 5.
The innovation process
Let D: = 9 ( A ) as in the previous sections. We define an LF(D)process B = {B,} whose increments {Bat}are given as follows:
where the integral is the Bochner integral. Definition 5.1. The S(D)-process B = {B,) is called the innovation process of X . This name will be justfied at the end of 8 8.
K. IT&
204
Theorem 5.1. The innovation process is a Wiener P(D)-process with characteristic norm b (b(f)2 = - 2p(Af,f),5 4). Proof. B = {B,}is obviously centered Gaussian. It is obvious that B J D ) c M:(x)
(5.2)
nM ; ( x ) ,
s
) L = m , , ( f ) m L , ( g ) IM;(X)N = 0 * Next we will check that
(5.5)
IIB8,(f)ll~= (t
- s>b(f)2*
To do this we will use Theorem 3.2: ( X s ( f > , X t ( d ) L = P ( f , St-sg)
9
t 2s*
Since there is no possibility of confusion, we will omit L and p in the proof below.
205
Omstein-Uhlenbeck Processes
- 2(f, Af) ( t - s) = (t - s ) b ( f y .
=
This proves (5.5). Now it is easy to derive the following from (5.4) and (5.5) :
( ~ ~ ~B(u. tm- )L= ~ KS,
(5.6)
ti n (u,~ I I ~ (g), . LL g E D
I
Hence B = {B,}is a Wiener 9(D)-process with characteristic seminorm b by Theorem 2.1. We denote by M;(dB)(resp. M:(dB)) the following subspace of
(L?II lid:
< u 5 2 , f €D} (resp. CLS{B,,(f): t 5 u < u , f e D } ) . CLS{B,,(f):
U
Similarly we define the a-algebras a;(dB) and a:(dB).
Theorem 5.2.
( i ) M;(dB) c M ; ( X ) ,
u,(dB) c a;(X) ,
M;(X) ,
u:(dB) 1a;(X) ,
(ii) M:(dB) where
1means orthogonality in (L, 11 /IL)
and IL means independence.
Proof. The statements for the subspaces of L are obvious by (5.2) and (5.3). Hence the statements for the a-algebras follow at once, because X is centered Gaussian.
K. IT8
206
Definition 6.1.
X d is called the deterministic part of X.
Hence X d is centered Gaussian. Proof. Use (6.3) and (3.2).
Proposition 6.4.
Hence X d is stationary by the centered Gaussian property of X d . Proof. Obvious from (6.7) and the definition of p d .
207
Omstein-Uhlenbeck Processes
Proposition 6.5. Mt(Xd)C M ( X d ) = M - , ( X ) = M , ( X d ) = M - _ ( X d ) .
(6.9)
Proof. The first inclusion relation is trivial. M t ( X d )C M - , ( X ) by the definition of X d . Hence M ( X d )C M-,(X), SO
c M ( X d )c M-,(X).
M - _ ( X d ) c M,(Xd)
Suppose that Y E M-,(X).
Then Y E M : , ( X ) , so we have
Projecting this vector to M - , ( X ) , we obtain
This implies that we can find Y, E M:.(Xd) such that J J Y- Y,JI Since Yk+, E M:.(Xd) for every k,
< n-'.
Y = lim Yk+, E M:,(Xd) k-m
for every n, so Y E M - , ( X d ) . Hence we have M-,(X) C M _ , ( X d ) , which completes the proof. Proposition 6.6. X d is Markov.
Proof. Because of the centered Gsussian property of X d it suffices to observe that for t > s
mxf)IM Q ( X d ) )
=X U ) =X3St-J) E
(by (6.9)) (by (6.7))
M,(Xd).
Proposition 6.7.
(6.10) (6.1 1)
Pd(StL Stg)
= P d ( Ag )
- w : ( f ) X : ( g ) ) = Pd(S,+uf,S,+ug)
for every u 2 max( - s, - t ) . Proof. By the definition of p d (Proposition 6.2) we obtain the first equality, which, combined with Proposition 6.4, implies the second equality.
K. IT6
208
Proposition 6.8. (6.12) llX1Z+h(f) - X?(f)IIL
5 IlXt+hW - X,(f)/lL + 0
(h
+
0;.
Proof. Obvious by the definition of X d . By Proposition 6.1, 6.4, 6.6, 6.8 and 6.2 we obtain the following: Theorem 6.1. X d is an OU 9’(E)-process with charactreistic seminorm p d and Xf( f)is a-,(X)-measurable for every t E T and every f E E. And so u(Xd)is independent ofa(dB) (Theorem 5.2 (ii)).
0 7. A special Wiener integral Let B = {B,} be the Wiener 9(D)-process with characteristic norm b introduced in 5 5. We denote its increments by {BSt},=
(7.1)
Y(Q,f>
We want to define an integral of the type:
First consider the case where Q, is a step operator-valued function of t vanishing outside of a bounded interval [a, b). Then there exists a decomposition of [a, b)
d:a=a, : = - (4g ) - (f,A g )
defines a Hilbertian seminorm on D = B(A). Hence we can construct a Wiener g(D)-process B with characteristic seminorm b (Theorem 2.1). Also we can assume that B and X d are independent. Define (9.7) We can easily prove that X = { X t } is an OU Y(E)-process whose deterministic part is X d and whose innovation process is B. Restoring the bar we denote this process by X. X is an OU 8 ( E ) process. Defining
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214
X( f ) : = x(8f ) (8: the canonical injection: E -+ E), we obtain the process X which is to be constructed. Proof of the uniqueness. Let X be any OU P(E)-process with characteristic norm p and characteristic operator A . p determines (E,p). A determines the semigroup { S t } , { S , } determines the covariance functional Vs. of the completetion X. V,, determines the finite joint distributions of X and so those of X .
9 10.
Continuous regular versions
In the previous sections we observed an OU P(E)-process X = { X , ( f ) } and its innovation process B = { B , ( f ) } which is a Wiener 9 ( D ) process. In this section we define their continuous regularizations to discuss the properties of these processes more thoroughly. c, cl, c2, stand for positive constants throughout this section. (a) A Gelfand triple {(&, rg), (D, I ) , ( K II ID}. In 8 4 we introduced a separable Hilbert space (D, r ) :
- -
D = N4, r ( f ) = P ( ( 1 - A ) f )
(10.1)
-
Let {d,} be an orthonormal base (abbr. ONB) in (D, r ) such that d, E 9 ( A 2 )for every n. Note that such an ONB exists because D(AZ)is dense in (D, I ) . Let {An} be a sequence such that
lAl":=cA;.
L>O,
(10.2)
n
Then K : = C Andn@d,
(10.3)
n
is a strictly positive definite Hilbert Schmidt operator in (D, r ) . Define a separable Hilbert space ( D K ,r,) as follows:
D,: = K(D) , r K ( f ) := r(K-I f ) =
(cr(f, dn)2A;2)1'2, n
rK(J;d:= r ( K - ' f , K-'g) =
C ~ ( fd ,M g , d n X 2. 7l
It is easy to check that (10.4)
The sequence
r(f)S
~rK(f),
c = SUP
{An}
. * >
The triple (10.5)
{(Dm y,), ( D , r ) , (H,
I1 ),1
*
is a Gelfand triple of Hilbert spaces. The ONB in ( H , 11 11) dual to {en} is denoted by {e;}, i.e.
(4,en> =
L 7 i
The a-algebra on H generated by the 11 11-open subsets (equivalently the a-algebra generated by the weakly open subsets or by the half spaces: { x E H : ( x , f) S a}, f E D,, a E R) is denoted by g ( H ) and is called the Borelsystem on H . We define Borel measurable functions on H , Borel measurable maps from H into H a n d H-valued random variables (abbr. Hvariables) on (Q, 9, P) with respect to 9?(H). ( x , y ) is Borel measurable in x E H for every f E D,. If Y(w) is an H-valued random variable, then ( Y(w),f ) is a real random variable. Let {S,} be the semigroup of linear contractions on ( E , p ) we have introduced in 5 3, and let A be its generator. When we restrict {S,} to D , we obtain a family of linear operators {Sf}. Since S,D c D and S,Af = A S , f for f E D , Sf carries D into D. Since
mf
r ( S f f 1 = P((Z 1 = P(SdI - A ) f1 SP((1- A ) f ) = r ( f ) , fED, Sf is a linear contraction on (D, r ) . Similarly we can prove the following:
Theorem 10.1. {Sf} is a strongly continuous semigroup of linear contractions on (D, r ) with generator A D := the A restricted to 9 ( A D ) : = 9 ( A 2 ) . The resolvent operator ( I - A ) - ' : E - t D gives an isomorphic map from (E,p ) to (D, r ) which transforms S, to Sf and A to AD. From now on we restrict our consideration to ( D , r ) and omit the index D in Sf and AD for notational simplicity.
K. IT6
216
(b) Stochastically r-bounded H-variables and r-bounded LRF's. Y(o) be an H-variable. Y is called stochastically r-bounded if E((Y,fY)SC:r(f)2,
Let
fEDK
Y is called centered Gaussian if ( Y,f ), f E DK form a centered Gaussian system. The family of all stochastically r-bounded H-variables is denoted by B ( H , r). B ( H , r) is a vector space with the usual linear operations. Let X( f ) be an LRF on D. X is called r-bounded if E ( X ( f I 2 )5 C$(f)Z
3
fE D.
The family of all r-bounded LRF's on D is denoted by 9 ( D , r). 9 ( D , r) is also a vector space. For Y E B ( H , r) we define
X : D,
4
L,
f-
X(f)(O)
=
(Y(o),f)
*
Since Y is stochastically r-bounded, X is a bounded linear map from (DK,r) into (L,(1 1 ) and so it is extended to a bounded linear map from (D, r) into (L, 11 11) because D, contains en,n = 1,2, . and so is dense in (D, r). The X , thus extended, belongs to 9 ( D , r ) . The map Y H X is a linear map from d ( H , r) into 9 ( D , r ) . This map, denoted by 'p, is injective. Suppose that 'pY = 0. Then ( Y , en) = 0 a s . for every n. Hence
--
P{( Y, en) = 0 n = I , 2, . . - } = 1 . Since Y(o) E H, this implies that P(Y = 0) = 1, proving the injectivity of 'p. In fact we have the following: Theorem 10.2.
'p:
g ( H , r) -+ 9 ( D , r) is bijective.
Proof. It suffices to prove that 'p is surjective, i.e. that for every X E 9 ( D , r) we can find an H-variable Y such that 'pY = X ; then Y EB ( H , r) follows automatically. Observing that
we obtain
C X(e,)z < n
Hence
00
a.s.
Ornstein-Uhlenbeck Processes
Y:=
C X(e,)eL
217
(in norm convergence in (H,
n
11 1 )
defines an H-variable. Observing that E(( Y ? f ) 2 )= E ( ( C x(efl>(eLf))')
c
4 E(C X(efl)z> (4,f ) z 5 c,"lJ12r,(fI2 n n for everyfc D,, we will check that (oY = X , i.e. (Y,f) = X ( f ) a.s.,
f~ D K .
This is obvious by the definition of Y for f = e f l and so for every f~ F: = LS (el, e,, . . .). For every f E DK we can find f f lE F such that (and so r ( f - f f l )d 0 by (10.4)) .
r K ( f - fn) --+ 0
Hence (Y,f) = 11 llL-lim(Y, fn) n
=
11
llL-limX ( f n ) = X ( f ) a.s. fl
Definition 10.1. For X E 2 ( D , r ) the H-variable Y : = q 1 X E g ( H , r ) is called a regular version of X . Theorem 10.3. Suppose that Y E 9 ( H , r ) . (i) ~(llY113< m, (ii) If Y is centered Gaussian, then (10.6)
E(llYll'")
s C(4E(IIYIlZY
s.
We can prove similarly as the above
E[IN,(x,1) - NAY, /1>lP1
s+ :
2 K’ It - sip" +
(
< K”(It - sp’z
+ luei - y - yue,~“Pdu ( x - y p + ( 1 - PI“”). [x
Then by Kolmogorov’s theorem, N,(x, 1) has a continuous extension at any ( t , x , 0). This means that ( f ( x ) , M ) , is continuously differentiable in x . Let 1 tend to 0 in (1.6), then we get
Repeating the argument inductively, we get the lemma. The following formula for changes of variables is a basic tool in our later discussions.
Theorem 1.4 (Ganeralized It6’s formula). Let F,(x), x E Rd, t E [0, T ( x ) ) be a local random field continuous in (t, x ) a s ., satisfying ( i ) For each t, F t ( . ) is a C3-mapfrom D,= { X I T ( x ) > t } into R’ a.s. (ii) For each x, F,(x), t E [0, T ( x ) ) is a continuous local semimartingale represented as Ft(x) = Fo(x)
+ 2 s’f..(x) j=1
0
dNi,
0
where N i , ’ . ., N r are continuous semimartingales, f j ( x ) , x E R d , t E [0, T ( x ) ) are local random fields continuous in (t, x ) a.s., C2 -maps from D , into R‘for each t as., and represented as fi’(x) = fl.’(x)
+
k=l
s’ 0
gJVk(x)dO:,
where Ot, . ‘ ‘, 0: are continuous semimartingales and g!sk(x),x B R d , x E [0, T ( x ) )are local random fields continuous in (s, x), C’-maps from D,into R‘. Let now M , = (M:, . -,M f ) , t E [0, T ) be a continuous local semimartingale and T’ = inf { t > 0; M , $. D,} A T. Then F,(M,), t E [0, T ’ ) is a continuous local martingale and is written as (1.7)
+ 5 [Lj(M.)
F,(M,) = Fo(Mo)
3=1
0
0
dNJ
+
Stochastic Partial Differential Equations
2.55
The proof is found in [5], Chapter I, Section 8 in case F,(x) etc. are global random fields. The extension to the present case is not difficult. The above is a differential formula for the composition of two stochastic processes. Writting m
dF,(x) = C j i j ( ~ o) d N j , j=1
the formula is written in short as
Hence it takes the same form as the classical differential formula for composite functions.
Q 2. Stochastic partial differential equation and the associated stochastic characteristic equation
A first order nonlinear stochastic partial differential equation of the parabolic type that we will consider is expressed as n
(2.1)
du, = C F(')(t,X , u,,au,) dW: 0
j=1
+ Fco)(t,X , u,,au,)dt .
Here, F(')(t,x , u,p), j = 1, ., n are continuous in (2, x , u, p ) E [0, m) x Rd X R' x Rd, continuously differentiable in t and C" +2-n-functionsof
>
( x , U,p ) fot some a 0. F(')(t, x , u, p ) is continuous in ( t , x, u, p ) and a C""."-function of (x, u , p ) . Here m is a positive integer greater than or
equal to 3. au, means the vector ( a p t , . ., adu,). W, = (W:, ., W;) is a standard n-dimensional Brownian motion. The equation may not have a global solution in general except for linear or semilinear equation. Hence we will define a local solution precisely.
Definition. Given a C1-function #(x), x E Rd, a local random field E [0, T ( x ) ) with values in R1 is called a local solution of equation (2.1) with the initial condition u,,(x, o) = $(x), if the following conditions are satisfied. ut(x); x E Rd, t
(2.2)
T = T(x,o)is an accessible, lower semicontinuous stopping time.
(2.3)
u,(x), 0
t
< T ( x ) is a local C'+-semimartingale and satisfies
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256
for all (t, x ) such that t < T ( x ) a s . If there is a solution of equation (2.4) such that the terminal time T(x,o)is infinite for all x a x , we will call it a global solution of (2.4). For convenience we put always W: = t. Then the above equation isiwritten as (2.4’)
u,(x) = #(x)
+ 5J‘ F(j)(r,x , u,(x), aur(x)) dW! . o
j=o
0
In the classical theory of deterministic partial differential equation of the first order, the characteristic curves associated with the partial differential equation play an important role. Our study of the stochastic partial differential equation of the first order is also based on the stochastic characteristic curves. As usual, we will use notations &,*‘ = a,F, F, = (F,,, . . ., FZd). Sim., F P J . For two vectors p , q, p . 4 ilarly, Fp, = aFIap, and Fp = (F,,, denotes the inner product. The stochastic characteristic equation associated with (2.4) is defined by
-
Stochastic Partial Differential Equations
257
+
where T ( x , u, p ) is the explosion time. SinGe F Y ) , FCJ)- Ff).p, FYI FLj)p,j 2 1 ( j = 0) are Cm+'~u(Cm*u)-functions of ( x , u, p ) , the solution has a modification which is a local C"Tfl-sernimartingale with any p less than a. Furthermore, for almost all w , the mapping ( f t ( . , w), v t ( ., w), C,(. ,w ) ) from the domain {(x,u,p)I T(x, u,p) > t } into R d X R'X Rd is a C"diffeomorphism for any 0 < t a s . See [ 5 ] , Chapter 11. Let $(x) be a function of C1+'+-classwith I m, which corresponds to the initial function of equation (2.4). We define local CIT#-semimartingales f,(x), q,(x), [,(x), t E [0, T ( x ) )where T ( x ) = T(x, $(x),a$(x)) as follows.
We shall study the process $,(x) in details in the remainder of this section. Processes T,(x) and [,(x) will be discussed at the next section. The map gt( -,w ) : { x I T ( x , w ) > t } into Rd is not a diffeomorphism in general, since the Jacobian matrix a,",(x) can be singular at some I less than T(x). Define
(2.8)
r(x) = inf { t
> 0; det aE,(x) = 0) A T ( x ).
It is an accessible, lower semicontinuous stopping time. Further, it holds lim,,,,,, det af,(x) = 0 if r(x) < T(x). We will show that f, defines a diffeomorphism if we restrict the map $, to the domain {r > t } . It is convenient to introduce adjoint stopping time of T ( X ) in the following manner.
> 0; Y 4 Et({r > f})}, where ,",({r> t } ) is the range of the set { x I r(x) > t } by the map F,. Lemma 2.1. ( i ) The map $, from the domain {r > t } into Rd is a C1-direomrophismfor all t a s . (ii) The inverse E;'(y), t < o(y) is a local C1-"kernimartingale and
(2.9)
a ( y ) = inf {t
satisfies
(iii) o ( y ) is an accessible, lower semicontinuous stopping time and satisfies that i f o ( y ) < 03
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258
Proof. Consider the following SDE for K , : (2.11)
drr, =
5
aEt(K,)-’F;j)(f,$, o c,,
4,
0
K,,
j=O
g, o c,) o d W i .
For any y E Rd, there is a unique solution K ~ ( Y ) such that ~ ~ (=y y) and K,(Y) E {x I ~ ( x ) t } for any t E [0,6(y)),where 6(y) is the terminal time of
>
It holds
c,(Y)which is accessible and lower semicontinuous.
We shall prove
E, o ~ , ( y=) y n
(2.13)
dtt(x) =
-CF j=O
for t
< d(y).
Note that E,(x) satisfies
L’) (6 $,(XI, at(x), Ux))
0
d 4
where the integrands FF) are local CzJ-semimartingales. Applying ItB’s formula to Faj), it turns out that they are represented as C;,OGif)(t, x)o dW2, where GLj)(t,x) are local Ct*fl’-semimartingaleswith p’ < p. Therefore we can apply generalized ItB’s formula to c,(x) and c,. Then it holds d($, 0 K,) = dct(Kt)
+ C a2$t(~,)dx: . 0
i
Observing (2.13), the first term of the right hand side is
In view of (2.1 l), the second term is
2a t t ( K , ) a ~ t ( C ~ ) - l F k ” ( f ,ct
0
K,,
0
K,,
j=O
n
=
C Fkj)(t, E,
0
K,,
9,
0
c,,
5,
0
5,
Therefore we have d(Et o K , ) = 0, which implies Define now
0
1)
,
If T = co, this means lim, &;‘(y)
= co.
0
dWi
.
Et o c,(Y) = y if t < Y y ) .
(2.14) ~(x)= inf {t > OlC,(x) 6 {6 > t } or ldet I)X(,$(,C~
< C(x).
Kt)
c,) dW{
j=O
We shall prove x, o E,(x) = x if t
0
= oo}Ar(x).
Since $,(c,(x)) = x, we see
S!ochastic Partial DiJ-fereritialEquations
259
~~,(K,(x))~K,(x) = identity matrix. Therefore, equation (2.13) is written as
We can apply generalized ItG's formula to the following SDE d ( ~0 ,$,) = dK,(C,) -I-
ca&,(ft)
0
lit
and
ct.
Then K , 0 f , satisfies
d;;
a
=
C ~ E , ( K o, $,)-'F:)(t, $,, 7, K , f , , T}into R d . We have thus proved (i) of the lemma. ) Now substitute x = t ; ' ( y ) to K , o E,(x) = x. Then we get ~ , ( y= c;'(y) if y E $,({I: > t } ) . Therefore $'; satisfies (2.10). The stopping time b satisfies that if 6(y) < 00
In fact, a$,($;'(x))i3E;I(x)
is the identity matrix and (2,12) is satisfied for Therefore what is remaining is to prove cr = 3. It holds t3Et(f;l(x)). {b > t } Et({r> t } ) since ~ , ( y = ) t;'(y) holds for any y E t,({r > t } ) . From the definition of K , , we have ~~((6 > t } ) C {I:> t } , so that we have f t o ~'((6 > t } ) c ,$,({I: > t } ) . Since Ct 0 K ~ ( x=) x holds on {b > t } , we get {b > t } c Et({r > t } ) . Therefore we have { b > t } = E,({r > t } ) for ) any y a s . The proof is complete. any t a s . This proves d(y) = ~ ( yfor
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260
Q 3. Existence and uniqueness of the solution We will now prove the existence and uniqueness of solutions of nonlinear stochastic partial differential equation of first order, using stochastic characteristic curves of the previous section. Main results are stated in the following two theorems.
Theorem 3.1. Let #(x) be a function of C1"*"-classon Rd with 2 I 5 m, a! > 0 and let t,)V L , 5, be processes defined by (2.7) and a(x) be the
stopping time defined by (2.9). Define u,(x) = q L ( & l ( x ) ) ) t E [0, u(x)). Then it is a local solution of equation (2.4). Further, the solution is a local C"-"~-semimartingalewith any B < a!. Theorem 3.2. Let u,(x), t E [0, T(x)) be a local solution of equation (2.4), where $ is a function of C1+lla-cIass.I f u,(x) is a local C2-'2"-semimartingale with 5 5 I 5 m, then it is represented as u,(x) = ?jt o E;'(x) for r E 10, T(x)Aa(x)). Since E, and 7, are uniquely determined by coefficients F ( J )j, = 0, . . .,m of equation (2.4) and the initial function #(x), the above theorem shows the uniqueness of the solution. It is conjectured that the solution u,(x), t e [0, a(x)) could not be prolonged beyond the time a@). We will
show this in case of quasi-linear equattion at the next section. In order to prove Theorem 3.1, we claim a lemma.
Lemma 3.3. It holds for i = 1, . . -,d
wit,,
(3.1)
a,q, =
(3.4
a,@, tF1) 0
=
[; 5;' . 0
Proof. For the proof of (3.1) set
We shall find a stochastic differential equation governing 8:. Observe the stochastic characteristic equation of q,. Since the integrands F") - F P( j ) are local C'*'-semimartingales, we can change the order of d and at and it holds da,q, = a,dij, by Theorem 1.2. Theorefore we have
261
Stochastic Partial Differential Equations
Similarly we have
+
d(5, *a$,) = d i f t 0 dSt
Ct
ddtCt
0
n
=
C {Fi".d,$, + F Y ' [ , . d , f , } o d W {
j=O
-
5(d,FF)(t,E,, C,,q t ) ) . C , o d W i .
j=O
Therefore, 8: satisfies the following linear stochastic differential equation n
d@ = C FLj'8; 0 dW{ j=O
.
Since f o ( x )= x,q,(x) = #(x) and C,(x) = d#(x), we have 8; = 0. Then the above linear equation has a unique solution 0: = 0. This proves (3.1). We will next prove (3.2). It holds
a(?, E;')(x)
= dT,($~'(~))d~,($;'(x))-'
= dT,(E;'(x))Jf;'(x)
0
.
In view of (3.1), the right hand side of the above equals C,(E;l(x))d$,(~;l(x))d~,(E;'(x))* = S,(E;'(x))
*
This proves (3.2). Proof of Theorem 3.1. We can apply generalized ItB's formula to Then we get
]7,(x) and $;'(x).
d(q,
0
$;'I
=
dq,(f;')
+ 2 d,~,(f;')
0
d(f;'Y
2=1
.
The first member of the right hand side is n
C F("(t, .,qt
o
f ; ' , g, o $;)'
o
dWi
3=0
n
-
C=o F y ) ( t , .,q1
o $;I,
3
g, o $;')c,
o
$';
o
dW/
because of (2.5). The second member equals n
C FY)(t, ., 7, C;', J=u 0
5,o
$;').d77,($,')d$,($;')-'
o
dWi
because of (2.10). We know from Lemma 3.3 that C,o
$';
=
a~,($;')a;',(,E;')-'
=
a(?, t;')= du, . 0
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262
Then we get du, =
5F(j)(t,.,7, t;',f, 0
0
f ; ' ) 0 dW5
3=0
=
5F'"(t,
3=0
, u,, au,) dW{ . 0
Therefore u, = 9, OE;' is a solution of equation (2.4). The fact that is a local Cz-'J-semimartingale is immediate. The proof is complete.
7,o E;'(X)
We will next consider the uniqueness of the solution. purpose, we require a lemma.
For this
Lemma 3.4. Suppose that u,(x)is a local C1-'~"-semimartingale. Then F(j)(t,x, u,(x), au,(x)) is a local C1-2'B-semimartingalewith /3 < a. Proof. By ItG's formula, it holds F'"(t, x, u , W , du,(x)) - F("(0, x, uo(x), auO(xN =
(3.3)
J: F W , x,u,(x), au,(x)> du,(x) + $ J: m3&x,u,(x), au,(x)) &us(x) 0
+ J p ( S , x, us(x),au,(x))dS. We will show that the first term of the right hand side is a local C1-',Bsemimartingale with ,B < a. For simplicity, we will write the integrand as F(s, x). Considering stopped process if necessary, we may assume that F(s, x), u,(x) and their derivatives up to ( I - 2)-times are all bounded and Lipschitz continuous. It suffices to consider the case that u,(x) is a Cz-',fl-martingale. Using Burkholder's inequality, we have
263
Stochastic Partial Differential Equations
Therefore we have the estimate
Then by Kolmogorov's theorem, tinuous in (t. x ) .
s:
F(Y,x)du,(x) is locally Holder con-
To prove the differentiability, we set for I NAX,
I) =
l -(J I
t
E
R'\{O},
+ Ie,)du,(x + Ie*) - J: ~ ( s x, ~ , ( x ) .}
~ ( s x,
0
Then we get similarly as the above
+ Ix - YIP + II - pup>
E[IN,(x,I ) - NLY, p)I"I I Z(It -
*
Hence N8(x,I ) has a continuous extension at I = 0, proving the differentiability of
s'
F(s, x)du,(x). Repeating this argument inductively, we
0
see that the integral is a Cz-2ij-martingalewith 19 < a. We can prove similarly as Lemma 1.3 that the joint quadratic variation ( F ( . ,x ) , u(x)), is a Cz-2~fl-process of bounded variation. Therefore,
s:
the Stratonovich integral
F(s, x ) o du,(x) is a C1-2J-semimartingale.
By a similar method, one can prove that the second term of (3.3) is a Cz- 27a-semimartingale. Hence we have the lemma.
Proof of Theorem 3.2. Let u,(x) be a local Cz-'~"-semimartingale solution of equation (2.4). Then in view of the above lemma, we can apply Theorem 1.2 and we get a,u,(x) = ai$(x)
+ 5 J &(F("(s, x, u,(x), au,(x))} Crws . 0
O'J
0
Then using I t 8 s formula to F ( j ) ( t ,x , u,(x),au,(x)), it is written as
F 1:GY)(s, x )
o
dW!, where GLj)(t,x ) are local Cz-2,8-semimartingales.
Consequently we may apply generalized It& formula to u, and we get
(3.4)
4%E J 0
=
(dUt,)
+ caP'(EJ i
O
dfE *
EL.
Then
H. KUNITA
264
The first member of the right hand side is
because of (2.1).
The second member is -
C= o au,(,E,) . F(i)(f, T,,< t ,
0
dW:
J
because of (2.5).
Therefore from (3.4) and (2.5), we have
We will regard that the above two equations (3.5) and (3.6) form a system Note that u o o ~= o To = 4 of SDE’s for u, o f , - T~ and d u , o;, and duo = to= 34. Then we see that 11, o $, - ?jL E 0 and au, 0 5, f 0 are the unique solutions of (3.5) and (3.6). We have thus proved the theorem.
c,.
0
$, 4.
et
Quasi-linear stochastic partial differential equation
A quasi-linear stochastic partiol difcwtitinl equntioiz is expressed as
Here, P;(t, x, ZL) and Q3(r, s,u ) , j = 1, . . ., n, are continuous in (t, x, u), continuously differentiable in t , C”‘+’+-funtionso f x and u for some a! > 0.
stochastic Partial DifferentialEquations
265
PXt, x, u) and Qo(f,x, u) are continuous in (t, x, u) and CmYs-functionsof x and u. Hence, using the notation of Section 2, it holds
which do not contain the variable p . Therefore in the stochastic characteristic equation for the quasi-linear equation, ( f t , 7 , ) satisfies a closed system of a stochastic differential equation n
dE, = -
(4.2)
cP,(t, cc, 7 , ) d W i O
9
j=O
n
dric = C Q,(t, E t , 71) dWl' 0
9
j=O
(5, is not involved). Let ( f t ( x , u), C,(x, u)), t E [O, Tl(x, u)) be the solution starting from ( x , u) at t = 0. For a Cziu-function$, we define similarly as (2.7), = Tdx, $ ( X ) ) ?
}
ft,
0
dW-l: 3
S J } odWZ' -
By ItG's formula it hods d { ( A t ,B,)(att,i3vt),} = 0. Taking the initial condition of (A,, B,) as (I,0) where I is the dXd-unit matrix, we get (A,, B,)(i3ft,i3qc))"= I. This proves that the rank of the matrix (a:,, a?,) is d for any t a.s. Therefore we have
lim {Idet 8$,($;'(x))l
t 1O ( X )
+ [aq,(f;l(x))l}> 0
if u(x) < 00
.
,
Since lim, o(x) ldet a$;l(x)I = 00, we have lim,, (.), det a$,($;'(x)) = 0. This implies limt,rt(x) Ii3qt(E;'(x))] > 0. Consequently we have
The proof is complete. Example (Communicated by Y. Yamato).
Consider a quasi-linear
Stochastic Partial Differential Eqriations
267
equation on R'. (4.5)
The associated characteristic equation is
It has a unique conservative solution v,(x, u) = u and f,(x,u) = x - u W,. Therefore, E,(x) = x - $(x) W , and q,(x) = $(x). Then it holds aE,(x) = 1 - $'(x)Wt and ~ ( x = ) inf {t > 0; 1 - $'(x) W , = 0). E;' and ~ ( x are ) not written explicitly in general. Consider the case $(x) = x. It holds a(y) = inf{t > 0; W,= I} if y # 0, = co if y = 0. Then $;'(y) = y(1 - W J - ' if t < ~ ( y ) .The solution of equation (4.5)is u,(x) = $&'(x)) = x(l - W,)-'. Hence we have lim, , o ( z ) Iu,(x)I = co if x # 0. Consider next the case $(x) = x2. It holds a(y) = inf{t > 0; 1 - 4yW, = 0) and $;'(y) = 2y(1 dl - 4W,y)-'. Hence the solution is u,(y) = 4y2(1 dl - ~ W , Y ) - ~ It .holds lim,, ,,(y) u,(y) = 4yz and lim,, .,(v) au,(y) = -co i f y # 0.
+
+
A quasi-linear equation (4.1) is called semi-linear if coefficients Pj(t, x, u) of equation do not depend on u. Then the characteristic equation of semi-linear equation takes the form
Hence the solution e,(x) defines a flow of diffeomorphisms of Rd. Since $,(x) = e,(x), Jacobian matrix i3ft(x)is always nonsingular and the inverse F;'(X) is well defined for all t, x. Therefore, we have the following. Theorem 4.2.
Semi-linear equation has a unique global solution.
We will finally consider a linear equation. called linear if coefficients Q,(t, x, u) satisfies Q,(t,
X, U) =
Qr)(r, X)U
A semi-linear equation is
+ Qy)(t, . X)
We will assume that QY),j = 0, . ., n are continuous in (t, x), Csta-functions of x (C4vafor Qo) and their derivatives in x are bounded. The process Tt(x)= v,(x, $(x)) satisfies a linear stochastic differential equation. It has a global solution and it is represented by
Consequently, the global solution of the linear equation n
du, =
C { P j ( t ,x)&,
j=O
+ Q(l)(t,X ) U , + Q(”(t,x ) }
o
dW{
with the initial condition uo = $ exists uniquely and is expressed by
We will rewrite the above formula using the backward integral.
0
For
< s 5 t, we denote by F,,,the least a-field for which W, - W,; s 5 u,
u 5 t are measurable. Now let t be a fixed time and fr be a continuous Then the backward Strabackward semimartingale adapted to (Sr,,). tonovich integral is defined by
where A = {s = to < . . . < t, = t } is a partition of [s, t ] and [dl = max [ t,+l - t , ] . We denote by E8,,(x)the solution of (4.6) starting from x at time s. Then the inverse f ; t ( x ) is a continuous backward semimarActually it satisfies the backward stochastic tingale adapted to (S8,t). differential equation
E;K4
=x -
c J: Pj(rY E;,*t(x)) JBi O
j=O
Furthermore, the following relation is satisfied
See [ 5 ] , Chap. TI,
0 7.
Similarly we can check that
Stochastic Partial Differential Equations
269
Therefore the solution (4.7) is represented by
QY)(s,c;;(x)) o d W!
X QP(s, E;t(x)) 0 dwt]
.
Acknowledgement. The author wishes to express his thanks to Y. Yamato for his suggestions and corrections to the first version of this article.
References [1] [2 ] [31 [4 ]
[51 [6]
171
J. M. Bismut, MBcanique alkatoire, Lecture Notes in Math., 866 (1981), Springer-Verlag. R. Courant and D. Hilbert, Methods of mathematical physics, 11, Interscience, New York, London, 1962. T. Funaki, Construction of a solution of random transport equation with boundary condition, J. Math. SOC.Japan, 31 (1979), 719-744. H. Kunita, Cauchy problem for stochastic partial differential equation arizing in non linear filtering theory, Systems and Control letters, 1 (1981), 37-41. -,Stochastic differential equations and stochastic flow of diffeomorphisms, 1’6cole d’CtC de probabilitks, Saint-Flour, 1982. S. Ogawa, A partial differential equation with the white noise as a coefficient, Z. W., 28 (1973), 53-71. -, Remarks on the B-shifts of generalized random processes, Proc. Inter. Symp. SDE Kyoto 1976, Kinokuniya, Tokyo, 1978. DEPARTMENT OF APPLIED SCIENCE KYUSHUUNIVERSITY 36 FUKUOKA 812, JAPAN
Taniguchi Symp. SA Katata 1982, pp. 271-306
Applications of the Malliavin Calculus, Part I Shigeo KUSUOKA and Daniel STROOCK*
9 0. Introduction This is the first in a series of articles dealing with the application of Malliavin’s calculus to various problems in stochastic analysis and the theory of partial differential equations. The present article is devoted, for the most part, to rather technical aspects of the basic theory. In section l), we begin with a resum6 of the Malliavin cluculus as it is developed in [ 8 ] . We then show how to incorporate the ideas of I. Shigekawa [6]into that framework. Section 1) closes with a rather careful examination of the regularity estimates which one can get on the distribution of functionals to which Malliavin’s procedure is applicable. In the second section, we show that solutions of It6 stochastic integral equations are “smooth functions” in the sense of Malliavin’s calculus. What is new here is that we allow the coefficients in our It6 equations to look into the past (i.e. the solutions need not be Markovian). Like section l), the second section is quite technical, but we see no way of justifying our conclusions without getting involved in such technicalities. At last, in the third and final section we begin putting all the machinery constructed in sections 1) and 2) to work. The main result of section 3) is that the distribution of the solution to a general It6 equation has very much the same regularity properties as that of a classical diffusion just so long as the coefficients of the white noise are non-degenerate. Although this result comes as no surprise, we do not know any other method of deducing it and believe it is a good example with which to illustrate the power of Malliavin’s calculus. Future articles in this series will concentrate on the Markovian case, where one can say much more. In particular, we will show how to complete the program, initiated by Malliavin in [4], of recovering Hormander’s renowned hypo-ellipticity theory for second order degenerate elliptic operators. Rather than attempting to describe here what we have been
* The research of this author was sponsored in part by N.S.F. Grant MCS 80-07300.
s. KUSUOKA
272
AND
D. STROOCK
able to do in this direction, we simply announce that we have not only recovered Hormander’s basic theory but also have been able to make certain extensions and improvements which may be a particular interest to probabilists. A preliminary version of our work can be found in section 8) of [9].
5 1.
The formalism of Malliavin’s calculus
The purpose of this section is to provide a brief review of the basic facts about Mallavin’s calculus. Some of this material is contained in [8] and therefore we will restrict ourselves to simply stating those results whose proof may be found there. Throughout, 0 will denote the space of continuous maps 8: [0, a) -+ Rd satisfying 8(0) = 0. Thinking of 0 as a Polish space (with the topology of uniform convergence on compact intervals), we use 27 to denote the Borel field over 0. For t 2 0, at denotes the o-subalgebra of a Finally, we use generated by @), 0 5 s 5 t. Clearly i% = o(Ulso g‘,). W to denote the standard Wiener measure on (0,g). That is,
r
for all t E [0, m), h > 0, and E g R a(the Borel field over Rd). with dense domain On L 2 ( Y f ) we consider a certain operator 9, Dom (P),called the Ornstein-Uhlenbeck operator. The operator 2’ is uniquely determined by the following properties : is self-adjoint; i) 2’ on Dom (9) 2 ’ admits a unique extension as a closed operator 9 on L * ( W ) ii) such that Dom (9) = {@ E Dom (9) n L ’ ( Y f ) : 2%) E L’(W)}; and F E CP(RD) iii) for any D 2 1, @ = (Q1, . . .,COD) E (Dom (2’))D, having bounded second order derivatives, F O@ E Dom (9). In particular, if V, V’ E Dom (Z),then V-K’ E Dom (9) and we may define the -+L ’ ( d f ) given by bilinear map (. , .)y: (Dom (2’))’ (1.1)
, =iv) for each 1 5 k 5 d and s 2 0, B,(s) E Dom (9) for all 1 5 I 5 d and t 2 0.
B,(s)/2, and (B,(s), B , ( t ) ) , = s A t
Applications of the Malliavin Calculus, Part I
273
To see that i)-iv) determine at most one operator is quite easy. Indeed, starting from iv) and making repeated use of iii), one sees that if such an 9 exists it must be the number operator (i.e. the space iWn) of nth order homogenious chaos is an eigenspace of 9 with eigenvalue -n/2). To prove that 9 exists (i.e. that the number operator has properties i)-iv)) requires some work. The interested reader should consult the first three sections of [ 8 ] . The bilinear map (., . ) 5 ' :Dom (9) X Dom (9) + L1(W)has some important properties. In the first place,
(0, O)z 2 0 , 0 E Dom (9).
(1.3)
From (1.3), it follows that
(1.4) where
From (1.4), we get ; (1.5)
-
Finally, as a bilinear map, ( -, (@-norm. In fact,
is bounded with respect to the graph
+ (2Z@)2]1/2Ew[V2 + (9V)z]1/2.
Ey[l(O,V),1] I EW[@'
(1.6)
For reasons which will become apparent shortly, it is useful to allow 9and to act on Hilbert space valued functions. To this end, let E denote a real separable Hilbert space. Given 0 E L7(W;E ) (we use L p ( W ;E ) to denote the measurable 0: 0 + E such that I l @ l l E E Lp($f)), we say that 0 E Dom (9; E ) if (0,e), E Dom (9) for all e E E and there is a V E Lz(W,E ) such that 9(0,e), = (V, e),, e E E. If 0 E Dom (9; E), we use 9 0 to denote that associated V. Next, given 0 E Dom (9; E) and an orthonormal basis {e,};. in E, define A,, = ((0,et),, (0,e,),),. Then, from (1.4), A:, < A,,A,, and so, by (1.6), EY[(C;,=,A:j)1/2] < E Y [ C L&,I EyQ@,et% (90, e,Y1 = E"[/I@IIL Il90ll"E. Thus, we can determine an element ((0,CD))~of L1($f;E 0 E ) by setting ((0,0))5' = Cz5=1Atjet @ e,. Making the usual identification of E @ E with H.S. (E; E ) (for Hilbert spaces El and E,, H.S. (El;Ez) is the Hilbert space of Hilbert-Schmidt operators from El to EJ, we see that ((0,O))s is a symmetric, non-negative definite, trace class valued map from 0 4 H.S. ( E ;E). We will use ((Q2 to denote Trace (((0,0))p)1/2. ( a ,
+
+
S. KUSJOKAA N D D. STROOCK
274
The fact that the operation (( -,.))Iis inherently quadratic is a source of difficulties. Thus it is useful to have a linear "square root" of this operation. The possibility of finding such a "square root" was first exploited by I. Shigekawa in [6]. We will adopt a variant of the Shigekawa approach. Denote by H the space of h E C([O, 03); R d ) such that h(0) = 0, h is absolutely continuous, and Jrn[ h'(t) 1' dt 0
Hilbert space under the norm llh![H=
1 and e:, . . ., e: E El, (yl, . -,yn) -+ F(et yje:) E Cm(Rn;E,), there is a continuous F(,) from El into the continuous multilinear maps Mn(E:; E2)for which
n;
+ C:
9
and for all n 2 0
II F(Ye:)ll.Mn(E;;E , ) I Cn(1 + IIe:llE1>rn for some C , < 00 and 7, < 00. (Here, and elsewhere, F(O) = F and Mo(E;; E,) = 4.1 In connection with the spaces C'p(El;E,), we need one more construction. Namely, given F E C;(El ; E,), and (& (Z!, e":)) E El 0*(EIY, note that, for any orthonormal basis {h,},", C; P ( e ~ ) ( Z ~ ( h.?;(A,)) j), converges in E, to an element which does not depend on the particular choice of {hj}?. Thus, we can define ( F ( ' ) ) : El -+ Mz(X(E1)Z; E,) to be given by the sum of this series. Clearly, (e:, e":, Pi) -+ (F(')(e:))(Z:,2:) can be thought of as an element of C;"(El0X ( E 1 ) ' ;Ez). Moreover, when F(')(e:)E H.S. ( E i ;E,) and therefore admits a unique extension F"@)(e:) as an element of Hom (El 0El ;E,),
(1.9) Theorem. If F E C;(El; E,) and G E C";E,; E J , then G O F E CT(El; E3). Moreover, if A : El 0 . . . 0En -+ E is a continuous nzultilinear map, then A E CT(El 0 . ' . 0En;E ) . Finally, if F E C";El; E,) and @ E X ( 9 ; El), then Fo @ E X ( 9 ; E,), D(Fo 0)= (F") o @)D@,and 3 ( F o @) = ( F " ) @(D@,D@)/2 (F'" 0 @)9@. 0
+
Proof. Only the final equation needs comment. First, note that if dim El < 03, then F @ )admits a unique extention F"(') as a map of El into Hom (El 6 El; E,) and ( F @ ) o) @(DO,DO) = (F"(') o @)(((@,@))). In par-
Applications of the Malliavin Calculus, Part I
277
0, set @* = (@* E')"'. Then 0
+ D((l/@JF) = - l/@D@Q 'P + l/@cDF
where 'P Q D@ is interpreted as an Z?(E,)-valued map by identifying P ( R ' ) with H and P ( E ) with H 0E. Using the closure property of the operation D, it is clear that this expression continues to hold after Q S is replaced by @ throughout. But then the induction hypothesis tells us that the resulting quantity on the right is in g n 0 ( 9Z?(E)). ; When no 2 (9; El. 1, a similar argument allows us to show that 9((1/@)?P)E Po-' Q.E.D. Remark. It should be pointed out that P. A. Meyer [5] has shown that the class g(9;E ) can be described without using the operation D. His idea is analogous to the well-known trick in finite dimensions which allows one to describe the Sobolev spaces entirely in terms of the Laplacian, without any mention of partial derivatives. Unfortunately, as with finite dimensional analogue, verifying that the L4(W;%"(E)) norm of
279
Applications of the Malliavin Calculus, Part I
Dn@ can be controlled by the Lq(W; E ) norm of entails a non-trivial singular integral argument when q # 2. For this reason we have chosen to avoid relying on his results. It will be useful to have some norms on the spaces Sn(9; E ) . In order to describe these norms, it is best to introduce in little notion. Let = 0 if IJ = 8 and IY] = n if 4 = {@}UU; ((0, l})n. For Y E 9, set Y E ((0, l})%. Also set [IJ]= 8 if IJ = 8, [Y] = {k:vk = 1) if Y # 8 I[Y]I = card ([IJ]),and l l ~ ~ = l l 211~1- I[IJ]I. Given @ E 9(9; E ) , define @(") = @ if Y = 9 and @("I = DylY'-yl.. D y n L P u n @if ~ I J J= n 2 1. Note that which we will often identify with E @ fPrVl1. takes values in fll[vll(E), For n 2 0, define III@IIlg),@ E P(9;E ) , by
and, for q E [2, w), set I I I @ I ] lg)E = 11 I [ I@I I I(En)jlLq(w). It is a quite easy matter to check that S n ( 9 ;E ) is complete with respect to {[[[. IIIgL: q E [2, m)) and in this way becomes a countably normed Frtchet space. Also, 9(9; E ) becomes of countably normed Frtchet space under (111 . I]&: n 2 0 and q E [2, w)}. I n addition to the preceding norms, we will have occasion to use the defined by norms 11
and, for q E [2, m), the associated norm obvious that the following relations hold;
11@11g)E= l ~ l ~ @ ~ ~ ~ ) ~ ~ L q ( v ) It.
is
Warning. When E = R', we will drop any mention of E in our R') will be written 9(9), 1 I I @ I I It?will be written notation. Thus, 9(9; I I I @I I I(,), etc. (1.14) Lemma. For each n 2 0 there exist constants C , f o r all m 2 2 , and A E M,(E, 0 . . 0E m ;E ) ,
<w
such that
s. KUSUOKA
280
AND D. STROOCK
In particular, for @, ZP E 9(9),
lK@, We-ll(")5 ~ n l l ~ @ l l ~ ~ l l ~ ~ l l ~ ~
(1.16)
*
Also, for non-negative @ E Y ( 9 ) satisfying l/@ E Y0(9)
II l/@ll(n)I Cn(ll ~ / @ l l ~ o ~ ~ n + l ~ l l @ l l ~ n ~ ~ "
(1.17)
*
Finally, given q, ql, . . ., qn
E
[2, w ) satisfying l / q = C 1"l/qL,
n lll@Llll~;E~ m
l I l 4 @ 1 9
*
. @m)lIlEL I . 9
C,m-lllAlldim(E1O...OE,;E)
(1.18)
II
4 @ 1 9
...
I
@m)lIEL
I
C,m-l I/A IIMm(EIO...@ E r n $ )
andfor 4, ql, q2 E [2, a) satisfying l/q = l/ql (1.19)
rn
I-I II
@1
Il&
;
+ l/qz,
I (@, We-llt-"'I ~7Ill~@ll~~Hll~~ll~~H II l/@ll?) I C"(ll ~ / @ l l ~ ~ + l ~ ~ l ~ n + ' ~ l l @ l l ~ ~ ~ ~ *
Proof. Given (1.15), (1.16), (1.17), (1.18) and (1.19) follow by Holder's inequality. Moreover, (1.16) is a special case of the second inequality in (1.15). To prove (1.15), it suffices to handle the case when m = 2 (the cases m > 2 follow by an easy induction argument). But, for n 2 1, D"(A(@,W)) =
2 ( 7 ) A(DL@,D-")),
1 =o
and so the second part of (1.15) with m = 2, follows immediately. T o prove the first part, we note that for w E 9,
where Ap,u,m E M2(21[p11(El) 0X1[v11(E2); X I [ " l 1 ( E ) )and has norm not exceeding 211"11 11 A [Idil(EIOEa;E). The proof of this representation is easily accomplished by induction on I w I ; one simply has to use =%4(@, 'y)>= &9@,
+ A(@,83p)+ ( A ) ( D @ ,DZP) ,
where ( A ) E M,(X(El) 0A?@); E ) is given by ( A ) ( ; ' , 2)=
5A(D'(h,),e"*(h,)) j=1
for any orthonormal basis {h,}," in H.
Applications of the Malliavin Calculus, Part I
281
In order to prove (1.16), we use induction on n 2 0 to prove that D"(l/@) =
~
Qrn
ml +
C ...
En(ml,
., m2)Dmi@0 . . . Q Dmn@
+ m n= n
-
where (En(ml, . ., m,)I 5 1 and we have identified X n ( R 1 )with H @ . Q.E.D. Clearly (1.16) follows easily from this. @)), and (1.20) Theorem. Let @ E g(9;RD) be given and set A = ((0, A = det A . Given a E ND\{0},there is a linear map .%?a: 9(9)+ 9(Y) such that
(1.21)
E ' [ ( ~ O @ ) ( A ~ ~ ~=~(-l)lulE~[(~~@)(~=T)] - ~ T ) ]
Hence, by Cramer's rule :
where A(") denotes the (ij)th cofactor of the matrix A . In particular, if ?€f E 9(9), then
S . KUSUOKAAND D. STROOCK
282
Using Lemma (1.14), one sees that there is for each n 2 0 a Cn(l) < 00 (not depending on @) such that (1.24)
I1 BiK ]I?) I Cn(l)(ll A I l t n g C _ l l ) ) q l ; R ~ ~ R ~ ) D - l x (11 IlglXCRD) + IISIRD>II Ilk")
+
for all q, ql,q2, q3 E [2, m) satisfying l/q = l/q, 1/q2 ticular, this proves (1.21) and (1.22) when ( a /= 1. Next let a E M Dso that
E
N o with n
=
la1 2 2 be given.
where i, = min {i: aim-.')> 0) . Given K 9(9)so that
KO= Y, Km = ABin-lYm-l + 2(n
E
+ l/q3.
In par-
Define a('),. . -,
9(2), define Yo,.
a ,
YnE
- m ) ( C A ( i n - l , j ) ( @ jA)2)€fm-l , j
for 1 5 m 5 n - 1, and Yn = Bin-lYn-l. Note that
Thus, by (1.23) :
if0
I m 5 n - 2, and
Therefore, (1.21) holds when we define BaK = Kn. Moreover, for 1 5 mI n - 1, one can combine Lemma (1.14) and (1.24) to find a C , < 00 such that for all q, ql, q2,q3E [2, m) satisfying l/q = l/ql l/q, l/q,
+
II Krn Ill."-"' I Cm(IlA l l k ~ x (I1 D@l;:;?b) for 1 5 m I n - 1 and
20-1
~
~
~
;
~
~
+ IIY@Il:rI$)
R
D
)
II v m - I l :-m+l)
+
Applications of the Malliavin Calculus, Part I
283
From these it is easy to deduce (1.22). (1.25) Corollary. Let @, A , and A be as in (1.20). Suppose that T E 9 ( 9 )has the property that T / A n E g ( 2 )for all n 2 1. Let YW denote the measure on (O,@’) given by (TW)(dO) = W(O)W(dB)and set ,u = ( T W ) 0Q-I. Then p(dy) = f ( y ) d y w i t h f e Y ( R D ) .Infact, $11 . I((m,n), M ,n 2 1, is the norm on Y ( R D )defined by
where A denotes Fourier transform, then Ilfll(rn,n)
I ll(1 + II@ II~~)m/2/1:0)lulI
= Er[(1 +
l @ l z ) m / z e f ( E(T/AZICI-I ~ ~ ) ~ >I
+ I yr)mlzei(f,v).All the assertions made follow easily Q.E.D.
When l/d E g0(S?);one can often obtain quite good estimates on the f and its derivatives. The basis of the analysis which we have in mind is the following simple real-variables lemma.
>
hasfirst (1.26) Lemma. Suppose that q D and suppose that g E L4(RD) order distributional derivative g , ] , . . ., g,= E Lq(RD). Then g E C,(R) and D
(1.27)
/lgllC,(RD,I c(q,
.I(&llg.llL*(RD))D’s
l l g l l Ll -CD( R / qD ) *
Proof. As a tempered distribution, g satisfies the equation
284
s. KUSUOKA AND D.STROOCK
where
This equation follows from the equation g = I G , * g - G,*(dg)
after integrating the second term by parts. As is easily computed, for q > D there are positive constants A(q, D ) < w and B(q, 0) w such that
\l?3
for all p E [ 1, w) and T
IIDt(t>ll:(E)
+ l19t(t)ll?3)Lp(W)
> 0.
r
(2.2) Lemma. Let a E 9(9; H.S.(Rd;E ) ) and Q, E F ( 9 ;E ) be given. Define C(T) = r a ( t ) d S ( t )
+
,k?(t)dt,T 2 0. Then [ E F , ( 9 ;E ) . In
fact,
f o r t 2 0 and h E H, and
+ j T9,Q(t)dt
9 E ( T ) = J T ( 9 a ( t )- a(t)/2)dB(t)
+ 3P-'TP-1 Proof. The proof of this lemma is eassentially the same as the proof
289
Applicaiions of the Malliavin Calculus, Part I
of Theorem (4.6) in [8]. That is, one first proves everything for the case when a and B, are simple. One then uses the estimates derived in this case, plus standard approximation results, to handle the general case. The required estimates follow easily from the stochastic integral expressions from D( and 2( together with Lemma (2.1). The only step that may cause problems, is estimation of the term
J:
a(t)h’(t)dt in the expres-
sion for D((T)(h). However, writing this term as (B(T),h)H, where
=I
B ( T ) :O+X(H.S.(Rd; E ) ) is given by B(T)(B) the X(H.S.(R; E))-norm of this term is ( J T
*AT
a(s, 8)ds, we see that
II(Y(~)~~L.~.(~~;~)
tribution of this term to the estimate for sup
IID((t)ilJP(E)is
the second
O 0. Given a compact metric space M and a separable Hilbert space E, let C X ( M ;E ) be the class of measurable E ; M X 0 + E such that ((Z) E X ( 9 ; E ) for all Z EM , E ( . ) : M - t E, DE(.): M + 2 ( E ) and YE(.): M -+ E are continuous for W-a.e. 8, and
Then the following is an easy consequence of Theorem (1.9).
s. KUSUOKAAND D.STROOCK
290
(2.5) Lemma. Suppose that M is a compact metric space, El and E, are separable Hilbert spaces, F E CSp(E,, E,) and C S ( M ; El). Then Fo E E C X ( M ;E,).
-
Given compact metric spaces MI, . -,M N and separable Hilbert let C"pC(Ml,El) 0 . 0 C(M,,>,); E") be spaces El, . -,EN and the class of continuous F : C(Ml, El) 0 0 C(M,, EN)+ E such that for any multi-index a = (a,. a,) with [ a1 = a, aN 2 1 and at:the map any g:!) E C(M,, Et), 1 2 i 5 N , 0 < j I a
---
a
( x p , . . ., xb:', xi,), . . .) X i , ) ,
--
+- +
. . . xi",) E R'"'
there is a continous 8""):C(M,, E l ) 0 . . . 0C ( M N E y N )+ Hom (C(M",E""); I?)
( M udenotes M,"' X . . .
X M B and EQa denotes EFalQ
. . . 0E
~ N )
for which
for g C i )E C(M,, Et), i = 1, . . y N . (2.6) Lemma. Suppose that F E C;(C(M,, El)0 . Then for any multi-index a, G,: C(Ml, El) 0 *
*
- . 0C ( M N ,E N ) ;I?).
- 0C ( M N ,E N )0 C(M", EQ")+ I?
291
Applications of the Malliavin Calculus, Part I
given by
G,
and g E C ( M " , E@'"),belongs to CT(C(Ml, El) 0 * * . 0 C ( M N ,E N )0 C(M", E@").E ) . Moreover,
if tiE C X ( M , , E6),i = 1 , . .
belongs to X ( 9 ; E"), N
(2.7)
N
p$-= C F('i) i=l
(2.8)
+2
N , then
[ Ey) 1
D$-= C F(ai) Z=1
a,
I:
L:)!
(ofi(**,)) , and
(f:?J [
W&(**i))
El(*,)
.- F ( a i f )
EN(*hJ
1
Here ai = (0, ., 0, i, 0, . . + , O ) , a i j = a, i f i j , and ( j , i ) i f i > j .
, +.
E(**))>Y
i,j=l
in E 0E uniformly on M X M. Thus, it is an easy matter to obtain (2.7) and (2.8) by passing to the Q.E.D. limit n +. 00 (cf. the last part of the proof of Theorem (1.9)). Given F: [O, m) X C([O, m), El)-+ E,, we say that F is a smooth, tempered, non-anticipating function if F is measurable: for each T 2 0 there is an P ( T ) E C3p(C([O, TI, El);Ez)such that F(T, +) = F(T)(+]LO,TJ for E C([O, w); El); and for each T > 0 and n 2 0 there exists C,(T) < 00 and T,(T) < w satisfying
+
293
Applications of the Malliavin Calculus, Part I
+
IIF(t )Y+) I1Horn ( C KO, t,n, Ep);E d I Cn(T)(1 + II II CCLO, t l ,E l ))Tn (0 for all 0 I tI T and
+
E
C([O, t ] ,El).
Warning. When dealing with smooth, tempered, non-anticipating functions F, we will use F(")(T,+), n 2 0, T 2 0 and E C([O,m), El), interchangeably with F(T)(")(+I c o , r l ) . Also we often use I[+~I c ( c o, r l , E l ) in and, in general, we think of E C(C0, m>;El) place of Il+l~o,~,llc(co.~lE,); E ) as being in as being in C([O, TI; El), and think of f E SC(9; CX(t0, TI; E ) .
+
+
(2.9) Lemma. Let a : [0, m) X C([O, m); RD)+ RD0Rd and b : [0, m) x C([O, m); RD)-+ RD be smooth tempered non-anticipating functions satisfying Ilb(n)(t? $)lIHom(C([O,t]n;
5
(2.10)
v 11 b ( n ) ( t$1~ /I
(RD)@ln);RD@Rd)
Cn(T)(l
+
+
~ o m ( C ( [ O , t ] n ;( R D ) @ ) ; R D )
II$IIC([O,tl;RD))"(')
0 and q E [2, co), there exists a K,(T) < co, depending only on C,(T) and C,(T) in (2.10), such that
for all0
0 and M,(x, R) and M,(x, R) are defined, respectively, as the supremum of ll4t, $F)IIE.S.W;;RD) and IlNt, +)IlRDfor 0 I t I T a n d
+
{$
E
c([o,
then there is a C,(T)
(3.12)
0 and y E RD. Replacing W in (3.1) by p,(X(t, x ) - x)W and using Ilpl.(X(t, x ) - X)Whn,+l)
I C,Ilp,(X(t, 4 - ~)lli;~l)ll~lli;:l)
Y
one gets (3.12) from standard estimates on YP'-(supoI,~,~X(s, x) - X I 2 r/2) Q.E.D. (cf. Theorem (4.2.1) in [lo]). (3.13) Remark. It is gratifying to see that the upper bounds just obtained on P ( t , x, .) are, at least qualitatively, the same as those which one would predict on the basis of the heat flow semigroup. Unfortunately, we have no good idea how to obtain corresponding lower bounds, although we are confident that good lower bounds exist. The best which we can say for sure is that for each ( t , x) E (0, w) X RDand each strictly positive W E 3(9),{ y E RD:pt(t, x , y ) = 0 ) contains no interior points. This inadequate observation is an easy consequence of the fact that if P, = W o X ( . , x)-' on C([O, w); RD),then supp (P,)= {W E C([O, w); R): W(0) = X} (cf. [7]). (3.14) Examples. 1) The basic example to which our results apply are non-degenerate, time-inhomogeneous diffusions. That is, suppose that (r: [0, w ) X RD-+ RD 0Rd and b : [0, w ) X RD3 RD are measurable functions such that: a(t, .) E Cy(RD;RD X Rd) and b(t, .) E C y ( R D ;RD)for all 't 2 0; and for each T > 0 and n 2 0, there exist C,(T) < w and Y,(T) < w for which
I Cn(T)(1
+ IIYIIRD)""~)
Y
E
RD.
Assuming that Y,(T)= 0 for all T 2 0, one easily checks that for each (s, x) E [O, w) x RD,there is a unique X s ( ., x) satisfying (3.15)
X'(T, X) = x
+ lTv'a(t, X'(t, x))dO(t) a
+
rv'
b(t, X s ( t , x))dt , T 2 0 .
8
Set P(s, x; T, .) = *T0 X 8 ( T ,x)-', (T, x) E [s, w) consequence of uniqueness, we have
w(x'(T,X) E r I g t )= ~ ( txyt, , x ) ; T, r)
x RD. Then, as a (as., w )
s. KUSUOKAAND D. STROOCK
304
for all 0 5 s < t < T and x E RD. In other words, A?(., x) is a timeinhomogenious Markov process with transition function P(s, x ; T,.). Moreover, from ItB’s formula, we see that for any q5 E C,”(RD),
where
Hence, P(s, x ; -,. .) satisfies (at least formally) the Fokker-Planck equation (3.17)
a -P(s,x; t, * .) = LFP(s,x;t, * * ) , at
t 2 s,
where L f is the formal adjoint of L,. If we now add the assumption that aa* 2 E I R D for some E > 0, then Theorem (3.5) and Corollary (3.9) apply, and we see that P(s, x ; T,dy) = p(s, x ; T,y)dy where p(s, x ; ., .) E C((s, m) X RD)andp(s, x ; T, .) E Y ( R D for ) each t s. In particular, we can now say rigorously that
>
<
0, then Theorem (3.5) and Corollary (3.9) apply to the solution I(., x), of
and we conclude that P(T, x, .) = YY o X(T, x)-' has a density p ( t , x, E Y ( R Dfor ) each T > 0. It is not clear to us how one could deduce such a result by any other technique. In some special cases, it is possible to embed X ( . , x) as an infinite dimensional Markovian system which is sufficiently tractable by more standard Markovian methods (cf. the last section of [9]). However. even in such cases the reasoning is quite involved; and it seems to us unlikely that such methodology can be made to work in general. a )
(3.19) Remark. It would not be difficult to show that p"(t, x,y) is smooth as a function of (x, y ) E RD x RD. The technique which we have in mind is essentially the same as the one used in section 7) of [XI. In fact, by being a little careful, one can show that
for O < t < T .
306
S. KUSUOKA AND D. STROOCK
References D. L. Burkholder, A geometrical characterization o Banach spaces in which martingale difference sequences are unconditional, Ann. of Prob., 9 (1981 ), 997-1011. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, N. J., 1964. S. Kusuoka, The nonlinear transformation of Gaussian measure on Banach space and its absolute continuity (I), J. Fac. Sci., U. of Tokyo, Sec. IA (Mathematics), 29 (1982), 567-596. P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, Proc. International Conf. on Stoch. Diff. Equ. at Kyoto (1976), Kinokuniya-John Wiley & Sons, Tokyo and N. Y., 195-263, 1978. P. A. Meyer, Quelques resultats analytiques sur le semi-groupe d'ornsteinUhlenbeck en dimension infinite, Theory and application of random fields, Proc. IFIP-WG 7/1 Working conf. at Bangalore, ed. by G. Kallianpur, Lect. Notes in Cont. and Inform. Sci., 49 (1983), 201-214, Springer-Verlog, Berlin. I. Shigekawa, Absolute continuity of probability laws of Wiener functionals, Proc. Japan Acad., 54 (1978), A, 230-233. D. Stroock, On the growth of stochastic integrals, Z. Wahr., 18 (1971), 240244. -, The Malliavin calculus, a functional analytic approach, J. Funct. Anal., 44 (1981), 212-257. -, Some applications of stochastic calculus to partial differential equations, Bcole d'&e de Probabilitks de San Flour 1981, ed. by P. L. Hennequin, Lec. Notes in Math., 976 (1983), 268-382, Springer-Verlag, Berlin. D. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, 1979. Shigeo KUSUOKA O F MATHEMATICS DEPARTMENT OF TOKYO UNIVERSTY TOKYO 113, JAPAN Daniel STROOCK OF MATHEMATICS DEPARTMENT UNIVERSITY O F COLORADO, BOULDER CAMPUS BOX 426 BOULDER, COLORADO 80309 U.S.A.
Taniguchi Symp. SA Katata 1982, pp. 307-332
Stochastic Flows of Diffeomorphisms Yves LE JAN and Shinzo WATANABE
0 1.
Introduction
As solutions of an ordinary differential equation (O.D.E.) on a manifold define a one-parameter subgroup of the group of diffeomorphisms called a dynamical system, solutions of a stochastic differential equation (S.D.E.) define a continuous random motion of diffeomorphisms called a stochastic dynamical system or a stochastic flow of diffeomorphisms. It is also called a Brownian motion on the group of diffeomorphisms since it is a continuous motion on the group with independent increments. The study of S.D.E. from this view point, especially that of the dependence of solutions on the initial value, was initiated by Soviet school e.g. Gihman and Skorohod [6] and Blagovescenskii and Freidlin [3] but it is rather recent that diffeomorphic property of solutions has been established by Bismut [4] and Kunita [9] (cf. also Elworthy [5], Malliavin [12] and Ikeda and Watanabe [S]). More recently, Brownian motions on the group of diffeomorphisms or homeomorphisms themselves have attracted attention and have been studied by e.g. Harris [7], Baxendale [2] and Le Jan [l I]. In these works, main problems are first to specify infinitesimal quantities which characterize Brownian motions (i.e. continuous motions with stationary independent increments) on the group of diffeomorphisms and secondly to construct Brownian motions from these infhitesimal data. It should be noted that the construction problem can be discussed by the method of S.D.E. but we need rather general S.D.E. based on Wiener processes (i.e. continuous Gaussian processes with stationary independent increments) on the space of vector fields. Also in the characterization problem, the infinitesimal data characterizing a Brownian motion can be given as data describing a Wiener process on the space of vector fields. The purpose of this paper is to discuss similar problems in more general framework of continuous random motions on the group of diffeomorphisms than Brownian motions. For example, if we consider the composite of two Brownian motions on the group of diffeomorphisms or the inverse of a Brownian motion, they are no longer Brownian motions, but are continuous processes which might naturally be called semi-martin-
Y. LE JAN
308
S. WATANABE
AND
gales on the group of diffeomorphisms. For such semi-martingales, we can associate uniquely a semi-martingale on the space of vector fields called the velocity field. Conversely, a semi-martingale on the space of vector fields generates a semi-martingale on the group of diffeomorphisms through S.D.E. so that it is the velocity field of the latter. Thus it is the main result of this paper to establish this one-to-one correspondence between a class of semi-martingales on the group of diffeomorphisms and a class of semi-martingales on the space of vector fields. A semi-martingale on the group of diffeomorphisms is a Brownian motion if and only if the corresponding semi-martingale on the space of vector fields is a Wiener process.
0 2. L.C.-system The main purpose of this section is to study a system of analytical objects (called an L.C.-system") which is necessary in characterizing infinitesimally a random motion of transformations of a state space and to describe this system in a geometric language when the state space is a smooth manifold. Let M be a topological space and C ( M ) be the space of all realvalued continuous functions on M . C ( M ) is an algebra under the usual operations and let 9 c C ( M ) be a subalgebra. We consider a system {L,( , ) x , , } of the following objects; 1) (f, g),,,, f, g E 9, x9 Y E M , which is a real bilinear form on 9 x 9 for each fixed x , y E M , 2) L : 9-+ 9, a linear operator on 9. We assume that the system {L, ( , )I,y} satisfies the following properties; a) b)
(f, g>x,, = ( s , f > y , x ?
vf, g
E
9
3
vx,Y
E
M,
(derivation property)
(LA,g>x,, = f i ( x ) ( f i >g>r,g + . A ( x ) ( L , g>x,,, VL7.h g
c)
(positive definiteness) m
*
L.C. is meant for local characteristic.
E
9, VX, Y E M ,
Stochastic Flows of Difleornorphisnzs
309
Definition 2.1. Such a system {L, ( , )I,y} is called an L.C.-system (with domain 9)on M. In this paper, we are mainly interested in the smooth case, that is, the case when M is a manifold and transformations are diffeomorphisms. So, from now on we assume that M is a smooth connected compact manifold of dimension d. We denote by C " ( M ) the algebra formed of all real C"-functions on M. Definition 2.2. An L.C.-system {L, ( , ) s , u } on M is called smooth if 9 = C m ( M ) L , : 9---f 9 is a differential operator and (x,y) + ( A g)%,, is C" in (x,y ) E M X M for everyf, g E 9. We rewrite a smooth L.C.-system in geometric language. In the following, we omit the summation sign according to the usual convention. First by b),
b)' ( A g>z,, = 4 x 7 Y " m 7 (d'),1 where A(x, y) E T,(M) 0 T,(M).(*) In a local coordinate,
a
A(x, y) = a y x , y ) -- 0 ax2
a ~
ayj
and the coefficients aij(x,y) are smooth in x and y by the above smoothness assumption. Next, L: C " ( M ) + C"(M) is a smooth second order differential operator such that L ( l ) = 0 ( 1 is the function on M identically equal to 1) as is easily seen from the properties b) and d). Then
in a local coordinate where the coefficients aii and pi are smooth in x. Now a), c), d) are equivalent to the following respectively: a)' c)' d)'
a"(x, y ) = a""(, x), aii(x,,x,)c~cj2 0, for every rn = 1,2, and cs E R, ati(x)= CZ"(X, x).
- - .,
xI,x2,-
-
0
)
Note that (a"(x)) is non-negative definite because of c') and d'). Definition 2.3. Set
*
T J M ) is the tangent space at x
6
M.
x, E M
310
Y . LE JAN AND S. WATANABE
in a local coordinate.
Lemma 2.1. For d E A , ( M ) (= the space of all diflerential l-forms on M ) , define ad. (V&(x) = u y x , x ) L ( x ) axk
(2.4)
+ ry(x)d,(x)
a
~
aYk [a%
Y)MY)l1 y = X
in a local coordinate, where R = di(x)dx'. rype (1.1).
Then V I is u tensor field of
Proof. It is not difficult to verify, by a direct calculation, that (Vd): obeys the transformation rule under a change of coordinates. More intrinsic proof may be given as follows. Let X E % ( M ) : = the space of all C"-vector fields, X(x) = X'(x)(a/ax') in a local coordinate and exp tX be the one-parameter subgroup of G = Diff(M) (=the group of all C"diffeomorphismsof M ) generated by X . Then hf(4
=
a""(, (exp tX>(x)>d,((exp(tX>(xNE T A M ) -h:(x)l d dt
and hence
E T,(M)
.
t=o
But
= X"x)(vI):(x) E T,(M) -hh:(x)) d t=o dt which proves that (VI)(x) E T ; ( M ) 0 T,(M) 2 2'(Tx(M)-+ T,(M)).
Corollary. (2.5)
1 L o ( f )= -(V(df))l 2
(contraction),
YE C"(M),
defines a second-order differential operator on M with the same main term as L. Hence (2.6)
B(f) =
af) - Jw), f Crn(M), E
is the first order differential operator on M , i.e., B E % ( M ) . When we regard B as a tensor field x -+ B(x) E T,(M), B(f)(x) is sometimes de] . we have obtained the following theorem. noted by B ( ~ ) [ ( d f ) ~Thus
Stochastic Flows of Difleomorphisms
Theorem 2.1. to giving a pair
To give a smooth L.C.-system { L , ( ,
a
(.(x, y ) = a y x , y)? ax
0-,a
B(x)
ay
311 )x,y}
is equivalent
=
with properties a'), c') such that both ( x , y ) -+ A(x, y ) E T,(M) 0 T J M ) and x + B ( x ) E T,(M) are smooth: (f, g ) x , v = 4 x 9 Y"f)x,
and
(4&l
+ W)[(df),l
L ( f ) ( x ) = Lll(f)(x)
where the second-order differential operator Lois defined by (2.5).
Thus, we may call such a pair { A ( x ,y), B ( x ) } as a smooth L.C.system. We shall study, a little more in detail, geometric aspects of the tensor field
a
a
A ( x , y ) = a"(x, y)0ax* ay5
.
If ai'(x) = a"(x, x ) is strictly positive definite for all x E M , then its inverse a,,(x) = a,,(x, x ) defines a Riemannian structure over M. In this case, T f j = -I':%z,,(x, x ) are the Christoffel symbols, i.e., coefficients of a linear connection, which is compatible with the Riemannian structure, that is the metric is preserved under the parallel translation. Indeed, the covariant derivative (VfX) of X = Xt(x)(a/axi)E B ( M ) with respect to this connection is given by
since
Y. LE JAN AND S. WATANABE
312
for X ( X ) = Xf(x)(a/axt)and Y ( x ) = Yi(x)(a/axt),
a
- X X m ( x ) Y n(x)am,n(x, x) ax5
a
X r n ( xyn(Y)am,n(X, ) Y)
=
____
=
(V!X)(x)a,,,(x,X)Y"(X)
aYt
Iy=i
a J % J ) y n ( x ) a m , n (x)l + ay" Y,
L!-x
+ v: Y ) ( x ) a m , * kx)Xrn(x>
and this proves the compatibility. It should be noted that this connection is usually not symmetric, i.e., torsion part does not vanish (cf. Example below). Finally, it is easy to verify that
where ( P f ) ( ( x )= a""(, x)(afiaxk)(x),that is, Locoincides with the LaplaceBeltrami opeator with respect to the connection defined by {i':j}. Example 2.1. Let M = Rd/2nZdbe a d-dimensional torus. A general form of smooth L.C.-systems on M which are invariant under translations is obtained in [ 141. Using the standard (Euclidean) coordinate,
a 0a ay
A(x, y ) = a y x , y)axi
-
and aij(x, y ) is the form at'(x, y )
=
CY + n EC {cos [ ( x - y)-n]c;!- sin [ ( x - y).n]dhj} Z d \ 101
where (c?) and (dkj) are real d X d-matrices for each n E Zd\{O} such that ~ ; = j c;l" = cYn, d;j = -dii = dYrL,c;jp,p, 2d;jp,R, chjR,R, 2 0 for every (p,),(It) E Rd and
+
C
n €Zd\iOl
(Ichfl + Idkj])Inlk< 03
Choose chj so that a"(x, x )
= cy
for all k
+
> 0, i , j = 1, 2, . . ., d .
+ CnEZd,iol~y = 8'. Then
Another characterization of the operator Lois given by Baxendale [2]. For the tensor field A given above, let H be the reproducing kernel Hilbert space (R.K.H.S.) of vector fields associated with A : to be precise, for
Stochastic Flows of Difleomorphisms
313
-
X = [xl,x,, . . .,x,] E M” and I = [A’, A,, . ., I”] where Rk = 2:(dxt)lXk E Tzk(M),k = 1, 2, m, for some m = 1, 2, . . .,
--
a ,
and H
=
{ X ( E , Ifor I all X and
completed by the Hilbert norm.
Then H
(2.8)
=
L,(f)(x)
-
I, m = I, 2, .
.}
f ( M ) ( * )and
trace, VfvZ
where VfgXis, for each x E M and f E C”(M), a bilinear form on H X H c_,S ( M ) X % ( M ) defined by V f * z ( X Y , ) = X ( Y ( f ) ) ( x ) . Note that the injection i : H-+ f ( M ) is Radonifying, i.e., if Y is the canonical Gaussian cylindrical measure on the Hilbert space H , then ib], which is a cylindrical measure on the FrCchet space % ( M ) can be extended to a a-additive Radon measure p on % ( M )so that (%(M),H, p ) is an abstract Wiener space. It is also known that every mean 0 Gaussian Radon measure on S(M)is obtained in this way from some A . Finally if p is the image measure of p under the map X 4 X B, then /z is a Gaussian measure on f ( M ) with mean B. Thus we can say that a smooth L.C.system ( A , B ) on M is in one-to-one correspondence with a Gaussian Radon measure /z on S ( M ) : B is the mean of p and A is the covariance of p, that is,
+
Cf. Baxendale [2] for details. In the subsequent sections, the following random L.C.-systems play an important role. Again we restrict ourselves to the smooth case. Let (Q, 9, P) be a probability space with a filtration (.Ft)tECO,w). We consider a system (AB+’(x, y), B8*@(x)) of smooth L.C.-systems which depend on (s,
0) E
to,
a)x
0.
Definition 2.4. system if
(A8+”(x, y ) , Bs-m(x)) is called a regular stochastic L.C.-
* The topology of %(M)is given by the uniform convergence of coefficients and their derivatives so that %(M) is a FrCchet space.
Y. LE JAN AND S. WATANABE
314
( i ) for each (s, OJ)
E
[0, co) X Q,
is a smooth L.C.-system, (ii)
for each x, y E M , (8, w ) -+ A 9 x , y ) E T J M ) 0 T,(M), (s, w ) ---f B 8 q x ) E T,(M)
are (FJ-predictable and
J SUP l D : D W s %
(iii)
0 X,V€M
for every t C"(M).
0 3.
s:>
y)[(df)x,(dg),l)lds
0, $ t o #;I is independent of .Faand #t 0 #;I is equally distributed as # t - s . In particular, it is a process with independent increments: for every to = 0 < t, t, < . < t,, n, are mutually independent. q5t, o $& k = 1,2
>
-
0, then we define
J: K,dX, to be the semi-martingale (4.4)
J: K,dX, =
s:
K,dM,
+
and call it the stochastic integral of K valued semi-martingale X,.
s:
=
K,(B".")ds
(K,) E K by the regular O ( M ) -
Remark 4.1. The space %(M)* is just the space of 1-currents on M : k E O ( M ) * is expressed formally, in each coordinate neighborhood, as k = k,(x)dx* where k,, . .,k, are Schwartz distributions on M defined in the coordinate neighbourhood which obey the transformation rule under the change of coordinates. To be precise, k(X), X E %(M),is given by k ( X ) = ( k , , Xi) if X ( x ) = X*(x)(a/ax')with support in a coordinate neighborhood and, in a general case, by using a partition of unity. Note that traceHs.&(')0 k @ )= ( W ,k(2))H,.0, k(QE %(M)*, 1 = 1, 2, is given by
C (ky)sz8k r ) ' J ,a,(x)aJ(y)a"(x,y , s, a)) I,J
where 1 = C , a,(x) is a partition of unity with Supp(a,)C V,, V, being a coordinate neighborhood and k") = kjz)2'(x)dxt in V,,
Aslw(x,y ) = aiJ(x,y , S, w)-
a a ay ax( 0__
in V , x VJ
k:'),' 0k y ) , Jis a tensor product of Schwartz distributions.
.
Y.LE JAN AND S. WATANABE
320
Finally we remark that the totality of regular %(M)-valued semi., X,,, be regular I ( M ) martingales forms a vector space. Let X,, X,, valued semi-martingales with local characteristics {A>,",x, y), B ~ " ( x )and } the semi-martingale decompositions
--
X k ( t ) = M,(t)
+
s'
.
B>@ds, k = 1, 2, . ., m,
respectively. Then if M,(t) = M:(t, x)(a/ax'),k = 1, 2, . . ., m, in a local coordinate, (MXf
4, MZ(t, Y)> =
j: a%, ,
Y , s, o)ds
with some function aX(x, y , s, o) and it is easy to see that (x,y ) --+ &(x, y , s, o) is smooth for each (s, w), (s, o)-+&(x, y , s, o) is (TJpredictable for each x , y E M and
for every multi-indices a! and /3.
Since
[ D:DCa;{(x, y , s, 0 )j2 I D;D,"a%(x, z, s,
Iz== x
0)
D,BD;aSlj(z,y , s, 0)] E = u
for almost all (s,o) (ds P(do)), we can find, for every xo,yo E M and coordinate neighborhoods Vzo,Vu,of x,, yo respectively, neighborhoods Uxo, Uu0such that xo E UzoC Vzo,yo E UuoC Vuoand
(J
(4.5)
a s . for every t
> 0.
)'"
sup {D:D;ap,(x, z, s, w)]I=x}ds
0 ZE U z o
If we set
a smooth tensor field A>,: is globally defined and, by (4.9, we can conclude that
Stochastic Flows of Diffeomorghisms
321
for every f, g E C"(M), smooth differential operators D:, Di on M and t > 0. Hence, if we set m
A"."(x,Y ) =
c
Y)
k,l=l
cB:."(x) m
Byx)=
k=l
then {A"."(x,y), B"."(x)} is a regular stochastic L.C.-system. It is now .. X , is a regular %(M)-valued easy to conclude that X = XI X , semi-martingale associated with these local characteristics. In particular, we have proved the following.
+ +
+
Proposition 4.1. The totality of regular %(M)-valued semi-martingales forms a vector space under the usual addition and scalar (=real) multiplication.
5 5.
Gvalued semi-martinglaes generated by %(&I)-valued martingales
semi-
In this section, a probability space (Q, S, P) and a filteration (9,) are given and fixed. Let X , = M ,
+
J-1
BSsads be a regular %(M)-valued
semi-martingale with the local characteristic {As+', B51."} which is, by definition, a regular stochastic L.C.-system.
+
Theorem 5.1. Let be an So-measurable G-valued random variable. Then there exists a unique regular G-valued semi-martingale $, such that
(i> $0 = +, (ii) for every x
E
M and f
E
Cm(M),
where K"(f ) E %(M)* is defined, for each x (5.2)
K " ( f )(X)= X ( f )(XI
E
X
Y
M and f E C"(M), by E
%(M)
and L:,"is defined, as above, by (5.3)
1 L;+(f ) = -Vs+(df): 2
=
1 trace,,.,Vf~" . 2
322
Y. LE JAN AND S. WATANABE
The local characteristic of $, coincides with {A8+',B*i*}. Proof. We assume 11. = id, i.e. +(x) = x : the general case can be obtained as #, 0 11. from the solution $, with $o = id. We embed M into a higher dimensional Euclidean space R N . Then $,(x) = (&(x), &(x), ., 4 f ( x ) ) and every X E % ( M ) is the restriction to M of a smooth vector field on R N ,that is, we may write
-
x = cF ( X ) -axta N
i=l
where x = (x', x2, . . . ,x N ) is the Euclidean coordinate. Set K;(X) = X i ( x ) and bi(x, s, w ) = (L;," B 8 , m ) ( f ) ( xwhere ) f ( x ) = xi, i = 1,2, . ., N . Then (5.1) is equivalent to the following:
+
(5.4)
&(x) = xi
-
+ [K!b(5)dM,+
1:
bt(q5,(x),s, w)ds ,
i = 1, 2, . . ., N. Note that, for any M-valued (9,)-predictable processes x , and y,, we have (5.5)
(1:
KSldM,,
J: KJ.dM,) = J: aif(x,,y , , s, o)ds
where A"*(x, y ) = aif(x,y , s, o),
a 0a ax%
ap
under the embedding M C R N . More generally, if K;,=E %(M)* is defined, for x E M , i = 1,2, . . ., N and multi-index a = (al, . ., aN),by
x = c X"X)- a N
Kt,.(X) = D ; X i ( x ) ,
i=l
ax6
E
X ( M ),
then, for any M-valued ($=,)-predictable processes x , and y,,
We shall now prove the uniqueness of the solution $,(x) = (+f(x))of (5.4) and diffeomorphic property of x 3 $,(x). For this, take any integer m > 0 and set
Stochastic Flows of Diffeomorphisms
323
(f)(x) XE .%M) 9
E
M , by
*
The local characteristic of Y , is obviously {$,*(As+),q5$(B"'")}. Debition 5.3.
This regular I(M)-valued semi-martingale Y , is
33 1
Stochastic Flows of Diffeomorphisms
denoted by
Theorem 5.6. Let {$](t)}and {Ibz(t)}be regular G-valued semi-martingales generated by regular S(M)-valued semi-martingales X l ( t ) and X z ( t ) respectivetly. Then $l(t) o q&(t) is a regular G-valued semi-martingale generated by the regular %(M)-valued semi-martingale
, G Proof. Let F f , x ( $ l$z):
x G +. R be defined, for f E Cm(M)and
x E M , by
D24Ff3”($l,$z)(X, Y) = [($;’)*(
Y ) I ( [ ( $ ~ l ) * ( ~ ) l ( ~ ) $z(x)> )($l * O
Then the assertion of the theorem can be easily deduced from Th. 5.5. Details are omitted. Corollary 1. Let X l ( t ) and X z ( t ) be regular %(M)-valued semimartingales and $‘(t) be a regular G-valued semi-martingale generated by X l ( t ) (uniquely determined up to the value $l(0)). Let
and $2(t)be a regular G-valued semi-martingale generated by $(t). Then #3(t)= $l(t)o $z(t)is a regular G-valued semi-martingale generated by the regular S(M)-valued semi-martingale Xl(t) X2(t).
+
Corollary 2. Let +(t) be a G-valued semi-martingale generated by a regular S(M)-valued semi-martingale X ( t ) . Then $ - l ( t ) is a regular G-
Y.LE JANAND
332
S. WATANABE
valued semi-martingale generated by the regular X(M)-valued semi-
1:
martingale -
$*(s)(dX(s)).
Corollary 3. The totality of all regular G-valued semi-martingales is closed under the composition and the inverse.
References P. Baxendale, Wiener processes on manifolds of maps, Proc. Royal SOL Edinburgh, 87A (1980), 127-152. -, Brownian motions in the diffeomorphism group I, Univ., Aberdeen (1982). Ju. N. Blagovescenksii and M. I. Freidlin, Some properties of diffusion processes depending on a parameter, Soviet Math., 2 (1961), 633-636. J. M. Bismut, MCcanique altatoire, Lect. Notes in Math., 860, Springer, 1981. K. D. Elworthy, Stochastic dynamical systems and their flows, Stochastic Analysis ed. by A. Friedman and M. Pinsky, 79-95, Academic Press, 1978. I. I. Gihman and A. V. Skorohod, Stochastic differential equations. Springer, 1972. T. E. Harris, Brownian motions on the homeomorphisms of the plane, Ann. Prob. 9 (1981), 232-254. N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland/Kodansha, 1981. H. Kunita, On the decomposition of solutions of stochastic differential equations, Stochastic Integrals, Lecture Notes in Math. 851, Springer, 1981. H. H. Kuo, Gaussian measures in Banach spaces, Lecture Notes in Math. 463, Springer, 1975. Y. Le Jan, Flot de diffusions dam Rd. C. R. Acad. Sci. Paris 294 (1982), Strie I, 697-699. P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proc. Intern. Symp. SDE. Kyoto ed. by K. ItB, 195-263, Kinokuniya, 1978. M. Metivier and J. Pellaurnail, Stochastic Integration, Academic Press, 1980. S. Watanabe, Stochastic flows of diffeomorphisms, Proc. Fourth Soviet, Japan Symp. on Prob., Lecture Notes in Math. 1021, Springer, (1983), 699-708. YVESLE JAN LABORATOIRE DE CALCUL DES PROBABILITIES UNIVERSITEPIERRE & MARIECURIE PARIS.FRANCE
SHINZO WATANABE DEPARTMENT OF MATHEMATICS KYOTOUNIVERSITY KYOTO606, JAPAN
Taniguchi Symp. SA Katata 1982, pp. 333-367
Some Recent Results in the Optimal Control of Diffusion Processes Pierre-Louis LIONS
5 0.
Introduction
We consider here optimal stochastic control problems where the state of the system is supposed to be given by the solution of a stochastic differential equation : (1)
dX, = u(X,, ct,)dB,
+ b(X,, a , ) d t ,
X,
=
x E RN
where B, is an rn-dimensional Brownian motion, u(x, a), b(x, a) are given matrix and vector-valued functions on RNX V -where V (called the set of the values of the control) is a separable metric space. We stop the state processes X,at the first exit time r from a given domain U and we consider cost functions given by
We next define the value of the control problem (also called the optimal or the minimum cost function):
(3)
u(x) = inf J(x,a,),
Vx E B
.
at
The heuristic argument of dynamic programming - due to Bellman [4] -indicates the u “should be related to the solution” of the following partial differential equation of elliptic type - called the Hamilton-JacobiBellman equation (HJB in short) -:
(4)
sup {A,u(x) - f(x, a)} = 0
in U
uEY
and u should satisfy on 8U (or on some part of aU): u = y. Here A , is the linear, second-order, elliptic (possibly degenerate) operator given by A , = -a,,(x, a)8,,- bt(x, a)8, c(x, a) and a = 1/2 ud-here and below
+
P. L. LIONS
334
we use the convention on repeated indices -. In addition, formally, if a “smooth” solution ii of (4) is known (with appropriate boundary conditions) then not only ii 3 u but also one can build optimal controls in the so-called feedback form (or markovian controls). This is why we will be interested in what follows by the relations between u and (4). Our purpose here is to review recent results on that equation mostly obtained by the author (see Lions [31], [32], [33]). In section I we briefly describe the controls problems we want to look at and recall the classical methods for those problems. Section I1 is devoted to various “elementary” properties of the value u such as continuity properties, characterization in terms of maximum subsolution, density of particular classes of control processes. In section I11 we consider various applications of the notion of viscosity solutions of HJB equations (introduced by Crandall and the author [ll], [12] in the case of first-order Hamilton-Jacobi equations): in particular we recall the observation (due to Lions [32], [34], [35]) that the value function u is always a solution of (4) in viscosity sense. In addition it is the unique solution of (4) in that sense -with prescribed values on
ao -.
Section IV is devoted to regularity results of u which extend and are of the same type then those initiated by Krylov [22], [23], [24], [25]. Those results are taken from [33] (see also [35], [36]). In section V we consider the problem of reflected processes and indicate briefly a few regularity results Finally section VI is devoted to the fundamental inequalities due to Krylov [26], [24] (cf. also Pucci [53] and the pioneering work of Alexandrov [l]). We propose here an approach to these inequalities which-even if not totally different from Krylov original proof-sheds some light- we believe- on the nature of these inequalities and the relations with control problems. We also derive from these inequalities some new estimates which insure, in particular, the existence of regular transition probability densities for homogeneous non-degenerate diffusion processes with bounded, Bore1 measurable coefficients. Those results have to be compared and complete the counter-example due to Fabes and Kenig [ 171. SUMMARY
5 I.
Position of the problem 1.1 The time-independent problem. 1.2 The time-dependent problem. 1.3 Classical methods.
Optimal Control
8 11.
8 111.
8 IV. 8 V.
8 VI.
335
Preliminary properties of the value function 11.1 Continuity of the value function. 11.2 Maximum subsolution. 11.3 Remarks on the control processes. Viscosity solutions of HJB equations 111.1 The Dynamic Programming Principle. 111.2 Uniqueness of viscosity solutions. 111.3 Various remarks. Regularity and Uniqueness IV.l Regularity results. IV.2 Uniqueness results. Reflecting boundary conditions V.l Description of the problem. V.2 Regularity results. Application to Krylov inequalities VI.1 Description of the main result. VI.2 Proof and remarks.
0 I. Position of the problem 1.1 The time-independent problem c, f
Let 0 be a smooth open set in RN (possibly RN itself) and let be given functions on RN x V- satisfying
+
0,
b,
where = otj, b,, c, f (1 < i < N , 1 < j < m). Finally to simplify the presentation we assume that (o E W s 9 3 O ) . We define an admissible system d as the collection of i) a probability space ( O , T , 2Ft, P) with the usual properties, ii) an m-dimensional 9, adapted Brownian motion B,, iii) a progressively measurable process at taking its values in a compact set of V- (depending on cu,). Because of (3, for each admissible system, there exists a unique continuous process solution of (1) (with initial position x in R N ) . We denote by 'c the first exit time of X , from d 'c
= inf { t
2 0, X , 4 6 )
(=+m
if X , E d for all t
0)
P. L. LIONS
336
and we will assume that inf{c(x,a))xER N ,a~ V }= R
(6)
>0.
In all the results below concerning the time-independent problem (or stationary problem) we will always assume (5), (6) and we will not recall it. We finally define the cost function for each x E d and admissible system d :
The optimal cost function or the value of the stationary control problem is given by
u(x) = inf {J(x,a)I d :admissible system}
(8)
for all x
E
8.
1.2 The time-dependent problem Let T > 0 be fixed, we denote by Q = 0 x 10, T [ . We now assume that the coefficients 0,b, c , f , depend on t E [O, TI and satisfy
I]+(*, t, a)IIwa.-(BN) < 03 +(. , - ,a) is uniformly continuous on RN x [0, TI, +(., t, a) E W Z , m ( R N ) ,SUP (I
€ V ,t
€[O,TI
uniformly in a E "Y . +(x, t, .) E C ( V )
for
+ = ot,, b,, c, f
(1
for all x E RN , t
E
[0, TI
,
< i < N, 1 <j < m);
(6')
inf { c ( x , t, a)I x
(9)
p E w332qi30 x ] 0, T [ ), p(
E
R N ,t
E
[0, TI, a E V }= R a ,
>-
03;
T ) 3 w2q0) .
In all the results below concerning the time-dependent problem, we implicitly make the above assumptions.
t
E
We define admissible systems exactly as before and for each x E R N , [0, TI we consider the solution X,( =X,(x, t ) ) of
(10)
dX, = a(X,, r
+
S,
a,)dB,
+ b(X,, t +
S,
a,)ds,
X, =
X.
337
Optimal Control
We then define the corresponding cost function
X
(71
+ X
and the value function of the time dependent control problem is given by (89
u(x, t ) = inf {J(x,t, d)[ d :admissible system} .
Remark 1.1. It is worth remarking that the time-dependent problem may be viewed as a special case of the time-independent problem by looking at the problem: 2, = ( X s , Y,) where Y,= t s, the domain Q = 0 X 10, T [ and remarking that 7 A (T - t ) is the first exit time from 0 of 2,.
+
1.3 Classical methods We just want to recall here two facts which can be found in Fleming and Rishel [18] (see also Bensoussan and Lions [5], Krylov [24], Lions [38]): if we know a priori that u given by (8) belongs to C’(0)- that is if the value function of the stationary problem is smooth - then u solves the HJB equation sup {A,u(x) - f ( x , a)} = 0
(4)
in 0 .
a€
l(Q)then u solves the equation (49
au -at
+ sup {A,u(x, t ) - f ( x ,
t, a)} = 0
in Q
.
Conversely if ii E C’(6) (resp. Cz,l(Q)) solves (4) (resp. (4’)) and satisfies: z2 = (D on a0 (resp. ii = p on a,Q = (a0 x [0, TI) U (6 x { T } ) , then u is the value of the associated control problem. Those results are easily proved by the use of ItB’s formula and are essentially verification theorems. Unfortunately they require either the a priori knowledge of the smoothness of the value or the existence of a smooth solution of the fully nonlinear degenerate elliptic-parabolic equations (4) or (4’). Of course when a does not depend upon a and when a is
P. L. LIONS
338
uniformly positive definite, (4) and (4') are only quasilinear uniformly elliptic-parabolic equations and it is possible to solve them directly by standard p.d.e. considerations (see 1181, [5] for more details). However in the general case considered here classical results in p.d.e. theory do not apply and furthermore easy examples show that u (the value) is not, in general, of class C'.
Q II. Preliminary properties of the value function II.1 Continuity of the value function We want to describe here a few conditions which insure that u given by (8) is continuous on d and that u = 9 on a0 (or at least on some part of a@). Of course some conditions are needed since it is well-known that even in the very special case when there is no control (i.e. when u, b, do not depend upon a) u corresponds to the solution of linear degenerate Dirichlet problem and may well be discontinuous. The condition we introduce is an extension of the condition: f(x, a)>O on b x 7". Let us first discuss this special case: remark first that r as B function of x is U.S.C. (a.s.) and thus quantities like
are also U.S.C. with respect to x (as.) iff is nonnegative. Next u being an infimum of U.S.C. functions is also U.S.C. We now introduce our main condition : we assume that there exists closed subset possibly empty, of = 8 0 ; a Bore1 bounded function w on 8 satisfying
r+,
( V d , vx E
r
8 ;P [ r < 03, X , 4
r,]= 0; l,,,,,(w - y)(X,) Q 0 a s . ,
Vx E r,, lim inf w(y) > ( ~ ( x,) u8d.Y-x
I
(is a StAr-submartingalesatisfying for any bounded stopping times
4 G 82: E"0,I 2 E"d
*
Theorem II.1. We assume (10). Then we have i) For all admissible system d , J( .,d)is U.S.C. in b and thus u is U.S.C. on 8. ii) u win 8, u (D on andfor a l l d , for all X E b
>
r,;
l~r
vx E a,vy E r / .
iJ' R < R,, r E (0,1) if R
= A,, Y = 1 i f
20.
Remark 11.3. In Lions [31], it is shown that this exponent is in general optimal. The fact that, for related problems, the modulus of continuity of u depends in an essential way on the ratio A/& was first remarked in Lions [39], [40]. We give below examples of situations where is closed and (15) holds: other examples may be found in [31]. Let us immediately notice that if 0 = RN (or if (12) holds with = 80) then = = q5 and all conditions are automatically satisfied. Thus in this case u E Co,'(RN)with the above value of 7.
r'
r-
r' r+
Corollary II.2.
(16)
W e assume (1 l), (12) and
0, 3a, > 0 , p ( x ) >, a,
if
Finally we set for (x, t ) E RN x [0, TI, E
>0
V6
dist (x, 8 ) 2 6 .
+ s, a,)
&(x, t ) = inf E [ r - ' h(Xs, t .d
where h = f - g. Using Krylov's result afore mentioned, one sees that ii*(x,t ) = inf J,(x, t, dM) dN
where J,(x, t, d)is the above cost function. Clearly as E J 0+, J,(x, t, d)decreases to
X exp
(-s'
c(X,, t
+ a, aJdg
)I ds
and we may pass to the limit in the infima to find ~ ( x t, ) = inf J(x, t , st) = inf J(x, t, d,). .d
.dM
Remarking finally that in view of Itb's formula, we have for every (x, t ) E Q and for every admissible system d
P. L. LIONS
346
we deduce ii(x, t ) = u(x, t ) - w(x, t ) = inf J(x, t, dM) - w(x, t ) ; d.u
and we conclude.
9 III.
Viscosity solutions of HJB equations
111.1 The Dynamic Programming Principle Recently, Crandall and the author [Ill, [12] introduced a new notion of solution of first-order Hamilton-Jacobi equations : with this notion general uniqueness, stability and existence results were proved (cf. 1121, [lo], [39]). In [39], the author pointed out the intrinsic relations between this notion of “viscosity solutions” of HJ equations with the Dynamic Programming Principle. It is very straightforward to adapt this notion to the case of second-order equations (cf. Lions [32], [34], [41]) and to preserve very general stability results (and existence results too) but the main open question for second-order equations is the uniqueness of viscosity solutions. This uniqueness question is solved in Lions [32] for the class of HJB equations and we recall below the typical uniqueness results (section 111.2 below). We first recall briefly here one possible definition of viscosity solutions of HJB equations and we show how the heuristic derivation of HJB equation is made rigorous by the use of viscosity solutions. Combined with the uniqueness results of ectison 111.2 below this shows that the value functions of stochastic control problems are always solutions of HJB equations in viscosity sense and that this characterizes those value functions. For more details and properties of viscosity solutioons we refer the reader to the works mentioned above. Definition III.1. Let u E C(0); u is said to be a viscosity subsolution (resp. supersolution) of the HJB equation (4)
sup {A,u - f ( x , a)} = 0
(4)
in 0
+
if the following holds for all E Cz(0): let x, be a local maximum point of (u - +) (resp. minimum point) then we have SUP
[-a&,,
+
a)&,+(x3 - bdxo, cu>W(xo)
C ( X 0 , .)U(XO)
- f(X0, a11
O
+ ++
where 6 = u(xo)- +(xo). Replacing by 6, we may assume without loss of generality that 6 = 0 and using ItB's formula (recall that E C2(b))we deduce
and letting t + 0 SUP {-a&,, a
+
+, we deduce easily
Or)&,+(xo)- bdxo, 4&+(xo)
+
Remark 111.3. One sees that, because of the very definition of viscosity solutions of (4), one takes advantage of the fact that x, is a minimnm point of u - to replace u by and to carry out the usual derivation of HJB equations since 11. is C2. Doing so we obtain only an inequality instead of the equality.
+
III.2
+
Uniqueness of viscosity solutions
Our main result is the following:
P. L. LIONS
348
Theorem IU.2. i ) Let ii E C(0) be a viscosity solution of (4). Then for all 6 > 0, for all x E Ga, we have
E(x) = inf E[S;df ( X c ,at)exp .d
(-1:
c)dt
+ i7(X,,) exp
ii) Let i7 E C(Q) be a viscosity solution of (4'). T - S,] we have
> 0 for all x E Bdl,t E [a,
Then for all S,,6,
Of course the main applications of the result are the following comparison-uniqueness inequalities : Corollary III.l. then we have
Let u, u E C,(B) be two viscosity solutions of (4)
sup (u - u)' I
where 00,
< sup (u - u)' r'+
,
r: is a closed subset of r such that for x E 8, and for all d :P(r'
+(x)}, u is a viscosity solution of (4)
sup {A,u(x) aEr
- f ( x , a)} = 0
Then using Theorem 111.2, we find
in 0 .
Optimal Control
351
and by the choice of 0
where ~(6)-+ 0 as 6 +- 0 u(x)
+.
And this yields
< P(x) + ~(6)
for all 6
>0.
The reversed inequality is obtained by the use of a standard penalty argument in the theory of obstacle problems and optimal stopping problems. Let E > 0, 6 > 0; we introduce ,6‘,a c C”(R) satisfying
/?:la < 0 on R,P J t ) /.,a
= 0 if I2 0, /?:,a(t)2 0 on R
1
converges uniformly, as 6 goes to 0+, to -(t A 0) = P 6 ( t ) . E
We then set
where the supremum is taken over all progressively measurable processes 6, satisfying: 0 6, < 1 a.s. Using the results of Lions [33], we see that one can find uI,a,yE C:(RN)satisfying
0, then for E Q E ~ 6, a,,(€), Y Y,(E, 6) we still have: maxRN(u.,a,v- u) > 0. Let x, be a point of maximum of u.,d,u- u. Using the definition of viscosity solutions with u,,a,” we find
Al which is in general necessary: see the example of Genis and Krylov [19]. Let us give a very simple example showing the necessity of this assumption: take 0 = ] - 1, 1[, V = {ao}, u ( x , a , , ) = O , b ( x , a o ) = +x in [-1 , +l],c(x,ao)=~,f(x,or,)=O,~=l. Then = = { - 1, + 1) and (30) holds, R1 = 2 and it is easily checked that u(x) = jxr in [ - 1, 11. Therefore (31) holds if and only if A 2 2
+
r rz
+
Optimal Control
353
i.e. A 2 A,! However in the nondegenerate case this assumption on b is not necessary: see Theorem IV.2 below. Using the fact that we know (section 111.1) that u is a viscosity solution of the HJB equation, one deduces easily from the above result: CorolIary IV.l. function u satisfies
Under the assumptions of Theorem IV.1, the value
and the HJB equation holds:
sup {A,u(x) - f ( x , a)} = 0
(33)
a.e.
in 0
=CY
Another application of Thoerem IV.l is the following: Corollary IV.2. Under the assumptions of Theorem IV.l and if we assume that there exist an open set OJ c 0, an integer p E { 1, . . ., N } , a constant v > 0 such that
{5 e,
V X E O J , I,&Y,, ~ ~ ~ ...,cw,
(34)
i=l
= 1,
2 oiakl(x,
ai>Ekf,
i=l
Then dlfuE L"(oJ) for 1 < i, j As we said before, R Theorem N.2.
(35)
3v
> 0,
~~:38,,...,8,~]O,I[suchthat
2v
for all
t E RN .
i=l
A, is necessary except in the following case:
We assume V ( X ,cw)
E
B x
V , a(x, a) 2 vJN .
Then u E W2,m(0)n C2ir(0)(for some 'i and the HJB equation holds.
11 a \ I m )
2 ti
E
(0, 1) depending only on 0, v,
Remark IV.2. Theorem IV.l and Corollary IV.1.2 are due to Lions [33], [34], [35]. The W2,"regularity in Theorem IV.2 is due to Lions [44], [45] and Evans and Lions [16], while the higher C2,rregularity was recently obtained by Evans [13], [14]: all those results being obtained by purely p.d.e. methods. Partial results were previously obtained by different methods by Krylov [22], [23], [24], BrCzis and Evans [7]; Evans and Friedman [15]; Lions and Menaldi [49], [50]; Safonov [54], [55]. The special case 0 = RN was treated independently by Krylov [25], [27] and Lions [46], [47] by different probabilistic methods.
P. L. LIONS
354
We first explain how the assumption on I naturally comes into the proof of (31) in the special case 0 = RN:indeed assume, to simplify, the coefficients u, b, c, f are smooth, then by standard results on stochastic differential equations we have for all t 2 0
for all d , for all x E RN,and for all t E R N ;\ E l = 1, where K depends only on SUP, {Ilullwa.llbllw~.-- IICllwa.I l f Ilw..-}Therefore if I > I ,
+
+
+
x [O, TI) ;
;
Diu E L2(U,X (0, T ) ) , - C
au < A,u < C + __ at
in 9‘(Q)for all (Y E V
355
Optimal Control
for all 6
> 0.
(39)
au -~ + sup {A,u(x, t ) - f ( x , t , a)} = 0
Furthermore the HJB equation holds a.e. in Q .
at
Finally if a, b, c, f a r e Lipschitz in t uniformly for x E 8, a E V then au/at E L"(Q) and thus supuE, I I A . u I ( ~ ~< ( ~co. ) IV.2 Uniqueness results
At this point let us recall a few results proved above: we have seen that the value functions are viscosity solutions of HJB equations and that in some cases the value functions have some regularity properties and the HJB equations hold "in an a.e. sense". But in view of the example mentioned below, there may be many solutions having this regularity and satisfying the HJB equations in that sense. To solve this nonuniqueness question, we already proposed the notion of viscosity solutions;we present now a different approach taking into account conditions like (31) which are satisfied by the value function.
0, u,(x) = (1 - ,de-^l"')/Ais smooth except at 0, u, E W'."(RN)and the HJB equations hold in RN - (0): lDuaI Theorem IV.4.
(20) (21)
+ Ru, = 1 .
Let ii E C(0) fl W:$(0) satisfy
< f( ., a) Aii < g
A,ii
in B'(O),
for all a E Y
in 9'(0), where g
E
L;Y,(O).
In particular (20) implies that pa = A& - f ( ., a) is a nonpositive measure and we assume that the HJB equation holds in the sense of measures: sup pa = 0 . UEV-
Then we have for all 6
> 0 and for all x E 6,
Of course exactly as Theorem 111.2 implies Corollaries 111.1-2, in the same way the above result implies various uniqueness and comparison results. The above example shows that some assumption like (40) is nec-
P. L. LIONS
356
essary and that Lzc is the "best" space since ug E W',m(RN) satisfies (20) and D2ugE Lp(RN)for all p < N (D2u, E M N ( R N ) ) . The proof of this result involves careful estimates on the (in) equations satisfied by convenient regularizations of u" and uses heavily the probabilistic estimates of Krylov [24] -see Lions [33] for more details -. A'weaker form of this result appeared in Lions [36], [37]. We now explain how this result is adapted to the case of timedependent problems : Theorem IV.5. aii --
(20')
at
3C
Let ii
+ A,ii
> 0,
3g
E
E
C ( Q ) satisfy D,ii
L;Y(Q) and
in B'(Q), for all
0
(where Supp 'p c BR). And this implies obviously
Proof of Theorem VI.2.
i)
Let p E C+(D),we consider u(x, t ) = > 0. Using the
&((P(-Y~)I(~~~)), and u(x, t ) = t " ' ~ ( x , t ) for x E D, t Markov property we deduce u(x, t ) = E , [ r A * ( N
+ l)(t - s ) ~ u ( X ,t, - s)dr] .
And using Krylov estimate [24] (the time dependent version of (45))
3 63
Optimal Con:rol
and this yields
and (48) is proved. ii) Next let ’p be a bounded Bore1 function on RN,we consider u(x, t ) = EZ(‘p(Xt)). Assume ‘p 2 0 to simplify, then we have
+
where is the first exit time of X , from x B,. Next, using the scaling invariance of the Brownian motion we see that C,
< CRZN+l,
In addition it is easy to show that for t P,(t
if R
1.
>0
> r,) < C exp ( - a R / J t )
(use for example Lemma 2.2 in Krylov [24]), for some C, a is fixed, assume that the support of ‘p is contained in x have for R 2 R, 0
C < u(x, t ) < FR*N+lIIyIILN
> 0.
If Ro > 0
+ BRo;then we
+ c exp (-uR/JT)II’pIILm.
This implies that P(t, x ; dy) = p ( t , x, y)dy with p ( t , x,J J )E L:(RN)and we find
In what follows we denote by C, a,Q, various constants (dependent of t ) independent of x; we denote by h(y) = 1(Z+BRo)(y)p(t, x, y ) and by p(s) =
364
P. L. LIONS
meas (h 2 s) for s > 0. We apply the above inequality with 9 = and we find
or
Using Young’s inequality, this yields
Therefore for all q
< N / ( N - l), we deduce p(s) < cs-*
and we conclude easily.
References A. D. Alexandrov, Uniqueness conditions and estimates for the solution of the Dirichlet problem, Amer. Math. SOC.Transl., 68 (1968), 89-119. Majorization of solutions of second-order linear equations. Amer. Math. SOC.Transl., 68 (1968), 120-143. Majorants of solutions and uniqueness conditions for elliptic equations, Amer. Math. SOC. Transl., 68 (1968), 144161. -, Dirichlet’s problem for the equation Det llzijll = Wzl, . . ., zn, Z, XI, ..., xn), I., Vestnik Leningrad Univ. Ser. Mat. Mekh. Astr., 13 (1958), 5-24, (in Russian). I. Bakelman, Generalized solutions of the Dirichlet problem for the Ndimensional elliptic Monge-Amphre equations, preprint. R. Bellman, Dynamic Programming, Princeton Univ. Press., Princeton, N. J., 1957. A. Bensoussan and J. L. Lions, Applications des intquations variationnelles en contr6le stochastique, Dunod, Paris, 1978. -, Contr8le impulsionnel et in6quations quasi-variationnelles, Dunod, Paris, 1982. H. Brtzis and L. C. Evans, A variational inequality approach to the Bellman-Dirichlet equation for two elliptic operators, Arch. Rat. Mech. Anal., 71 (1979), 1-13. S. Y. Cheng and S. T. Yau, On the regularity of the Monge-Ampsre equation det (aau/axiaxj) = F ( r , u ) , Comm. Pure Appl. Math., 30 (1977), 4168.
J. M. Coron and P. L. Lions, to appear. M. G. Crandall, L. C. Evans and P. L. Lions, Some properties of viscosity
Optimal Control
11 11
1171
P11 1221
1271 281 1291 1301 1311
t321
1331
3 65
solutions of Hamilton-Jacobi equations, to appear in Trans. Amer. Math. SOC.(1983). M. G. Crandall and P. L. Lions, Condition d'unicitt pour les solutions gtneraliskes des Cquations de Hamilton-Jacobi du premier ordre, ComptesRendus Acad. Sci. Paris, 252 (1981), 183-186. -, Viscosity solutions of Hamilton-Jacobi equations, Trans. Arner. Math. SOC.277 (1983), 1 4 2 . L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-364. -, Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators, preprint. L. C. Evans and A. Friedman, Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Trans. Amer. Math. SOC., 253 (1979), 365-389. L. C. Evans and P. L. Lions, RBsolution des Cquations de Hamilton-JacobiBellman pour des opirateurs uniformiment elliptiques, Comptes-Rendus Acad. Sci. Paris, 290 (1980), 1049-1052. E. B. Fabes and C. Kenig, Examples of singular parabolic measures and singular transition probability densities, Duke Math. J., 48 (1981), 845856. W. H. Fleming and R. Rishel, Deterministic and stochastic optimal control, Springer, Berlin, 1975. I. L. Genis and N. V. Krylov, An example of a one-dimensional controlled process, Th. Proba. Appl., 21 (1976), 148-152. N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland, Amsterdam, 198 1 . K. It& t o appear. N. V. Krylov, Control of a solution of a stochastic integral equation, Th. Proba. Appl., 17 (1972), 114-131. -, On control of the solution of a stochastic integral equation with degeneration, Math. USSR Izv, 6 (1972), 249-262. -, Controlled diffusion processes, Springer, Berlin, 1980. -, Control of the diffusion type processes, In Proceedings of the International Congress of Mathematicians, Helsinki, 1978, 859-863. -, Some estimates in the theory of stochastic integrals, Th. Proba. Appl., 18 (1973), 54-63. -, Some new results in the theory of controlled diffusion processes, Math. USSR Sbornik, 37 (1980), 133-149. -, On the selection of a Markov process from a system of processes and the construction of quasi-difEusion processes, Math. USSR. Izv., 7 (1973), 691-708. N. V. Krylov and M. V. Safonov, An estimate of the probability that a diffusion process hits a set of positive measure, Soviet Math. Dokl., 20 (1979), 253-255. -, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR Izv., 16 (1981), 151-164, (in Russian). P. L. Lions, Optimal control of diffusion processes and Hamilton-JacobiBellman equations, Part 1, The Dynamic Programming Principle and applications, Comm. P.D.E., 8 (19831, 1101-1174. -, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part 2, Viscosity solutions and uniqueness, Comm. P.D.E. 8 (1983), 1229-1276. -, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part 3, Regularity of the optimal cost function, In Nonlinear Partial Differential Equations and applications, Colllge de France Seminar, Vol. V, Pitman, London, 1983.
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366 1341 -,
Optimal stochastic control of diffusion type processes and HamiltonJacobi-Bellman equations, In Advances in Filtering and Optimal Stochastic Control, Ed. W. H. Fleming and L. Gorostiza, Springer Lecture Notes in Control and Information Sciences, Berlin, 1982. -, ContrBle optimal stochastique et tquations de Hamilton-Jacobi-Bellman, Comptes-Rendus Acad. Sci. Paris, 295 (1982), 567-570. -, Equations de Hamilton-Jacobi-Bellman dCgtnBrCes, Comptes-Rendus Acad. Sci. Paris, 289 (1979), 329-332. -, Equations de Hamilton-Jacobi-Bellman, In Se'minaire GoiilaouicSchwartz 1979-1980, Ecole Polytechnique, Palaiseau. -, On the Hamilton-Jacobi-Bellman equations, Acta Applicandae, 1 (1983) 17-41. -, Generalized solutions of Hamilton-Jacobi equations, Pitman, London, 1982. -, Existence results for first-order Hamilton-Jacobi equations, Ricerche Mat., 1983. -, Fully nonlinear elliptic equations and applications, Proceedings of the Function Spaces and Applications Conference in Pisek, Teubner, Leipzig, 1982. -, Sur le tquations de Monge-Amptrre, I. To appear in Manuscipta Math. 41 (1983) 1-43. -, Sur les tquations de Monge-Amtrre, 11. To appear in Arch. Rat. Mech. Anal. (1983). -, Rdsolution des probltrmes gtntraux de Bellman-Dirichlet, ComptesRendus Acad. Sci. Paris, 287 (1978), 747-750. -, Rtsolution analytique des problkmes de Bellman-Dirichlet, Acta Math., 146 (1981), 121-147. -, ContrBle de diffusions dans RN, Comptes-Rendus Acad. Sci. Paris, 288 (1979), 339-342. -, Control of diffusion processes in RN,Comm. Pure Appl. Math., 34 (1981), 121-147. P. L. Lions and J. L. Menaldi, Optimal control of stochastic integrals and Hamilton-Jacobi-Bellman equations, I, SIAM J. Control Optim., 20 (1982), 58-81. -, Optimal control of stochastic integrals and Hamilton-Jacobi-Bellman equations, 11, SIAM J. Control Optim., 20 (1982), 82-95. -, Problkmes de Bellman avec le contrde dans les coefficients de plus haut degr6, Comptes-Rendus Acad. Sci. Paris, 287 (1978), 503-506. P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, preprint. A. V. Pogorelov, On the Minkowski multidimensional problem, I. Wiley, New York, 1978. C. Pucci, Limitazioni per soluzioni di equazioni ellitiche, Ann. Mat. Pura Appl., 74 (1966), 15-30. M. V. Safonov, On the Dirichlet problem for Bellman's equation in a plane domain, Math. USSR Sbornik, 31 (1977), 231-248. -, On the Dirichlet problem for Bellman's equation in a plane domain, 11, Math. USSR Sbornik, 34 (1978), 521-526. D. W. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second-order and their associated diffusions, Comm. Pure Appl. Math., 25 (1972), 651-714. -, Diffusion processes with boundary conditions, Comm. Pure Appl. Math., 24 (1971), 147-225. -, Multidimensional diffusion processes, Springer, Berlin, 1979.
Optimal Colltrol UNIVERSTE PARIS IX-DAUPHINE, CERE MADE PUCE DE LAITRE DE TASSIGNY 75775, PARIS, CEDEX 16 FRANCE
367
Taniguchi Symp. SA Katata 1982, pp. 369-386
Implicit Functions in Finite Corank on the Wiener Space Paul MALLIAVIN
0 0. Introduction In R",integration relative to the Lebesgue measure may, sometimes, use the differentiable structure of the space; for instance disintegration along a family of submanifold, method of stationary phase and critical points. A definition of smooth functions on the Wiener space have been worked out from sometimes [4], [6], [7]. The advantage of this definition is that Wiener functionals coming from the integration of S.D.E. are smooth. The main application of this smoothness have been to prove regularity result for the law of some Wiener functionals. It could seem now appropriate to push forward some kind of differential calculus on the Wiener space in order to obtain some insight in the disintegration of Wiener measure by finite dimensional valued Wiener functional. One difficulty to apply standard implicit function theorem is that the Wiener functionals are not continuous for any Banach norm. Some kind of redefinition is therefore necessary. We shall introduce a scale of capacities and the notion of slim set, that is a set of capacity zero for all the scale of capacities. Then smooth function could be redefined outside a slim set, and outside this set there are continuous (with their derivatives) relative to the Banach space norm. Furthermore the projection of a slim set by a linear projection of finite corank is a slim set on the image of X . With these ingredients an implicit function theorem will be proved in the last section of this work. (Further developpements in [8], [9]) I am very grateful to the warm hospitality of my Japanese colleagues during this conference and also to the generous support of Taniguchi foundation which makes possible this meeting which has been for me an important occasion of exchange of ideas. 1.
2. 3. 4.
Contents. Smooth functions on the Wiener space. Pseudo-direct sum decomposition, Sobolev spsces and slim sets. Redefinition of smooth functions. Implicit function theorem.
P. MALLIAVIN
370
8 1.
Smooth functions on the Wiener space
We shall recall in this section known results. We denote by X the space of continuous applications of [0, I] in a finite dimensional space, vanishing for r = 0. The Brownian motion defines on X a Gaussian measure p. Finally we shall take on X the uniform topology and we shall call ( X , p) the Wiener space. We denote by H the Cameron-Martin subspace of X :
We denote
H, = { x E H ; x”(r) is a measure} . For h E H, we define ( 4 h ) = x( 1). h’( 1) -
s’
~ ( 7 h”(t)dr ) .
.
Then we shall obtain in this way a continuous linear form on X . Given an orthonormal basis e,, . . ., en, . . of H, such that e, E H,, we consider
hn(x) =
Il hni((x, ei))
>
--
where h, are the Hermite polynomials on R,n = (n,, -,n,, . . .) is a multi-index with n, 2 0 and In I = C n, < 00. We denote by Sqthe space generated in L 2 ( X )by the h, with In I = q. Then we have the Wiener’s chaos orthogonal decomposition :
+
L y X ) = 02Pq. P
We denote by B the finite linear span of the Sq, then B is a dense vector subspace of L2(X). We denote by 8,the totality of elements of 9 of degree < q. We define the Ornstein-Uhlenbeck operator 9 on X qby
9 h n = -InIh, and on B by linear combination. More generally if G is a separable Hilbert space, g,, . . .,g , , . . . an orthonormal basis of G , i f f € L2(X,G), we shall write
f=C h . Then
Implicit Functions in Finite Corank
For p
=
011
the Wiener Space
371
2 we have lIfll2L2LY,G)
=
c
Ilfill2ll(X)
.
We denote by B,(G) the space of functionsfc L2(X,G ) such that
fi E 9,
for all I .
Then i f f € P,(G), 2 Y f ~ Lp(X,G). We denote B(G) = U, B,(G) and we define aP(S?, G) as the domain of the closure in Lp(X, G) of 9 defined G) is defined. Given h E H, and f c on B(G). In the same way gP(.Yr; B(G)the derivative off along the vector h is well defined : ( V , f ) ( x ) = lim e - ' { f ( x
+ eh) - f ( x ) } .
Then +(
VhfXX)
is a linear map of H into G. We shall denote by [llVfllaBQits HilbertSchmidt norm. We define in the same way 9p(V;:G):asthe domain of the closure of V for the norm
We can consider the Hilbert space G,=G@H
with the Hilbert-Schmidt norm. Then
Of: X + G , . We can iterate the process and define VZf = V ( V J ) ,
Vrf = V ( P - l f )
and the norm
.
where H , = symmetric tensor product of H 0 0H ; we define the domain g P ( V r ;G'), as the closure of V r defined on B ( G ) under previous +
P. MALLIAVIN
372
norm. (The Hilbert-Schmidt norm defines on H 8 G an Hilbertian structure). Theorem. (Meyer's inequalities [5]). Let 1 < p
Z)dPz(4 Z
Now we integrate in y , by Fubini’s formula
< c{Ii
I~Pz~IILP(x;G)
ri7fII~~(X,~~@~)}
Using again, but in reverse way, the Meyer’s inequalities we obtain Il2ZfllL.p
0, p absolutely continuous relative to dp contradiction which proves the theorem.
References
141 [51
[61 71
V. P. Havin and V. G. Mazilia, Non-linear potential theory, Uspehi Math. Nauk, 27 (1972), No. 6, 67-138, (Russian Math. Survey 27 (1972), NO. 6,71-148.) L. I. Hedberg, Spectral Synthesis in Sobolev Spaces, Acta Mathematica, 147 (1981), 237-264. M. KrCe, PropriCtt de trace en dimension infinie, d'espaces du type Sobolev, C.R. Acad. Sc. Paris, 279 (19741, 157-164; Bull, SOC.math. France, 105 (1977), 141-163. P. Malliavin. Stochastic Calculus of Variation and Hypoelliptic Operator, Proc. Int. Symposium on S.D.E., 1976, Kyoto, ed. K. Kinokuniia, Tokyo, 1978. P. A. Meyer, Quelques results analytiques sur le semi-group d'OmsteinUhlenbeck en dimension infinie, Theory and Application of Random fields, Proc. IFIP-WG, Working Conference, Bangalore, 1982, Lect. Notes in Control and Information Sci., Springer-Verlag, Berlin, 201-214; Note sur les processus #Omstein-Uhlenbeck, SBminaire de ProbabilitCs XIV, 1980/81, Lect. Notes of Math., Springer-Verlag, Berlin, 95-133. I. Shigekawa, Derivative of Wiener functionals, J. Math. Kyoto Univ., 20 (1980), 263-289. D. Stroock, The Malliavin calculus and its application to second order parabolic equation, J. Math. System Theory, 14 (1981), 25-65. I. M. Bismut, Large deviations and the Malliavin Calculus, Birkhauser 1984. A. B. Cruzeiro, Equations dserentielles et Formules de Cameron-Martin non-IinBaires, Journal of Functional Analysis 1983, (54) p. 206-227.
fis,
10 RUE SAINTLOUIS VILE 75004, PARIS
FRANCS
Taniguchi Symp. SA Katata 1982, pp. 387-408
Conditional Laws and Hormander’s Condition Dominique MICHEL
Introduction After the first papers of Zakai [38] and Fujisaki-Kallianpur-Kunita [l 11 on the filtering equation, a number of results on conditional laws and, especially, their regularity properties have been obtained using the filtering equation. Following the work of Pardoux (in his thesis [28]) on the stochastic partial differential equations (S.P.D.E.) under ellipticity conditions, Krylov-Rozovskii [ 161, Pardoux [29] and Rozovskii-Shimizu [31] proved the regularity of conditional laws under a uniform partial ellipticity hypothesis and assuming the existence of a density for the initial conditional law. Using a backward S.P.D.E. and a stochastic FeynmanKac formula, Pardoux [30] extended these results to prediction and smoothing. In an other direction, Davis [7] and Eliott-Kohlmann [lo] used the filtering equation and the works of Doss [9] and Sussmann [36] to get a robust version of the conditional law in some particular cases. More recently, without using the filtering equation, Michel [27] and Bismut-Michel [4] applied the Malliavin calculus to get regularity results for the conditional laws under a local hypoellipticity condition of the Hormander type. In [4],they also show that, after a transformation by a stochastic flow, the conditional law is the law of a semi-martingale whose characteristics are explicitly calculated. Chaleyat-Maurel-Michel [6] obtained the regularity result of [4] under a global hypoellipticity hypothesis using extensions of Kohn [151 technics to the probabilistic aspect of the problem and applying them to the filtering equation. Kunita [19], [20], [21] obtained results of the same type in the particular case where the coefficients of the Zakai equation do not depend on the observation. We want to present here the main ideas of Bismut-Michel [4] and Chaleyat-Maurel-Michel [6]. In the first stage, we use stochastic flows and Girsanov’s transformation to reduce the problem to a simpler one. The second step is to get some results on the simplified problem and the third step is to use these results to study the initial problem. But we shall see that, in the regularity problem, we get better results working directly with the initial problem than by first tackling the simplified one.
D. MICHEL
388
The paper is organized as follows: in the first part, we show how to perform a transformation by a stochastic flow and a Girsanov transformation to get a simplified problem. In the second part, we give some results on the structure of the conditional laws. In the third part, we obtain a first theorem of regularity using a Hormander theorem for a heat equation whose coefficients are not regular in time. In the fourth part, we expose the main ideas of Malliavin' calculus in the Bismut' formulation and apply it to the regularity problem. Finally, in the fifth part, we give a stochastic Hormander theorem whose proof uses the construction of a class of test semi-martingales and the definition of Sobolev norms adapted to S.P.D.E. The author wishes to thank Professor Ito and Professor Ikeda for their invitation and their kindness during the workshop at Katata, as well as Taniguchi Foundation for financial support.
5 1.
Stochastic flows and Girsanov transformation
1.1. Notations Let us denote by SZ (resp. 6) the space V(R+, R") (resp. %(R+,RP)). The generic point of D (resp. 6) is denoted by w (resp. I?) and P (resp. P") is the Wiener measure on 0 (resp. 6). If Xis a stochastic process defined on D x 6, 33': is the a-field generated by X until time t . In particular, = 33':~". If X,is a semi-martingale on (0x 6, 9, 9,, let us define 9, P 0P") SX,(resp. dX,) is its Ito (resp. Stratonovitch) differential (cf. [26]); S,X, (resp. SJ,) is the martingale (resp. bounded variation) part of SX,. Let us consider m 2p 1 vector fields X,,. . ., X,, XI, . . ., X,,Z,, . . Z,, on R" X RP such that
+ +
X,(x,z)
-
-
I
a ,
= X i ( x , z)-
a
ax,
-.
X,(x, z) = X:(x, z)Z,(X, z) = Z { ( Z ) - ,
a
a=,
,
a ,
ax,
0
