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1, is a consequence of Dynkin's proof and Theorem 2.3 of Adler and Lewin (1991).
Adler & Lewin: Intersection local times of superprocesses
9
which we refer to as the time and space coordinates of v. V_ denotes the set of entrances for our graph, and if v E V_ we set s = 0 and yv = x,,. If v is
the exit labelled by j, i < j < n, we set (sv, yv) = (t.i, zj)
Finally, Vo denotes the set of internal vertices, i.e. those vertices that are neither entrances nor exits. We shall, in what follows, be interested only in third and fourth order moments. In the latter case, the set D4 of (2.1) consists of the following six basic graphs, and their various combinatorial rearrangements. The contri-
bution of graph - i.e. the integral appearing in (2.1) - is written after the graph. Since we shall be concerned only with finiteness of moments, we shall not bother with the combinatorial factors associated with each graph.
Graph 1 (O'X 1)
(O,X 2)
(t1,Z1)
(t2'Z2)
4
(O,X 3)
(O,X 4)
(t3,Z3) (t4 ,Z4)
fjP0,ti(Xi,z,),P;(z,)dx;dz;
10
Adler & Lewin: Intersection local times of superprocesses
Graph 2 (O,x 2)
(t 1,Z 1)
%
(S2,Y2)
,YJ)
(t 2,Z 2)
(t 3,Z 3)
(t 4,Z 4)
f dxldx2 f dyidy2 f dslds2ps (xi,yl)p 2(x2,y2) 4
X f pt,-s, (yl, zl )pt2-s, (yl, z2)pt,-s2 (y2, z3)pta-s2 (y2, z4) II y,(z,)dz=
Graph 3 (0, x
(t 3,Z 3)
(t4,Z4)
f dxldx2dx3 f dsl f dyl ps, (x3, yl ) 4
X
f pt,(xi,zi)pt2(x2,z2)pt,-s,(yi,z3)pt,-s,(yi,z4)11cp(z,)dz;
Adler & Lewin: Intersection local times of superprocesses
11
Graph (O,x 1)
(O,x 2)
(s1,y1) (t 4,z4)
J
dxldx2
J
ds1 f dy1 ps (x1, y1)
J
ds2 f dye P32-s, (y1, y2) 47
x J pt,-'1(y1' zl)pt2-s2(y2,z2)pt3-32(y2,z3)pt4(x1,z4) jj(p(zi)dzi icl
Graph 5
J
dxl
J
dsl
J dyl p2 (x1, yl) J
ds2ds3
J
dy2dy3 p 2-31(yl, y2)ps -s, (yl, y3) 4
x f pt,-,,(Y2, zl)pt2-s2(y2, z2)pt3-33(y3, z3)pt4-33(y3, z4) J1 (P(zi)dzi i=1
Adler & Lewin: Intersection local times of superprocesses
12
Graph 6 (O,x 1)
1
021Z 2)
(S1,yi)
(S2,y2)
(S3,y3)
2, and note that (p, Xt), where Xt is either a super Brownian motion or super stable process, is a continuous semi-martingale (cf., for example Roelly-Coppoletta (1986) for this result for a more restricted class of ep, and Adler and Lewin (1991) for cp E W2,n fl W2'1). The associated finite variation process is given by
Xt), and the martingale part by (p, Zt), with associated increasing process [(o,Z)]t = fo(cp,X,)2ds, (cf.(1.2)).
It thus follows from a standard Ito formula (e.g. Karatzas and Shreve (1988)), that for' E C2(ll + x fit) and cp E W2 P fl W2'1, the formula (1.11)
of Lemma 1.3 holds, with the singular exception that the last term there (remember that k = 1) is replaced by JtP
J0
(s,(p,X,))d(p,Z,).
(3.1)
To complete the proof, we have to show that the above expression is equivalent to t
f
0 J 82d
Wx(s, (cp, X,))cp(x) Z(ds, dx).
Recall that, by assumption, there exists an integral a > 0 such that Wx(s, x) < C(s)IIxIla for all x E Rd. It thus follows from Dynkin's moment
formulae that E(Wx(s,(p,X,)))2 < oo. Hence the integral (3.1) is also in ,C2(P).
We can therefore approximate %Fx(s, (cp, X,)) in £2(P) by a simple function of the form n
Wi` `
L L. CA;IA,(W)I(8,,t;)(s) i=1 A;
where (si, ti] n (si, t7] = 0
if i # j, Ui(si, ti] = R+, and Ai E .F,;, UA; Ai = Q. Thus the integral (3.1) can be approximated in £2(P) by E E CA, I(o,tl(ti) [(gyp, Zt+) - (tP, Z3, )]. i
A;
(3.2)
14
Adler & Lewin: Intersection local times of superprocesses
But, according to Walsh's (1986) formulation of stochastic integration with respect to martingale measures, if we also take an approximation via simple functions to '(x) = Wx(s, (p, X,))cp(x), we immediately have that (3.2) is also an £2(P) approximation to
ff t
0
Wx(s, (4,, Xs)) yo(x) Z(ds, dx).
Std
Going to the £2(P) limit on both sides of this equivalence establishes the required result.
(b) Proof of Lemma 1.4. As in the previous proof, one can apply a regular Ito formula for the continuous semi-martingale (p,Xt) to the non-anticipative functional T (t, x) = x f0 (p, X,) ds, and evaluate it at T(t, (0,Xt)). Noting the various identities for' and its derivatives following (1.12), we thus obtain (1.13) but for the fact that the last term is replaced
f
by
It J ds (V,(x), X3) 0
Zt).
As before, we have to rewrite this as a stochastic integral against the martingale measure Z(dt, dx) to obtain the required form of (1.13). The argument, however, is precisely as before.
(c)
Proof of Lemma 1.5. We commence with functions 'DN(x, y) of the
form 4 N(x, Y)
_
N / E Ok(x)`Vk(y)
(3.3)
k=1
where both cp, i E W2 'P n W2'1. It follows immediately from Lemma 1.4,
with the notation of our lemma, that
f
(I'N(x, y), Xt(dx) XT(dy)) dt
f
T dt
0
t
J0
ds (L Sf
+j
y), X,(dx) Xt(dy)) dt t
+j T Jtd J
N(x, Y), X,(dx) Xt(dy))
ds
y), X,(dx)) Z(dt, dy).
(3.4)
This, of course, is the required (1.14) but for the restriction of the form (3.3). We need (3.4) for ' (x, y) E C2(Rd X Rd) of compact support.
Note, however, that every such function can be represented as the limit of functions 'PN of the form (3.3), with convergence in the supremum norm
Adler & Lewin: Intersection local times of superprocesses
15
be as required, and let {41iN}N fed dx dsl fed dy1 ps, (x, y1)
f
T
x
f ds2
dye p32-s1 (y1, y2) f823d pt, -31 (y1) zl )pt2-s2 (y2, z2)pt3-s2 (y2, z3)
(3.13)
dzldz2dz3.
(There are, of course, a number of contributions of this kind with the roles of the various pairs of times and test functions interchanged.) As in the previous case, use the symmetry of pt to integrate out x, and ChapmanKolmogorov to integrate over yl. After rearrangement, (3.13) becomes
IVd N(zl, x
z3)
IT T dt3
t3
f dtl It' dt2
IT
T dsl f T ds2
JRd dy2 pt,+s2-2s, (y2, zi )
x pt2-s2(y2, z2)pt3-s2(y2, z3)dzidz2dz3.
(3.14)
Bound 'DN(z1, z3)IN(z2i z3) by II(DNII'00, and then perform the z3 integral over all of Jd rather than just D. Then integrate out Y2 via ChapmanKolmogorov, to bound (3.14) by T
t3
T
t3
1I4NII00 f dt3 f dt1 f dt2 f ds1 f 0
0
0
0
DxDp(t,+t2-2s,)(zl, z2) dzldz2.
(3.15)
18
Adler & Lewin: Intersection local times of superprocesses
Integrate z2 over Rd, and then z1 over D to finally find a bound of the form IDI as required. All other terms can be handled similarly.
As before, to complete the proof, we must show that the £2(P) limit, assured by Cauchy convergence is, in fact, the stochastic integral IJOT
JBtd JOt
ds (4)(x, y), Xt(dx)) Z(dt, dz).
(3.16)
This, however, is relatively straightforward by a subsequencing argument.
Implicit in the above calculations is the fact that, for every y E Dy = {y y), Xt(dx)) is Cauchy in £2(P). Thus (x)y) E D} and t E [0,T], there is an a.s. convergent subsequence ('Nk(x, y), Xt(dx)), which, since Xt(Dx) < oo, where Dx = {x : (x, y) E D}, and the convergence of'kN to is uniform, implies (-I)Nk(x, y), Xt(dx)) $ y), Xt(dx)) as k - oo. Furthermore, both ('kN(x, y), Xt(dx)) and (4k(x, y), Xt(dx)) are uniformly :
continuous in y E Dy and t E [0, T]. Hence, along the subsequence {Nk}k>1 we must have JOT
(x, y), Xt(dx)) Z(dt, dy)
,/d lot ds
T0d 0
y), Xt(dx)) Z(dt, dy).
ds
Thus the £2(P) convergence along the full sequence must be to the appropriate limit.
Proof of Theorem 1.2 We can now, finally, turn to the proof of Theorem 1.2. We shall concentrate on establishing the evolution equation (d)
(1.10). Since the proof is of an G2 nature, the convergence of -y' (T, gyp), which
forms the first part of the theorem, will be proven en passant. To start, let {GE},>o be a sequence of Cc functions of bounded support such that G, + Ga as a -> 0. Take the evolution equation (1.14), and replace the function -1)(x, y) appearing there by G,(x - y)cp(x). Subtract l dt Jot ds (G,(x - y)cp(x), Xt(dx) Xt(dy))
from both sides of the resultant equation. Note the definition (1.9) of y, (T, cp), as well as (1.8), to obtain -tE (T, tP)
IrT r 0 I'd + AfT
IT
/'t
ds (GE(x - y)cp(x), Xt(dx)) Z(dt, dy)
0 dtIt
dS
(G,(x - y)cp(x), X3(dx)Xt(dy))
(G,(x - y)cp(x), Xt(dx) XT(dy)) dt.
(3.17)
Adler & Lewin: Intersection local times of superprocesses
19
If we now show that each of the three terms on the right-hand side of this equation converges in G2, as e --> 0, to a limit that is independent of the sequence {GE}, then the same must be true of the left-hand side, and so the central Theorem 1.2 will be established.
The proof hinges on the moment formulae given in Section 2, and very often closely parallels similar calculations in Rosen (1990). Because of the fact that we have three terms to consider, each one involving a number of graphs, and because the examples given in Rosen's paper show how these calculations must be carried out, we shall now carry out only one of the moment calculations in detail.
The term that we choose is the stochastic integral with respect to Z. There are two reasons for this. Firstly, Rosen's calculations do not involve an expression of this kind, and secondly, it will become clear during the calculations where the main conditions of the theorem (i.e. d = 4,5 in the Brownian case and d/3 < cti < d/2 in the stable case) come from. Thus, we now turn to proving that for any sequence {GE}, of the form described above, converging in G' to Ga,
{jT Eli o E
lods ((GE - GE)(x - y)(x), X3(dx)) Z(dt,
J
d)}= 0 . (3.18)
To save on notation, fix e, e' > 0 and set D(x) = G,(x) - G,, (x). From Section 2, three different graphs arise in computing (3.18), described there as (a)-(c). We claim that (a) is easy, and so leave it to the reader. A typical term arising from (b) is J d dxl
4
J T dsl 1
d
dyi ps (xi, yi) f T dti fed dzi pig-s, (yi,
dt2 f d dz2 pts-,, (yi, z2)D(zl - z2) XJ
/T
d
dx2 f dt3 fed dz3 pt3(x2, z3)D(zl - z3) .
(3.19)
(If this expression is not "obvious", then write D(x, y) as E;_1 dj (x)d'j (y),
Adler & Lewin: Intersection local times of superprocesses
20
and consider a term in this expansion corresponding to the graph 2
dj (z1) dk (z1)
_ 0; Q2 = 1.
Aldous: The continuum random tree II
29
Strict t-ary (0 or t offspring): po = 1 - 1/t, pt = 1/t; az = t - 1. Unary-binary (0, 1 or 2 offspring): PO = Pi = P2 = 1/3; az = 2/3. unordered labelled trees. Unrestricted degree: Poisson distribution pi = el/i!, i > 0; az = 1. Unary-binary: po = 2+,727 P1 = z+2zp3 = 2+72; az = z+ z Strict t-ary: same as ordered case.
Remark. I have slid over one issue: in the combinatorial story the trees are regarded as rooted and labelled, i.e. the n vertices are distinguishable. The distinction between labelled and unlabelled is irrelevant for ordered rooted trees (because the ordering serves to distinguish vertices anyway) but relevant for unordered trees. The model "all unordered unlabelled trees equally likely" does not fit into this set-up, and no simple probabilistic description is known.
2.2 Ordered Trees and Brownian Excursion With a finite rooted ordered tree t on n vertices we can associate the following two sequences (the terminology is not standard).
The height profile (h(j);j > 0), where h(j) is the number of vertices at distance j from the root.
The search depth (x(i); 1 < i < 2n - 1) defined as follows. At each vertex v, suppose the edges at v leading away from the root are ordered as "first", "second", etc. Then depth-first search of the tree is the following deterministic
walk (v(i) : 1 < i < 2n - 1) around the vertices. Let v(1) = root. Given v(i) choose (if possible) the first (in the ordering) edge at v(i) leading away from the root which has not already been traversed, and let (v(i), v(i + 1)) be that edge. If not possible, let (v(i), v(i + 1)) be the edge from v(i) leading towards the root.
This walk terminates with v(2n - 1) = root, having traversed each edge exactly once in each direction. Finally, define the search depth x(i) = distance from root to v(i).
There is a connection between the two sequences: for j > 1
h(j) = number of upcrossings of (j - 1j) by the sequence x(i).
(9)
For a random tree distributed as CBP(n) these become random sequences
30
Aldous: The continuum random tree II
(H(j)) and (X(i)), say. Define the rescaled cumulative height profile process
H(j), t > 0
Hr = n-1
(10)
j oo (H,,; s > 0) -+ (Ho,/2i s > 0)
(12)
34
Aldous: The continuum random tree II
where
H. =
f
0
1
l(w, 0) - (ZlO8/2;s > 0) where
_ 1°
d
r
ds Jo 1(w 0) are independent copies of standard BES(3).
It turns out that we can construct a realization of R from a realization of 2B, analogous to the construction of S from 2W in section 2.2. In brief, we construct a tree labelled by it : t > 0} from (2Bt; t > 0) and separately construct another tree labelled by {-t : t > 0} from (2B-t; t > 0); then we join the trees by identifying (for each b > 0) the points labelled T and T, where
Tb = max{t : 2Bt = b}, T = min{t < 0 : 2Bt = b}. This becomes the point b on the baseline at distance b from the root. In figure 2, we regard positive-time BES(3) as tracing out the part of the tree above the baseline, and negative-time BES(3) tracing out the part below the baseline.
Here is a verbal description of how R arises as a limit of rescaled CBP(n). Rescale edges to have length 1/a(n), where throughout this section
a(n) -> oo, a(n) =
o(n1/2).
Let v be the measure putting mass 1/a2(n) on each vertex. Then the rescaled random set T of vertices of CBP(n) converges in distribution to R, and the random measure v converges to v. From this limit procedure (or from the BES(3) construction) we see that the SSCRT has a self-similarity property: multiplying distances by a constant c doesn't affect the distribution of the random set R, though it does take the measure v to c-2v. These results could be formalized and proved in the same way as was done in [3] for the CCRT. Here are the concrete results analogous to Theorem 2 and Corollary 3.
Reconsider the search depth (x(i); 1 < i < 2n - 1) associated with a tree tin section 2.2. The search starts and ends at the root: x(1) = x(2n - 1) = 0. For present purposes we want to center at the root, so we define (x*(i); -n < i < n) by
x*(i) = x(i),1 < i < n; x*(-i) = x(2n - i),1 < i < n
38
Aldous: The continuum random tree II
with x*(O) = -1. For a random tree distributed as CBP(n) this becomes a random sequence (X*(i)); we also have the height profile process (H(j)) as in section 2.2. Rescale as
11 = a-2(n) E H(j), s > 0
(15)
j 00 (k,,,; s > 0)
- (Qo8/2; s > 0)
where
(17)
00
Q8 = f
1(B, oo, (a-1(n)H([a(n)s]); s > 0) - (Zgo8/2; s > 0)
(4 q8; s > 0)
where dQ8 8
ds
is total occupation density for 2-sided BES(3). Kolchin [33] Theorem 2.5.4 and Kennedy [30] Theorem 1 have given the 1dimensional convergence results implicit in Conjecture 7, but I have not seen the full weak convergence result published explicitly.
Aldous: The continuum random tree II
39
It is well known that the total occupation density of one-sided BES(3) is the diffusion with drift and variance rates µ(x) = 2,
o'2(x) = 4x,
or equivalently IB2(s)12, where Bd is standard d-dimensional Brownian motion. It follows that (q,) is IB4(s)12, or equivalently the diffusion with
µ(x) = 4, 0,2(x) = 4x. The marginal distributions are Gamma(2, ): x
.fq(e)(x) =
x
x exp( 2s).
(18)
See Pitman and Yor [40, 41] for extensive accounts of related properties of Bessel processes. As mentioned above, this Gamma limit distribution for generation size in conditioned Galton-Watson processes was known, but analytic proofs give little insight into why this particular limit distribution holds. The BES(3) representation gives one: the positive-time and negative-time occupation densities are obviously i.i.d. exponentials. 2.6 Discrete limits of CBP(n) As a final piece of background, one can take limits in CBP(n) without rescaling edge-lengths. In this setting, the limit process T. (described below)
depends on the entire distribution of . This result is in Grimmett [24] and in [2] Theorem 2, in the special case of Poisson(1) offspring; and the general case is implicit in Kesten [31].
The discrete infinite tree T,,. For each k = 0, 1, 2, ... create independently branching processes, whose first generation size has distribution but whose subsequent offspring distribution
is . Regard these as trees with root ik and other vertices unlabelled. Then connect i0, i 1, i2.... as a path, deem io the root and delete labels.
This is a convenient place to introduce the idea of random re-rooting. A random tree T distributed as CBP(n) is normally considered as rooted at the progenitor of the branching process. We may, however, choose another vertex v of T and declare that to be the root. (To avoid discussing ordering of the re-rooted tree, regard trees as unordered). If v is chosen uniformly at random from the n vertices, call this procedure "random re-rooting". In the combinatorial model "uniform random labelled unordered tree", i.e. the Poisson(1) special case of CBP(n), it is immediate from the combinatorial
Aldous: The continuum random tree II
40
description that random re-rooting does not change the distribution of the random tree. For general CBP(n), the distribution does change. However, the discrete limit distribution T* is almost the same as T,,O above, except for one change:
the branching process rooted at io has first generation
instead of .
offspring distribution
(19)
This result, implicit in earlier work, is given explicitly in Aldous [1]. As an aside, the idea of taking discrete limits in randomly re-rooted trees works for almost all the larger class of "height O(log n) trees" mentioned in the introduction, whereas for those trees looking at limits around the original root is not interesting - this topic is the subject of [1].
2.7 Symmetries of Trees, and the Arrow of Time We now have four ways to look at the CCRT S (Brownian excursion, the global construction, limits of CBP(n), and the random f.d.d.'s (13)). An audience from theoretical stochastic processes is likely to concentrate on the first way, and think the whole subject is just a corner of Brownian excursion theory. But I hope to show that misses the point: all four ways are useful in doing calculations.
As an illustration, the fact that in a special case CBP(n) is exactly invariant under random re-rooting implies immediately that the distribution of the CCRT is invariant under random re-rooting. (20) By considering the search depth process, we could write this as a statement about Brownian excursion W. Fix u and define inf
u 0 it is useful to write b for the point on the baseline at distance b from the root, and Rb for the part of R connected to the initial segment [root, b] of the baseline. The projection process. This is the process (Z(b); b > 0), where Z(b) = v(Rb), the total "weight" of Rb. There are two interpretations of the process as limits in CBP(n), using either the original root or the random re-rooting procedure of section 2.7, and the latter is more interesting. Let V. be a uniform random V the ba(n)'th generation vertex of CBP(n). Let before V, and let be the total number of descendants of Then from Theorem 5 (for re-rooted CBP(n)) Z(b).
In [2] section 7 the global construction was used to prove
Lemma 10 (Z(b), b > 0) is the positive stable (1/2) process, that is
Eexp(-9Z(b)) = exp(-b 29) Z(b) -A b 2Z(1).
It is convenient to record an easy calculation here:
jb(b - s)Z(ds)
d
(4/9)b3Z(1).
(44)
One can alternatively obtain Lemma 10 from the BES(3) representation. The last exit time process (Tb ; b > 0) for (2Bt) is a positive stable (1/2) process (see e.g. [40]) and then Z(b) d Tb +Te because v is the measure induced by 2B from Lebesgue measure.
Aldous: The continuum random tree II
53
Remark. One could construct BES(3) by starting with the last exit time process (T+) and then filling in excursions above levels b. In the global construction, each bush attached to the baseline represents such an excursion.
The depth process. Write Fb for the height of Rb, considered rooted at the original root, and write Db for the height of Rb, considered re-rooted at b. Think of Db as the "depth" of b. We can give Db an interpretation as a limit in CBP(n), using as above a uniform random vertex V of CBP(n). Let be the number of generations until extinction, for the process of descendants of the ancestor of V in the ba(n)'th generation before V. Then Db.
A symmetry property baseline reversibility which is obvious from the global construction is the following: the distribution of Rb is invariant under reflection of the baseline segment [root, b] about its midpoint. In particular Db
d
Fb
for each b.
(45)
But (Fb) and (Db) are different as processes, e.g. because Db - b is nondecreasing in b whereas Fb does not have that property.
Lemma 11 For each b,
P(Fb g) = 2g-2(1 - g)(g + (1 - g) log(1 - g)), 0 < g < 1. To see (a), consider the point process P = {(s*, h*)} recording the heights h* and positions s* of bushes branching off the baseline in the global construction. Then P is a Poisson point process of intensity p(s, h) = 2h-2. This fact comes out of the argument in [2] section 6, or alternatively from the BES(3) construction using excursions from last exit times. Plainly
P(Ci > c) = P( no points (s, h) of P with s < c and h > 1 - s) exp(- j 0
p(s,h)dhds) 1-e
giving (a). For (b), we can use the 2-sided BES(3) description to rewrite G1 as follows. Consider local time measure on Is : B, = 1}; pick at random two times from this measure (normalized to a probability measure), and write T(1), T(2) for the order statistics of these two times; define G, = mineT(,) Bt minT(l) 0) taking values in the CCRT (S, y) or in the SSCRT (R, v) (by default, random processes start at the root). The discussion is necessarily heuristic, because no rigorous proof of existence of Xt has been written down. Such processes may be interesting as counterparts to the recent rigorous theory of Brownian motion on regular fractals developed by Barlow and Perkins [8], Lindstrom [34] and others. Loosely, our particular CRTs have "dimension 2", inherited from Brownian sample paths: in view of the rigorous construction of continuum trees from general continuous functions in [3], one can certainly construct continuum trees of any fractional dimension. Despite the large physics literature, rigorous study of diffusions on non-regular fractal sets seems difficult. But obviously a tree structure is a great simplification, and a rigorous theory of Brownian motion on rather general continuum trees seems a natural next step. I outline below the shape that such a theory might
take, but do not intend to pursue the topic myself.
It is important to regard S and R as spatial trees, and downplay their constructions from Brownian excursion and BES(3). For simple symmetric random walk on a discrete tree there is an elementary formula for the mean first passage time from one specified vertex to another. Using these formulas it is easy to see that in CBP(n) (with v2 = 1 for simplicity) the mean passage time between random vertices l2 n3/2. Results of this type go back to Moon [36]. Thus one way to think of Brownian motion on S is as a rescaled limit of simple random walk on CBP(n): make the edges have length n-1/2 so as to get the limit S, and make the time between steps be n-3/2. Similarly, we could start with simple symmetric random walk on the discrete infinite tree of section 2.6: the same rescalings should lead to Brownian motion on R. The latter discrete setting was studied in Kesten [31], who refers to his
unpublished work proving the existence of a limit n-3/2T,,,/2 - T for the first passage time T,,,/2 from the root to the point at distance n1/2 along the infinite path. I interpret T as the time taken by X on 7Z to first hit the point on the baseline at distance 1 from the root. The ingredients of a proof of the
Aldous: The continuum random tree II
57
existence of X via such a weak convergence construction would plainly be similar to this unpublished work of Kesten. In the next section I suggest an alternative "sample path" construction which I believe will handle more general continuum trees. But my main purpose is to exhibit concrete calculations of distributions (mostly expectations) associated with X, and this is the subject of sections 5.3 and 5.4. 5.2
An Occupation Density Construction
In this section let us work with the precise definition of a (deterministic) continuum tree (So, µo) given in [3] section 2.3. Essentially, So is a set which is topologically a tree, i.e. has a unique non-self-intersecting path between any pair of points, and po is a probability measure on So, which should be thought of as a "uniform measure". So contains a "root" denoted by 0. (As a technical remark, for our purposes here we also assume So = support(Fto), though in [3] a weaker condition was used.)
We now define Brownian motion on a compact continuum tree (So, µo) (the
locally compact case needed for R is similar) to be a So-valued process (Xt; t > 0) with the following properties. (i) Continuous sample paths. (ii) Strong Markov.
(iii) Reversible with respect to its invariant measure µo. (iv) For each path [[a, b]] C So and each x E [[a, b]],
P.(T. < Tb) = d(a,
bb>
(47)
where d denotes distance.
be the mean occupation measure for the (v) For points a, b E So let process started at a and run until it first hits b. Then ma,b(dx) = 2d(b; c(b : a, x))po(dx), x E So
(48)
where c(b: a, x) is the point at which the paths [[b, a]] and [[b, x]] diverge.
The skeleton Sao of So is the set of points x which are in the interior of some path [[a, b]]. A consequence of the precise definition of continuum tree is that
Aldous: The continuum random tree II
58
the skeleton has go-measure zero. Thus the process spends Lebesgue-0 time in the skeleton.
In contrast to the setting of general fractal subsets of Rd [34], there is no difficulty in proving uniqueness here, because of the explicit formulas (47,48)A sledgehammer proof could be based upon the result about general Markov
processes being determined up to random time-change by their exit place distributions. To outline an existence argument, let V1i . .
.,
Vk be picked independently from
µo and let R(k) be the subtree of So spanned by the root and the points V1, ... , Vk. Then R(k) is a (random) tree consisting simply of a finite number
of edges with positive edge-lengths. Recall (14) that in the particular case of the CCRT S, this R(k) is distributed as the tree produced by the first k branches in the global construction of section 2.3. Let µk be the natural induced Lebesgue measure on R(k). Assume the following regularity property (local homogeneity): there exist deterministic ak -> oo such that
-ILk --> µo a.s. as k -g oo
(49)
ak
The CCRT has this property with ak = vl'2--k (this is essentially Theorem 3 (ii) of [2], but also follows easily from the Brownian excursion representation).
On each R(k) we can define ordinary Brownian motion Xk(t) started at the root 0. One way of viewing X is as the weak limit of these ordinary Brownian motions as we "fill out" the tree and speed up time:
Xk(akt) - X(t) as k -3 oc. In fact we can do better and use a sample path construction. The key idea is: if we measure time by a suitable "local time" rather than absolute time, then we can make these Brownian motions consistent as k increases. Fix r > 0. Run ordinary Brownian motion Xk(t) on 7Z(k) until the local time at the root reaches T. Regard the accumulated local time as an occupation density lk: fT
1(Xk(t)EA) dt = fA lk(T,w, x)pk(dx), A
C 1(k).
(50)
We can construct 1k+1 in terms of lk. For R(k+1) consists of R(k) plus a new edge (Bk+1, Vk+1) attached at a point Bk+l E R(k). The occupation density lk+1(T,w,x) coincides with lk(T,w,x) for x E R(k), while on the new edge its conditional distribution given lk(T, w, Bk+l(w)) = 1 is the occupation density for 1-dimensional Brownian motion on a line segment of length d(Bk+l, Vk+l)
Aldous: The continuum random tree II
59
started at 0 and run until the local time at 0 reaches 1. Continuing for all k, we get a function l,,(T, w, x), x E S,, defined on the skeleton S,,, of So which satisfies (50) for all k. If we can show l,, (T, w, ) extends to a bounded continuous function on So,
(51)
then by (49) l,, is the k --- oo limit time-rescaled occupation density for Brownian motion on R(k), and this can be regarded as occupation density for some process (X(t) : 0 < t < t(r)) on So run until its density at the root reaches T, i.e. until absolute time
t(r) =
w, x)po(dx). ISO
In other words, we are trying to construct X by specifying its occupation density up to times T. The argument is rigorous up to (51); and it remains to put together different T's and show that X has the properties (i)-(v). I conjecture that very little more than (49) is required - perhaps only some weak "metric entropy" condition on So.
5.3 Easy Distributional Properties We now return to the special cases of the CCRT (S, µ) and the SSCRT (7Z, v), and assume that Brownian motion X exists as a weak limit of rescaled random walks as in section 5.1 and also via the construction in section 5.2.
The SSCRT case is somewhat more tractable, because it inherits from R a self-similarity property. (X"; t > 0)
d
(c'/3Xt; t > 0).
(52)
One explanation of the "1/3" comes from the rescaling in section 5.1 of discrete random walk. Another comes from a scaling (53) of first hitting times, which we now derive. As in section 3.5, for b > 0 write b for the point on the baseline at distance b from the root, and write Z(b) = v(Rb) = the total "weight" of the part 1Zb of R connected to the initial segment [root, b] of the baseline. Appealing to (48)
E(TbIR) = f b 2(b - y)Z(dy) where we write to mean conditioning on the realization of the random tree (R, v). Now Lemma 10 says that Z(.) is the positive stable 1/2 process, and appealing to (44)
E(TbIR)
d
8b3Z(1). 9
(53)
Aldous: The continuum random tree II
60
Of course EZ(1) = oo and so the unconditional first hitting time T6 has infinite expectation: this is a disadvantage of working with the SSCRT.
Turning to the CCRT (S, µ), consider first hitting times Tx on arbitrary x E S. Immediate from (48) is Ex(TyI S) = is 2d(y, c(y : x, z))µ(dz).
(54)
Another consequence of (48) is a simple formula for mean round trip times between leaves (here the root counts as a leaf) EE(TyIS) + EY(TXIS) = 2d(x, y) for any leaves x, y E S.
(55)
These have nothing to do with the particular structure of S, but reflect elementary general identities for simple random walks on discrete trees. Now
let V and Vl be independent random points of S chosen according to µ (which puts mass 1 on the leaves - here continuum trees are simpler than discrete trees!). Using invariance (20) under random re-rooting, Ed(V1, V) _ Ed(root, V) and Ev1Tv = ETv(= ErootTv). It follows from (55) that
ETv = Ed(root, V) =
,/2 by (34)
(56)
which is the Brownian motion analog of the result of Moon [36] mentioned earlier. A slight elaboration is provided by
E(TvJ h(V)) = h(V)
(57)
where h(V) = d(root, V). This is based upon another symmetry property of S. Recall from (14) that we may regard [[root, V]] as arising from the initial line-segment [0, C1] in the global construction of S. Treating this initial line segment as a baseline, we can define a "projection process" Z(b) analogous to the SSCRT case (section 3.5). That is, Z(b) is they-mass ultimately attached to the first b units of the initial segment. The global construction makes clear the reversibility property (2 (b); 0 < b < h(V))
(h (V) - b); 0 < b < h(V)).
Rewriting (54) as
E(TvIS) = f (58) implies (57).
h(V)
2(h(V) - b)d2(b)
(58)
Aldous: The continuum random tree II
61
It is immediate from (55) that, given S, the v which maximizes the mean round trip time from the root to v and back is the vertex V* at maximal distance from the root. For future reference,
ErootTv* + Ev*Troot = 2V'.
(59)
by (22). On the other hand, one can show that the v which maximizes the one-sided hitting time Eroot(TvIS) is not V*. One could calculate variances of hitting times in similar ways, starting from formulas in the discrete setting. Moon's result in [36] Corollary 7.3.1 (whose proof relies on the special structure of the uniform random unordered tree) implies 32
var(Tv) = 32 15
In principle this variance could be decomposed as the sum of three components - contributions from the choice of S, the choice of V given S, and the choice of Brownian path given S and V - but the computations look messy. 5.4 Hard Distributional Properties The distribution of baseline hitting times for Brownian motion on the SSCRT turns out to be related to the distribution of inverse local time in the CCRT.
Here is our best shot at describing these distributions, though it doesn't qualify as "explicit". Consider 1-dimensional Brownian motion, started at 0 and run until its occupation density at 0 reaches a. It is well known ([43] VI.52) that its occupation density Z, at position s > 0 behaves as the Ray-Knight diffusion
Zo = a, dZ, = 2Zdf s > 0,
(60)
where (,Q,) is another 1-dimensional Brownian motion. Now consider the global construction of S via the cut-and-join points (C;, Jz). Fix a > 0 and define (Z8 , s > 0) by
Zo = a dZe = 2 Ze d/3, on C;_1 < s < Ci
Zc; = Za. Clearly (Ze , 0 < s < Ck) describes the occupation density over 1Z(k) for
Brownian motion on the tree 7Z(k) constructed from the first k cuts, the
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Aldous: The continuum random tree II
motion run until the local time at the root 0 reaches a. It is not hard to argue (c.f. the urn model of [2] sec. 4) that the limits so
La =x0-,00 limso1- f Zeds
(61)
0
exist a.s. From the construction of Brownian motion X on S in section 5.2 we see that the process (La, a > 0) is the "inverse local time at the root" process for X. (This construction should be regarded as "unconditional on S"). Of course, conditionally on S the process (La) is a subordinator, and E(LaIS) = a. Getting explicit distributional information about (La) seems the most important potentially solvable open question in this area. Now consider the SSCRT. As in section 2.7 (relation 2) we may regard the SSCRT as a semi-infinite baseline with "bushes" attached, the bushes being rescaled copies of the CCRT. More precisely, let (Se, µc) denote the CCRT (S,,u) after scaling by multiplying lengths by c and making the total measure = c2. Then mark the baseline [0, oo) according to a marked Poisson process, c 2 dc. Wherever a mark c appears marks in [c, c + dc] appearing at rate on the baseline, we attach a copy of (SC, µ0).
Now consider Brownian motion X on the SSCRT, run until the time Tl it first travels unit distance along the baseline. This has an occupation density (w.r.t. v) process (Z(x), 0 < x < 1) on the baseline, which is just occupation density for 1-dimensional Brownian motion on the half-line started at 0 and run until first hitting 1. Specifically, (Z(1 - x),0 < x < 1) is the other Ray-Knight diffusion with drift rate u(z) = z and variance rate o, 2(Z) = 2z.
We are interested in the occupation density (over all R) for X at time Ti. What happens on a bush depends only on the occupation density Z(x) at the point x where the bush attaches to the baseline. A scaling argument shows that inverse local time La for the Brownian motion on (Sc, µ0) scales as (LQ, a > 0)
d
(c3La/c, a > 0).
Thus the amount of time spent in a bush with scaling factor c attached at x is distributed as c3L2(x)/0. Adding over bushes gives
Tl A
E
c' L(''(xt)/0t
(62)
where POIS is the Poisson process of points (x, c) E (0, 1) x (0, oo) with rate J71c'2 and where Lj') is distributed as at (61), independent as i varies. This
Aldous: The continuum random tree II
63
is the advertized connection between SSCRT first passage times and CCRT inverse local time. Changing topics, another quantity which (surprisingly) can be calculated explicitly is related to the cover time
C = inf{t : S = Uo C : Xt = root}. Clearly C+ is at least the time to visit the furthest leaf and return, which by (59) has mean 2 In fact I assert
EC+ = 6
(63)
I do not know any simple symmetry argument for the factor of 3, though it is tempting to seek some overlooked symmetry. Pedestrianly, one can study random walk on unconditioned Galton-Watson trees and set up a recursion for the analogous cover-and-return time. Write C, for this time, conditioned on the size of the tree = n. In the case of Poisson offspring (i.e. the uniform unordered random labelled tree) it is proved in [6] that if EC+ - cn1/2 then c = W 2-7r.
(64)
This is an intuitively convincing argument for (63). At the rigorous level, a major issue is that, even granted weak convergence of rescaled random walks
on discrete trees to the limit Brownian motion on S, this does not imply convergence of cover times.
6 SUPERPROCESSES This section is directed at readers already familiar with the subject of superprocesses. I have no technical knowledge of the subject, but am merely aiming to set out in conversational style some remarks about what is intuitively obvious, given a "random tree" background.
There is an underlying nice continuous-time Markov process (Xt; t > 0) taking values in a space E. The associated superprocess takes values in the set M(E) of non-negative measures on E, starting at time 0 with (say) the unit mass at a point xo. It can be constructed as a weak limit of finite-population branching Markov processes, or directly via martingale characterization. Recently attention has been given to the associated "historical process" which
64
Aldous: The continuum random tree II
gives the family tree of the limit population: see e.g. Dawson and Perkins [15], Dynkin [18, 19], Le Gall [23]. My viewpoint is to take this as the starting place: think about tree-indexed E-valued processes rather than time-indexed M(E)-valued processes.
Given a rooted tree t with a finite number of edges of positive length, let t be the point-set of all points of the tree, i.e. the points in the edges as well as the branchpoints and endpoints. Given a starting position xo for the underlying Markov process, there is an obvious construction of a tree-indexed
Markov process (X,; s E t), as follows. Put Xroot = xo and then define X on one edge at a time, working away from the root. For an edge [[a, b]] for which Xa = Xa has already been defined, we define (X s E [[a, b]]) to be distributed as the underlying process started at xa and run for time d(a, b), independently of the previously-defined parts of the tree-indexed process. Now consider a general continuum tree (So, µo) (e.g. constructed from some function f as in section 2.2). By the Kolmogorov extension theorem we can define a tree-indexed process (X,; s E So) such that for each finite set of points V 1 ,-- . , vk in So the process restricted to the subtree t spanned by (v;) is distributed as specified above. Of course such constructions by extension are unsatisfactory because different "versions" may have different sample path properties. To get a cleaner construction, suppose the underlying process has cadlag paths. Then we can construct (X,; S E skeleton(So)) such that every realization is cadlag at each point in the skeleton. For leaves x E So one can seek to define by continuity: Xx = lim{X, : s E [[0, x]], s -> x} In general the limit will exist outside a po-null set of leaves, on which we must let X be undefined.
With each s E So we associate the "intrinsic time" t(s) which is just the distance from the root to s: t(s) = d(root, s).
We can then define a time-indexed M(E)-valued process
OtO
= ISO 1(X. E-) 1(1(-):5t) /to (ds).
(65)
If sufficiently smooth in t we can differentiate to get et(') =
(66)
Aldous: The continuum random tree II
65
Some notation: for a positive measure 0, write 11011 for its total mass.
Now consider this construction of a tree-indexed process X in the two special cases where the indexing tree is the CCRT (S, µ) or the SSCRT (R, v). I claim these are the same as the usual superprocess, but with different conditionings. In the second case (the immortal superprocess) we are conditioning on the process being created (with infinitesimal mass) at point x0 at time 0, and on the process surviving for ever. In the first case (the superprocess excursion) we are conditioning on the process being created (with infinitesimal mass)
at point x0 at time 0, and on the total (i.e. integrated over time up to extinction) population size being equal to 1. Arguably these conditionings are more natural in many biological applications than the usual "start with unit mass" model. To fit this usual model into our set-up, we use as index the continuum random tree pictured on the bottom in figure 3, i.e. the part of the SSCRT which branches off the first unit of the baseline, measuring "time" as "distance from baseline". In all these examples, what is usually called the superprocess is the M(E)-valued process (9t) at (66).
The assertions above are intuitively obvious from the interpretation of S and R as limits of family trees in critical branching processes. As with the development of historical processes, the point is to "decouple" the family tree structure from the Markov motion in the space E. In the next section I make some observations about superprocesses based on this "continuum random tree" viewpoint. 6.1 Five Observations 1. Computer simulation. Suppose you want to estimate some distribution associated with some explicit superprocess by computer simulation. The naive way would be to simulate a discrete-time critical branching process. Starting
with 50, say, individuals and running until extinction would require order 502 = 2, 500 calls to the random number generator just to simulate the "family tree". It is much more efficient to use the global constructions in section 2.3 and simulate the first 50, say, branches of the CCRT, which requires only
2 x 50 = 100 calls to the generator (and then finally simulate the Markov process along the edges).
2. Different models for random trees. Consider the IMST(n) model of section 4. This is a random tree on n vertices, where we pay no attention to the order in which edges were added in the construction. Now imagine one vertex placed at 0 E Rd and the edges having length n-1/2 and independent uniform random directions in Rd. Let 4 be the empirical distribution of
Aldous: The continuum random tree II
66
the vertex positions. Granted the conjecture that this model also rescales to S, it is intuitively obvious that converges in distribution to O, the total occupation density (65) associated with the superprocess excursion built over d-dimensional Brownian motion. The point is that the discrete IMST(n) model has no notion of "Markovian branching" or even of "time", but still leads to a superprocess. 3. A connection between superprocesses and Brownian motion on continuum trees. One quantity of interest in the latter context was L° at (61), the time at which Brownian motion on the CCRT has accumulated local time a at the root. It is intuitively clear that L°
d fS X8
p(ds)
where X(a) is the S-indexed process (i.e. superprocess excursion) built over the Ray-Knight diffusion (60). 4.
Symmetry properties. The symmetry properties of S and R. we have
discussed lead to symmetry properties of superprocesses. Here is one example. Suppose the underlying Markov process is ergodic with stationary distribution
ir. Choose X0 from ir, then run the superprocess excursion starting at Xo. Let O. be the total occupation measure (65) and pick X* according to O,,,. If the underlying Markov process is reversible then (Xo,
X*) A (X*, 000, Xo)
In words, the distribution of O,,,, relative to a random individual in the population is the same as its distribution relative to the progenitor: so the progenitor is not special. This property is intuitively clear from the "random re-rooting" property (20) of the CCRT.
5. The immortal superprocess. This has recently been studied rigorously by Evans and Perkins [21, 20]. From our viewpoint of the superprocess as the Markov process X indexed by the SSCRT, some elementary properties are obvious.
(a) The "population size at time t" 10t1 has Gamma(2, 2/t) distribution, by Corollary 6 and (18). (b) If the underlying Markov process has absorbing states then (because the SSCRT consists of bounded bushes attached to an infinite baseline) the superprocess gets absorbed with the same probabilities as for the underlying Markov process ([20]).
Aldous: The continuum random tree II
67
(c) If the underlying Markov process converges from any start to a unique stationary distribution it then et/IBtI 4 7r.
(67)
For using a standard exchangeability fact (that for exchangeable sequences, pairwise independence implies independence) to prove (67) it suffices to prove dist(Xv,(t),Xv2(t)) -> 7r x it
(68)
where V1(t) and V2(t) are picked uniformly from the population at time t. The individuals V1(t) and V2(t) had last common ancestor at time Gt, say, and
P(Xvt(t) E A, X vz(t) E BIGt = 9, XG, = x) = Px(Xt_9 E A)Px(Xt_9 E B).
By the self-similarity property of the SSCRT we have Gt d tGl, and then (68) follows easily from the convergence assumption on the underlying Markov chain [21].
In fact, the exact distribution of Gl was calculated in Proposition 12, as was
the distribution Cl of the time of the last common ancestor of the entire time-1 population. References
[1] D.J. Aldous. Asymptotic fringe distributions for general families of random trees. 1991. To appear in Annals of Applied Probability, May 1991. [2] D.J. Aldous. The continuum random tree I. Ann. Probab., 19:1-28, 1991.
[3] D.J. Aldous. The continuum random tree III. In preparation. [4] D.J. Aldous. A random tree model associated with random graphs. Random Structures and Algorithms, 1:383-402, 1990.
[5] D.J. Aldous. The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Discrete Math., 3:450-465, 1990.
[6] D.J. Aldous. Random walk covering of some special trees. J. Math. Analysis Appl., 157:xxx, 1991. To appear.
[7] D.J. Aldous and J.M. Steele. Asymptotics for Euclidean minimal spanning trees on random points. 1991. Unpublished.
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[8] M.T. Barlow and E.A. Perkins. Brownian motion on the Sierpinski gasket. Probab. Th. Rel. Fields, 79:543-624, 1988. [9]
R.N. Bhattacharaya and E.C. Waymire. Stochastic Processes with Applications. Wiley, 1990.
[10] P. Biane. Relations entre pont et excursion du mouvement Brownien reel. Ann. Inst. Henri Poincare, 22:1-7, 1986. [11] P. Biane and M. Yor. Valeurs principales associees aux temps locaux Browniens. Bull. Sci. Math. (2), 111:23-101, 1987. [12] K.L. Chung. Excursions in Brownian motion. Arkiv fur Matematik, 14:155-177, 1976.
[13] E. Csaki and G. Mohanty. Excursion and meander in random walk. Canad. J. Statist., 9:57-70, 1981. [14] D.A. Darling. On the supremum of a certain Gaussian process. Ann. Probab., 11:803-806, 1983.
[15] D.A. Dawson and E. Perkins. Historical processes. 1991. Mem. Amer. Math. Soc., to appear. [16] R. Durrett, D.L. Iglehart, and D.R. Miller. Weak convergence to Brownian meander and Brownian excursion. Ann. Probab., 5:117-129, 1977. [17] R. Durrett, H. Kesten, and E. Waymire. On weighted heights of random trees. J. Theoretical Probab., 4:223-237, 1991. [18] E.B. Dynkin. Branching Particle Systems and Superprocesses. Technical Report, Cornell University, 1989. [19] E.B. Dynkin. Path Processes and Historical Processes. Technical Report, Cornell University, 1989. [20] S.N. Evans. The Entrance Space of a Measure- Valued Markov Branching Process Conditioned on Non-extinction. Technical Report, U.C.Berkeley, 1991. Canad. Math. Bull., to appear.
[21] S.N. Evans and E. Perkins. Measure-valued Markov branching processes conditioned on non-extinction. Israel J. Math., 71:329-337, 1990.
[22] P. Flajolet and A. Odlyzko. The average height of binary trees and other simple trees. J. Comput. System Sci., 25:171-213, 1982.
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[23] J.-F. Le Gall. Brownian Excursions, Trees and Measure- Valued Branching Processes. Technical Report, Universite P. et M. Curie, Paris, 1989.
[24] G. R. Grimmett. Random labelled trees and their branching networks. J. Austral. Math. Soc. (Ser. A), 30:229-237, 1980. [25] V.J. Gupta, O.J. Mesa, and E. Waymire. Tree dependent extreme values: the exponential case. J. Appl. Probab., 27:124-133, 1990.
[26] W. Gutjahr and G. Ch. Pflug. The Asymptotic Contour Process of a Binary Tree is Brownian Excursion. Technical Report, University of Vienna, 1991. To appear in Stochastic Proc. Appl. [27] J.-P. Imhof. On Brownian bridge and excursion. Studia Sci. Math. Hungar., 20:1-10, 1985.
[28] K. Ito and H.P. McKean. Diffusion Processes and their Sample Paths. Springer, 1965.
[29] D.P. Kennedy. The distribution of the maximum Brownian excursion. J. Appl. Prob., 13:371-376, 1976.
[30] D.P. Kennedy. The Galton-Watson process conditioned on the total progeny. J. Appl. Prob., 12:800-806, 1975.
[31] H. Kesten. Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincare Sect. B, 22:425-487, 1987.
[32] F. B. Knight. On the excursion process of Brownian motion. Trans. Amer. Math. Soc., 258:77-86, 1980.
[33] V.F. Kolchin. Random Mappings. Optimization Software, New York, 1986. (Translation of Russian original).
[34] T. Lindstrom. Brownian motion on nested fractals. Memoirs of the A.M.S., 420, 1989.
[35] A. Meir and J.W. Moon. On the altitude of nodes in random trees. Canad. J. Math., 30:997-1015, 1978.
[36] J.W. Moon. Random walks on random trees. J. Austral. Math. Soc., 15:42-53, 1973.
[37] J. Neveu and J. Pitman. The branching process in a Brownian excursion. In Seminaire de Probabilites XXIII, pages 248-257, Springer, 1989. Lecture Notes in Math. 1372.
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[38] A.M. Odlyzko and H.S. Wilf. Bandwidths and profiles of trees. J. Combin. Theory Ser. B, 42:348-370, 1987.
[39] R. Pemantle. Choosing a Spanning Tree for the Integer Lattice Uniformly. Technical Report, Cornell University, 1989. To appear in Ann. Probability. [40] J.W. Pitman and M. Yor. Bessel processes and infinitely divisible laws. In Stochastic Integrals, pages 285-370, Springer, 1981. Lecture Notes in Math. 851.
[41] J.W. Pitman and M. Yor.
A decomposition of Bessel bridges.
Z.
Wahrsch. Verw. Gebiete, 59:425-457, 1982.
[42] A. Renyi and G. Szekeres. On the height of trees. J. Austral. Math. Soc., 7:497-507, 1967. [43] L.C.G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales. Wiley, 1987.
[44] P. Salminen. Brownian excursions revisited. In Seminar on Stochastic Processes 1983, pages 161-187, Birkhauser Boston, 1984.
[45] J.M. Steele. Gibb's measures on combinatorial objects and the central limit theorem for an exponential family of random trees. Prob. Engineering Inf. Sci., 1:47-60, 1987. [46] G. Szekeres. Distribution of labelled trees by diameter. In Combinatorial Mathematics X, pages 392-397, Springer-Verlag, 1982. Lecture Notes in Math. 1036.
[47] L. Takacs. Limit theorems for random trees. 1991. Unpublished.
[48] L. Takacs. On the total height of a random planted trivalent plane tree. 1991. Unpublished.
[49] L. Takacs. Queues, random graphs and branching processes. J. Appl. Math. and Simulation, 1:223-243, 1988.
Harmonic Morphisms and the Resurrection of Markov Processes P. J. Fitzsimmons* Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112 USA
1 INTRODUCTION Our purpose in this paper is to carry out the program, outlined at the end of §5 in Csink, Fitzsimmons & Oksendal [CFO], of extending the stochastic characterization of harmonic morphisms given in that paper to the case where the constant functions are not necessarily harmonic. Various consequences of this characterization were established in [CFO]; they remain valid, with only minor changes, in the present context. The reader can also consult [CFO] for references to the earlier literature on harmonic morphisms.
We shall now describe the main result in detail. Let (E,U) be a 'p-harmonic
space in the sense of Constantinescu & Cornea [CC72]. In particular, E is a locally compact, second countable Hausdorff space, and U: G --> U(G) (G open in E) is the sheaf of positive U-hyperharmonic functions. Equivalently, (E,U) is a balayage space in the sense of Bliedtner & Hansen [BH] which satisfies the local truncation property [BH, p.125], and which has no absorbing points.
1.1 Definition Given two q3-harmonic spaces (E, U) and (F, V), a continuous function p: E -> F is a harmonic morphism provided u o cp E U(cp-1(G)) whenever u E V(G) and G C F is open. *Research supported in part by NSF grant DMS 8721347
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Fitzsimmons: Harmonic morphisms
The notion of harmonic morphism seems to be due to Constantinescu and Cornea [CC65]. Actually our definition is somewhat broader than that used by previous authors, since the positive hyperharmonic functions have been substituted for the superharmonic functions. Clearly if the pullback u ra uo(p preserves (local) superharmonicity then it also preserves (local) harmonicity, since u is harmonic if and only if u and -u are superharmonic.
Roughly stated, our main result asserts that if X and Y are Markov processes associated with (E, U) and (F, V) respectively, then a continuous mapping cp : E -+ F is a harmonic morphism if and only if cp carries the sample paths of (a time change of) X into those of Y. To associate a Markov process with (E,U) we must assume that the constant function 1 is in U (E). Then there is a bounded continuous, strict potential
p > 0 in U(E); see [CC72, Prop.7.2.1]. Let X = 9t, Xt, Pr) be the Hunt process associated with (E,U) and p. The state space of X is E U {o}, where 0 is the cemetery point adjoined to E as the point at infinity if E is noncompact, and as an isolated point otherwise. Writing (Pt) for the semigroup of X and U = f °° Pt dt for the associated potential kernel, we have
p(x) = U1(x) = 1000 Ptl(x) dt = Px((),
where C = inf{t : Xt = Al is the lifetime of X. A different choice of the strict potential p would result in a Hunt process related to X by time change. The kernel U is strong Feller (i.e. U f is continuous for any bounded Borel function f) and consequently X has a reference measure; cf. Blumenthal & Getoor [BG, p.197]. The paths of X are continuous on [0, C[, and evidently
Px(( 0: Xt
XtG =
Xt, 0 t}, and let it be a Borel function on E with 0 < 7r < 1. Fix an open set G C E and put T = TG. Define C C S x [0,1] by C = {(w,u) : r(w) = ((w) and 7r(XS_(w)) > u, or r(w) < ((w)}. Consider the stochastic process p(XB(t)(w)), Zt(w, w, u) =
1 t-Ar(w)(J'), O,
0 < t < Ar(w); t> A.(w), (w, u) E C; t > Ar(w), (w, u) C;
under the law Px defined on S2 x f2 x [0,1] by Px(dw, dw, du) = Px(dw)Q`°(Xr-("))(dw) du,
where f2 is the path space and 0 the cemetery for Y. Roughly speaking, Z is constructed by continuing cp(XB(t)) beyond time Ar by tacking on a copy of Y started at y = lp(X,), provided Xr E E. (Aside from the initial condition
Yo = y, Y is independent of X.) Also, if Xr = A but Xr_ E E, then we toss a coin whose probability of heads is 7r(Xr_) and continue (now with y = cp(X,_)) if a head is tossed. This "randomized continuation" accounts
for any differences in the "killing rates" k and .£ of X and Y. We call (Zt,Px) the 7G-splicing of cp(XB(t)) and Y with continuation probability 7r.
1.2 Definition A continuous mapping cp: E --+ F is X-Y path preserving with time change B and continuation probability 7r provided the rG-splicing of cp(XB(t)) and Y with continuation probability 7r has the same law (under Px) as (Yt, P`p(x)), for all x E G and all precompact open sets G C E.
Fitzsimmons: Harmonic morphisms
75
Here is our main result.
1.3 Theorem Let (E,U) and (F, V) beT-harmonic spaces, and let p E U(E) and q E V(F) be bounded continuous strict potentials such that (2) and (3) hold. Let X and Y be the associated Hunt processes. If cp: E -4F is continuous, then the following three assertions are equivalent: (i) cp is a harmonic morphism (of (E, U) into (F, V)); (ii) there is a CAF A = (At) of X and a Bore] function 7r on E with 0 < 7r < 1 such that
Vf(,p(x)) =P'
f
r(G)
f(ca(Xt)) dAt + HG(V f ° )(x) r(G)
0
+
Px
(5)
Vf-co(Xt)ir(Xt) k(Xt) dt 0
for all bounded positive continuous functions f on E, x E G, and precompact open sets G C E; (iii) there is a CAF (At) of X with inverse B and a Bore] function it on E with 0 < 7r < 1 such that cp is X-Y path preserving with time change B and continuation probability 7r.
1.4 Remark It should be noted that the CAF A of Theorem 1.3 need not be strictly V(X1) is increasing. In fact, one consequence of the theorem is that t constant on any interval of constancy of t H At; cf. [CFO, Prop. 3.1]. This is consistent with the continuity of t H cp(XB(t)). See Remark 2.5 for a reasonably explicit characterization of the ingredients A and 7r in the decomposition (5).
1.5 Remark For a better understanding of (5) one should note that
V (G) Px(9(Xr(G)-)) = HGg(x) +Px
g(Xt)k(Xt) dt
.
(6)
This identity, which follows from results of section 2, makes the implication
(iii)=(ii) in Theorem 1.5 quite transparent.
Fitzsimmons: Harmonic morphisms
76
1.6 Example The need for the "splicing" in Theorem 1.3 is easily illustrated. Let F be the complex plane and E the cut plane F\] - oo, 0]; let Y be complex Brownian motion, and let X be Y killed at the first hitting time of ] oo, 0]. The mapping cp : E -> F defined by cp(z) = z2 is analytic, hence a harmonic morphism . Indeed, by a famous theorem of Levy, if we put At = fo IW'(X,)I2 ds, and B(t) = A-1(t), then Yt = XB(t) , 0 < t < AS, is equal in law to Y killed at a certain random time o. The catch is that v is not a stopping time of Y and k is not itself a Markov process. Only after continuing k beyond o with an independent copy of planar Brownian motion do we obtain a Markov process.
If 1 is U-harmonic then we can take k - 0; also, PX (TG < c) - 1 in this case so the function ir is irrelevant. This special case of Theorem 1.3 is the main result (Theorem 1.1) of [CFO]. Our proof of Theorem 1.3 amounts to showing that the present situation can be reduced to that of [CFO] by a process of "resurrection."
For if 1 is not U-harmonic, then Pr(X(_ E E, C < oo) > 0 for some x E E;
see Lemma 2.1. One might say (with tongue in cheek) that on {X(_ E E, c < oo}, X has died an unnatural death. We can remedy this situation as follows. Consider the f2-valued Markov chain (X."; n = 0, 1, 2, ...) with initial law Px and transition kernel 11(w,w') = Px(w)(dw'),
where x(w) = X(-(w). Note that P° is the unit mass at the constant path t u--* A. Let C" denote the lifetime of X", and define a process X by
Xt=Xt,
if
(0+...+C"-1 0. If also M is a Px-supermartingale for all x E E (equivalently, if Px(MM) < 1 for all x) then the formula Pt f(x) = PZ(f(Xt)Mt) defines a subMarkov semigroup (Pt*). One can associate with (Pt) a Markov process X*. For example, if X is a right Markov process then so is X*. We refer the reader to [BG] and Sharpe [S] for details on the transformation of processes by multiplicative functionals. In the sequel it will be convenient to indicate the dependence of X* on M by refering to X * as (X, M).
78
Fitzsimmons: Harmonic morphisms
Both of the processes X and X will be realized as the coordinate process on the canonical sample space Q of paths w : [0, oo[ -> E that are continuous on [0, ((w)[ and absorbed in A at time ((w). The two processes are distinguished by their respective laws {Px; x E E} and {Px; x E E}. In particular, a (perfect) CAF of X can be viewed as a CAF of X. Likewise, at times a CAF of XG will be thought of as a CAF of XG.
2 PROOF OF THEOREM 1.3 We begin this section with the rigorous definition of the resurrected process
9. Since X is a Hunt process, hence quasi-left continuous, the accessible part of the lifetime ( is predictable, and the totally inaccessible part of ( is given by
t; = Sl
C,
XS_EE,( U(x,1G) = Px
1G(Xt)
(10
dt)
,
W
soPx(TG 0 be bounded and continuous, and such that liminf Wj{x}
u(x) - Hwu(x) > 0, Pr(T_W)
-
(W open) Vx E E.
Fitzsimmons: Harmonic morphisms
81
Then u E U (E).
Proof. For e > 0 define uE = u + eUl. Then
uE - Hwu = u - Hwu + EP'(Tw), hence
lim]
f
E(x)
uE(x)
> E > 0.
Thus for each x E E there is a neighborhood Wx of x such that uE(x),
V open W C Wx.
As is well known [BH, 111.4.4], this together with the continuity of u implies
that uE E if (E). The lemma follows upon letting e 10. 0
2.4 Proposition Suppose that cp is a harmonic morphism between (E, U) and (F, V). Then it is also a harmonic morphism between (E, 0) and (F, V). Proof. Let N be an open subset of F. We must show that if v E V(N), then u := voce E U(cp-1(N)). Fix X E -1(N) and e > 0. Let N' C N be an exit set (for Y) containing cp(x). Then as a consequence of Proposition 2.2 there is a sequence (v,,) of bounded continuous YN'-potentials increasing to v on N'; once we show that vnocp E U(cp-1(N)), the analogous inclusion will obtain for v by a passage to the limit. Thus, there is no loss of generality
in assuming that v is bounded and continuous. Let G C N' be an open neighborhood of cp(x) such that G C {Iv - v(p(x))l < e}. Let c = u(x) - e and note that v - c E V(G) since the constants are V-harmonic. Since V(G) C V(G), it follows that u - c E U(cp-1(G)). Now let W C W-1(G) be any exit set (for X) with x E W, and put r = Tw. Then by (13)
u(x) - c > Hw(u - c)(x) = Px(u(XT) - c; r < () = Px(eKr (u(XT) - c); T < () - Pz((eKr - 1)(u(Xr) - c); ,r < = fl (u - c) - Px((eKT - 1) (u(XT) - c); T < (). But Pz((eK, - 1) (u(XT) - c); T < () < ePx((,K, - 1);,r
oo and then N T G through a sequence (Nj) we obtain (22) since rNj T rG, {TN, < rG, Vj} C {rG < C}, and since h°cp is bounded and continuous on W-1(D) D G.
Now fix a bounded positive Borel function f on E. Then h := V f - V * f is Y-harmonic on G. Using (22) and (17), we see that on G
Vf°cp=h°cp+V*f°cp
= Ux((Vf -V*f)°cP)-UA((e(Vf -V*f))°cp) + HG((Vf -V*f)°cp)+V*f°cP = UK(Vf°cp) - UA((e' Vf)°cp) + HG(Vf°cp)
- U( V *f °cp) + UA((e . V *f )°cp) - HG(V *f °cp) + V *f°cp
= UA(f°pp) + Ux(Vf°cp) - UA((e . Vf)°cp) + HG(Vf°cp) Thus
Vf°W = UA(f°p) + UK(Vf°cp) - UA((e Vf)°W) + HG(Vf°cP)
(23)
3° By the obvious adaptation of [BG, p.240] there is a sequence (W,,) of exit sets (for X) with U,,Wn = E, and we can certainly assume that for each x E E and each e > 0 there are infinitely many indices n such that Wn is of diameter < e and contains x. It is then easy to see that the (increasing) sequence of iterated exit times defined by TO = 0 and Tn.+1 = Tn +
°8T
(24)
almost surely. Likewise there is a cover (Wn) of F by Yexit sets. A compactness argument shows that we can arrange that Wn C satisfies Tn T
Fitzsimmons: Harmonic morphisms
86
cp-1(Wn) for all n, and that for each n there is an open set Dn C Wn with Wn C cp-1(Dn). Thus the considerations of steps 1° and 2° apply. We now call to the reader's attention the fact that the CAF produced in those steps depended implicitly on G. But if G1 and G2 are two exit sets (for X) then it follows easily from (23) that
Pi
2 = Px(AGG1nGz)),
(25)
X E G1 f1 G2i
where we have written AGI to emphasize the dependence on G. The compatibility condition (25) allows one to use the "piecing together" argument of [BG, V,§5] to produce a single CAF A of X such that Atnr(G) = Atnr(G) ,
Vt > 0, a.s. Px,
Vx E G.
Thus (23) holds with this CAF A for all (sufficiently small) X-exit sets G. Moreover, as noted following (16), x H Px(Ar(G)) is bounded by IIgIIoo < oo
for any such exit set G. Let G be a precompact open subset of E. It follows that if we define
stoppings time S. in the same way as the Tn but with Wn replaced by G,,:= Wn fl G, then a.s.
Sn T TG
(26)
Now fix a bounded positive Borel function f on F. Write T = TG, Tn = TG., and
Ct =
f
t
v . Vf)°cP(X9)] dA, +
0
f
dK,.
0
Then C is a (signed) CAF and clearly x '- a Px (I Cr I) is bounded. By (24) and (26), P'(Csn P'(Cr) = ) n>1
-
Csn-1
_ n>1 E PX(Crn °OSn-1) 1: HG1 ... HGn-1 (X, P (Crn )), n>1
and because of (23) the nth term of this series is equal to HG1 ... HGn-1(V f °(P - HGn V f °cP)
(27)
Fitzsimmons: Harmonic morphisms
87
so the last sum in (27) telescopes to yield PZ(Cr) = V f ocp(x) - lim HG1 ... HG, (V f ocp)(x) n
= V f op(x) - lim Px(V f
(28)
n
= Vfocp(x) - HG(Vfocp)(x),
since Sn T rG and {Sn < rG,Vn} C {rG < (}, and since V f o(p is continuous on E. Clearly (28) implies (14), so the proof of Theorem 1.3 is complete, modulo Proposition 2.2.
2.5 Remark We can extract from the above proof the following characterizations of A and it. If V1 is bounded, then the (bounded) potential of A is simply UA1 = u - U(ku), where u is the X-potential part of V loco. In the general case let G and G be exit sets for X and Y respectively such that G C Put
(jr(G)
exp(jt
vG(y) = Ql(r(G)) = QY
£(Y$) ds) dt
rr(G)
aG(x)
PX(Ar(G)) = UG(X) - Px
J
(kuG)(Xt) dt
0
(It is easy to verify directly that the third term in the above display does not depend on the choice of G.) Now let (Wn) be a sequence of exit sets covering E and such that the stopping times defined by (24) satisfy Tn T . Then the (bounded) potential of A is given by
UA1=P(AS)=EHw1...Hw-1aw n>1
Finally note that once A is known, it is determined up to a set of potential zero by the relation Uk = UA(Qocp) + U(kr).
Fitzsimmons: Harmonic morphisms
88
3 PROOF OF PROPOSITION 2.2 3.1 Lemma The lifetime of X is predictable.
Proof. Since Kt < t for all t > 0, the local martingale L is reduced by the sequence of constant stopping times Rn = n; that is, (L.nn) is a uniformly integrable martingale (relative to each law Px) for each integer n. Fix x E E. Let (p the predictable part of C, and choose an increasing sequence
of (.Ft+)-stopping times such that S. < (p for all n and limn Sn = a.s. Px. Set Tn = Sn A n, and T :=T limn Tn. Then since LT = 0 {C< a)
,
L± (,ya(t)) are independent
random variables, with L+ (Ya(t)) + L- ('>a(t)) = ya(t)
(10)
,
and the above result is equivalent to lEa
{ e M'±(7a(t)) {
exp{-tf(1-d's)da(s)}
,
(11)
0
for each
> 0, va being given by Eq.(8). We establish the above results by
finding the distribution of tix(a) = inf (s > 0 : X(s) = a I X(0) = x) in terms of the quantum mechanical Hamiltonian H.
2 LOCAL TIME AND EXCURSIONS FROM THE ORIGIN FOR O-U PROCESSES We begin with a more or less standard result. (See Ref.(1)).
Proposition 2
The quantum mechanical harmonic oscillator Hamiltonian H = 2"1dx ( -a2 +x2) is
limit point at too , the unique L2 eigenfunction near ±-, corresponding to
Hawkes & Uhlenbeck: Statistics of local time
94
eigenvalue ? being Dx? (±42x), respectively, D being Weber's parabolic 2
cylinder function.
Proof
d 2
+ V(x)) dx near 00 is that a positive differentiable function M(x) and positive constants k1, k2 and a can be found such that (i) V(x) > -k1M(x) for x > a ,
The well-known sufficient condition for limit point behaviour of (2-1
00
3
(ii) M'(x) I (M (x)) 2 < k2 for x > a and (iii)
J (M(x)) 2 dx diverges (See a
Ref.(6)). Taking e.g. M(x) = x2 proves that the harmonic oscillator Hamiltonian is limit point at oo , and - co can be treated similarly. Laplace's method of solving the o.d.e. gives DI 1 (± J2x) as the unique L2 eigenfunctions of
2
2 (-2"1
2+
2"1 x2) near too , with
D (x)
a
4x2r(2X) 276
t 2
f exs C2 s s 2
ds
,
(12)
Y
Y being the contour starting just below the cut on the negative reals at - oo , looping once around the origin and finishing just above the cut at - oo . //
Corollary For the O-U process 1fa
{
e
exp
I
exp {
P)L(a,a)
}
r(a) D_X(I2a) D_)L(-42a)
}
00
p )L(x,y)
=J a Xs ps(x,y) ds being the Laplace transform of ps(x,y), the transition
density, D Weber's parabolic cylinder function.
Hawkes & Uhlenbeck: Statistics of local time
95
Proof The first part of the above identity follows because X is a stationary Markov process with transition density (explicitly) given by
a 2t)) 2 exp
Pt(x,y)
(y xe2 t)2 } (1 h
{-
(14)
,
whose Laplace transform pA.(x,y) (resolvent kernel) is bounded and continuous in (x,y). Because of the operator identity: 2
_(2-s2 -xdx) = ex2'2 (_2
2
d2
_
2+ 2 _2) ex
2/2
a fE(x) = fE (x) (H-E)
(15)
fE being the ground state of the Hamiltonian H, E the ground state eigenvalue 2f2
(fEx) = n 4 e x
,
E=
1
in our case) , we obtain
1 -1 (a,a) = p),(a,a) = (H + - 2)
2 D_,(12a) D_,(-42a) W
,
(16)
W being the Wronskian of D_),(42x) and D_),(-42x). W is a constant and so it can be evaluated at the origin. By using Hankefs formulas , we obtain
) =7
D_X(0
qn
( 17 )
22 r(2+2> and
An D40) _ -1X
,
(18)
22-2r(2) so by the duplication formula (see Ref.(1))
W _ 24n (19)
giving the desired result. II Hankel's formula states: 1' 1(z) = (2ni) 11 et tz dt , each z E C , y being the contour above, 7 t z being defined as e z lnt In being principal value of the logarithm. 1
96
Hawkes & Uhlenbeck: Statistics of local time
We now come to the main result in this section - an explicit formula for the Poisson-Levy excursion measure vol
Proposition For the O-U process, for X, t > 0,
{ -t
a-),?(t) } = exp
%
f
(1 - e XS) dv0(s)
0
}
(20)
,
where v0 s) _ 2 e2s (e 2S 1)_23
ds
(21)
qX
Proof Using the above results in Eqs. (17) and (18) and the duplication formula gives
;
{ -2t1-V
{ e X?(t) } = exp
}
r(2)
(22)
Therefore, setting H(u) = v0 ( [u,-)), we wish to find H(u) such that
0
eAu H(u) du
(23)
or
eu r(2).+1)
H(u) du
(24)
J0
or
= "n
B(2 ()L+1),
2)
JOOe ).u 0
H(u) du
(25)
.
Therefore 00
t 2f 0
e)Lx e -x (1 - e -2x) 2 dx = 47t J e-Xx H(x) dx 0
.
(26)
Hawkes & Uhlenbeck: Statistics of local time
97
Finally t
H(x) _
T(e
2x
- 1) 2 and dvo(x) = -If (x) dx
We shall generalise this result in the next section, where we find a connection between the Poisson-Levy excursion measure v and the quantum mechanical Hamiltonian H.
3 GENERAL EXCURSIONS FOR O-U PROCESSES We need an elementary consequence of the Feynman-Kac formula (see section (25), Ref.(7)):
Proposition 2
Let H = 2"1(_ +x2) , with corresponding ground-state eigenfunction dx2 2h
fE(x) = 7t 4 e -x
and corresponding eigenvalue E = 2 . Define
H+ = lira (H + 2x +) ,
(27)
XT. x± being the characteristic function of { x : x >< a) , so H+ is the Dirichlet Hamiltonian with a Dirichlet boundary condition at x = a. Then for a.e. x >< a respectively,
P (tix(a) > t) = fE (x) (exp (-t (H+ - E)) fE(x))
.
(28)
Proof
Set c (x,N) _ -cx(N) nix(-N) . Then, because 1 y fE (u) du diverges as y -±°°, ,c (x,N) T- µ a.s. as N T co (see e.g. Ref. (2) p.159). Thus, for each a. > 0, 1£ {exp (-? f tx
(Xx(s)) ds) } = Jim
11i{
NT
x {ti(x,N) > t)
t exp (A x + (Xx(s)) ds) 1, a
X (ti(x,N) > t)
being 1, if 'c (x,N) > t , and 0 , otherwise.
(29)
98
Hawkes & Uhlenbeck: Statistics of local time
x+
By the Girsanov-Cameron-Martin theorem, for
Brownian
motion starting at x, with TB(x,N) = inf (s > 0 : I Bx(s) I = N} , using Ito's formula (see Ref.(8) and (9)),
r.h.s.= lim )£ { x (tiB(x,r)
NT-
>t
} exp { (-
fE(Bx(t)) fE (x) }
t
O
VO+),x
+E)(Bx(s))ds } (30)
,
for VO(x) = 2-1 x2 . Using the local integrability and positivity of (VO+? X +)
'
the Feynman-Kac formula gives via the dominated convergence theorem, for a.e.x.,
r.h.s. = ff1(x) (exp {-t (H + Xx
+
E) fE(x))
(31)
(see section 25 in Ref.(7)). Since for x >< a , respectively,
P (tix(a) > t) = lim R{ exp (A J tx { (Xx(s)) ds) }
,
(32)
,Too
the desired result follows. // For our one-dimensional time-homogeneous process
(x,a) _ ff1(x)(H + x -
)£{e
(H + ? -
(a,a)
E)-1
(x,a)fE(a)
E)-1(a,a)
(33)
where pA(x,a) = Joe Xs ps(x,a) ds. Multiplying both sides of the last equation 0
by fE(x) and integrating by Fubini's theorem, we obtain for the Poisson-Ldvy excursion measure va 00
x-1 { (H + X -
E)-1
(a,a) J 1
-1
=
J e-Xu va { [u, oo) } du 0 00
=
fE ( a ) J. -2
f 2(x) IE{ aXtix(a) } dx
(34)
for each ? > 0. Combining this result with the above proposition we obtain:
Hawkes & Ulilenbeck: Statistics of local time
99
Proposition 2
For H = 2-1( .
4
1
+x2), E = 2 , fE(x) = x 4 e-'
2/2
and H+ = lim (H + 2x +) Too
,
XT being the characteristic function of (x : x >< a ), respectively,
1(H + - E)-1
I
5(1e ) dva(s) 0
(a,a)1-1
=
(35)
,
where Va { [s, °O) } = fE (a) {
(fE , (H+ E) a s(H+ + (fE , (H_ - E) a
E)
fE)L2
s(H_ E)
fE)L2 I
(36)
We can now establish the main result of this section:
Proposition
Let a = an
be the successive values of a for which D_a(+'2a) = 0,
respectively. Then
dva(s)
ds
dv a(s) =
ds
dva(s) (37)
ds
+
with
+
d a(s) ds
+
+ 2 (a2+so n) =F(an) e
an
{
f
2
D_ +(+"/2y) e -y
y > T, which are absolutely continuous with square integrable derivative. We equip 7-1(T) with the norm
IIX112 = f IIX(t)112dt + f 1
0
0
dX(t) 1
dt
2
dt.
It is straightforward to check that 71(T) is a Hilbert space. We define 7-1°(T)
to be the codimension 2d subspace consisting of those X with X(0) _ X(1) = 0.
Jones & Leandre: LP-Chen forms on loop spaces
107
Now let w be a path in M and let X be a path in l(T,,,o) and assume that parallel transport r,(w) is defined. Then define T(w)X by (r(w)X )3 = r9(w)X3.
Thus r(w)X is a vector field along w. Now define xw to be the space of all vector fields along the path w of the form r(w)X where X E ? l(Two ). There is a linear isomorphism r(w) : ?{(Two) -> xw so we make xw into a Hilbert space using the inner product inherited from ?{(Two ). The spaces ?{w form a measurable field of Hilbert spaces tangent to PM. This means that xw is defined for all w except for a set of measure zero and each of the spaces xw is a linear subspace of the tangent space of the Banach manifold
PM at the path w. Suppose now that w is a loop based at x E M. Then define HU, to be the space of vector fields along w of the form r(w)X where X E ?-l°(Two). Since ri(w)(0) = 0 it follows that the vector field r(w)X along the loop w starts and ends at 0 E TxM. In this case we get an isomorphism T(w) : ?'l°(Two) -p H°iu
and the spaces ?-l° form a measurable field of Hilbert spaces tangent to LxM.
To define the corresponding field of Hilbert spaces on LM let hw = Tl (w) : Two --+ Two be the holonomy around the loop w. Then the path r(w)X is a vector field along the loop w if hw(XI) = Xo. We define Kw to be the subspace of ?- (Two) consisting of those paths which
satisfy this holonomy condition and then define Tw to be the subspace of 'Hw given by r(w)X where x E X. In this case 7-(w):1Cw-+ Tw
is an isomorphism and the spaces Tw form a measurable field of Hilbert spaces tangent to LM. If w is a loop in M
7l,° cTwCrlw and each space has codimension d in the next. The space ?-lw consists of paths in the tangent manifold TM which cover the loop w, Tw consists of loops in TM which cover w, and ?-l° consists of loops in TM which cover w and start and end at 0 E Two M.
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Next we construct a basis for the Hilbert spaces f,,,. First we consider the space 71 = ?l(R). We start with the linearly independent set if n > 0 Sl cos(2irns), if n < 0. ( sin(2irns),
an(s)
This set is an orthogonal basis for 'H and, for a suitable constant C, Ilan 112 = 1 + Cn2.
We define an
bn
IlanIl
so bn is a basis for 'H,,, and there is a constant C such that C sup Ibn(S)I
o(w)(en1 i1 , ... , enr,ir )
where the sum is taken over (nl,...,nr) E Zr,
1 < 21,...,ir < d.
In view the definition of the en,i, given in (1.3), there is no difficulty in defining the stochastic integrals o(w)(en, ,i1, ... 1 enr,ir )w, for almost all w. It remains to show that the series (1.8) converges almost everywhere and this requires an estimate for o(W)(en, p, , ' ' 1 enr,Yr )w' Fix X E M and let fl, ... , fr : [0,1] -> TTM be continuous functions which have bounded variation. We need some hypotheses on the fi since we will use Stratonvitch integrals and these hypotheses are adequate for our applications. Presumably, it is possible to get away with much weaker conditions on the fi. Let f = (f11...,fr)
Ii = jr, +...+ri+l,...,rl +...+ri+ri+1}.
Jones & Leandre: LP-Chen forms on loop spaces
112
and define (1.9)
A(f; w) = Jon w1 (dw, r(fr, ))9, ... wn. (dw, T (frn )),n
.
Choose a constant C(f) such that
Ilffll.R
is defined for almost all loops w E LM. Finally we restrict to the subspaces Tw
?(w)w:Tw x... xTw -+ R to get a measurable form on LM. An exactly analogous argument shows how to restrict a(w) to the based loop space to get a measurable form on L, ;M.
A Chen form on LM is defined on Tw for almost all loops w. However, from the definition, it has a canonical extension to f1,,,, for almost all loops w. For the purposes of calculation it is often convenient to use this extension and we sometimes use this trick, and its analogue for L,rM, without making it explicit. We can construct a larger class of forms on PM and LM using the o(w) as follows. Let 0 be an r-form on M and for t E [0, 1] define a measurable form Ot as follows et :
?w x
... x Nw
R,
et(bi, ... , br) = 9(bl (t), ... , br(t)).
Now let w be as above and suppose we are given forms 0 and cp on M, then we define (1.18)
a(9; w; yo) = 9o A a(w) A p,.
This is a measurable form on PM. On the loop space LM we need a form 0 on M and then we define (1.19)
Q(8; w) = 9o A o(w).
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Definition 1.20. The spaces of Chen forms C(PM), C(LM), and C(LXM) on PM, LM, and L2M are defined to be the linear span of the forms o(e; w; p0),
o'(9; w),
mo(w)
respectively. We write C' for the space of Chen forms of degree r. Notice how the relation between the spaces of Chen forms mimics, at the level of differential forms, the inclusions
L2M C LM C PM. One reason for the importance of Chen forms is that the cohomology of PM, LM, and L2M, and also the equivariant cohomology of LM can be computed, algebraically, from the spaces C(PM), C(LM), and C(LXM), see [17]. Indeed in [17, §3] the relation between these spaces of Chen forms and various constructions from homological algebra, in particular the bar complex, is described in detail.
Recall the multiplicative formula for Chen forms [17, §4]. Given -0 = b1 ® ... ®0n. and w = w1 ®... ®w,,,. then the shuffle product S(0, w) is defined to be (1.21)
S(O,w) _ E S,r(01 ®... ®On ®w1 ®... ®wm).
Here the sum is taken over all (n, m) shuffles of {1, ... , n + m}; S,,. is, up to a sign, the permutation it of the components. The sign is determined by the following convention: if we interchange a and /3 then we introduce the sign (-1)(deg a-1)(deg 0-1), the exponent involves deg a - 1 to take account of the interior product in the definition of Chen forms. It follows that (1.22)
a(S(1G, w)) = o'( )mo(w);
the proof is identical to that given in [17] since we are using Stratonovitch integrals. Note that if w = w1®... Own where deg w; = 1 and 0, cp are functions on
M then v(9;w;V) is a function on PM; indeed C°(PM) is the linear span of elements of this form. Similarily we can look at the spaces of functions C°(LM) and C°(L2M). Theorem 1.23. The spaces C°(PM), C°(LM), and C°(L2M) are dense in all the spaces of LP functions on PM, LM, and LxM respectively. Proof. We give the proof for PM. The proofs for LM and L2M are almost identical. By Ito's formula in the Stratonovitch context we see that if g is
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117
a smooth function on M then g(wt) = g(wo) +
J0
t
(d9(w,,), dw.)
= 9(wo) + o(as)
with a, = dg if s < t and a, = 0 if s > t. Thus, if 0 < t < 1, the function w i-4 g(wt) is a linear combination of Chen forms. Now (1.22) shows that all finite products gl(wt,) ... gk(wt,, ), where g; is smooth and 0 < t; < 1, can be expressed as finite linear combinations of Chen forms. Therefore, for any smooth functions 90, ... , gk+1 on M, 9o(wo)91(wt,) ... 9k(wt,, )9k+1(w1) E Co.
and since the space spanned by these products is dense in all the LP spaces this proves the lemma. We finish this section by showing that a Chen form is completely determined by its restriction to the space of paths, or loops, with finite energy. We use the notation o(1 2)(w) for the restriction to the space of paths or loops with finite energy.
Theorem 1.24. a(1,2)(w) = 0
i a(w) = 0
Proof. We write out a(w) in terms of the basis ail of forms defined in (1.4)
a(w) =
aI(w)QI
where the al(w) are functions. From (1.6) al(w)w = a(w)(EI)w
is defined for all paths cp of finite energy. We will show that al(w)w = 0 for all paths cp with finite energy if and only if al(w),,, = 0 for almost all w. To prove this we consider the measure µ on M defined by
[t(f) = Ex [aI(w)v,f(wl )] where the expectation Ex is taken over all paths starting at x. This measure has a density q(y) = Ez,y [0.1(w) ,] p1(x,y) and since p1 (x, y) > 0 the result for the path space follows from [21, Theorem
8.1] and for the loop space and the based loop space it follows from [5, Theorem II.1].
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Jones & Leandre: LP-Chen forms on loop spaces §2 LH,P,a-CHEN FORMS
Recall that if N is a Hilbert space then the natural inner product on the exterior power A'f is given by the formula (xi A ... A xr, yl A ... A yr) = det((xi, y.i))
This makes Ar(H) into a Hilbert space and we get a Hilbert space norm II - II H on Arf. In our context this inner product defines a pointwise norm IIaIIH,w on measurable forms a on PM, LM, and LxM. We want to allow infinite sums of homogeneous forms with some prescribed decay conditions and to do this we weight the pointwise norm IIaIIH,,,, by a function a(r, p) where r = deg a. The properties we require of this weighting function are: (2.1) (2.2)
(2.3)
(2.4)
a(r, p) < a(r, q),
if p < q,
ra(r - 1,p)2 < Cl(p)a(r,2p)2,
r! a(r, p)
C2(p)r, 2r+sa(r + s, p)2 < C3(p)a(r, 2p)2a(s, 2p)2
Examples of suitable weighting functions are given by
a(r, p)
2" _ (r - ap)!
where a is an integer. Now define the random (H, p, a)-norm of the homogeneous measurable r-form a by (2.5)
IIQIIH,p,a,w =
a(r,p)II aII H,w.
If or is not homogeneous then we define the inner product and the norm by making the different homogeneous components orthogonal. Thus the random (H, p, a)-norm of an inhomogeneous form or is defined by a(r,p)2II arII H,w
IlallH,p,cr,w =
r
where ar is the homogeneous component of a with degree r. The norm of a is defined by
LH,p,«-
IIQIILN.n, _ (E
This expectation E is taken over PM, LM, or LxM, according to the case, using the measures described at the beginning of §1. There are infinite
Jones & Leandre: LP-Chen forms on loop spaces
119
sums of homogeneous terms with finite LH,p,'-norm. We are particularly interested in such infinite sums so from now on we will assume that a measurable form a on PM, LM, or LZM is not necessarily homogeneous unless explicitly stated otherwise. The H in the notation for these norms is there to make it clear that we are dealing with the random (p, a)-norm and the LP,°'-norm for measurable forms based on the Hilbert norm II - 11H. This will be implicitly assumed for the rest of this section so it will cause no confusion if we drop the H from the notation. So if o is a measurable form, then, for the rest of this section, IIall will mean IIaIIH, IIaIIp,,,, will mean IIaIIH,p,«, and
IIaIILP,.
will
mean IIaIILH,p,a.
If p < q, it follows from (2.1) that IIaIIp,. "-norm of ib(w).
Lemma 2.9. Let b be a measurable vector field and a a measurable form on PM, LM, or LxM; then IIib(a)IILP,a < Cl(p)IIbIIL2p IIaIIL2p,a.
Proof. Once more we only give the proof for PM. We work with the basis en,i for measurable vector fields on PM given in (1.3) and the basis /j1 for measurable forms given by (1.4). In terms of these bases
a = Ea(I)0j.
b = E An,i,-n,i, Then, since
if (n,i) E J
iln'i (QJ) _ ±/3J\(n,i), = 0,
if (n, i)
J
it follows that the coefficient of PI in the formal expression for ib(a) is
fAn,ia(In,i)
/1I =
(2.10)
(n,i)I where In,i = I U {(n,i)}. We must prove that the sum in (2.10) converges almost everywhere: E [ IuIIP71/P C E
IAn,il la(In,i)I
(n,i)I
J
p/2
E
la(In,i)I2
IAn,iI2
(n,i)I
(n,i)I 1/2P
F
E Ian,i12 ((n,i)I )P1
P
E
la(In,,)I2
(n,i)fI
1/2p
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121
If a has finite L2p,« norm and b has finite L2p norm then the right hand side of this inequality is finite and so (2.10) converges almost everywhere. We now compute the LP'" norm of ib(a). Since
IµuI2
Ht(I)dot(w, J) (2.23)
(I,J)
Ho(0) = 1
if J#0.
Ho(J)=0,
This system can be solved inductively on III. Let Xt and Yt be two continuous scalar processes. Then the solution of the equation, in the Stratonovitch sense, dUt = UtdXt + dYt
is given by 1
'Pt
iO-IdY,
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129
where cp, is the solution of the homogeneous equation starting from 1. This solution cp, can be computed by the Picard method. This allows us to solve (2.23) inductively in III.
A priori it is not clear that the Picard series, which is only defined for a fixed t, defines continuous processes Ht(I) which are solutions of the equation (2.23). But in fact it follows from the Kolmogorov criterion that this equation does define a continuous process. In this case we have a true semi-martingale for the term Xt; we do not get an anticipative condition since dXt is given by the degree 1 component wl of w and in the term wl (dwt) no final condition in parallel transport appears. Thus we get a unique family of continuous processes t u- 4 Ht(w, I) indexed by I. It is also possible to show that the Picard series of (2.23) converges for
all t < 1, instead of just t = 1, in all the Lp'a and so we get a form expw(w)(t)
of infinite degree for t < 1. Moreover the function t r-a expw(w)(t)
is continuous in all the LP'a, by using the basic estimate Lemma (1.10). By using Lemma (1.10) and the Kolmogorov criterion, it is also possible to show that if, for any continuous process Ht, we define Ht(w, I) by
expw(w)(t) = E Ht(w, I)/3r then the Ht(w, I) are solutions of (2.23). §3 LB'p'a-CHEN FORMS
If 1 is a Hilbert space then there is a natural operator norm on the space A''(fl*) of alternating multilinear functions on R. Given 0 E Ar(1H*) then II/IIB is defined to be the smallest real number such that II#(xl, ... , Xr)II p
Cq,a C Cp,a,
and we define p
The proofs of the next two theorems are easy modifications of the proofs of (2.15) and (2.16).
Theorem 3.13. If a, r E C.,,, then the exterior product a A r is also in
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Jones & Leandre: LP-Chen forms on loop spaces
Theorem 3.14. Suppose that o is in C...,o and that b is a vector field which has finite LP-norm for all p; then ib(o) E Coo,a.
We also get the analogue of Theorem (2.18) using the Lp'' -norm based on 11 - 11,u rather than 11 - 11H. We formulate the result for the based loop space L,,M since the proof we given below uses the inequality (3.11) and
we have only established this in the case where of the based loop space. However, as we noted after (3.6) the extra terms introduced by working on the free loop space LM or the path space PM are not difficult to estimate and, as in Theorem (2.18), the analogous result is valid in these cases. Theorem 3.15. Let wi,n, where 1 < i < n, be differential forms on M and define wn = wi,n 0 .. ®wn,n. (1) Suppose that the power series 00 In
E n=1
i=1
n I I "i, n I I k, oo VfnTI
has infinite radius of convergence for all k < 2d + 3 where d is the dimension of M; then 00
1: o(wn) E Coo,,,(LxM). n=1
(2) Suppose that for all u the diffusion w,(u) is hypoelliptic and that the power series 0o
n=1
In
zn
II IIwi,nhIl,o V/n-! i=1
has infinite radius of convergence; then 00
1: o(wn) E Coo,a(LxM). n=1
Proof. It is enough to consider the case p > 2. Let ri,n = deg wi,n - 1 and
rn = rl,n +- + rn,n From (3.11) we know that Io(wn)(bl,...,brn)I < rn
1/2
IA(wn);s,J)I2dsl ...dsn
IlbiII
s=1
(J;I1,...,In)
[0,1]n
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135
and therefore IIa(Wn)IIL-
0 and Lemma (5.45) to derive the third inequality. This completes the proof of (4.15). In fact (4.15) is true for all v under the hypothesis of Remark (5.47). From (4.10) - (4.14), we get the following estimates for all v: (4.18)
IE: '[J2(l,s)]I < d
K2 N
0
l,t)]I
IEt
i-1 (\
Xi(r), (u, r, v)11/
drdt
IEe'v[J3(l,s)]I 0, to get estimates for Ex"x[J(n,1)], it is enough to get nice estimates of qn(y). In order to do this we will apply the Malliavin calculus.
Let us introduce Y, a system of vector fields on M such that the family Y(x) span TxM at each point x E M. If, for all multi-indices a (4.36)
P. [Y'f] < Cn(lai)IIJ II.
we will get the following estimate:
sup Ign(y)I < K SUP Cn(lal) lal 0, and v > 0 such that
P {)i(t, x) < K
(5.25)
C exp (-cK"))
Proof. Since
oo
t/K
d
(S,
2
X`(we(x))
x9)
ds
is bounded below by
it follows that (5.26)
d
jt/K
i1
2
jt/K
ds -
e(t) (, X(a,i)(x))
(, R(s, x))2 ds i=1
II«IISto-1
where the 8a are given by [27, Appendix]. The ea satisfy (5.27)
9Xi(w9(x)) _ E ea(s)X(a,i)(x)+Ri(s,x) 8x) II«II (x) «#o,ll«II- 0 define the following stopping times for the canonical process w of C([0, oo); D): TK;' TK
= =
inf{u > s : w(u) ¢ K} inf{u > 0 : w(u) O K}
under the convention, infima of empty sets are interpreted as oo.
We summarize the relationship between r-martingales and convexity in a theorem.
THEOREM 2.2. The probability law P renders the canonical process w as a r-martingale if and only if for all t > s > 0, all closed geodesic balls B C N, all convex 0 : B -, [0,1]
E P [cb(w(TB" A t)) IF.]
>
¢(w(s))
P -almost surely.
(2)
Kendall: Convex geometry and nonconfluent r-martingales I
169
Moreover to test for the r-martingale property it suffices to establish the inequality only for smooth convex 0, only for a dense subset of all possible closed geodesic balls B (under the topology induced by the parametrization B ' -+ (centre, radius)), and only for a dense subset of possible t > s > 0.
We give no proof, as this theorem follows immediately from results of Darling (1982) and Emery and Zheng (1984). Darling discusses the case of smooth convex ¢. The essence of the matter is to apply Ito's formula via (1) to 4(w),
and to use the rich supply of locally strictly convex functions furnished by exponential coordinates additively perturbed by multiples of x i-+ dist(x, z)2 for x near to z. Emery and Zheng discuss the case of general (nonsmooth) convex functions. Finally, it is enough to test only for a dense subset of geodesic balls because the I'-martingale property is a local property, and a straightforward convergence argument shows that a countable dense subset of t > s > 0 suffices. Our next task is to show that the space of all I'-martingale laws is closed in the topology of weak convergence of probability measures.
DEFINITION 2.3. The space of r-martingale laws. Let V C N be as above. Define rM(D) to be the set of probability laws on the path-space C([0, oo); D) making the canonical process t H w(t) into a r-martingale. Furthermore define rMZ(D) by
rm,(D) = {P E rM(D) : P [w(0) = z] = 1} . THEOREM 2.4. r-martingales and weak convergence. The space of probability measures IM(D) is weakly closed in the space of probability measures on C([0, oo); D).
Proof. Suppose P" P with P" E rM(D) for all n. We have to show the canonical process w is a r-martingale under P. From P " E rm (D) it follows Ep"[O(w(TB;,
A t)) )IF3]
>
O(w(s))
P "-almost surely.
(3)
for all t > s > 0, all geodesic balls B = ball(z, R) in N and all convex 0:B
[0, 1]. The inequality persists in the limit precisely when f B;s.t : w H
Kendall: Convex geometry and nonconfluent F-martingales I
170
q5(w(TB1° At)) is continuous at w for P -almost all w. Now this latter function is in general only lower-semicontinuous. However for each fixed z, for all but
a countable number of R (the choice of which depends on z, t, and s), the function f eaii(z,R);s,` is continuous at w for P -almost all w (see Stroock and Varadhan, 1979, Lemma 11.1.2).
Thus for a countable dense subset of t > s > 0, for a countable dense set of geodesic balls B C N and all convex ¢ : B --+ [0, 1] it follows E"[4(w(TB" A t)) IF,]
>
0(w(s))
P -almost surely.
(4)
This establishes the theorem. o
THEOREM 2.5. r-martingales and tightness. Suppose D is compact and {Z" : n > 1} is a sequence of r-martingales in D all begun at z E V, with Z" having r-parallel transport " and r-development Mn. Suppose further that for some constant A > 0 the differentials of the "intrinsic time" measures r" of the Z" satisfy the bound drr"
=
tr (( " (& "") d[M", M"])
= Ed [f ((8")TU',dM"), f(( =n)TUi' dM-)]
1} form a tight sequence of probability measures.
Remark: Because of the compactness of V the choice of reference metric (which intervenes in the choice of the {u'} and thence the definition of the trace) does not alter the condition (5).
Proof: We employ the tightness criterion described in Stroock and Varadhan (1979, Theorem 1.4.6). Adapted to our situation this is as follows: embed D smoothly in Euclidean space. For each nonnegative smooth real function f
Kendall: Convex geometry and nonconfluent r-martingales I
171
there should be a constant Af such that the process t --, f (Z'(t)) + Aft is a submartingale for each n. The Stroock and Varadhan criterion applies when A f can be chosen to be the same for all translates of f. By compactness of D this is automatic if Af depends continuously on f and its derivatives and the connection coefficients of r.
Apply Ito's formula via (1) to f (Z"). The Ito differential of f (Z") turns out to be
dr(f(Z"))
=
(grad f, =' d1M') +
(Hess f (E", E'), d[M", M"])
(6).
2
Here Hess f is the second covariant derivative of f, defined of course by using
the connection F. The second terns is bounded by Af = 11 Hess f 11 A dt, applying the intrinsic time bound given in the theorem. Notea the definition of 11 Hess f 11 depends on the choice of reference metric. Now 11 Hess f 11 is
finite since f is smooth and D is compact. Furthermore Hess f depends continuously only on f and its derivatives and the connection coefficients of F. Hence the tightness criterion applies, yielding the theorem. o
3. Stochastic Control and r-martingales. Consider, for a given compact region 13 as above, the value of the stochastic control problem V(x, y)
=
sup {P [X(oo) = Y(oo)]
:
(X, Y) E FM( )(B x B)}
(7).
(Note one or both of the limits may not exist, in which case we take them to be not equal.) A standard argument shows that V(X, Y) defines a supermartingale for any I'-martingale (X, Y) E FM(s. )(Bx13) (note (X, Y) should more properly be referred to as a "product-connection-martingale", but the slight abuse of terminology causes no ambiguity). Now X and Y can be produced by composing I'-geodesics with real-valued continuous martingales. Hence Q(x, y) = 1 - V(x, y) is a convex function, since this consideration shows it yields (conventional Euclidean) convex functions when composed with product-connection geodesics in B x B (which are simply products of I'-geodesics from B). So Q is a (product-connection) convex function Q on
BxB.
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Kendall: Convex geometry and nonconfluent r-martingales I
It is clear that Q vanishes on the diagonal A of B x B. It therefore supplies convex geometry for B precisely when it vanishes nowhere else. Suppose now that uniqueness holds for problem (B), so that I -martingales in B are uniquely defined by their terminal values. We shall show the supremum in the definition of V above is always attained if V(x, y) = 1, and hence in such a case x = y. It follows Q vanishes precisely on A, and so B has convex geometry.
Here is a preliminary lemma showing the effect on I -martingales of a strictlyconvex-supporting property for B.
LEMMA 3.1. Suppose the compact region B supports a strictly convex and smooth function f : B -+ [0, 1]. Let h2 > 0 provide a lower bound for the
Hessian off via Hess f
>
(8)
ic2 g
where g is the metric tensor for the reference Riemannian structure. Suppose Z is a I -martingale in B with I'-development M and I'-parallel transport EE. Then the expectation of the total intrinsic time of Z is bounded by
E [ftr((E®).d[M,M])]
1} form a sequence of r-martingales in B x B such that Xn(0) = x, Y"(0) = y and P [X'(oo) = Y"(oo)]
1.
Let En, '" be the r-parallel transports, and Mn, Nn the r-developments, of Xn, Yn respectively.
The problem is unaffected by time-changes and so we may suppose the differential of the intrinsic time measure of (X", Y') is given by 1[0,T-Jdt, where Tn is a Markov time for the filtration for (X", Y") and 1[o,Tnl is 1 when
0 < t < Tn and 0 otherwise. In particular both Xn and Yn stop moving after time Tn. Applying the I -martingale tightness criterion of theorem 2.5, and selecting a subsequence if necessary, we may suppose (Xn, Yn) converges weakly to a I -martingale (X, Y). The differential of the intrinsic time measure of (X", Y") is bounded above: 1[o,T"Idt
=
0, be a simple symmetric nearest neighbor random walk on V. P2, is the probability measure governing random walk paths which start at x; Ex denotes expectation with respect to P. For any finite set of vertices B define
T(B)=inf{n>0:SnEB}, HB(x, Y) = PX{ST(B) = y}
if y E B,
*Research supported by the NSF through a grant to Cornell University.
Typeset by AA,}S-T X
Kesten: Multiple contact diffusion-limited aggregation
180
PB(y) =slim P,{ST(B) = y I T(B) < oo} -00 lim
HB(x' y)
'°° EHB(x, z)
if y E B.
zEB
Both HB(x, y) and PB(y) are taken as zero when y B. The existence of the limit in the definition of PB was proved by Spitzer ([19, Theorem 14.1 and Prop. 2.6.2]). For d = 2 one does not need to condition on T(B) < 00 since this event has probability 1. For d > 3 PB(y) equals (cf. [19, Prop. 26.2], [6])
EsB (y)
E ESB(z)' ZEB
where
EsB(y) = Py{S
B for all n > 1}.
(ESB is the so-called escape probability from B.) We now describe the two procedures.
Procedure 1. This procedure is related to the method of noise reduction or multiple contact DLA, which is used widely in simulations ([5], [12], [13], [14], [17], [20], [21, Sect. 9.2.2]). When A,, has been determined, we associate with each z E t9A1, a score, s(z). We start all these scores at zero and, in a sequence of steps increase one stochastically selected score by one. In each step the probability of increasing s(z) by one (and leaving the other scores alone) is P8A.(z). Thus we can view the sites in OA,, as urns. In this description each step corresponds to "releasing a random walk at infinity
and letting it run until it falls in one of the urns." s(z) counts how many random walkers have been collected in the urn for z. We continue such steps until one of the scores for the first time reaches a predetermined value m(n). In our version we have in(n) - Co log n with a Co to be determined
later (cf. (2.72)). If S(Zk) is the first score to reach the value m(n), then we take y,, = Zk and A,,+l = A U {Zk}. Next we reset all scores to zero and also attach scores zero to the new boundary sites (i.e., the points of 0An+1 \ We then follow the same procedure to select the next point y,i+1, etc..
In the above process, in(n) random walks have to land on one site before
it is selected for addition to the aggregate. The original Witten-Sander
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181
model of DLA is the special case m(n) - 1. For the simulations one has so far only considered m(n) = m, a constant. However, Meakin and Tolman
([12], [14]) have done simulations with m = 100, n < 105 (even some simulations with m = 104, n = 5175 in [14]). Unless the constant Co in m(n) has to be taken much larger than we expect, then 100 > Co log(105), so that one cannot distinguish simulations done so far with large constant m from simulations with m(n) > Co log n. There is, however, a very significant difference between our version and the one usually simulated. In the latter versions the scores s(z), z E 8A/z,
are not reset to zero when yn has been chosen. Only the scores of the new boundary points (points in 8An+1 \ OAn) are started at zero for the selection of yn+1. The other scores start with their values at the time yn was chosen, when yn+l is to be selected. For small n both versions (with or without resetting the scores to zero) behave similarly with all particles being added at the tips only (see [1]) and Theorem 1 below.
Procedure 2. This is a version of the so-called q-model for dielectric breakdown (cf. [16], [4], [11]). In this procedure we take
P{yn = yI An} =
Zn1(11(n))[paA,(y)]n(n)
and
0
otherwise
for
y E 8An
,
where (1.2)
Zn(rl(n)) _
[PaAn(Y)]o(n)
yEBA
Again the usual version is to take i7(n) constant. We will let 77(n) grow with n so that 77(n) - Co log n for a constant Co to be determined later (see (2.72)). Taking i7(n) in this fashion is akin to taking the limit 17 --r 00 before n goes to oo. This kind of limit was already mentioned in [16]. The effect of requiring m(n) contacts in Procedure 1 or of raising µaA to a power i7(n) in Procedure 2 is to make it more likely that the new particle yn is added near a "tip" of An, where 100A is relatively high. In fact, in both procedures we will choose the proportionality constant Co in the growth rate of m(n) and i7(n), respectively, so large that with overwhelming probability yn has to be one of the tips of An itself. To make the notion of "tip" precise, we restrict ourselves to a special class of A's which consist of intervals of some of the coordinate axes. We say that A is a generalized plus sign if d
(1.3)
A = U {kei : -k(i, -) < k < k(i,+)} i=1
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for some k(i, e) > 0, with k(i, e) > 0 for at least two choices of (i, e) E {1, ... , d} x {-, +}. (The terminology is of course inspired by the case d = 2 and k(1, -) = k(1, +) = k(2, -) = k(2, +), which represents an ordinary plus sign.) We denote the set A of (1.3) by A(k(i, E)). Note that we allow k(i, e) = 0 for various (i, e). Our principal result says that with the exception of d = 2 or 3 generalized plus signs with more or less equal arms occur with positive probability.
THEOREM 1. Let S be any subset of 11,... , d} x I-,+} with (SI > 2 (I S1 denotes the cardinality of S). Let E(S) be the following event: (1.4)
E(S) = {For all n, An is a generalized plus sign, A(k(i, e, n)) say, with k(i, e, n) = 0 for (i, e) and
k(i, e, n) -+
1 A
S for (i, e) E S}.
n
Then for sufficiently large Co (cf. (2.72) below)
P{E(S)} > 0
(1.5)
for Procedure I if d > 3, and for Procedure 2 if d > 4. For both procedures (1.5) continues to hold in d = 2 or 3 if S has the special form {(i, -), (i, +)}. In the case where S = {(i, -), (i, +)}, E(S) says that An is a line segment along the i-th coordinate axis, asymptotically symmetric with respect to the origin. We suspect that in dimension 2 these are the only E(S) with positive probability. We have no proof of this for ISI = 3 or 4, but the next theorem shows that An cannot have an asymptotic L-shape. THEOREM 2. For d = 2 and both procedures with any Co > 0
P{E(S)} = 0 when S = {(1, ei), (2, e2)}(ei E {-, +}).
(1.6)
Remarks. (i) We do not know whether P{E(S)} > 0 for Procedure 2 in dimension 3.
(ii) Theorem 1 shows that with strictly positive probability An never contains a point off all coordinate axes, and grows only (and at more
or less equal rate) along the half-axes indexed by S. One expects that there is also a strictly positive probability of obtaining finite
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perturbations of this. More specifically, let F be a finite connected lattice set containing the origin and define (1.7)
E(S, F) _ For all large n, An = A(k(i, e, n)) U F for some k(i, e, n) with k(i, e, n) = 0 if (i, e)
and n k(i, e, n) -
I isi
S
if (i, e) E S. }
We can then indeed prove that
P{E(S, F)} > 0
(1.8)
for any finite connected F containing 0 in the following cases: (1.9) (1.10)
d = 3,
Procedure 1 and
d= 2,
{(i, -), (i, +)} C S for some
i.
Procedure 1 or 2 and S = {(i, -), (i,+)}.
It is even possible in dimension 2 to obtain a "halfline with a finite side branch". Specifically, for k0 and CO sufficiently large and d = 2, we have (1.11)
P{For all large n, A _ {jei : 0 < j < n - k0}
U {ee3_i : 0 < t < ko}} > 0, for both Procedure 1 and 2 and i = 1, 2. (iii) The proof of Theorem 2 will show that if for some n (1.12)
An = {jei : 0 < j < k(1,+,n)} U {jet : 0 < j < k(2,+,n)}, then either at some later time rn, An will no longer be of the form (1.12), or if An has the form (1.12) for all m > n, then k(i, +, n)
k(3 - i, +, n)
-0 fori=1or2
i.e., one arm will grow much faster than the other. If CO is sufficiently large, then one arm will even stop growing at some time. In this case
k(1, +, n) or k(2, +, n) will be constant for some time non, and An behaves as in (1.11).
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We summarize in the table below which configurations are possible or impossible.
d
Procedure 1
Procedure 2
>4
Any generalized plus sign
Any generalized plus sign
possible
possible
Any generalized plus sign. Even finite perturbation possible if plus sign contains a symmetric line segment (i.e. as in (1.9)).
Symmetric line segment with finite perturbation
Symmetric line segment with finite perturbation, and halfline with finite side branch possible. L-shape not possible.
Symmetric line segment with finite perturbation, and halfline with finite side branch possible. L-shape not possible.
3
2
possible.
Table of possible and impossible asymptotic shapes.
Note added in proof: Greg Lawler has shown that for Procedure 2 in d = 3 certain L-shapes are possible.
2. Proof of Theorem 1. We shall concentrate on Procedure 1 in dimension 3. The case d > 4 is considerably simpler, while much of the preparation for d = 2 will have to he left for another publication ([8], [9]). We define the tips of the generalized plus sign A(k(i, e)) to he all the points of the form
(t;(i,+) + 1)ci
if
kr(i,+) > 0,
and
-(k(i, -) + 1)(,,i
If
(See Figure 1.) Note that, we defined the
k(i, -) > 0.
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185
w
.v
0
U
Figure 1. A generalized plus sign in dimension 2. In this example k(1, -) = 0 and the tips are u, v and w.
tips to be points of DA, rather than of A itself. We also introduce the notation
B= B U 8B = {v : v E B or v is adjacent to B}, for any lattice set B. Its hitting and return time are defined as (2.1)
T(B) = inf{n > 0 : Sn E B} and R(B) = inf{n > 1 : S E B},
respectively, and its escape probability is (2.2)
ESB(v) = Pv{Sn
B
for all n > 1}.
We shall use Ki to denote strictly positive, finite constants whose precise value is unimportant for our purposes. In fact, their values may vary from appearance to appearance. a A b= min{a, b},
a V b= max{a, b}.
The idea of the proof of Theorem 1 is to prove two types of inequalities. The first one (see (2.4), (2.6), (2.61) and (2.67)) says that if An is a generalized plus sign with arms of more or less equal length, then is
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approximately the same for all tips t of A,,. The second inequality says that µaA (w) is less than a fixed multiple (1 - a) < 1 of µ8A (t), whenever w is not a tip and t is a tip (cf. (2.5), (2.62), (2.69) and (2.70)). Since our procedures shift mass to the points with high paAn, the second inequality can be used to show that for large Co this effect is sufficiently strong to guarantee that A only grows at the tips forever. The first inequality is then used to show that with positive probability the different (non zero) arms grow at the same asymptotic rate. PROPOSITION 1. Let d > 3. Then there exist constants a > 0, KI, K2 < oo, depending on d only, with the following properties (2.4), (2.5): For any generalized plus sign A(k(i, E)) which satisfies (2.3)
for some k > 0 either
k(i, c) = 0 or k < k(i, e) < 2k f o r all
11,
. . .
I-,+},
we have (2.4)
µ8A(12) µ'8A(t1)
-1
Kl(logk)-I
K21 and also (2.5)
µ8A (/w)
µ83(t)
K2 (w is otherwise arbitrary in OA). If d > 4, then (2.4) can be sharpened to (2.6)
Iµ8A/(12)
1
at : S,, belongs to the first coordinate axis}.
Thus, IS,,} is the imbedded random walk on the first coordinate axes. We set TZ = first coordinate of S,,,
(2.9)
Y1+1 = Tt+i - TZ.
Note that the Yt, e > 2, are i.i.d.. We write Y for a random variable with the same distribution as these variables. If S. is on the first coordinate axis, then even Y1 has the same distribution as Y.
We begin with calculating the characteristic function of Y, '(O). For simplicity we take S. = 0; then Y has the distribution of the first coordinate of S. Let 0 < T1 < 7-2 < ... be the successive values of t for which
St+1 - St E {±e2, ±e3}. The STj}1 - S,rj , j > 1, are i.i.d. The part of STj+1 - STS in the second and third coordinate direction equals ±e2 or ±e3, each possibility occurring with probability 1/4. On the other hand, the first coordinate of ST,+1 - STS
is the sum of a geometrically distributed number of steps of a symmetric
simple random walk on Z (and independent of the second and third coordinate). From this it is easy to find (2.10)
((O)
.=
E{exp i9(ST,+1,1 00
t _1
2 3
-
1 Y-1 (cosO)t-1
j>1.
3
Furthermore, S can return to the first coordinate axis for the first time, either at the first step (when S1 = ±e1), or at the time T,,, where v = min{j : S,r,+1 E first coordinate axis}
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(when S1 # ±e1 and T1 = 0). In the first case Y = ±1, each event occurring with probability 1/6. In the second case, given that v = k, the conditional characteristic function of Y is [C(B) ]k-1. In total we find 2
1
00
'Y(O) := Eea8Y = 3 cos 0 + 3 E P9 { a two dimensional simple k=1
symmetric random walk returns first to 0 at time 2k
}[((8)]2k-1.
Let VO = 0, V1, ... be a simple symmetric two-dimensional random walk, starting at the origin, and set
f2k = P{first return to 0 by V is at time 2k}, U(S) _
U2k = P{U2k = 0},
00
u2k S2k
0
Then standard renewal theory [2, Theorem XIII.3.1] and the above show that cos0 + 3 cos0 +
3((8)
{1 -[U(s(e))]-1}.
Now it is easy to see from (2.10) that
1-((9) - 482,
9,0.
Also u2n - (zrn)-1 ([2, Sect. XIV.7]) and the Abelian theorem ([3, Theorem XIII.5.5]) show that U(s)
- loglls
srl.
Consequently (2.11)
1 - *(0) ^
3
{logj]',
This takes the place of (2.17) in [7].
0 -. 0.
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189
The same kind of argument shows that 00
P{Y > n} = P{Y < -n} = 3
(2.12)
f2kP{W2k_1 > n},
n > 1,
where W2k_1 is the sum of (2k - 1) independent random variables, each with characteristic function C. In particular, by the central limit theorem, P{W2k_1 > n} --r 2
(2.13)
as
k/n2.00.
Moreover, CO
Es fL =
s
k
k=1
-S
L-k
1-fs 00
L
1 1
s U(s)
1
(1 - S) log
7r
f
l
1-S
ll
}
8t1.
S Hence, by the Tauberian theorem ([3, Theorem XIII.5.5]
r2k := E fL
(2.14)
ir(logk)-1,
k
oo.
L-2k
We can now find the asymptotic behavior (as n oo) of (2.12). For any e > 0 the sum over k < n2/e is, by Chebyshev's inequality, at most 23
k
n} = P{Y < -n} ti 6 (log n)-1,
This replaces (2.27) in [7].
n -*oo.
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190
We now proceed as in [7]. v(n) is defined by k+1 03
1` v(n)rn' = expt o l exp
( E y;)
1
r o0
kr
YE >
2 k+1
2
1
f-
{=o}
k+1
2
1
+"
47r
,
1-r2 1 + r2 - 2r cos B
log[1 - W(B)] dB}
By (2.11), when z = eie and 0 - 0, e`a)-11
(1 - V,(8))I1 + log(1 -
.
JJJ
= (1- 1(8))I1 + log(1 -
z)-1
_, 4n 3
Therefore, by Poisson's formula and Fatou's theorem (cf. [18, Theorems 11.30b with proof and 11.23] 1/2
lim 1 + log
1 1
rj1
r
00
> v(n)r" 0
limexp{- 1 ril 47r
+x
1-
r2
1+r2-2rcosO
log[(1-''(9))11+log(1-eie)-1I] d9}
-
47x1/2 3
Equivalently, 00
7r v(n)rn r [(4ir)
3log 1 1
r
11/2
0
and, again by the Tauberian theorem ([3, Theorem XIII.5.5]) (2.16)
> v(k) 0
(C 3) log
n)1/2 ,
n -r oo.
Now assume So = jet, a point of the positive first coordinate axis, so that To = j (see (2.9) for Ti). Define
p = min{n : Tn < 0},
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191
G(j, t) = E{number of 0 < n < p with Tn = eITO = j} jAl (2.17)
_ > v(j - n)v(f - n) n=1
(cf. [19, Prop. 19.3]). Then
P{S,, IkISO=jell = P{TPoo.
n=O
(use (2.16) and partial summation for the last step; compare [7, Lemma 6]).
Now we first prove (2.7) for v = jel, j > 0 fixed. Clearly
Eslk (je1) < P{S,o (s,)3/2 (2.18)
2
Ik I SO = je1 }
j-1
v(n)(log k)-lie.
n=0
On the other hand, for any constant A
ESI,,(jel) > P{S0 ,
IAk
and
Sn V Ik
for
n > vPISO = jel}
co
P{TP=-rITO=j} r=Ak+1
(1 - P{Sn visits Ik at some time ISO = -rel }) .
But, if (2.19)
&,y) = Er{number of visits to y by S. }
is Green's function for simple random walk on Z3, then it is well known ([19, Props. 25.5 and 26.1], [10, Theorem 1.5.4]) that (2.20)
9(SnnT{y}, y)
is a positive martingale
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192
and that
Therefore,
y-x->00
PX{S. ever visits y} - C2Iy-xI-1, for C2 = 3{2ag(0, 0)}-1.
Hence for r > Ak k
3C2
P_, el IS. ever visits Ik} < 2 C2 E
Ir-1 jl -< A- 1* j =0
Thus
Eslk(jel) ?
AC211 1
C1 -
P{S,P V IAkISO = gel}
(,)3/2
3C2 A
2
n=0
v(n) (log Ak)1/2.
It follows that
( 3/2 j-1
(2.22)
lim (log k)1/2ES1, (jet) = 2 \ )
v(n)
koo
n.=0 2
(_ 4 (3) {log j}1/2 for large
)
This proves (2.7) for v = jet, j > 0. For general v one can prove (2.7) by a decomposition with respect to SR where
R = min{n > 1 : Sn E first coordinate axis}, more or less as in Lemma 6 of [7]. We skip the details. We will, however, prove (2.8) and this proof illustrates most of the ideas.
Let v = -jet, 0 < j < k/2. Then most of the previous considerations remain valid. However, this time we replace (2.18) by
Eslk(x) < P {So1 0 Ik, SoP
Ik} 00
=eel}Ple,{SoP
Ik}
e=1
= P{YP{Y= j+e}l:G(e,r)P{Y 1 00
G(t, r)P{Y < -r - k) = P£el {Scp
Ik } < 1.
r=1
Finally, with another partial summation 1 00 Eslk (v) < K6 (log k)+ EP{Y = j + P} min 2K5 ( log( + 1)
log k
£=1
K6{109(i
)
1/2 1
+2).logk}-1/2
This proves (2.8) when 0 < j < k/2. The case k/2 < j < k follows now by symmetry with respect to the midpoint of Ik.
Unfortunately we have to work with escape probabilities from 7k rather than from Ik itself. The next lemma shows that Es Ik is of the same order as Eslk .
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194
LEMMA 3. There exists a constant C3 > 0 such that
kli(log k)1/2Es7k (ej) = C3.
(2.23)
Proof: We use a last exit decomposition (with respect to TO :
Esjk(je1) = Es7k(jel)+Piei{S. visits Ik at some time > 1, but not Ik 1.
(2.24)
The second term here may be decomposed as (see (2.1) for R)
1: Piei{R(Ik) < oo, SR(7k) = v}Esjk(v). VE7k\Ik
We want to show that (2.25)
klim (log
Pie,{R(Ik) < oo,SR(Ik) = v}Esrk(v)
k)1/2
VEjk\jk
exists and can be calculated by taking the limit inside the sum. The main step for this is to estimate, for large L k
(log k)1/2)7 Pie,{R(Ik) < oo,
(2.26)
e-L
SR(7k) = -eel + e2} Esjk (-eel + e2).
In view of (2.8) this sum is at most (2.27) k
K3 EPie, {R(70 < oo, SR(jk) = -eel + e2} {log((e A (k - e)) + 2)}-1/2 £-L
k
< 6K3 EPie,{R(Ik) L}}.
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195
By taking L large we can make the right hand side of (2.27) small. This gives an estimate for (2.26) which is uniform in k.
Now Ik \ Ik consists of the points e1, (-k - 1)e1, -eel ± e2, -ee1 ± e3, 0 < e < k. The sums over v = -eel - e2 or v = -eel ± e3 are equal to the sum over v = -eel + e2. The term (logk)'I2Pjei {R(Ik) < oo, SR(7k) = -(k + 1)ei} EsJk(-(k + 1)e1) is negligable, since, by symmetry,
Eslk(-(k+1)el) = Eslk(el) -
C1(e1)(logk)-1/2.
These estimates justify interchanging the limit and sum in (2.25). If we set (2.28)
R = inf {n > 1 : S E {-jet : j > 0}} ,
then we obtain from (2.24) (2.29)
lim (logk)1/2Es7k(je1) koo
C1(je1) - 4 E Pjei {SR J SW = -eel +e2}
C1(-eel + e2).
e>o
Thus the limit in (2.23) exists and we merely have to show that it is strictly positive. Of course it suffices to show (2.29) strictly positive for some large j, since
(2.30) Es7k(el) > P {S. visits jet before returning to Ik} Eslk(jel) 1
and the first factor in the right hand side here is bounded away from 0 as k
00.
To show the positivity of (2.29) we go back to (2.24). Note that for v = -eel ± e2 or -fe1 ± e3
EsJk(-eel) > P-ee,{S1 =v}Eslk(v) = 6 Eslk(v). Thus, by (2.7) and (2.8), there exists a constant K7 such that
(log k)1/2Eslk (v) < K7 for all v E Ik \ Ik Thus, by (2.24) (2.31)
(log k)1/2Esyk(jel) > (logk)1/2Esrk(je1) - K7.
Now (2.22) shows that the right hand side here is strictly positive for large j.
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196
LEMMA 4. Let A = A(k(i, e)) be a generalized plus sign. Assume that d = 3 and that for some k > 0 (2.3) holds. Let i = s(k(i, s) + 1)e1 be a tip of A and w = 0 or w = -efeE ± ei for some fixed f > 0, j # i, so that t + w E OA. Correspondingly let w = 0 or w = -Ce1 + e2. Then for some constants K; (which may depend on w but not on k)
IEs.r(t+w) - Eslk(el +w)I
K3(log k)-1/2 for some K3 = K3(w) > 0. Together with the first inequality of (2.32) this will then also give the second inequality of (2.32). Thus it suffices to prove the first inequality of (2.32). To this end we introduce the notation J(i, +) = { jeE 0 < j < k(i, +)},
J(i, -) = {jeE -k(i, -) < j < 0}. Also, just as in the case d = 2 in [7], let C(s) be the discrete sphere of radius s
C(s) _ {v E Zd : IvI > s, but v is the endpoint of an edge whose other endpoint, v', satisfies I v' < s}. To begin we prove
Pz {R(Ik) < oo} < logk 1: g(z, v),
(2.34)
VElk
where, as in (2.19), is Green's function for simple random walk. (2.34) rests on the observation (2.35)
E g(z, v) = Ez{number of visits by S. to Ik} VElk
= I [Z E 7J + 1:
Px{R(I,k) < 00, SR(7k) - u}
UE7k
E {number of visits by S, to 70
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197
Now it follows from (2.21) that for any u E Ik
Eu {number of visits by S. to Ik } = E g(u, w) W Elk
> K5 i
1 > K4 log k.
1<j K5. The reader can check that (2.56)-(2.58) take care of all w E OA which
are not tips of A. Note that even the point -e, for a j with k(j, -) = 0 is taken care of, since this is of the form -ej +O.ei for some i with k(i, +) > k
or k(i, -) > k, i # j (recall that a generalized plus sign has at least two arms, i.e., k(i, g) > 0 for at least the choices of (i, s) E { 1, ... , d} x {+, -}). We therefore have (2.5) for K2 = max{K4, K5}.
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205
Again it is easy for d = 3 to generalize (2.5) with A replaced by A U F
for some fixed finite set F, provided that for some i, k(i., +) > k and k(i, -) > k. In this case we can estimate EsauF(w) for w "close to the origin" by using the fact that {Pei : -k < f < k} C A. This, together with (2.8) gives for jwl < k/2 (2.59)
EsTuF(w) < Es{te;:-k 0, Ki < oo and a E (0, 1), and sufficiently large k (2.61)
(2.62)
Pa(AUF)(tf) ... C1{k(i, +) + k(i, -)}-1/2,
/ba(AUF)(t+) -
µa(AUF)(t_)l
< Ki{k(i,+) + k(i, -)}-5/2
and (2.63)
µa(AuF)(W)
µa(AUF)(t±)
< 1 - a,
where tE stands for the tip e(k(i, e) + 1)ei (e = + or -) and w is any point of 8(A U F) which is not a tip. (b) Assume that A is L-shaped, i.e., for some el, e2 E {-, +}, k(1, e1) > 0, k(2,e2) > 0 (2.64)
A = {eijei : 0 < j < k(l,ei)} U {e2je2 : 0 < j < k(2,e2)}.
Writeti for the tip ei(k(i,ei)+1)ei. Then there exist C2(/3) > 0, C3()3) > 0 such that (2.65)
1/2
lim [k(1, e1) + k(2,,-2)]
NaA(t1) = C2(3) > 0,
Kesten: Multiple contact diffusion-limited aggregation
206
1/2
lira [k(1, ei) + k(2, E2)]
(2.66)
µaA(t2) = C3(13) > 0
as k(i, Ei) - oo, i = 1, 2 in such a way that k(l, el) , 3E[ 0 ,0 0 1-
(2 . 67)
k(2, e2)
There also exists a constant C4 > 0 such that (2.68)
k(1, E1)
- k(2, E2) PaA(ti) - PoA(t2) = (C'4 + 0(1)) [k(1,Ei) + k(2,c2)1312
as k(i,e2)- 00, i = 1, 2 in such a way that (2.69)
k(1,E )
- 1.
k(2 e22)
There exists a constant a E (0, 1) such that (2.70)
µ8A(w)
uaA(ti)
k(3 - "'-3-i), w E OA not a tip and k(1, el) A k(2, E2) sufficiently large.
Finally, for all e > 0 there exists an a(E) E (0, 1) such that for all sufficiently large k(1, E1) and k(2, E2) > (1 + -)k(1, El) and all w E 8A other
than w=t2, (2.71)
µaA(w) PaA(t2)
< (1 - a(e)) .
Note that w = ti is included in (2.71), but that w # tl,t2 in (2.70). The proof of part (a) is very similar to that of Proposition 1. However, part (b) requires new techniques and has a very long proof. The proof of Proposition 2 will therefore be given elsewhere ([8], [9]). Here we only show how to obtain Theorems 1 and 2 from Propositions 1 and 7.
The next step shows that we can take Co in our procedures so large that if A is a large generalized plus sign, then y,, is essentially chosen uniformly from the tips of A,,, according to each of the procedures 1 and 2. ISI will denote the cardinality of S.
Kesten: Multiple contact diffusion-limited aggregation
207
PROPOSITION 8. Let An = A(k(i, E, n)) be a generalized plus sign for which
(2.3) holds for some k > 0. For d = 2 assume that An has the form (2.60) or (2.64). Let a > 0 be such that (2.5) holds (respectively (2.63) when An has the form (2.60), and (2.70) when An has the form (2.64)). Without loss of generality take a < 1/2. If (2.72)
Co > 31log(1 -
a)l-1
V 24a-2,
then if yn is chosen according to Procedure 1 or 2 (2.73)
P{yn = w I An} < Kln-3 for all w E OAn which are not a tip of An .
Also, with S = Sn = {(i, E) : k(i, E, n) > 0}, (2.74)
P{yn = t I An} - ICI < K2(logn)-1/2 if t is any tip of An, I
provided d > 4, or An has the form (2.60) when d = 2, or that Procedure 1 is used when d = 3.
In order not to interrupt the proof we first give a lemma about urn schemes which will be needed to prove (2.74) for Procedure 1. LEMMA 9. Let U1, U21 ... , U. be s urns and assume balls are successively
and independently put into these urns. The probability of any given ball being put into Ui is pi, 1 < i < s. Let s
Pi
= s
+ Ei,
Ei = 0.
Define
T(m) = first time one of the urns contains rn balls,
(2.75)
r(c, rn) = P{ the T(m)-th ball is added to U1 } = P{at time T(?n), U1 contains m balls and for 2 < i < s, Ui contains fewer than 771 balls).
Kesten: Multiple contact diffusion-limited aggregation
208
Then there exist constants K1 = K1(s) and Mo = M0(s) such that 7r(E, m) - 1
(2.76)
for m > Mo.
< Ii1 f max I Ei ti
Ifs=2,E1>0=-e2,then asm-+ooandEi10such that el in _,0 7(6, m) -
(2.77)
1
2
ti
2
E1 .
VrIF
On the other hand, if for some fixed 6 > 0 pi < (1 - b)p1,
(2.78)
2 < i < s,
then for 6 < 1/2 one has
ir(E, m) > 1 - (s - 1) exp (-
(2.79)
m62 8
Proof: We begin with (2.76). We set
p=
_in max Ie
I.
I
It then suffices to prove (2.76) only for small p and large m. Indeed it is easy to see that Iir(E, m) -1/sl = O(p) for fixed in, and once we have (2.76) for p < po it follows for all p > 0 by taking K1 > po 1. Let B = B(rn) be the collection of (s - 1) tuples (k2, ... , k3) with 0 < ki < m - 1. Then
(E' m)
(m-1+k2+ +k3)!
= B
(1n - 1)! k2! ... ks!
m k2 k, Al p2 ... ps
(compare with the standard derivation for the negative binomial distribu-
tion when s = 2). By symmetry ir(s, m.) = 1/s when ei = 0 for all i. Therefore, if we set
P(k2'...'ks)
-
(m- 1+k2+...+k,)I !
!
(m - 1 ) . k2.... k3.
1
m+k2+...+ks
s
then
r(e) -
s
=
P(k2,... ,k,){(sp1)m(Sp2)k2 ...(sps)ke B
Kesten: Multiple contact diffusion-limited aggregation Now
1
S
(Spl)m(Sp2)k2
209
... (Sps)" = {fJ(Spj)}m j=1
JJ(spj)k3_m
j=2
and, since spj = 1 + sej, we have for some 02 E [-1,+1] log { (spl )1b (sp2 )k2 ... (Sps )k, } s
s
(kj - m.) (sej + B,+j s m{Sej + Bj s2e } + j=l j=2 s
E(kj
- m)sej + 2©s3p2
j=2
(recall (2.80)
0 and 0 < kj < m). Thus ir(e, rn)
T, P(k2,... ,k,)
s
B S
I exp{
J(kj - m)sej + 20s3p2} -
1
j=2
To estimate the right hand side of (2.80) we write the sum as an expectation. Let Q be the probability measure governing the following process. Balls are put successively into U1,... , U,, each ball being uniformly distributed over the s urns. We continue until r(m), the time when Ul first contains m balls, and define (2.81)
fj = number of balls in Uj at time r(m).
We write I for the indicator function of the event that
{$j < m = $l,
2 < j < s}. Then the right hand side of (2.80) equals
Ji Iexp{ 1: '
(2.82)
9=2
Note that under Q
- ms nL
e2-rn rin
mej +20s3p2}
- l1 dQ.
£, - rn
7"M-)
is the sum of m i.i.d. vectors and hence converges in distribution (as m -> oo) to an (s - 1)-dimensional normal distribution with mean zero
210
Kesten: Multiple contact diffusion-limited aggregation
and some covariance matrix E. Thus we expect that, as in
oo and
p - 0, (2.82) will behave asymptotically as E{J
(2.83)
s
J-2
0, 2 < j < s). has an N(0, E) distribution and J = Once we show that the difference of (2.82) and (2.83) is o(p) as p 1 0, uniformly for m greater than or equal to some M1, (2.75) is immediate. where
To justify the above asymptotic equivalence we introduce the abbreviation Pi = Vm--Ej and the events
E(j,ij) = { IPp (m - fj) I
>-
It then suffices to prove that for some M1 and for all il > 0 we can find p0 (independent of m) such that (2.84)
IiE(i,77)
[1+exp{
Spi E j=2
C 7p
?7L
for pM1. But by Holder's inequality (2.85)
4(0))
spi (m - Pj)I }dQ
eXP{I j.2
< [liJ exp{l Pms(s - 1)(m -
}dQ]1(s
?j)l
j#i
[J
s(s - 1)(7n - fi)I}dQJ
E(i, rl) exp{I
Under Q, (m - £j) has generating function (2.86)
Je9(m_i)dQ =
eB
m
2-e-0
e29
m
2e0-1
(see [2, Ex. XI.2.d]). Taking 0 = ±pjs(s - 1)m-1/2 we see that the product over j # i in the right hand side of (2.85) is for small p bounded by 2s-2 exp(2ps(s - 1))2. The last integral in (2.85) is at most
e-PJexp {
(iis -1) + i) m-
} dQ
< 2e-'P2 +5 for ps(s - 1) < 1 and in > M1
Kesten: Multiple contact diffusion-limited aggregation
211
(by (2.86)). By replacing pi by 0 in the last estimate we see that also 5
1dQ < 2e-a,
Thus (2.84) holds and the difference of (2.81) and (2.82) is o(p) as p 1 0, uniformly in m > Ml .
This proves (2.76). We leave it to the reader to derive (2.77) for s = 2 by dropping the absolute value signs in (2.80).
Lastly, (2.79) is a standard large deviation estimate. Indeed, for any
9>0
1 - 7r(E, m)
3!log(1 - a)1-1
i
We also obtain from (2.88) that n(n)
Zn(y) -
[
t a tip of A.
=
w not a tip of A.
YaA
(w)ln(n)
< n(1 - a)n(n)Zn(7]) < n-2Zn(i) Thus if t1, ... , tlsl are the tips of An, and d > 4, then by (2.6), for some 8i E [-1,+1]
P{yn = tjA0}
7(n)
[Zn(rl)]-1 Is'
_ (1+01n-2)-1
n(n)
-1 µaAn(te)]n(n)
[PaAn(ti)]
j.1
(1 + Bin-2)-1(1 + s
02K1n3-d)n(n)
/
l
{E-1[µaAn(tP)]n(n)I
1 [lUaAn(tC)]+l(n)
7
= I1 (1+0171- 2)-1 exp(283K1n3-dCo log n). Thus (2.74) holds if d > 4. This completes the proof for Procedure 2 when
d>4.
For Procedure 1 and d = 3 we again begin with proving (2.73). Let t be a tip of An and w E 9An not a tip. Then by (2.5)
µaAn(w) < (1 Now, in the notation of Procedure 1, (2.89)
P{yn = wIA0} < P{s(w) reaches the value m(n) - COlogn, before s(t) reaches m(n)j.
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213
In estimating the last probability we can ignore all particles which land at boundary sites other than w or t. In other words, we can calculate the probability as if in each trial we either increase s(t) or s(w) with probabilities I
NaA(t)+P An(w) respectively. Thus, by (2.79) with b = a, s = 2 the right hand side of (2.89) is at most n-3
exp
for Co > 24a-2, and (2.73) follows.
For (2.74) we again denote the tips of An by tl,... , tjsj. We modify Procedure 1 a bit. Instead of taking yn equal to the first z E c9An for which
s(z) = m(n), we ignore all sites which are not tips. We simply take yn equal to the first tip whose score reaches m(n). By what we proved already this change affects yn with a probability at most IOAnIn-3 < 2dn-2. This can be ignored since we allow a much larger error in (2.74). Now in the modified procedure we can again ignore all sites of 0An which
are not tips and assume that at each trial the score s(td) of the tip tj is increased by 1 with probability
['Au]'AiI. s
By (2.4) and (2.87)
< Kl(logk)-1 < 2K1(logn)-1 and (2.76) now shows
Plyn
1
tj I An}
IS
< K2(Co log n)112(log n)-1 I
for large n, and 1 < j < ISI. Thus (2.74) is proven. Once again no change of proof is necessary when An is replaced by An U F
for a fixed finite set F and w E O(An U F), w not a tip of An, in the cases
Kesten: Multiple contact diffusion-limited aggregation
214
d = 3 with Procedure 1 when {(i, -), (i, +)} C S for some i (cf. (1.9)) or d = 2 with A1, of the form (2.60) (and either procedure). We are finally ready for the
Proof of Theorem 1.
We fix Co > 31 log(1 - a)I-1 V 24a-2 and take m(n) - Co logn in Procedure 1, respectively q(n) - Co log n in Procedure 2. Also fix S C {1, ... , d} x {-, +} with at least two elements; when d = 2 take S = {(i, -), (i, +)} for i = 1 or 2. Define the following events, random variables and stopping times: F(n, ij)
{A,, = A,, (k(i, e, n)) is a generalized plus sign
with k(i, e, n) = 0 for (i, e) jk(i, E, n)
S and
- n/jSjI < un for all
(i, E) E S}.
It is necessary to extend the definition of k(i, E, n) to a general A which is not necessarily a generalized plus sign. In general we p ut
k(i, E, n) = max{k : ekei E An } , (i, E) E { 1, ... , d) x {+, -}. Let T. be the o--field generated by A1, ... , A, and n
Zn = Zn(i, E, e) = k(i, e, n)-k(i, E, e)- E{k(i, e, r+1)-k(i, E, r) I r}, r-f
e+ 27(r-t) for all r> e
G = G(e) = fiZr(i,e,e)j < 2
G(e, n) = {l Zr(i, e, e)I
0 for some (i, e) 0 S
Kesten: Multiple contact diffusion-limited aggregation
215
of = inf{r>e:IP{yr=e(k(i,r)+1)erl Fr) -1/ISI1 > 4 for some (i, e) E S} .
It should be noted that for r < r, k(i, e, r+ 1) - k(i, e, r) = 1 or 0, according as yr equals e(k(i, e, r) + 1)e1 or not. Thus, for r < r
E{k(i, e, r + 1) - k(i, e, r) I.Tr} = P{yr
= e(k(i, e, r) + 1)ex I.77r}1
and forn e and 0 < y
V. Finally we estimate
P{ G(t) fails I F(e,17) }.
This is done by means of standard exponential bounds for martingales. Indeed {Zn(i, e, f)},,>L is a martingale. Since k(i, e, n) - k(i, e, n - 1) equals 0 or 1 the increments of Z,, are at most 1 in absolute value, and we conclude from [15, pp. 154, 155]
P{G(f)fails I F(f, ij)}
< E P{JZ,,(i,e,f)l> 2P-1-4(r-f) for some r>f} (i,e)ES
< 4dexp (-\2f) , where A = A(77) > 0 is such that
Thus, for 0 < i < (61SI)-1 and all sufficiently large .f
P{G(f) occurs, 7 = oo and F(r, 3q/4) occurs for all r > 4f 1 F(e, i )}
> 1 - 2K3f-1. It follows that for large £
P{For all q,F(r, > 1 - 2K3
(4)grl)
00
for r> 4qf F(f,ij)} (4qf)-1 > 1
q=0
-8 3
Since clearly P{F(e, rl)} > 0 we have
P{ n {F(r, (3)qf) for r > 4qf} } > 0. 111 q>0
111
This proves (1.5) since
n {F(r, (3)qf) for r > 4qE} C E(S). q>o
1
2
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We leave it to the reader to check that one also obtains (1.8) in the cases (1.9) and (1.10). The necessary minor modifications have already been indicated in comments immediately following the proofs of Lemmas 4, 6 and Proposition 8.
3. Proof of Theorem 2. To simplify notation we take e1 = Ez = + so that S = {(1, +), (2, +)}. We also write k(i, n) instead of k(i, -{-, n) so that we are dealing with An of the form (3.1)
An = { jet : 0 < j < k(1, n)} U
n-)
with
k(l,n) + k(2, n) = n - 1.
(3.2)
The tips of such an An will be written as
ti = ti(n) _ (k(i, n) + 1)ei. Even though it is not needed for Theorem 2, we point out that if a is as in (2.70), and Co satisfies (2.72), then we may assume that for all n, An is of the form (3.1) and yn is one of the tips ti(n), i = 1, 2. In fact, the proof of (2.73) shows that if for some £, A, is of the form (3.1), then (3.3)
P{An is of the form(3.1)foralln>fIAQ} > 1-K3P-1
(compare (2.96)).
The preceding observation is not needed for (1.6), for E(S) fails in any case as soon as (3.1) fails for some n. If there is a first time when yn is not
a tip of An, then E(S) can no longer occur. We may therefore stop the construction of An, m > n, once An is no longer of the form (3.1). If An is of the form (3.1) we write y(i, k(., n)) for the conditional probability that yn = ti(n), given yn E {tl(n), t2(n)}. Thus y(i, k(', n))
P{yn = ti(n) I An} P{yn = t1(n) I An} + P{yn = t.,(n) I An}
Kesten: Multiple contact diffusion-limited aggregation
219
Define an auxiliary Markov on the event {A,, is of the form (3.1) }. chain Y(n) = (Y1(n),Y2(n)), n > 1, with state space Z+ and transition probabilities
(3.5)
P{YY(n + 1) = YY(n) + 1, Y3_i(n + 1) = Y3-i(n) I 1'(n) n)), i = 1, 2. = k(j, n), j = 1, 2} = y(i,
We may view the vector process (ki(n), k2(n)) as a realization of the Y(n) process which is stopped at the (random) time when A7, is of the form (3.1) for the last time (this time may be infinite). In order to prove (1.6) it therefore suffices to prove (3.6)
P{ lim
Yi(n) Y2(n)
= 0 or oo } = 1.
For (3.6) will imply that w.p.1 either A,, is not of the form (3.1) for some n < oo or k(1, n)/k(2, n) converges to 0 or oo. To attack (3.6) we first need to extract some information about the 7(i, n) from Proposition 7 b). Write
7ri = ii (k(., n)) = if A,, has the form (3.1). Then for Procedure 2 ( 3.7 )
y(i, k (., n)) =
[7ra]n(n)
[li]n(n) + [ir2]"(n)
It is easy to see in this case (from (2.68), (2.65) and (2.66) and symmetry
considerations and (3.2)) that for each constant D > 0 there exists an 0 < Eo < 1 and an no = no(D,EO) such th at (3.8)
k(, n)) > 2 + n [k(i, n) - k(3 - i, n)]
on the set (3.9)
k(3 - i, n) < k(i, n) < (1 + EO)k(3 - i, n),
n > no.
Note that this relies only on q(n) - oo, not on the precise behavior of ij(n). In addition, by (2.71), on (3.10)
k(i, n) > (1 + 16) k(3 - i, n),
n > no.
220
Kesten: Multiple contact diffusion-limited aggregation
we have 7r3_a
0
y(i, k(., n)) > 1-n-'
(3.12)
Here we do use the fact that rj(n) - Co log n. We also see that we can make t > 1 by taking Co large. For Procedure 1 we do not have as explicit an expression as (3.7) for the
y(i, k(., n)). Now 7(i, k(., n)) is the probability that the urn UU receives m(n) balls before the urn U3_ti, in the setup of Lenima 9 with s = 2 and Pi
r2 (k(.1, n))
=
7r1(kQ, n)) + 7r2 (k(.7, n))
We again conclude from (2.68), (2.65), (2.66) and symmetry and monotonicity considerations that on (3.9)
Pi >
1
+ Inl (k(i, n) - k(3 - i, n))
for some K1 > 0. We now conclude from (2.77) that (3.8) holds also for Procedure 1 on the set (3.9). Similarly, by virtue of (3.11) p3_j < (1 - a(c0/16))pti
on (3.10).
(2.79) now shows that also for Procedure 1 (3.12) holds under condition (3.10).
From our previous remarks we see that (1.6) is a consequence of the following proposition, which has some independent interest. We are happy to acknowledge some help from R. Durrett with its proof.
PROPOSITION 10. Let {Y(n)l,,>1 be a Markov chain on 72 in which at each time exactly one of the coordinates increases by 1. Let y(i, k(., n)) be the transition probabilities as in (3.5). Assume that for some D > 2 and 0 < so < 1, tc > 0, (3.8) holds on (3.9) and (3.12) holds on (3.10). Then P{lien Y1 (n)
- 0 or oo} = 1.
Kesten: Multiple contact diffusion-limited aggregation
221
If , > 1 in (3.12) then even P{Y1(n) or Y2(n) remains bounded} = 1.
(3.13)
Proof: We break the proof up into several steps. We define
X11 = max Yi(n) - min Y (n) i-1,2
i-1,2
= j Yi(n) - Y2(n) I
to+1 =
,
X12+1-X'.
Then An+1 takes on the values +1 or -1. In step (i) we show that limsup Xn/ f = oo w.p.l. This comes about by more or less random fluctuations. We then show that once X,,/.,fn- is large, (3.8) provides enough upwards drift to give sup Xn/n > e0(2 + ee0)-1 w.p.1 (steps (ii) and (iii)). Finally then (3.12) takes over to give (3.6).
Step (i).
Let Fn denote the o--fields generated by Y(1),...,Y(n).
Then (3.8) shows that for sufficiently large n (3.14)
P{On+1 = 11 fin} > 2 +
on {Xn no and let (3.16)
o- = inf{n > no :
n
n -> e0(2+260)
In this step we prove (3 . 17)
P{o. < oo or
lim
n Xn
= oo }
=1
.
222
Kesten: Multiple contact diffusion-limited aggregation
Let A be a (large) fixed number and e, -y, n1 be strictly positive numbers such that
2D(1 - e) > 1,
(3.18)
(1 - e)(1 + 7)1/2 < 1 - 2 ,
2D(1 - E)(1 + 7)-1/27 ? (1 + 47)(1 + 7)1/2 - 1 + 47.
A < 8n1/2
EA < Assume that also
ni 1'2Xn1 > A but n1 < Q.
(3.19)
Now consider the martingale nno
V. := Xnno --Y., -
E{Oj I
lj-1}, n> n.1.
ni+1
Note that for j < v, D (3.20) E{oj I ,Pj_1} 1> (+x j j J
D
(1 1
j
2D
Xj JJJJ
j
> 0,
by virtue of (3.14). Therefore, on the event (3.19), (3.21)
Xn < (1 - e)An1/2 for some n. E [n1, n1(1 + 7) A o]
implies that for this n also
Vn < Xnno- Xnl < (1-E)AVn- -Afl r
E4 n11 /2 + A 7 E{o, I j-1} 4 ni n,+1
for some n1 n1(1 + y) then (cf. (3.18)) (1+7)ni n1+1
E{o, 1 F 1} > An 1/2{(l + ?')(1 + 7)1/2 - 1 + 4 y 4
Therefore, on (3.19)
P{(3.21) fails, r > n1(1 + y) but 1
X(1+7)ni < (1 +
n1(1 + y) and {Xn < (1 - -)An 112 for some n E [n1, n1(1 + y)] or X12,(1+ti) < (1+ 47)A[(1+y)n1]112) I Pni}
< 2exp(-C1A2),
224
Kesten: Multiple contact diffusion-limited aggregation
with
2
C1
16y
64'
We can now repeat this estimate with A and n1 replaced by (1 + a 7)3A and (1 + y)3 n1, respectively, for j = 1, 2, .... This results in (3.22)
P{o- = oo and Xn < (1 - e)An1/2 for some n > n1
2exp{-Cl(1+y)2jA2}
< j=0
on the event (3.19). Since, for every A, Xn1 > AnVV2 for infinitely many n1 (by step (i)), (3.17) follows. Step (iii). We now improve (3.17) to
P{o- < oo} = 1.
(3.23)
To prove (3.23) we introduce a further submartingale. Let Wn
log{
n
1 or
Xnno }
By definition of a, and the inequality Xk+l -Xk < 1, Wn is bounded above. Again assume that (3.19) occurs, and let
r = r(A) = inf In > n1 : Xn < (1 - e)An112}, with e as in step (ii). We saw in (3.22) that (3.24)
P { c = oo and r < oo l .F',b
1
} , 0 as A -> oo.
On the other hand, on n < v A r we have E{Wn+1 I fin} = E{log(X,, + On+1) I Tn} - log(n + 1)
= log{iXn}+E{log(1+,Yn1An+i)Ijn}-log('+n)
> Wn +
>W
+ n +l
Xn1E{on+l
2n (1 - E 46
A2
1J7n} )-2
-
1
'n} - n
Kesten: Multiple contact diffusion-limited aggregation
225
(by (3.20) and A2n+1 = 1). If A and n1 are chosen large enough, then with 6 = 2(2D - 1) > 0 (recall D > 1 by assumption) 6 2D 1 > (1
n+1
2n
- 2 E )-2A-2 -
n+l
n
for n > n1. Therefore nnanr-1
WnAT -
6
E
j+1
0
is a submartingale, which is bounded above, and has a finite limit with probability 1. However, on o, A r = oo, the limit is (by (3.16)) at most log
n+1 = -oo.
E0(2+2E))-1-
0
Thus we must have a A r finite with probability 1. In view of (3.24) this proves (3.23).
Step (iv). Finally, we prove in this step that (3.6) holds, as well as (3.13) when tc > 1 in (3.12). Choose no so large that steps (ii) and (iii) apply and let a be as in (3.16). Assume that o- < oo, and for the sake of argument, let
Y1(o) > Y2(o-) + Eo(2 + 2e0)-1o.
Without loss of generality assume that no/2 > 8 and no ? (E0/16)(1 + IY1(1)I + IY2(1)I). Define -0--"/2)}.
v = inf{n > o, : Yi(n) aandforo- 1 - 2n-", Q2{On+1 I .fin} < 4n-' (by virtue of (3.12) and Y1(n) + Y2(n) = n - 1 + Y1(1) + Y,(1). Thus, if o, < v < oo, then v-1
[1 i(v)-Y2(v)] - [Y1(o) - Y2(o)] - > E{On+1 I Tn} 0
< -1 E0(2 + 2eo)-lo - (v - o,)(1 - 2o--" - 1 +0--,/2)
< -2
Eo(2+2E0)-lno
-
0-1
0,K/2
01
a
2{.,n+1 P7n}.
226
Kesten: Multiple contact diffusion-limited aggregation
Once more by the exponential bounds of [15, pp. 154, 155] we obtain on the event In, < oo}
P{v < oo I F. } < exp{-
l (2+2,-0)-'n0).
Since this estimate holds for every large n.o, there exists with probability 1 for every n1 < oo an n2 < oo and an i E {1, 2} such that
Yi(n) > Y3-i(n) + (n - n2)(1 - n2 "12) for all n > n2. Since Y1(n) + Y 2(n) = n + 0(1), this implies lim inf
(n) > Y3-i(n) i
2- -," f2 n2
K
/2
> 2nr/2 - 1.
(3.6) follows because nl can be taken as large as desired. If K > 1 in (3.12), then on {Yi(o) > r0(2 + 2E0)-113_i(o)} we have
P{Y;(n + 1) = Yi(n) + 1, Y3_i(n + 1) = Y3-i(n) for all n> o IPo }
> fJ (1 - n-") > exp{-Keno-"}. n>no
(3.13) follows immediately.
To conclude we note that (1.11) is proven by (3.3) and (3.13).
REFERENCES 1. J.-P. Eckman, P. Meakin, I. Procaccia, R. Zeitak, Growth and form of noise-reduced diffusion-limited aggregation, Phys. Rev. A 39 (1989), 3185-3195.
2. W. Feller, "An introduction to probability theory and its applications," Vol. I, 3rd ed., John Wiley and Sons, 1968. 3. W. Feller, "An introduction to probability theory and its applications," Vol. II, 2nd ed., John Wiley and Sons, 1971. 4. Y. Hayakawa, H. Kondo, M. Mathshushita, Monte Carlo simulations of generalized diffusion limited aggregation, J. Phys. Soc. Japan 55 (1986), 2479-2482.
Kesten: Multiple contact diffusion-limited aggregation
227
5. J. Kert6sz, J., T. Vicsek, Diffusion-limited aggregation and regular patterns: fluctuations versus anisotropy, J. Phys. A 19 (1986), L257-L262.
6. H. Kesten, How long are the arms in DLA?, J. Phys. A 20 (1987), L29-L33.
7. H. Kesten, Hitting probabilities of random walks on V, Stoch. Proc. and their Appl. 25 (1987), 165-184.
8. H. Kesten, Relations between solutions to a discrete and continuous Dirichlet problem, in "Festschrift in honor of Frank Spitzer," (R. Durrett, H. Kesten, eds. Birkhauser-Boston, 1991.
9. H. Kesten, Relations between solutions to a discrete and continuous Dirichlet problem II, preprint. 10. G. Lawler, "Intersections of random walks," Birkhauser-Boston, 1991. 11. M. Matsushita, K. Honda, H. Toyoki, Y. Hayakawa, H. Kondo, Generalization and the fractal dimensionality of diffusion-limited aggregation, J. Phys. Soc. Japan 55 (1986),2618-2626.
12. P. Meakin, Noise-reduced diffusion-limited aggregation, Phys. Rev. A 36 (1987), 332-339.
13. P. Meakin, The growth of fractal aggregates and their fractal measures, in "Phase transitions," (C. Domb, J. L. Lebowitz, eds.), Academic Press, 1988, pp. 335-489. 14. P. Meakin, S. Tolman, Fractals' physical origin and properties, (L. Pietronero, ed.), Plenum.
15. J. Neveu, Martingales a temps discrete, Masson and Cie.
16. L. Niemeyer, L. Pietronero, H. J. Wiesmann, Fractal dimension of dielectric breakdown, Phys. Rev. Lett. 52 (1984), 1033-1036.
17. J. Nittman, H. E. Stanley, Tip splitting without interfacial tension and dendritic growth patterns arising from molecular anisotropy, Nature 321 (1986), 663-668. 18. W. Rudin, "Real and complex analysis," 3rd ed., McGraw Hill Book Co., 1987.
19. F. Spitzer, "Principles of random walk," D. van Nostrand Co, 1984. 20. C. Tang, Diffusion-limited aggregation and the Saffman-Taylor problem, Phys. Rev. A 31 (1985), 1977-1979.
21. T. Vicsek, "Fractal growth phenomena," World Scientific, 1989. 22. T. A. Witten, L. M. Sander, Diffusion- limited aggregation, a kinetic phenomenon, Phys. Rev. Lett 47 (1981), 1400-1403.
Limits on Random Measures and Stochastic Difference Equations Related to Mixing Array of Random Variables
H. Kunita Department of Applied Science, Faculty of Engineering, Kyushu University, Fukuoka 812 Japan 0.
Introduction
Let {t;jn, j E N}, n = 1, 2, ... be a sequence of stationary stochastic processes
with values in R' converging in law to a stationary
j E N}.
Denote the laws of i and e1 by 7rn and -7r respectively. Suppose that there exists an increasing sequence of positive numbers {cn} tending to infinity such that the sequence of measures {µn} defined by µn (F) = n,7rn(cnF) converges vaguely to a measure p on R'" - {0}. Define two families of random measures by 1
[nt]
B' (t, E) = - >(XE(en) - 7rn(E)), j=1
NT (t, F) = >1 XF(cn ),
(0.2)
j=1
where E, F are Borel sets in R' such that F C R' - {0} and XE, XF are indicator functions of the sets E, F. Then for each n, ((Bn(t, .), Nn(t, .)), t E [0, oo)) may be regarded as a stochastic process with values in the space of (signed) measures on R"n x (R' - {0}), cadlag (right continuous with left hand limits) with respect to time t. In the previous paper [7], the author has shown the weak convergence of the sequence {(Bn(t, .), Nn(t, .))}n in the space D with respect to Skorohod's Ji-topology, assuming that j E N are independent random variables for any n. He proved that (1) the limit B(t, E) is characterized as a Brownian random measure, i.e., it is a Gaussian random measure for each fixed t and is a Brownian motion for each fixed E, (2) the limit N(t, F) is a Poisson random measure with the intensity tp(F), and (3) these two are independent processes. The first object of this paper is to extend the above limit theorem to the case where , j E N are not necessarily independent but have some mixing propery. Let {ok, k E N} be the uniform mixing rate of the stationary process
{; j E N}. (For the definition, see (1.2) in Section 1). We show that if it satisfies Ek 1(supn ok)1t2 < oo , then the sequence {(B"(t), N' (t))}n converges in law in the space D and the limit (B(t), N(t)) has properties (1),(2),(3) mentioned above. See Theorem 1.1.
230
Iiunita: Random measures and stochastic difference equations
Another object will be to discuss the weak convergence of the solutions of stochastic difference equations represented by
+fn(i-1) + ngn(0ij = 1,2,...
(0.3)
where fn(x, A) and gn(x) ; x E Rd, A E R' are continuous functions with values in Rd converging to f (x, A) and g(x) respectively as n -* oo. Let be the solution of (0.3) with the initial condition ')o = xo and set cpt = a[nt] Then for any n the process Ot satisfies
bjn
t
Ot
= XO+0 f fAJCnM
+
f
A)Nn(ds, dA)
`["]/n
+
(0.4)
{gn(tPnr_) +
0
where M > 0 and fn(x,A)
=
,fn
bM(x)
1 fn(x,CnA),
cnM
f
n(x, A)7fn(dA).
(0.5)
(0.6)
Assume that the sequences { f"(x, A)}n and {b y(x)}n converge to f (x, A) and bM(x), respectively and limM-.o bM(x) = bo(x) exists. A question is that whether the limit (pt exists or not and if it exists, it satisfies t xo + Jo
+ +
ftf 0
Rn' f (Ps
Rn`-{OJ
a)B(ds, d)
f (cp,-, A)N(ds, dA)
ft{g
+ bo}(p,_)ds.
(0.7)
0
We will prove that the sequence {cpt In converges in law. However equation (0.7) is false in many cases. In general the limit process (pt can not be represented by the above two random measures B(i, E) and N(t, F). We need another Brownian random measure Ko(t, G), which is concentrated at the
boundary of R'n (homeomorphic to m - 1 dimensional sphere Sm-i). We show in Theorem 2.1 that the stochastic differential equation governing the limit process ot is given by
t
Wt = x0+ Jo
JRm f (to,
A)B(ds, dA)
Kunita: Random measures and stochastic difference equations
+
+
h(cp,_, oo, 0)Ko(ds, d0)
o
sm-
0
o
231
f
A)(N(ds, dA) - dsp(dA))
f( o_, A)N(ds, dA)
+ j{g + b1 + c}(,_)ds.
(0.8)
f (x, r, 0)/p(r), where Here the function h is defined by h(x, oo, 0) (r, 0) is the polar coordinate of A and p(r) is a positive nondecreasing function such that limp(cnr)/V/n- exists. Further, c(x) is a certain correction term. Limit theorems for stochastic difference equation similar to this paper have been studied by Fujiwara [1], [2] and Kunita [8]. In the papers [1], [2] he appear linearly in equation discusses the case where the driving processes (0.3), i.e., the coefficients f"(x, A) are of the forms f"(x, A) = f"(x)A. On the other hand, in the paper [8], the case of nonlinear coefficients f"(x, A) are studied but the limit process is restricted to a diffusion process. 1.
Convergence of random measures
j E N}, n = 1, 2, and Let N be the set of all positive integers. Let {e j E N} be stationary stochastic processes with values in the common P). We set space R' defined on the probability space 7rn(E) = P(C E E), ir(E) = (1.1) E E). These measures do not depend on j since they are stationary. We will introduce three conditions to {q}. Any finite dimensional distributions of {}, n = 1, 2, Condition converge weakly to the corresponding finite dimensional distribution of Then the sequence {?fn} converges weakly to ir. The second condition is concerned with the mixing property. Let h, k(h
0 be a strictly positive continuous nondecreasing function such that lim n-oo
P Cn r n,
= j(r)
(1.9)
exists and is a continuous function of r. Then lim,..o i7(r) = 0 holds. Associated with the function p(r) and the sequence {cn}, we will define another sequence of random measures on IV by
K: (t, G) =
(1.10)
JG P(IAI)X[o,cne](I AI)Bn(t, dA),
and discuss the weak convergence of the sequence {(B' , Nn, Kr)}n. For this purpose, we assume two more conditions. For a positive constant M, we define the truncated random variable of by j'M = gX[o,cnM](Iq I )
Condition (e.4) For any j and M, the sequence {E[pae;MI)2I.r_i]} is uniformly integrable, i.e.,
n
11
E[p(I '
lim sup n
C]
= 0.
(1.11)
Given e > 0 and n > 1, define a measure on R'n by v`
(G) =
J
(1.12)
Kunita: Random measures and stochastic difference equations
234
Then sup. vn (Rm) < oo holds by Condition (e.4). Condition (e.5) For anye > 0, the sequence of measures {vn }n converges weakly to a measure v, on Rm.
The family of the measures {v,}, decreases as a decreases. We set
v = limy,. r-o
(1.13)
Now we will define the law of (Bn, Nn, KE ). Let G(Rm) = {Gk} be a ring of Borel sets of Am such that it generates the Borel field of Rm. We assume that any element G of G(Rm) satisfies of (G) -+ v,(G) for any e > 0. Let M(Arm) be the set of all additive set functions on G(R.m). Then we may define the law of (Bn, Nn, KK) on the space D = D([0, oo); M(Rm) x M(Rm - {0}) x M(R.m)). We denote it by Pr. If the sequence {P, }n converges weakly as n -> oo, the sequence {(Bn, Nn, Kn)}n is said to converge in law.
Theorem 1.2. Assume Conditions (e.1) - (e.5) for {qjn, j E N}, n = . Let e be a positive number such that {JAI < e} is a p-continuity set. Then the sequence {(B' , Nn, KK)}n converges in law. Let (B, N, K P,-) be its limit law. Then the followings hold. (1) (B, N) has the same property as in Theorem 1.1. (2) For any G1i ..., G (K, (t) Gl),..., K, (t, G,)) is a Levy process. Further, set 1, 2,
Ko(t, G n ORt) - K,(t, G) - JGn(Rm-{o})
AI)B(t,
JGnR"' p(I
d\)
?i(IAI)X(o,eJ(IAI){N(t, d)) - tp(dA)}.
(1.14)
Then (Ko(t, G1 nORm), ..., Ko(t, G, n oRm)) is a Brownian motion with mean 0 and covariance
E[Ko(t, G, n OR-)Ko(t, G; n OR')] = tv(G, n GG n 8R').
(1.15)
For any t, Ko(t,.) can be extended continuously to a Gaussian random mea-
sure on OR. (3) B, N, Ko are mutually independent. The proof of these theorems will be given at Section 4. Remark. Instead of Condition (e.4), we introduce the following.
Condition (e.4)' For any j and N, the sequence
is uni-
formly integrable.
Clearly Condition (e.4)' implies (e.4). Further, under and the sequence of measures {vn}n converges weakly. The limit measure v, is
Kunita: Random measures and stochastic difference equations
235
supported by Rm and is represented byf v(G) = ve(G) =
p(IAI)2ir(da).
JG
Therfore the Brownian random measure Ko of Theorem 1.2 is identically 0. As an example, consider the case where does not depend on n, which we denote by {Z;1}. If E[p(eM)2] < oo holds for any M > 0, Condition (e.4)' is satisfied. Hence the Brownian random measure KO is 0. S'n-1 Remark. Let Rm be another m-dimensional Euclidean space and let Sm-1 be the unit sphere in Rm, i.e., = {x E Rm; Ixi = 1}. Let I'1 = {x E Rm;
Ixi < 1} and r2 = {x E Rm; Ixi > 1}. Then Rm is splitted into two S-1
domains F, and 12 and is the splitting boundary of the inner domain F1 and the outer domain F2. Further F1 is homeomorphic to R'n and r2 is homeomorphic to R'n - {0}. Therefore we may regard that the Brownian random measure B(t, E) of Theorem 1.1 is defined on the inner domain I i and the Poisson random measure N(t, F) is defined on the outer domain r2Further the Brownian random measure Ko(t, G) is defined on the splitting Sm-1 boundary 2.
Convergence of solutions of stochastic difference equations.
Let us return to the stochastic difference equations (0.3) or (0.4). We will introduce assumptions to coefficients { f n(x, A)} and {gn(x)} of the equations. Let Ck= Ck(Rd; Rd) be the space of k-times continuously differentiable maps from Rd into itself. For f E Ck we define seminorms II 11k ,N, k = 1, 2 by
IIfiIk,N= sup I=I
= f t (Jsm-t
8)v(i7,-, 9)v(d9)) ds.
(2.10)
238
Kunita: Random measures and stochastic difference equations
Theorem 2.1. Assume Conditions (e.1) - (e.5), (f.1),(f.2) and (g.1) for equations (0.4). Then the sequence {(B", N", Kn, con) In converges in law. Let (B, N, K co, Poo) be its limit law. (1) (B, N, KK, Poo) has the same property as in Theorem 1.2. (2) cPt is represented by t
If fR. f A) B(ds, dA) tf h(co,_, oo, 8)Ko(ds, dB) +f
c2t = xo +
s
0
t
+
o 0. Assume further h(x, oo, 0) = lim
f((x, Ian, 0)
Ian-- (1 + IAI)a
(2.15)
exists and is a continuous function on S. (1) If ,(3 < 1/2a, the sequence {(Bn(t), K(t), cpt)}n converges in law. The limit (B(t), K,(t), cpt) satisfies
i
Tt
=
x0
+
+
JO
rt
J0
R. f (c03, A) B(ds, dA)
f
Sm_,
h(tp oo, 0)Ko(ds, d0)
+ f t{g + b1 + c}( M})
(3.5)
O<s 0 and N > 0 there exists a sequence of non-decreasing (.Fnt]) adapted cadlag processes {Dt } such that
IIE[fn M(SI'M)I]II2,NIIfk M(Sk'M)II1,N [ [ n k=[nsl+11=k+1 1
[nt]
n k=[ns]+1
M)II1,N)2 < Dt - D
(Ill
for any 0 < s < t and that its compensators {Dn,p} are C-tight. (2) {Ilfn'MII1,N}n is bounded for every M > 0 and N > 0.
Condition (A.II) (1) For any bounded continuous function h, N > 0, 0 < 6 < M and 0 < s < t, n
[nnnt`]
f
limEI'sup n-*oo l
-(t - s)
(f
-[ns1+l
L-10) h (f (X, A)) X(a,Ml (IIf II1,N) µ(dA)Il = 0.
(2) For any N> 0and0 <s
f (cod
dx)) ds
(Iw' f (ws
ft
= +2
(fRm
f (GPs-, A)27r(dA)) ds
A') V(dA, dy')) ds.
Jot (f JRmxRm f(,ps-,
(4.30)
Further by (4.23)
< Mr-, Jt fpm h(cp3_, ))K,(ds, d)) > o
=
It I
h(,p,-,
A)d, < M,, K,(s, d\) >
o
ft
=
A)os(cP,-, dA)) ds
(f
=
ds
Jot (fRm f (ca3_,
A')V(d\, dy')) ds
+ Jot (f fRmxRm f (cps-, t
+ f0
(fSm-' h(cp,_,
oo,
0)2v(d0)) ds.
(4.31)
Since Ko(ds,dA) is defined by (1.14), we obtain from (4.30) and (4.31),
< Mt , f0 ISm-t h(cp,-, oo, 0) Ko(ds, dB) > t
= f 0(fsm.i h(
,
)
9)2(d9)) ds.
(4.32)
Kunita: Random measures and stochastic difference equations
253
Also we have by (4.27) t
< Mt , f f
f (tp,_, A)1V (ds, dA) >
0
Jo
t
ff 0
t
$m-1
h(tp,_, oo, 0) Ko(ds, d9) >
+ < Mt, f JR"`-{ol f ((P°-' A)N(ds, dA) >=< Mt > .
(4.34)
Therefore we get < Mt - Mt >=< Mt > -2 < Mt, Mt > + < Mt >= 0, proving Mt = M. Then (4.18) and (4.29) imply
/
ft
x+
c°t
JRm f (cp,_, a)B(ds, d A) t
+ f fm-I h(cp,_, oo, B)K0(ds, dA) o
s
t
+ / (bM + c +
f
It t
f(w3-, A)N(ds, dA)
O M}).
(4.35)
s1 satisfies UT, then there exists a filtration (Ft) such that X is an (,Ft)-semimartingale and f H,"- dX, converges in distribution in the Skorohod topology to f H,_ dX,. That is, the sequence (X" )n>1 is good.
The condition UT is sometimes difficult to verify in practice. An alternative condition is given in Kurtz and Protter [2]. To subtract off the large jumps in a Skorohod continuous manner, define h6: [0, oo) -+ [0, oo) by h6(r) 6/r)+, and J6: DR- [0, 00) -+ DR- [0, oo) by
J6(x)(t) = E h6(IOx,I)Axs, O<s 0, there exist stopping times T"'a such that oo. a) < « and supE{[M"'6,M"'6]tATn.- + JonT^
P(T",a
1 is a good sequence but that UT does not hold. Then there exists a sequence (H" )n>1, H" EH,, and a sequence c" tending to co such that for some e > 0, lim inf P" { n-*00
J
H"_ dX; > C" } > e.
But this implies (1)
lira inf P" {J n-.oo
1H,"_ dX; >- 1} > e Cn
as well. Since IHn I < 1, we have that n H" converges in distribution
(uniformly) to the zero process. Since X" is good by hypothesis, then f , H; dX, converges in distribution to f OdX, = 0. This contradicts (1), and we have the result. We can employ the argument in the previous proof to show that the property of goodness is inherited through stochastic integration. Theorem 4. Let (X" )n>1 be a sequence of R'"-valued semimartingales, X" defined on "En, with (X")n>1 being good. If H" defined on EEn are cadlag, adapted, Mk,-valued processes, and (H",X") converges in distribution in the Skorohod topology on DMkm x Rm [0, 00), a good sequence of semimartingales.
then Yt" = fo H; dX; is also
Proof: Let k = m = 1. By Theorem 1, it is sufficient to show that (Yn) satisfies UT. Suppose not. Then, as in the proof of Theorem 3, there exists
258
Kurtz & Protter: Weak convergence of stochastic integrals
a sequence (Hn) with Hn E 9-l n, a sequence cn tending to oo, and e > 0 such that
liminf Pn{J H; dYn n-oo
Cn} > E.
and equivalently
liminfPn{ n-.oo
rft _H,n_ dX; >- cn} > e.
which implies
- -
liminf Pn{J 1CnH,n_H,n_ dXa > 1} > e
(2)
n-oo
But, as before, the goodness of (Xn) implies that the stochastic integral in (2) converges to zero contradicting (2) and verifying UT for (Yn). 0 As an application of the preceding, we consider stochastic differential
equations. Suppose that for each n, Un is adapted, ckdlag and Xn is a semimartingale on En, (Un, X n) = (U, X), and X n is a good sequence. Let Fn, F be, for example, Lipschitz continuous such that Fn --+ F uniformly on compacts, and let Zn, Z be the unique solutions of t
Zt = UU +
= U+
J0
j
Fn(Z,"_)dX,n
F(Z-)dX,.
Then combining Theorem 4 with Theorem 5.4 of [2] (or similar results in [3] or [5]) yields that Zn is also a good sequence converging of course to Z. The preceding holds as well for much more general Fn, F; see [2]. In particular
by taking Fn(x) = F(x) = x, we have that Xn a good sequence implies that Zn = £(Xn) is a good sequence, where E(Y) denotes the stochastic exponential of a semimartingale Y.
REFERENCES
1. Jakubowski, A., Memin, J., and Pages, G.: Convergen suites d'integrales stochastiques sur 1'espace D' de Skorokhod. Probab. Th. Rel. Fields 81, 111-137 (1989).
Kurtz & Protter: Weak convergence of stochastic integrals
259
2. Kurtz, T.G. and Protter, P.: Weak limit theorems for stochastic integrals and stochastic differential equations. To appear in Ann. Probability.
3. Memin, J. and Slominski, L.: Condition UT et stabilite en loi des solutions d'equations differentielles stochastiques. Preprint (1990).
4. Protter, P.: Stochastic Integration and Differential Equations: A New Approach. Springer-Verlag, New York (1990).
5. Slominski, L.: Stability of strong solutions of stochastic differential equations. Stoch. Processes Appl. 31, 173-202 (1989).
STOCHASTIC DIFFERENTIAL EQUATIONS INVOLVING POSITIVE NOISE
Tom Lindstrom, Bernt Oksendal and Jan Uboe Dept. of Mathematics, University of Oslo, Box 1053, Blindern, N-0316 Oslo 3, Norway
Contents
§1. Introduction and motivation §2. The white noise probability space §3. Generalized white noise functionals §4. Functional processes §5. The Hermite transform H §6. A functional calculus on distribution valued processes §7. Positive noise §8. The solution of stochastic differential equations involving functionals of white noise
§1. Introduction and motivation An ordinary stochastic differential equation may be viewed as a differential equation where some of the coefficients are subject to white noise perturbations. Perhaps the most celebrated example is the equation (1.1)
Xtt
= (r + a W1)X1;
Xo = x,
which is the mathematical model for, say, a population Xt whose relative growth rate has the form r + aWt, where r, a are constants and Wt denotes white noise, which represents random fluctuations due to changes in the environment of the population.
Quite often, however, the nature of the noise is not "white" but biased in some sense. For example, if Xt represents the size/weight of a tree in a
262
Lindstrom et al: SDE involving positive noise
random environment then a suitable stochastic differential equation for Xt would be dXt
dt = (r + a Nt)Xt;
Xo = x
where Nt is a stochastic process representing "positive noise" in some sense. As a second example, consider incompressible fluid flow in a porous medium.
Combining Darcy's law and the continuity equation we end up with the partial differential equation (in x for each t) div(k(x) V p(x,t)) = -f(x,t)
for x E Dt
p(x, t) = 0
for x E ODt
k(x) > 0 is the permeability of the medium at the point x E R3, f (x, t) is the given source/sink rate of the fluid at the point x and at time t and p(x, t) is the (unknown) pressure of the fluid, while Dt denotes the set of points x where the fluid has obtained the maximal degree of saturation at time t. (See e.g. Oksendal (1990) for details). Because the permeability k(x) is hard to measure and varies rapidly from point to point we find it natural to propose a stochastic mathematical model where k(x) is regarded as a 3-parameter positive noise process.
There are of course several ways to interpret "positive noise process". For example, many papers have been written about (1.3) with the assumption that k(x) is some ordinary stochastic process parametrized by x, i.e. k(x) _ k(x, w); w E St, assuming only nonnegative values. In particular, in Dikow and Hornung (1987), the following situation is studied: (1.4)
k(x) = ko(x) expC(x, w),
where ko(.), w) are (uniformly) Holder a-continuous for each w, C is a bounded process and ko is bounded and bounded away from 0. Actually, under these assumptions the operator in (1.3) becomes uniformly elliptic and one can approach the corresponding boundary value problem pathwise (i.e. for each w) by known deterministic methods.
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In Oksendal (1990), it is shown how to solve (1.3) for more general types of k: It suffices that k(x) is an A2-weight in the sense of Muckenhoupt, i.e.
that (1.5)
suup(IBI
J
k(x)dx)(-
JBI
J k( x )dx) < 00
the sup being taken over all balls B C R3, dx denoting Lebesgue measure and JBI = f dx being the volume of B. In particular, this allows k to have zeroes.
B
Here we propose a stochastic approach which is radically different from the methods mentioned above:
We represent the stochastic quantities involved by a suitable functional of multi-parameter white noise Wi; x E R". For example, in equation (1.3) we put k(x) equal to some positive functional white noise (see §7). Then we look for a solution of the corresponding stochastic partial differential equation in some generalized/distribution sense. Finally explicit information about the solution can then be obtained by taking averages of the distribution solution. The philosophy behind this approach is in a sense similar to the philosophy behind ordinary stochastic differential equations involving white noise: It
is better to calculate/solve the equation first and then take averages rather than take the average first and then calculate. In this report we explain the first step in such a program: Here we will consider only 1-parameter equations, the multi-parameter case will be discussed in future papers.
To summarize, the purpose of this paper is to construct a (1-parameter) noise concept which is general enough to include good models for positive noise from real life situations, yet at the same time allows us to use calculus to solve differential equations involving this noise.
The key concept in our approach is the functional process, a distribution valued process which we construct in §4, after giving some background in §2-3. In §5 we introduce the Hermite transform H, which associates to each functional process a (deterministic) complex valued function on
264
Co = {(zl, z2,
Lindstrom et al: SDE involving positive noise
); zi E C, 3N with zi = 0 for j > N}. This transform is
fundamental for our subsequent calculus on functional processes.
Even though a functional process is distribution valued we show in §6 that one may define a functional calculus on such processes, in the sense that one can compose them with analytic functions which do not grow too fast at oo. In §7 we introduce the concept of a positive functional process and we prove that the (Wick) product of two positive processes is again a positive process.
This justifies the use of this concept to model non-negative growth in a random environment. Finally, in §8 we illustrate our method by solving the 1-dimensional version of (1.3). We also give an estimate for the (mean square) error that we make if we replace the noisy equation (1.3) by the equation where k is replaced by its (constant) mean value.
Our main inspiration has come from the works of Ito (1951), Hida (1980) and Kuo (1983), but it has gradually become clear to us that also many other authors have discussed more or less related subjects, for example in connection with mathematical physics (renormalization etc.). We have included those that we know about in the reference list, and apologize to those that should have been mentioned that we are not yet aware of. After this paper was written we have learned that ideas similar to ours based on the S-transform (which is related to our Hermite transform, see §5) and the Wick product have already been adopted by Kuo and Potthoff (1990).
§2. The white noise probability space Since white noise is so fundamental for our construction, we recall some basic facts about this generalized (i.e. distribution valued) process: let S(R") be the Schwartz space of all rapidly decreasing smooth (C°°) functions on R". Then S(R") is a Frechet space under the F o r n = 1, 2,
Lindstrom et al: SDE involving positive noise
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family of seminorms
ilflIN,a = sup (1 + IXIN)Iac.f(X)I, zER"
where N > 0 is an integer and a = (al,
, ak)
is a multi-index of nonnegative integers aj. The space of tempered distributions is the dual S'(R'S) of S(R"), equipped with the weak star topology. Now let n = 1 for the rest of this section and put S = S(R), S' = S'(R). By the Bochner-Minlos theorem (see e.g. Gelfand and Vilenkin (1964)) there
exists a probability measure µ on (S', B) (where B = 8(8') denotes the Borel subsets of S') such that r et<w,O>d1z(w)
(2.1)
= e-J110112 for all ¢ E S,
So
where II0II2 = II¢IIL2(a) and < w, ¢ >= w(¢) for w E S'. It follows from (2.1) that (2.2) J
f(< w, 0 >)d/A(w) = (27
S
110112)
' Jf(t)eiiii dt; q E S, P.
for all f such that the integral on the right converges (It suffices to prove (2.2) for f E Ca (R), i.e. f smooth with compact support. Such a function f is the inverse Fourier transform of its Fourier transform f and we obtain (2.2) by (2.1) and the Fubini theorem). In particular, if we choose f (t) = t2 we get from (2.2) (2.3)
Eµ[< w, ¢ >2] =110112; 0 E S.
This allows us to extend the definition of < w, ¢ > from 0 E S to 0 E L2(R) for a.a. w E S', as follows: (2.4)
< w, 0 >:= k rn < w, Ik > for 0 E L2(R),
where .bk is any sequence in S such that Ok -+ 0 in L2(R) and the limit in (2.4) is in L2(S', µ).
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In particular, if we define (2.5)
w, X[0, t] >
then we see that (Bt, S', µ) becomes a Gaussian process with mean 0 and covariance E1[Bt(w)B3(w)] =
f
< w, X[o t] > < w, X[o ,] > dp(w)
St
= JX[o,t](x) X[o,s] (x)dx = min(s, t), using (2.3). R
Therefore Bt is essentially a Brownian motion, in the sense that there exists a t-continuous version Bt of Bt: p({w; Bt(w) = Bt(w)}) = 1
for all t.
If u E L2(R) we define, using (2.4) 00
f 6(t)dBt(w) =< w, ¢ >
(2.5)
00
which coincides with the classical Ito integral if suppq C [0, oo). If we define the white noise process W# by
W.(w) =< w, ¢ >
(2.6)
for ¢ E S,w E S'
then WW may be regarded as the distributional derivative of Bt, in the sense
that, if 0 E S
< d Bt(w), 0 >= - f l'(t)Bt(w)dt = f O(t)dBt(w) 00
= o mo
4(ti)(Bt;+ - Be,) = otimo
lim < w,
At;-0
_00
4(ti) < w, X(t;,t;+l] >
0(tj)X(t;,t;+l] >=< w, 0 >= Wo(w), .i
where the second identity is based on integration by parts for Ito integrals.
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§3. Generalized white noise functionals In this section we summarize Hida's concept of generalized white noise functionals. They will be a natural starting point for our definition of functional
processes in §4. For more details see Hida (1980) or Hida-Kuo-PotthoffStreit (1991). By the Wiener-Ito chaos theorem (Ito (1951)), we can write any function f E L2(µ) (= L2(S', µ)) on the form 00
f = > 1 fndB®n,
(3.1)
n=0
where
In E L2(Rn,dx),
(3.2)
i.e. fn EL 2 (Rn,dx) and fn is symmetric (in the sense that fn(x,1)x,,, xon) = f (xl, , xn) for all permutations a of (1, 2, , n)) and fn(u)dB®n
J .fndB®n = J R:n
(3.3)
00
= nt
Un
ns
rig
r ( r ... r ( f fn\ul, ... un)dBnl)dBu7 ... dBun-1)dBun -00 -00
-00 -00
for n > 1, while the n = 0 term in (3.1) is just a constant fo. For a general (non-symmetric) f E L2(Rn) we define (3.4)
J
fdB®n :_ I fdB®n
where f is the symmetrization of f, defined by (3.5)
f(ul,...,Un) =
'
E f(u,,,...,21on),
the sum being taken over all permutations a of (1, 2,
With f, fn as in (3.1) we have 00
(3.6)
11f112
n!IIfnhIL2(Rn)
L2(µ) n=0
,
n).
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Note that (3.6) follows from (3.1) and (3.3) by the Ito isometry, since
E[(I fndB®n)( r fndB®m)] = 0 fore Rn
m
Rm
and
E[(J .fndB®n)2] = (n!)2E[( f ... (I fn(ul, ... 00
us
-00
-00
r
Rn 00
_ (n!)2 .
, un)dB1) ... dBu,. )2]
as
r ... (J f!(Ul, ... un)dul) ... dun = nI -00
J Rn
-00
fndx
Here B. (w); u > 0, w E S' is the 1-dimensional Brownian motion associated with the white noise probability space (S', p) as explained in §2.
For s E R we define the Sobolev space H8 = H°(Rn) by (3.7)
H8 = {0 E S'(R");
II0IIH.(Rn)
IyJ2)°dy < oo},
J
Rn
where
denotes the Fourier transform of 0. Then the dual of H° is simply
H-°, for allsER. The Hida test function space (L2)+ = (L2)+(µ) is the subspace of L2(µ) consisting of all functions f E L2(µ) of the form
f=
J fndB®n n=0
where fn E H
`- (R') and 00
n!IIfnIIH
IIfII(L2)+
Rn
n=0
H2 (Rn) being the space of symmetric functions in the Sobolev space H 2 (Rn). Note that by the Sobolev imbedding theorem each fn has a continuous version, so we may - and will - assume from now on that each fn is continuous.
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The Hida space (L2)- of generalized white noise functionals (see Hida (1980), Kuo (1983)) is the dual of the space (L2)+. Formally we may represent an element F of (L2)- as follows
F=
(3.10)
n=0
f FndB®",
where
Fn E H-'(R.") for all n,
(3.11)
and 00
(3.12)
IIFII(L2)- = E n!IIFnhI n=O
the action F(f) =< F, f > of F on an element f E (L2)+ being given by 00
F(f) = >2 n!Fn(fn)
(3.13)
n=0
Note that in the special case when F E L2(µ) then (3.13) coincides with the result of taking the inner product of g and f in L2(µ), since, by (3.5) and (3.6), (3.14)
E[(r FndB®")(J fndB®")] _
E[Ff] _ n=0
R"
Rn
n! J Fn(u) fn(u)du n=0
Rn
00
1: n!F,(fn) n=0
More generally one can consider the space (S)* of generalized white noise functionals consisting of elements of the form E f FndB®n where Fn E S(R."), equipped with a certain topology. See Hida-Kuo-Potthoff-Streit (1991).
In addition to the natural vector space structure the space (L2)- (and (S)*) also has a multiplication called Wick multiplication o, defined by (3.15)
(J FndB®") o (J G,ndB®'") _ j Fn®G,ndB®("+m), Rn
R-
Rn+m
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270
where Fn®Gm denotes the symmetrized tensor product of F,, and Gm. (If Fn and Gm are functions, this means that we first form the usual tensor product F. ®Gm(z) = F. (D Gm(xi, ... , -En, yl) ... , ym)
=
Fn(x1,...,xn)Gm(yl,...,ym)
and then symmetrize the result, so that
- (n + m)! 1
Fn®Gm(z)_
Fn ® Gm(xol, ...) ZOn}m ))
n+m). This definition extends in the usual way to tempered distributions Fn, G.). the sum being taken over all permulations a of (1,
Note that if F. E 1
(Rn), Gm E ft _
,
(Rm) with n < m then
Fn®Gm E H-(Rnxm) and (3.16) IIFn®GmIIH-MPI(Rnxm)
:= F4 := F(¢). Then (4.2)
YO(w) =
J
F4'(u)dB®"(w)
R^
is defined for a.a.w as an L2(S')-limit in the usual way for each 0 E S = S(R"). Thus Y is a random linear functional on H', in the sense that Y(C1 01 + c202)(w) = c1YO1(w) + c2Y¢2(w) for a.a.w,
for all 01, 02EH'andall c1,c2ER.
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273
It follows that Y has a version with values in H-', if n
r>s+j
See Walsh (1984), theorem 4.1. In the following we will assume that this H''' version is chosen, so that we may regard (4.3)
F E H-°(R'; L2(R"))
J
FLn
as an H-''(R")-valued stochastic process. For notational simplicity we put
H-°° =
(4.4)
I
l
I H-k
k=1
so that if F E H-O° then F E H-k for some k and then we write
IIFIIH-- =
IIFIIH-k
DEFINITION 4.1. A functional process
is a sum of distri-
bution valued processes of the form (4.5)
Xo(w) = X(¢,w) = E
J "=ORn
F("1(¢®")dB®"(w); ¢ E S,w E St
where
F(")
E H'°°(R' ; L2(R")) for all n > 1
and
F(01(.) E H-°°(R). Moreover, we assume that
> (4.6)
EX
2=
n! "=o
r < F("), 0®" >2 (u)du < 00 Rn
for all 0 E S with II0IIL2 sufficiently small.
Lindstrom et al: SDE involving positive noise
274
To make the notation more suggestive we often write the functional process
X(¢,w) on the form
Xt(w) _
(4.7)
J
J F(n)dB®n,
i(u)dB®"(w) _
n=OR
n=ORn
where each F(n)(u) is really an LZ-valued distribution in the t-variable, t = (t1, , t,). The distributional derivative of Xt with respect to t is then defined by 00
dd (w) _
(4.8)
where
dtF( n=O Jn
d _ dt t t ' t =
.>'t(U)dBon(w)
n 8F(n)
(z
xj
)z=(t,...,t),
J=1
az, denoting the usual distributional derivative with respect to x;, i.e. ,gF(n)
< ax7
,
> for d = &(x1, ... , xn) E S(Rn).
10 >_ - < F(n), 1
EXAMPLE 4.2. The white noise process Wt can be represented as a functional process as follows: 00
(4.9)
Wt =
J -00
bt(u)dB
where bt(u) is the usual Dirac measure, i.e. < bt(u), O(t) >= 4(u)
To see this note that, according to the definition above, (4.9) means that (4.10)
WO(w) =
J
which is just a reformulation of (2.6).
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275
EXAMPLE 4.3. By the previous example together with (3.15) we can represent the square of white noise as follows: (4.11)
Wt
= Wt o Wt = J bt(u)bt(v)dBudBv,
*2
R2
This means that as a distribution valued process Wt *2 can be written
0ES
Wo2(0 (& 0,w) = f R2
In particular, 0ES
Wo2(c6 ® 0,w) = f 4(u)cb(v)dB R2
so a more correct notation for Wt 2 would be Wt2.
EXAMPLE 4.4. The functional process
Xt(w) = f
Bt 2(w)
where
lt(u)
10 1
if u>toru R is a bounded function depending only on finitely many variables yl, , yn. (The formula (5.6) defines A as a premeasure on the algebra generated by finite products of sets in R and so A has a unique extension to the product v-algebra (Folland (1984)). Combining (5.3)-(5.6) we obtain
X0 = W E ca(®a)dB®n =00E E C. C'O
n=0
lal=n
n=0 jal=n
00
= E E Ca[J (x + 2y)adA(y)]a= f cdB n=0 lal=n
J
(®adB®n
Lindstrom et al: SDE involving positive noise
279
where (x + iy)a = (xl + iyi)a` (x2 + iy2)a' ... (xm + iy+n)am if a= (al, .. , am) and we evaluate the right hand integrals at xi = f (jdB,
j = 1,2,.... 00
DEFINITION 5.2 Let X4, = E E ca f C®adB®n be a functional pron=0 lal=n
cess represented as in (5.2). Then the Hermite transform (or f-transform) of X0 is the formal power series in infinitely many complex variables z1, z2, given by 00
f(X0)(z) = XO(z) = 1: >2 caza =
(5.8)
caza
a
n=0 lal=n
where by (5.3).
Remark. It turns out that there is a close relationship between the Hermite transform and the S-transform as defined, e.g., in Hida-Kuo-Potthoff-Streit (1991); see Theorem 5.7 below. We claim that the sum in (5.8) converges for all z E Co , where
C0 = {(z1 i z2,
); only finitely many of the zjs are nonzero}
More generally, X4(z) is defined for all z = (zl, z2,
) E CN such that
-(zal2 < 00
a
In fact, for such z the series in (5.8) is absolutely convergent, since 00
00
>(>2 IkaI
a! ca)3(
Izal) n=0 jal=n
n=0 lal=n 00
lal=n
00
< (>2 L2 a!c2)I n=0 jal=n
IzaI2)j
.
(2 > n=0 lal=n
IzaI2)4 < 00
N (otherwise za, = 0). For such a with jal = n there must exist an a3 > j and hence 1
1 a!-[n]!,
denotes integer value. This gives
where
Izol2
00
1: E n0 I a= a!
0 a.s. Computer simulations of Exp[WW] for 3 choices of "averages" 01, 02,03 with
supports [t, t+ ], [t, t+ s ] and [t, t + loo ], respectively are shown on Figure 1 = 1). 1 (In all cases IIciIIL' Returning to the examples (1.2) and (1.3) in the introduction, we see that a crucial question for the usefulness of the concept of positive functional processes is the following:
If X0, Yo are two positive functional processes and X0 o YO is defined, is X0 o Y# also a positive functional process?
The main result in this section is an affirmative answer to this question, together with a characterization of positiveness for functional processes in terms of positive definiteness of its Hermite transform. This characterization
is analogous to Theorem 4.1 in Potthoff (1987), but there the setting, the transform and the methods are different.
THEOREM 7.4 Let X be a functional process and fix ¢ E S. Then Xo(w) > 0 a.s. if and only if 9n(y) = X0 (Zy)e
2
Y E Ro
is positive definite for all n, where X 1(x) is defined by (5.10).
Lindstrom et al: SDE involving positive noise
291
Figure 1 A plot of Exp[WW] showing all values. Supp[c] = [t, t + 1/10].
A plot of Exp[W,] showing "fine"structure. Supp[4] = [t, t + 1/25]. Exp[W]
0.0035
1
Ii
0.003 0.00251
0.002 0.0015 0.001
0.0005
A plot of Exp[WW] showing "micro"structure. Supp[q] = [t, t + 1/100]. Now
the graph is torn completely apart. Exp[W]
-
4. 10 -18
It
Lindstrom et al: SDE involving positive noise
292
Remark: Condition (7.3) means the following:
For all positive integers m and all y('),..., y('n) E Ro , a = (aj) E Co
we
have m
- y(k)) > 0.
>ajakgn(y(j) j,k
Proof. Recall that F(z) := Fn(z1, the variables z1, zn.
,
zn) := X )(z) is analytic in each of
Let
J5r(x + iy)dA(y)
Hn(x) = Hn(x1, ... , xn) = (7.5)
iy)e-Iy'(2ir)-,dy
= J F(x + where y = (y1,
,
dyn. We write this as
yn), dy = dy1
(7.6) a-; X2 . r F, (z)e . Z. e-ixy(2ir)- 2 dy = e 2
x2(2ir)
j
2 J G(z)e-'xydy,
n
where z = (z1, ... , zn), z2 =
z2+. ..+Z2, XY =
j=1
F(z)e2
xjyj and G(z) = Gn(z)
is analytic.
Consider the function (7.7)
.f (x;
)=
f
G(x + iy)e-ifydy; x, C E Rn
By the Cauchy-Riemann equations we have Of
49G
87x1 = ,l ax,
Now
00
(-i)aG
.
a -`Eydy
J(_i
)e _1d
y
00
et'Y1dyl = i
-00 and this gives
=
J G(z)ei6Y'(-iSt1)dy1 l
-00
of 57x1
=S1Ax; c )
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293
Hence f (X1, x2, ... , xni b) = f (O, x2, ... , xni
and similarly for x2i (7.8)
S)ettxi
, xn. Therefore
Ax;() = f(O; )etx = etz f
G(iy)e-'tydy
We conclude from (7.5)-(7.8) that (7.9)
Hn(x) =
Jk(x + iy)dA(y) = e.z2(2)_2 J..)(iy)e_312e_iz3ldy 1gn(x), x2
= e2
where y'n(C) = (27r)-n/2 f gn(y)e-'{ydy is the Fourier transform of gn.
Note that gn E S and hence gn E S. Therefore we can apply the Fourier inversion to obtain gn(C) _ (27r)-'
Jn(x)e1dx
(7.10)
= Je12Hn(x)dX(x), C E R. Hence, if ((1), ((2) ... E RN and a = (aj) E Co we have E ajakgn(S(j) - b(k)) = j,k
J
I7(X)12Hn(x)da(x),
where y(x) is the vector [aje'{(')x]1<j<m We conclude that (7.11) ajakgn( j,k
U)
-
(k)
)=
2 E[I7(f (dB)I (f X'kn(f (dB + iy)d)i(y))]
-> E[I7(f (dB)I2XO(w)] as n -> oo.
So if XO(w) > 0 for a.a.w we deduce that m (7.12)
t lim \ ajakgn(((j) - (k)) > 0 n-.oo j,k
Lindstrom et al: SDE involving positive noise
294
But with (1),
(m) E RN fixed gn(f W_&)) becomes eventually constant
as n --> oo so (7.12) is equivalent to (7.4).
Conversely, if gn is positive definite then H,,(x) > 0 for y - a.a.x and if this holds for all n we have Xo(w) > 0 a.s. by (7.5).
COROLLARY 7.5 Let X, Y be two functional processes such that X oY is a functional process. If X > 0 and Y > 0 then X o Y > 0. 2
Proof. For ¢ E S consider X(n )(iy)e-2Y as before and similarly 2 YY")(iy)e- 2V . Replacing 0 by p¢ where p > 0 we obtain from Theorem 7.4
that Xo(ipy)e-Y2
is positive definite,
g,(,P)(y) :=
hence
on(y) := X
(7.13)
and similarly -/n(Y) :=
)(iy)e_( 0. EXAMPLE 7.6 To illustrate Theorem 7.4 let us check that the exponential of white noise, Xt = Exp(Wt), is positive: Choosing ¢ = p(, we have
Xt(x) = exp(pzl) so that 1
g(y) = exp(zpyi - y2). 2
Now
ajakg(y(7)
=
- y(k) ) _
a1.iake
*1eh1)
e-i'yik) a-i(v
)(akeiPylk))e_3(Y')_Y(k))2 > 0,
W- (k))2 y
Lindstrom et al: SDE involving positive noise
295
since f (y) = e- z v' is positive definite. Hence g is positive definite and therefore Xt > 0.
EXAMPLE 7.7 Returning to equation (1.2) in the introduction, we see that Corollary 7.5 is necessary for our concept of a positive noise to make sense in such a model: If for example r = 0, a = 1 so that dXt dt
=NtoXt,A = 1
with Nt a positive noise, then Corollary 7.5 gives that ddt is indeed positive if Xt is.
§8. The solution of stochastic differential equations involving functionals of white noise. As pointed out in the introduction it is important that our generalized noise concept leads to stochastic differential equations which can be handled
mathematically. In this section we show that this is indeed the case. The main idea is to transform the original noise equation into a (deterministic) differential equation for the Hermite transform. By applying the inverse Hermite transform to the solution of this deterministic equation we will under some conditions - obtain a functional process which solves the original equation.
To illustrate this method we look at the 1-dimensional version of equation (1.3), i.e. (8.1)
(k(t)Xt)' = 0
where
k(t) = Exp(eWt),e > 0 constant, is our model for the positive noise k(t), and the product is interpreted as a Wick product o. Taking fl-transform we get the equation
exp(eWt) Xt = Cl (Cl constant)
Lindstrom et al: SDE involving positive noise
296
From (5.12) we have
E S
So
±.'O(z) = C1 exp(-e E(O, (i)zi) which gives
X(x) = C1 ' k [
f
k
(i)zi)dA(y)]x= f tda
eXP(-e
i=1 k
2
(by formula 6.2) = C1 klim [exp(-e E((O, (j)xi + 2 i=1 OdB
= C1 exp(-e J
(,)2))].=f
tdB
2
- 2 1I¢1Ii2)
In other words, 00
XO = CiExp(-eWW) = C1 E
(8.2)
(-1)e
n=0
n!
J
as we would have obtained by direct formal computation from (8.1).
Note that if we define G : S(R2) - L2(R2) by 00
(8.3) < G, O®O >= 2 110(u+-,)V,(v+s)+O(v+s)tk(u+s)}ds*xtu>o,v>o) 0
then if supp 0 C R+, supp z/4 C R+ we have
0.
(8.4)
dt Gt = Similarly we see that if we more generally define (8.5) 00
G(n) :=< G(n), 00n >:= f O(ul + s O(u2 + s ... 0(un + s)ds 0
then
= ®n if
-
d G(n)
supp0CR+
In other words
T Gtn)
(8.6)
= b®n fort > 0.
Using this in (8.2) we conclude that (8.7) (-1n)!nen
Xt = Xt`) = C2 + C1t + C1 . 00 1:
J
n=1
G(tn)(u)dBr (C2 constant)
is the solution of (8.1).
We want to write this on a more transparent form. A test function ¢ should be substituted for t, and we then generate a sample path inserting Bt(u) = ¢(u - t). This gives (-1)' en
X(C) = C2 + C1t + C1
00
n.
n=1
f
G(()(,tt)dB®n
Using the definition (8.5) of G(n), we get
X ') = C2 + Clt + Cl
L
(-1) en
00
n=1
f
00
nEon
fX(O,o)(h1)(tL1 + s - t) ... X[o,eo) (un)q5(un + s - t)dsdB®n 0
(-1)!
= C2 + C1t + C1 f 0
I
n=1
f X(o,oo)(u)Ot-,(u)dB(u)J
L
= C2 + Clt + C1 J{ExPE_eWx(Ot_.] - 1) ds 0
ds
Lindstrom et al: SDE involving positive noise
298
If we assume that 0 has support in the interval [0, h], the integral is zero unless s < t + h. After a change of variables r = t - s, we therefore get t
Xle = C2 + Clt + Cl J {Exp[-eWX[o ,c,.] - 1}dr h
If r > 0, then X(o,)Or = Or. Hence tr
x (e)
00
= C2 + Cl J {Exp[-eWX(o )0,] - 1}dr + C1 J Exp[-eW jdr h
0
The first two terms are just constants. So we get tf
X -l = Cl
(8.8)
J0
Exp[-eW.,r]dr + C2(e, c6, w)
which is nearly what we would get from a formal solution of the equation (8.1).
Computer simulations of this solution with C1 = 1 and C2 = 0 and various choices of e are shown on Figure 2. There we have chosen q(u) = qt(u) _ n X[t,t+h] (u) with h = 0.1, so Xt really means X4 . The following question is of interest: How does the stochastic solution Xt E) differ from the no-noise solution Xi0) = C2 + Clt? How big error do we make if we replace the stochas-
tic permeability k(t) = Exp(eWt) by its average k0(t) = 1? The answer depends heavily on the ratio between e2 and the support h of the averaging function 4(u) = h X[t,t+h](u):
THEOREM 8.1 For e > 0 let (8.9)
0
Xi`l = t + n=1
t
(- (-We"
JG"(u)dB?" =
I 0
Exp [-eW,r]dr + C(e, 0, w)
Lindstrom et al: SDE involving positive noise Figure 2 Integral Positive Noise 1
0.8 e = 0.1
0.6} 0.4 0.2 t
Integral Positive Noise 1.2 1
0.8 e = 0.2
0.6 0.4 0.2
Integral Positive Noise 0.51
e=1
299
Lindstrom et al: SDE involving positive noise
300
be the solution of (8.1) with C1 = 1, C2 = 0. Then
E[Xi`)] = t
(8.10) and
(8.11) max{e2(t + h/3),
th
(e4 - 1)} < E[(Xg - t)2] < (t
2h)h
(e* -1)
Proof. (8.10) is straightforward from (8.9). Consider E[(Xef) (8.12)
- t)2] = E[(E ( n=1 00
6n n)[
,l
Gin)(u)dB®n)2]
°O 2n
= n=1 L n' J(GNu))2du Now
O(u1+s)... (un+3) 54 0 only ifu;+3 E [t,t+h]foralli. Hence t+h
J
(Gtn)(u))2du =
[J h-n X( ui+-E[t.t+hl,vi}d3]2du 0 t+h
J R+
[J [O,t+h]x...x[O,t+h]
t+h
(8.13)
h-nX{t-min(ui)o,(xt)t>o,(PZ)ZEEI associated with (EA,D(£A)) , that is for every bounded B(E)-measurable function u on E and all t > 0
f u(Xt)dPz
=e
tLA u(z)
µ a.e. z E E.
LA is the generator of (£A,D(£A)) on L2(E,p) , i.e. the unique negative definite self-adjoint operator on L2(E,p) such that
£A(u,v) = (-LA u,v)p for all u E D(LA)
, v E D(£A)
.
With respect to (£A,D(£A)) we have the usual notions such as (1-)capacity Cap , q.e. etc. The reader can find the corresponding details in [3] (and of course [5] ). Let Pp be the probability measure on (0,7) defined by (1.7)
Pp(.) = f Pz(.)µ(dz) . E
E' C L2(E,p) , it is easily shown as in [3, Lemma 5.1] that E' c D(£ A) . Hence by [3, Theorem 4.3] the following decomposition holds: Since
(1.8)
k(Xt) - k(Xo) = Mt + Nti
,
t > 0 , Pz a.s., q.e. z E E E.
Here (Mt)t>0 is a square integrable continuous martingale additive functional of ii of finite energy, (Nt)t>0 is a continuous additive functional of 1l of zero energy (cf. [3], [5]) . When A(z) = IdH ("collecting" the components of the decomposition above) it was proved in [3] that there is an E-valued (7t)t>0 Brownian motion (W)t>0 with covariance given by < , >H and an E-valued continuous process (Nt)t>O both starting at 0(E E) such that for q.e. z E E (1.9)
Xt = z + Wt + Nt and E, E =Mk
,
t > O , Pz a.e..
Let o(z) be the positive square root of A(z). Note that z,-4a(z) is again
continuous from E to L'(H) (cf. e.g. [9] ). In this paper, using the stochastic integral in [6] we prove an extension of (1.9) in case E is a Hilbert space.
Assume from now on that (1.10)
E is a Hilbert space with inner product < , >E and H c E is
Hilbert-Schmidt . The stochastic integrals below are in the sense of [6] (see (2) in Section 2
324
Rockner & Zhang: Dirichlet processes on Hilbert space
below).
Theorem 1. There exist continuous E-valued (It)t>0 adapted processes (Wt)t>0 and (Nt)t>0 starting at 0(E E) such that (i)
for q.e. z E E under P. ' (Wt)t>o is an (Jt)t>0 Brownian motion
on E with covariance < >H
t (ii)
Xt = z + f o(Xs)dWs + Nt , t > 0 , Pz a.s., q.e. z E E 0
(iii)
t
If Mt := f Q(Xs)dWs , t > 0 , then for all k E E' 0
E, E = Mt and E, E =N , t > 0 , PZ a.s.
t
for all z E E outside some capacity zero set (possibly depending on
k) where Mk , Nt are as in (1.8). Theorem 2. Fix T > 0 . Let 3t = o-(XT-s,sH starting at 0(E E) such that
t
T
Xt - XO = f o(Xs)dWs - f o(XT-s)dWs for 0 < t < T , 0
T-t
The proofs will be given in Section 2 below.
2. PROOFS Proof of Theorem 1. (1) Construction of the Brownian motion (Wt)t>0. Since H C E densely by a Hilbert-Schmidt map we can find an orthonormal
Rockner & Zhang: Dirichlet processes on Hilbert space
325
Co
basis {en I n E M} of (H,< , >H) and an > 0 , with I An < +w such n=1
that {en/An I n E II} is an orthonormal basis of E , and
<en,h>E = an <en,h>H for all h E H (C E)
(2.1)
,nE
N .
Define in E E' by en(z) = <en/An,z>E , z E E E. Observe that by (2.1),
en(h) = <en,h>H if h E H , i.e. en corresponds to en E H under the embedding E' c H . We can find S c E with Cap(S) = 0 such that (1.8) with k = en holds for all z E E\S and all n E QI . By [3, Proposition 4.5] we also know
that for all i,jEII,zEE\S,
t
(Mei,Mei)t
(2.2)
f (A(Xs)ei,ej)H ds
Now fix z E E\S , and let
01-1 (z)
t > 0 , PZ a.s.
,
denote the inverse of Q(z) on H . Fix
also k E E' , and for n E IN define a square integrable martingale (Mt)t>0 by the following stochastic integral n
e
= i=1I 0f ('1ce) dMsl
Wt 'n
(2.3)
,
t>0.
H
Under Pz for n > m , t > 0 <Wk,n _ Wk,m, _ Wk,m>
t
' n
n
f
1(X)kei) dMsl , H
i=m+10 n
Y,
I ft
1(Xs)k'ei)
1: Y f (r'(X5)k,ei) i=m+1 j =M+1 0
tl
_ f \a(X 0
n s)rl
i=m+1
l
\ dMsl ) H
t
(r'(X5)k,e)H (A(X5)e,ej)H ds
(01
i=m+1 j =M+1 0 n n
_
i=m+10
t
n
f KU_i(X)c,ej)
H
\ l
1(Xs)k,e.>
Co,(Xs)ei,o,(Xs)ej)H ds
H r
1(Xs)k'ei)eiJ' u(X5) l
i=m+1
(rl(X5)k,e)
1\
I )ds
J/
Rockner & Zhang: Dirichlet processes on Hilbert space
326 1
0 can be defined such that. for all z E E\S , it is a square integrable (7t)t>0 martingale under Pz and for all t > 0 ,
Wt = lim Wt'n in
(2.4)
L2(S2,Pz)
.
n-+w
Now we compute the bracket of Wk . From the definition, for t > 0 , we have that in L2(1l,Pz)
<Wk,Wk>t <Wk'n,Wk'n>t
= lim it-to
n
n tnom
(A(Xs)ei'ej)ds \(Xs))H ((:'5''11
l
= lim f (A(X5) r ii-to
0
n
(6'(X5)k,ej)
l
H ei] ' i=1
i=1
\ ei) ds .
l H
H
-1 Since A(X8),a (Xs) E L°D(H) , by (1.5), and the dominated convergence theorem we conclude that
t (2.5)
<Wk,Wk>t = f (A(X5)r'(X5)k,r1(X5)k) ds H
0
= tllk,12
From the construction of Wk
,
,
t>0.
it is easy to see that for fixed
k - Wt is Pza.s. linear on E' In addition, (2.5) implies that .
t>0,
Rockner & Zhang: Dirichlet processes on Hilbert space
327
k
Ez[eiWt}
= exp[-2 tIIkIIg,
(2.6)
,
t > 0 , k E E'
Consequently, (the proof of) Theorem 6.2 in [3] applies to (Wt )t>0 and we conclude that there exists an E-valued (7t)t>0 adapted continuous process (Wt)t>0 on (1,7) such that, for all z E EMS , under Pz , (Wt)t>0 is an (1t)t>0 Brownian motion starting at 0(E E) with covariance < >H Furthermore,
E,E=Wt , t>O,PZ a.e. for all kEE' ,and all ZEE
(2.7)
outside some capacity zero set (possibly depending on k ).
(2) Stochastic integral with respect to (W)t>0
Below for a linear operator T on E we denote its restriction to H by T . Define finite dimensional orthogonal projections Q. , n E (N, on E as follows. n
n 11
<ei/\i,z>E ei/Ai]
E' <ei,z>E ei
(2.8) Qn z
.
i=1
i=1
Then by (2.1), Qn , n E
,zE E
IN
, are also orthogonal projections on H and both
(Qn)nEIN , and (Qn)nEIN converge strongly to IdE on E , IdH on H respectively. Denote the set of all Hilbert-Schmidt operators from H to E by L(2)(H,E) ; the usual norm in L(2)(H,E) is denoted by Below again we fix z E E\S . Let C be the space of all (7t)t>0 adapted processes 11-112.
t (((t))t>O with state space L(2)(H,E) such that Ez f II((s)I12 ds < +m 0
for each t > 0 . Using (Qn)nEIN , for any (E C the stochastic integral of
t w.r.t. (Wt)t>0 denoted by JC(t) = f C(s)dWs , t > 0 , can be defined as 0
in [/6]
.
JS\ is a continuous E-valued (7t)t>0 martingale (i.e. all components
E E' , are martingales) with the property \
(2.9)
t
/
Ez JC(t) = 0
,
EzIIJC(t)IIE
= Ez f IIS(s)I12 ds . 0
Rockner & Zhang: Dirichlet processes on Hilbert space
328
[YW m
Clearly, the embedding H C E has II'II2 norm equal to
l 1/2
.1
2n,
and
n=1
for each T E LT(H) we have that OD
r
T E L(2)(H,E) with IITII2
(2.10)
IITII Lm(H)
A2 ]
I
n=1
1/2
J
In particular, it follows that (o(Xt))t>0 E C and by the above discussion, we get a continuous E-valued, (Ft)t>0 martingale (Mt)t>o defined by
t f o(Xs)dWs
Mt
(2.11)
,
t>0.
0
(3) Identification. In this step, we will show that
E, E = Mk , t > 0 , PZ a.e. for q.e. z E E\S
t
(where Mt is as in (1.8)). We need the following simple Lemma.
Lemma. Let Tn , T E L(2)(H,E) , n E DI , such that
IIT-TnQnII2n Proof. For n E IH
IIT-TnII2
n
0 , then
w0.
,
frT-T11 Qn)ej0I2 j=1
E
jII E
I
j=1
IIT e;IIE
+ j=n+1
1
Take a sequence Tn = {0 = to < tl < t2
... }
of partitions of [0,w[ such
that S(Tn) = Max [01+1 - t i --- 1 0 as n --+ m . For each n > 1 we define a J
process Sn E C by (2.12)
an(t)=Q[Xti nI
if toE in L2(S1,PZ) as n -4 m for all j E Vi . Fix j,n E IN . Now we calculate ECej,JCn(t)>E . For notational convenience
the index n is omitted in the following. By definition of J(. if tN_1 < t < tN we have that PZ a.s. N-1
(j,JC(t)) E
d
j'((ti)Q[W(ti+1)-W(ti)])E
i=1
+
(e, C(tN) [w(t)
i)E
and
EICej,C(ti)Q{w(t±1)_w(t)] n
E (ek,W(ti+l)_W(ti))
_
I
e.,C(ti)ek>E
[Wk(ti+l)_Wek(tj)]
Kej, (t)ek)
k=1
n
E'
w
t.l+1
f
K0.1(X)ek,eL)B (C(ti)ek,ej) dMsl
k=1 1=1 ti
Y t i+1 1
Y
(r'(X)eC(t)ej> dMsl H
1=1 tt
where we used (2.4) and the symmetry of o,-1 (Xd and C(ti) . Hence
E'Cej'Jn(t)') 'D
ft K4(Xs)en Cn(s)ej)g dMsl
1=1 0
where the sum converges in L2(SZ,PZ). Define Tn(s)
-1
Cn(s)Qn v 1 (Xs)
Rockner & Zhang: Dirichlet processes on Hilbert space
330
t f <ej,ej>H dMs' , we obtain that
W
- Id1 , s > t , then since MtJ =
1=1 0
\
2
Ez (E, \ej,Jsn(t) )E - MtJ] t
OD
/
Ez(I fC £=1 0 m
OD
ee 2 s J
H
t
Efds
f (T)t,) lTn(s)em,e) (A(Xs)ei,em)
Ez(I:
1=1m=10
JH
ds]
(A(Xs)Tn(s)ej,Tn(s)ej
-0
since Tn(s)
//H
as n --.m,
n
0 strongly on H and by the dominated convergence theo-
rem. Hence
E' <ej,Mt>E = "m
n-+m E'
\ (e.,J (t)) J
e
/E
Sn
= MtJ
,
t>0,P za.e.
,
and consequently, if k E E' is a finite linear combination of elements in
{ejIjEI1}, (2.13)
E, E =
For general k E E'
Mk
,
t
t > 0 , Pz a.e.
by approximation we also get (2.13) Pz a.e. for q.e. z E E\S (cf. [F80, Corollary 1(ii), p.139] ). Combining (2), (1), (3), and let,
ting Nt = Xt - Xo - Mt , t > 0 , we finish the proof of Theorem 1. Proof of Theorem 2.
Since E' C 2 (EA) we know by [7], [8] that for all k E E'
(2.14) k(Xt) - k(Xo) = 2 Mt
$
k T - T-t, P- a.e. for 0 < t < T.
( 19
Here Mk,Mk is a square integrable It martingale, Tt martingale, respectively. Moreover,
Rockner & Zhang: Dirichlet processes on Hilbert space
C=
331
t
= 0f
f0 CA(Xs)kl,k2)ds
CA(XT)kl,k2)
ds
g
for 0 < t < T
.
As in (2.4), we can define OD
=
t
f (1(c5)i,e)
dMsi
i=1 0 and OD
Wt =
t
f CdIWs
e.
i=1 0
By the same procedure as that in (1), there exists an E-valued 7t Brownian
motion Wt , and an E-valued 7t Brownian motion Wt such that for all
kEEl E, E = Wt
,
E, E = Wt Pµ a.e.
t and the stochastic integrals Mt := f o(Xs)dWs ;
t f Q(XT-s )dWs
'Wt
0
0
are well defined. Furthermore, for all k E E'
E, E =
Mk
t
t Set
,
E, E = 1GIt for 0 < t < T , P- a.s. T
Zt =1 f a(Xs)dWs - if f Q(XT_s)dWs ; then T-t 0
Z
is a continuous
E-valued process such that
k(Zt) = k(Xt Xo) Pt a.e. for all k E E' By the continuity of Xt Xo , we finally have that
t Xt - Xo =
T
f o(Xs)dWs -1 f u(XT--s)dWs T-t 0
,
0 < t < T , P- a.e..
332
Rockner & Zhang: Dirichlet processes on Hilbert space
REFERENCES
[1]
Albeverio,S., Kusuoka,S., Rockner,M.: Partial integration on infinite dimensional space and application to Dirichlet forms. J. London Math. Soc. 42, 122-136 (1990).
[2]
Albeverio,S., Rockner,M.: Classical Dirichlet forms on topological vector space - construction of an associated diffusion process. Probab. Th. Rel. Fields 83, 405-434 (1989).
[3]
Albeverio,S., Rockner,M.: Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Preprint Edinburgh 1989. To appear in Probab. Th. Rel. Fields.
[4]
Albeverio,S., Rockner,M.: Classical Dirichlet forms on topological vec-
tor spaces - closability and a Cameron-Martin formula. J. Funct. Anal. 88, 395-436 (1990).
[5] [6]
Fukushima,M.: Dirichlet forms and Markov processes. AmsterdamOxford-New York: North-Holland 1980.
Hui-Hsiung Kuo,H.H.: Gaussian Measures in Banach Spaces. Springer-Verlag, New York 1975.
[7]
Lyons,T.J., Zhang Tu-sheng: Decomposition of Dirchlet processes and its application. Preprint Edinburgh 1990. Publication in preparation.
[8]
Lyons,T.J., Zheng, Weian: A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Colloque Paul Levy sur les processus stochastique, Asterisque 157-158, 249-272 (1988).
[9]
Reed,M., Simon,B.: Methods of modern mathematical pyhsics. Vol. I. Functional Analysis. New York, Academic Press 1975.
[10] Rdckner,M., Schmuland,B.: Tightness of C"p-capacities on Banach space. Preprint (1991), publication in preparation.
[11] Schmuland,B.: An alternative compactification for classical Dirichlet forms. Stochastics 33, 75-90 (1990).
A Supersymmetric Feynman-Kac Formula Alice Rogers Department of Mathematics, King's College, Strand, London WC2R 2LS
1. INTRODUCTION Over the years a number of non-commutative generalisations of Brownian motion have been developed. One such generalisation is the concept of fermionic Brownian motion, where paths in a space parametrised by anticommuting variables are introduced, following the original idea of Martin (1959), in order to study the evolution of quantum-mechanical systems which have fermionic degrees of freedom. As with conventional Brownian motion, the original motivation was physical, but the approach is also valuable in handling analytic questions on manifolds, extending standard probabilistic methods to spin bundles and differential forms. The aim of this article is to introduce probabilists to this extended family of physical and geometrical ideas, and particularly to the role of supersymmetry. Recently physicists have given much attention to so-called supersymmetric systems. There are two (equivalent) ways of characterising such a system. One is to describe a system as supersymmetric if its quantum Hamiltonian operator fl is the square of a Dirac-like operator Q. (More details of both Hamiltonians and Dirac operators are given below.) The other characterisation is that there should exist an invariance of the classical action of the theory under transformations which intertwine bose and fermi degrees of freedom- it is this intertwining which makes the symmetry super in the usual physicist's terminology. Standard techniques in classical mechanics show these two characterisations to be equivalent, but for our purposes it will be most appropriate to concentrate on the first idea, and consider supersymmetric quantum mechanics.
The study of supersymmetry has lead to a variety of mathemati-
334
Rogers: Supersymmetric Feynman-Kac formula
cal constructions which are, loosely, Z2-graded extensions of conventional constructions. Such objects are referred to by the prefix `super', which in this context has no connotations of superiority. For instance, superspace is a space parametrised by both commuting and anticommuting variables, while a supermanifold is a manifold which locally resembles superspace.
Before introducing the non-commuting, fermionic aspects of supersymmetric quantum mechanics, a brief summary of simple quantum mechanics and its relation to conventional Brownian motion will be given.
This material is standard, but is presented so that the role of analogous constructions using anticommuting variables should be clear. Thus section 2 of this article describes conventional Brownian motion and bosonic quantum mechanics, while section 3 introduces the analysis of functions of anticommuting variables. At this stage, only finite-dimensional spaces will be considered. Section 4 describes how fermionic quantum mechanics may be formulated in terms of operators on spaces of functions of anticommuting variables, and in section 5 fermionic Brownian motion is introduced and its use in fermionic quantum mechanics established. In the following section fermionic Brownian motion is combined with conventional Brownian motion and thus paths in superspace are considered. The appropriate stochastic calculus for these paths is described. Also superpaths, which are parametrised by both a commuting time t and an anticommuting time r, are introduced, leading to a supersymmetric Ito calculus. In section 7 a supersymmetric Feynman-Kac formula for the evolution operator exp(-ftt - QT) of a supersymmetric quantum-mechanical system in flat space is first described. Solutions to stochastic differential equations in superspace are used to construct more general superpaths, which are then used to extend the supersymmetric Feynman-Kac formula to a waider class of operators, particularly those which relate to curved superspace.
2. Brownian motion and quantum mechanics In quantum mechanics the observables of the system are represented by selfadjoint operators on a Hilbert space, the space of states of the system. The particular choice of space and operators determines the physical content of
the system, and is determined from the classical physics of the system by
Rogers: Supersymmetric Feynman-Kac formula
335
a process called quantisation. The quantisation process involves replacing classical observables such as position and momentum by quantum operators in such a way that the Poisson bracket algebra of the classical system is preserved as the commutator algebra of the observables. For a single particle with no intrinsic spin and moving in one dimension two fundamental observables are position i and momentum p corresponding to the classical canonical coordinates x and p. Since x and p satisfy the canonical Poisson bracket relation {p, x} = -1, (2.1) the quantum operators must obey the canonical commutation relation
Li, i] = -iii.
(2.2)
In the Schrodinger reresentation of these operators the states (or wavefunctions, as they are often called in this context) belong to the Hilbert space L2(R) of complex-valued square-integrable functions of the real line, and the operators p and i are defined by
Pf(x) = -ihOf (x), if (X) = x f(x). The dynamics of the system is defined by the Hamiltonian, which is the energy operator, and usually denoted H. Classically, the Hamiltonian, which is a function of the canonical phase-space variables x and p, generates infinitesimal time translations. According to the standard quantisation prescription the quantum Hamiltonian is obtained from the classical Hamiltonian by replacing the phase space variables x and p by the quantum operators i and p. (Since classically px = xp while in quantum mechanics
pi # ip this prescription may not be unambiguous.) In the Schrodinger picture the Hamiltonian determines the evolution of the system via the Schrodinger equation iii
= Hf,
(2.4)
(where units have been chosen such that the constant Ii = 1). This equation may be solved if one can construct the evolution operator exp ift. Feynman
Rogers: Supersymmetric Feynman-Kac formula
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proposed a path integral approach to the construction of this operator for certain Hamiltonians; this is non-rigorous, involving ill-defined oscillatory integrals, but if one replaces t by it, using imaginary time, the heat equation is obtained, and for many useful Hamiltonians Brownian motion and Wiener integrals provide representations of the solutions in the form of FeynmanKac formulae. For instance, if H = p2 + V (.i), the Feynman-Kac formula z
is f
( exp
-Ht) f (x) = J dy exp (-
f
V(x + b,)ds) f (x + b)
(2.5)
where dp denotes Wiener measure, bt is a Brownian path and units have been chosen so that Planck's constant h = 1. While this is of course standard material for a probabilist, it has been spelt out in detail here to emphasise a point which is crucial in the generalisation to fermionic Brownian paths. This is the dissection of the Hamiltonian into two parts, the part p2
catered for by the measure and the part V(x) which appears explicitlyz in the integrand. Physically, 2p2 is the Hamiltonian of a free particle (in other words, simply its kinetic energy) while V(x) is the potential energy. Even in more complicated situations, such as that of a particle moving in curved space, an analogous dissection exists, and the fermionic Wiener measure constructed below incorporates the free Hamiltonian in a similar manner.
3. CALCULUS IN SUPERSPACE In this section the analysis of functions of anticommuting variables is briefly
described. Such functions carry a representation of the operators used in fermionic quantum mechanics which is an analogue of the Schrodinger representation for bosonic particles; this allows fermionic and bosonic variables to be treated in on an equal footing, which is clearly desirable in any supersymmetric theory. From a more mathematical point of view, representations of Clifford algebras can be built from differential operators on these spaces,
which makes them the natural arena for various geometric constructions, such as Hodge de Rham theory on a Riemannian manifold. Let 61, ... , 972 be n anticommuting variables. There are various interpretations of this statement (Berezin and Leites (1975), Kostant (1977), Batchelor (1980), Rogers (1980)): one can take a concrete approach, consid-
Rogers: Supersymmetric Feynman-Kac formula
337
ering anticommuting variables to be functions with values in the odd part of a Grassmann algebra; alternatively one may think of the 9' as generators of an abstract algebra over the complex numbers, with relations
i, j = 1, ... , n.
9'9j = -0'91"
(3.1)
In either approach, one finds that the most general polynomial function, or element of the algebra, is of the form
f(9) = E fµ9µ µEM
where µ is a multi-index u = µl ... µk with 1 < it, < ... < µk < n, Mn denotes the set of all such multi-indices (including the empty multi-index), each fµ is a complex number and 91' = 91l ... 01'k. For instance, if n = 2, the most general polynomial function has the form fl91
(3.3)
+ f202 + f129192.
f(0) = fe +
Differentiation with respect to 9' is defined by 8 9µ _ 092
(_l)r+1eµ1 =0
...
9µ,-,eµr+l ...9µk
if µr = 2 otherwise.
Integration is carried out according to the Berezin prescription where, if f(9) = fl...n91 ... 0n+ lower order terms, then dne f (o)
J
(3.5)
= f1...n.
This allows any linear operator M on the space of polynomial functions to be represented by an integral kernel M(9, 0) with
Mf (9) = f dnOM(e, 0)
f (O).
(3.6)
(Here both 0 = (91, ... , 9n) and 0 = (41, ... , On) consist of variables all of which anticommute with one another. Indeed, throughout this paper,
338
Rogers: Supersymmetric Feynman-Kac formula
any anticommuting variable is defined to anticommute with any other anticommuting variable.) One particular example is the S-function, which as usual is the kernel of the identity operator I: S(9, q5) = I(9, )
_ (9r1 - 01) ... (9n -
on)
= f dnic exp [-iK`(9`
- 0_)].
(3.7)
The Berezin integral is defined formally, and does not have any measuretheoretic interpretation as, for instance, the limit of a sum. The analysis of functions of purely anticommuting variables is simply the analysis of a certain family of finite-dimensional vector spaces. It becomes useful when combined with functions of ordinary variables. (The notational convention will be that commuting variables are represented by lower case Roman letters, anticommuting variables by lower case Greek letters, while Roman capitals are used for quantities which may have either
parity, or, in the case of multicomponent variables, both parities. Also the standard Einstein summation convention, where repeated indices are summed over their range, will be used.) Given any class of functions on R' (or on some subset of Rn`) there is a natural way to extend this to (m, n)-dimensional superspace. For instance L2(Rm) may be extended to the space L2'(Rm'n) by defining this space to consist of functions of the form
f(x()eµ 11
... , xm, e1,..
.
en)
fµx
=
,
(3.8) 1
,uEM.
where each of the coefficient functions fµ is in L2(Rm). Other spaces, such as Coo'(Rm,n) can be defined in a similar way. Differentiation and integration may be defined by combining traditional operators with those described above for anticommuting variables. Each of these extended function spaces has a natural Z2-grading, with an even function consisting entirely of terms containing the product of an even number of 9's and an odd function having
terms which contain products of an odd number of 9's. There is a corresponding grading of differential operators on these spaces, with an operator
Rogers: Supersymmetric Feynman-Kac formula
339
said to be odd or even according as to whether it does or does not change the parity of a function. An important example of an odd operator is the supercharge Q of a supersymmetric system, as will appear below.
The particular case of calculus on a (1,1)-dimensional superspace (parametrised by a real variable t and an anticommuting variable r), will now be described. Functions on this space will be termed superpaths, and play a considerable part in the supersymmetric stochastic calculus which follows. First, the superderivative DT is defined to be the operator DT
aT
(3.9)
+r 5j-
This operator may act on a function of the form F(t, T) = A(t) + rB(t) where A(t) has a time derivative. (As the notation implies, A and B may have either Grassmann parity.) One then has
DTF(t, T) = B(t) + T at (t).
(3.10)
Where applicable straightforward calculation then shows that DT = at As will be seen below, in supersymmetric quantum mechanics there is a square root of the Schrodinger equation where the time derivative is replaced by DT and the Hamiltonian by its square root, the supercharge. Integration in (1,1)-dimensional superspace is exceptional in that one may define an integral which includes both commuting and anticommuting limits, in the following way (Rogers (1987a)):
JdJd F(s, Q)
r
J
r dcr
ds F(s, r).
(3.11)
0
The even integral on the right hand side of this equation is evaluated by regarding fo ds F(s, o) as a function of u, and evaluating this function when
u = t + or by Taylor expansion about u = t. (Here both o and r are anticommuting variables, that is, they anticommute with one another and with all other anticommuting variables.) Integration with respect to the anticommuting variable o is then carried out according to the usual Berezin prescription. Thus if F(t, r) = A(t) + rB(t) one has
T0 do J0 t ds F(s, o) = rA(t) + Io t B(s)ds.
(3.12)
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Rogers: Supersymmetric Feynman-Kac formula
This definition of integration leads to a supersymmetric version of the fundamental theorem of calculus in the form T
t
J0do
ds DSF(s, o) = F(t, T) - F(0, 0),
(3.13)
0
and hence the natural rule for the superderivative of such an integral with respect to its upper limits is obtained as rT
t
DTI do 0
I0
ds G(s, a) = G(t, r).
(3.14)
The stochastic version of these integration results gives a supersymmetric Ito formula which in turn leads to the supersymmetric Feynman-Kac formula of section 8.
4. FERMIONIC QUANTUM MECHANICS Intrinsic spin is a property of a particle in quantum mechanics which has no classical analogue. It may take integer or half-integer values, particles with integer spin being known as bosons and those with half-integer spin as fermions. To describe n fermionic degrees of freedom one requires operators b , i = 1, ... , n satisfying canonical anti-commutation relations `z%ij +
2.
An analogue of the Schrodinger representation can then be constructed by letting the Hilbert space of states be the space of polynomial functions of n anti-commuting variables 81, ... , on and setting
=e`+ae,,
i = 1,...,n.
(4.2)
The dynamics is then specified by setting H = V(b). For a free particle the fermionic operators do not contribute to the Hamiltonian, and so in building
the appropriate Wiener measure one splits the Hamiltonian H = V(ii) into H = 0 + V(t%) and forms the measure from the kernel of the identity operator. Despite its apparently vacuous nature, this measure, which is described in detail in the following section, has useful properties and can be used to build a Feynman-Kac formula.
Rogers: Supersymmetric Feynman-Kac formula
341
This purely fermionic quantum mechanics only describes the spin degrees of freedom of a particle, and a full treatment must include position and momentum, so that the appropriate Hilbert space is a space of functions of both commuting and anticommuting variables, with the Hamiltonian a function of i, p and . In particular, supersymmetric quantum mechanics in flat space is based on the free supercharge Qo =%2` ay, which satisfies
2o - Ho
(4.3)
where Ho is the Hamiltonian of a free particle, 10
a 2
==1
a
(4.4)
ax' ax'
Qo is the simplest example of a Dirac operator; a more general Dirac operator is described in section 7.
5. FERMIONIC BROWNIAN MOTION Let I be the real interval [0, t1) (where t f may possibly be infinite). Then, for any positive even integer n, fermionic Wiener measure is defined on the infinte-dimensional superspace (R°'2")I by specifying its finite distributions. As with conventional Wiener measure, these are built from the kernel of
the evolution operator exp -Hot of the free Hamiltonian. Thus, if J = {t1, ... , tN} C I with 0 < tl < ... < tN, the distribution function f j on
Ro2' is defined by fj(10'.
N
_ 7r(0
1e
1
N 1P)ir(1e 20,2p)
... 1r(N-1 B
Ne
NP)
where r9 = (01r'... , 9' ) for r = 1,. .. , N and ir(e, 0, ti) = exp [-ik'(e` - 0')].
(5.2)
The space (R°'2' )' equipped with this measure will be called fermionic Wiener space. Fermionic Brownian motion is then defined to be the (0, 2n)dimensional process 9i , pt, j = 1,. . . , n, with 9o and po defined to be zero. For the purpose of representing the operators 9' + a/a9' the process t = et + ipt
(5.3)
Rogers: Supersymmetric Feynman-Kac formula
342
is also useful. The principle property of this measure is that if V(t) _ µ' ....%/Lk with 1 < pi < ... < ack < n, and f is a function of n anticommuting variables 91, ... , on,
(V( ))f(O) = f
dpf(Bµi +
91)...
(9µk +b9k)f(9+9,).
This result allows one to establish a Feynman-Kac formula in the expected form. That is, if f is a polynomial function of n anti-commuting variables and H = V(t) where V(z&) = > EMn Vµz%µ, the Vµ being complex numbers,
(exp-Ht)f(9) =
J
dyfexp(-J tV(9+e,)ds) f(9+9t).
(5.5)
0
Details of the limiting process needed to define the integrand in this expression, together with other aspects of fermionic Brownian motion, may be found in Rogers (1987b).
When considering bosons and fermions together, one may use the super Wiener space of paths (bt, Ot, pt) in (m, 2n)-dimensional superspace, with super Wiener measure dp, being the product of bosonic Wiener measure dpb and fermionic Wiener measure dp f. This leads to a Feynman-Kac formula for Hamiltonians of the form H = z p2 +
6. STOCHASTIC CALCULUS IN SUPERSPACE As in the purely classical, bosonic case a much wider class of differential operators can be included if one considers solutions of stochastic differential equations, but first the appropriate stochastic calculus must be developed by incorporating fermionic paths into the Ito calculus. This stochastic calculus is outlined in this section; a more detailed account may be found in Rogers (1990).
Classically Wiener measure, Brownian motion and stochastic calcu-
lus can be formulated in a number of different ways. Only one of these approaches seems compatible with the formalism of fermionic Brownian motion, that of defining the measure from its finite-dimensional marginals, using the Kolmogorov extension theorem. A further restriction is that random variables must be defined as the limit of a set of functions IN defined
Rogers: Supersymmetric Feynman-Kac formula
343
on (Rm2n)JN where the JN are finite subsets of I, and the expectations of the fN tend to a limit. (The details may be found in Rogers (1987b).) A process At corresponding to sequences of functions At(N) on (Rm2n)Jt(N)
is said to be adapted if, for each t E I and for each positive integer N, Jt(N) C (0, t].
It is not necessary to define an integral along fermionic Brownian paths because the fermionic measure already allows one to handle differential operators of any order; this contrasts with the classical, commuting case, where Ito integrals are needed to handle operators which contain first order derivatives, while stochastic differential equations are required for general elliptic second order differential operators. The case of fermionic Brownian paths is different because the phase space or fourier transform variable is
retained in the paths. In fact it does not seem possible to define integrals along fermionic Brownian paths, because they are extremely irregular. This irregularity is most easily seen if one considers the Fourier mode expansion of 6t given in Rogers (1987b). However, it is still necessary to give a meaning to fo A,db, where A. is an adapted process on super Wiener space. Provided that it possesses suitable regularity properties, such a process can be integrated along a Brownian path according to the following definition.
Definition 6.1
For a = 1,.. . , m suppose that Ft is an adapted stochastic process on (RL'2n)I. the super Wiener space For each N = 1,2.... let JN = {tl, ... , t2N_I} with tr = J for r = 1,...,2 , ... , 2N N - 1. Then the process Fi is defined by sequences of functions NFt on (RL'2n)J'. Let GN E L21 ((RL'2n) JN, C) with
GN(1 x,..., 2N-1 m 2N-1
N Fa
X10
l
,..., 2N-1 ---, r
1
P) ...,
to,..., r
2N-1
1
P)
rp)(r+1 a
r a)
(6.1)
a=1 r=1
Then the sequence (JN, GN) defines a random variable which is denoted
tm
JF:db:. a=1
Rogers: Supersymmetric Feynman-Kac formula
344
It is also useful to define a stochastic integral to be a process Zt such that if 0 < tl < t2 < t f,
Zt' -Zt, =
I tz l
11t2 m
A,ds+EC,db' l a=1
where A, and C, are adapted processes on the super Wiener space, again obeying suitable regularity properties. Full details of these definitions may be found in Rogers (1990). A key theorem, underpinning the development of supersymmetric stochastic calculus and the supersymmetric Feynman-Kac formula, is the following generalised Ito theorem.
Theorem 6.2 Suppose that Z, (j = 1, ... , p + q) are stochastic integrals on the super Wiener space (RL2n)(°'t) with m
dZ,j = As
dt +
C;' dba(s)
(6.2)
a=1
and that Zj is even for j = 1, ... , p and odd for j = p + 1, ... , p + q. Also suppose that E(IA;I)2 is bounded for each positive integer r.
Let F E C5'(R(P,q), C). Then, with NZj denoting the Nth term of the sequence defining the random variable Ze , the sequence F(NZ,) defines
a stochastic process denoted F and F, is also a stochastic integral with p+q m
dF, = E >B,F(Z,)C;jdb, +
j=1 a=1 p+q a j=1
m p+q p+q
C,kC,'ajakF(Z,)\
Ids.
F(Z')As+ a
a=1 j=1 k=1
This theorem, which is proved in Rogers (1990), shows that the fermionic Brownian paths do not affect the Ito correction term. In particular, for the simple case where Ct J = f(bt, Ot, pt) and A; = 0, the correction term takes the familiar form z ;=1 ae, '571 f (bt, et, pt).
Rogers: Supersymmetric Feynman-Kac formula
345
The next step is to develop a stochastic version of the (1,1)dimensional integrals fo do fo ds, and so obtain a supersymmetric Ito theorem with a correction term of the form (Qo - 2THo) f (b(t, T), (t, T)). Here b(t, T) and (t, T) are superpaths, which will now be defined.
Definition 6.3 For 0 < t < t f and r an anticommuting variable let ZT, CT denote the processes (6.4)
+ \/2iobt . (Here the notation bt is formal; it is to be interpreted in combination with dt as bt dt = dbt. The placing of (7. in formulae will be such that only this combination will occur.) Before giving the supersymmetric Ito formula, a time integral for superpaths must be defined. CT = 4
Definition 6.4 Suppose that G, and K, are suitably regular adapted stochastic processes on super Wiener space. Then, if Fs = G, + oK r
J0
t
t
do
J0
ds Fs =de f
J0
ds K, + rGt.
(6.5)
The following supersymmetric Ito formula may then be proved by expanding
each side as a series in r and applying the classical Ito theorem together with (6.3).
Theorem 6.5 Suppose that f E CS'(RP.m). Then
f(zT, et) - f(0, 0) = Jf dst [Qof zs, 0") -2oHo f (zs, 9,)] +
j
(6.6)
r
t
do
J0
db; oaaf(zs, e').
This Ito formula will be used in the following section to prove a supersymmetric Feynman-Kac formula. A further result in this extended stochastic
Rogers: Supersymmetric Feynman-Kac formula
346
calculus is the following theorem which establishes the existence of solutions of a useful class of stochastic differential equations. The full details of the regularity conditions on the functions Aa and B' require an excursion into the theory of functions of anticommuting variables which, together with a proof of the theorem, may be found in Rogers (1990).
Theorem 6.6 Let p, m and n be positive integers. For j = 1,. . . , p and for a = 1, ... , m let AQ and B' be suitably regular functions on RP,2' x R+. Also let A be a fixed element of CP. Then, f o r j = 1, ... , p, there exists a stochastic process Z9 such that dZt = Aa (Z.4, 0., p, s) db9 + B' (Z 89, P9, s) ds
Zo =A
(a) (b)
(6.7)
(where (b 0 p,) are Brownian paths in (m, n)-dimensional super Wiener space). These solutions are effectively unique, in a sense which is defined in Rogers (1990).
By using solutions to carefully chosen stochastic differential equations a more general supersymmetric Ito formula may be proved, leading to a supersymmetric Feynman-Kac formula for a wider class of supersymmetric Hamiltonians. 7.
EVOLUTION IN SUPERSYMMETRIC QUANTUM ME-
CHANICS In supersymmetric quantum mechanics the Hamiltonian ft is the square of the supercharge Q. The supercharge is always an odd operator, which means that the Schrodinger equation
8f at
-fif
has a square root (Friedan and Windey (1984))
DTI = Qf
(7.2)
Rogers: Supersymmetric Feynman-Kac formula
347
Here f is a function of m + 1 commuting variables x1, ... , x', t and n+1 anticommuting variables 91, ... , 9', r. Since, by definition, Or = -r9, one finds that Q anti-commutes with DT, and hence, recalling that DT = at one finds that (7.2) implies (7.1). One may also check by direct calculation that, provided k is suitably regular,
DT(exp(-Ht - QT)) = Q exp(-Ht - Qr) _ (exp(-Ht - 07-)) (Q - 27-H).
(7.3)
Thus exp(-Ht - Qr) is the evolution operator for the square root of the Schrodinger equation.
For a supersymmetric particle in flat, Euclidean space, wavefunctions are defined on an (m, m)-dimensional superspace and Q has the form
+O(x,0)
(7.4)
where 0 is an odd function. By applying the supersymmetric Ito formula (6.6) to a carefully chosen function one obtains the following supersymmetric Feynman-Kac formula.
Theorem 7.1 Let F E C°°'(Rm'm). Then
exp(-Ht - Qr)F(x, 0) rr
r
=Jdµ,exp{J
t
0
xF(x+
ds¢(x+`j2T9+zs,0+(s)}
do,
(7.5)
0
r9+zT,9+9t).
Proof Let Ut,, be the operator on L21 (Rm,m) defined by the completion of Ut,, : --. L21(Rm,m) with C"(Rm,'z)
Ut,, G(x, 0)
jjr
E
exP
do,
f
ds O(x + /2 Q9 + zs, 9 + (s)
xG(x+ Zr9+ZT,9+9t)
(7.6)
Rogers: Supersymmetric Feynman-Kac formula
348
for G E Co-'(Ri'm). Then, applying the supersymmetric Ito formulae, one finds Ut,T G(x, 9)-G(x, 0) rT
= J do I'ds U,,QQG(x, 9) - 2QU,,,fTG(x, 9).
(7.7)
0
DTUt,,G(x, 9) = Ut,T (Q - 2rH)G(x, 9),
(7.8)
and hence, using equation (7.3) and appealing to the uniquness of solutions to differential equations such as (7.8), one may deduce that
Ut,TG(x, 0) = exp(-Ht - Qr)G(x, 9)
(7.9)
as required.
These methods can be extended to more general theories, where the supercharge has the form (7.10)
(with the function 0 linear in Vii) by introducing modified superpaths. The Hamiltonians of such theories takes the form
-2g(x) a
8
9 xi 8 xj
+ lower order terms
(7.11)
where gi3(x) = El 1 e,(x)ea(x). As in the case without spin operators, paths which satisfy stochastic differential equations are required.
Definition 7.2 For a, i = 1, ... , m let
(a and
=;+
\12io b;
178 = 6a,i (x,).
(7.12)
Rogers: Supersymmetric Feynman-Kac formula
349
where x; satisfies dx, = ea(x,)db, + 2(s° + e ;)ea(x,)C9jeb(x,)ds
(7.13)
xo = 0.
In terms of the superpaths z;, (;, one has the following supersymmetric Feynman-Kac formula for the supercharge Q of (7.10):
Theorem 7.3 If F E C51(RL'm) and (x, 0) E RL'",
exp(-Ht - Qr) F(x, 9)
r rr
= E exp{ J ll
0
t do,
J0
ds ¢(x + zs +
F(x+zT+ 2ri7t,0+9t)
9+
(s)} (7.14)
.
This theorem is proved in Rogers (1990), using a generalisation of the supersymmetric Ito formula (6.6). For geometrical applications, such as the investigation of the Dirac operator of a spin bundle or the Hodge de Rham operator on a Riemannian manifold, it is necessary to formulate this approach in a globally valid manner on a supermanifold constructed over a conventional manifold. As in the classical treatment of path integration on manifolds described in Elworthy (1982) and Ikeda and Watanabe (1981), the approach depends on stochastic differential equations which transform covariantly under change of coordinate, as will be described in a forthcoming paper (Rogers, in preparation).
8. Conclusion This article describes some new stochastic techniques for handling Dirac-like operators. It is primarily addressed to probabilists, and it seems appropriate
to conclude with a brief indication of the motivation for this work. Physically, symmetry in a model is desirable both as an aesthetic criterion for
350
Rogers: Supersymmetric Feynman-Kac formula
selecting models, and for the practical reason that models tend to become more tractable as they become more symmetric. One great hope of supersymmetry is that cancellations between the fermi and bose sectors will remove the infinities which plague most models in quantum field theories. So far this hope has not been realised, but neither has it been abandonedcurrently superstring models are receiving most attention as possible candidates for a unified theory. Supersymmetry is essential to these models, but the technical demands are enormous, and it will be some time before a full assessment of them can be made. However it is clear that rigorous, rather than purely formal, manoevres with anticommuting variables should play a part in handling the fermionic aspects of these theories, and this is one motivation for the work described in this article.
Mathematically, supersymmetry has had some interesting consequences. Use has been made in various contexts of the positivity of H implied by the formula ft = Q2, but of particular relevance to the material of this article are the supersymmetric proofs of the index theorem due to Alvarez-Gaume (1983)and Friedan and Windey (1984). These proofs use fermionic path-integral manipulations which, while standard for physicists, are not entirely rigorous. The techniques descrbed in this article allow one to make these arguments rigorous, as will be shown in a forthcoming paper (Rogers, in preparation).
References Alvarez-Gaume, L. (1983) Supersymmetry and the Atiyah-Singer index theorem. Comm. Math. Phys. 90 161-173
Batchelor, M. (1980) Two approaches to supermanifolds. Trans. Amer. Math. Soc. 258 257-270 Berezin, F.A. and Leites, D. (1975) Supervarieties. Sov. Math. Dokl. 16 1218-1222
Elworthy, K.D. (1982) Stochastic differential equations on manifolds. London Mathematical Society Lecture notes in Mathematics, Cambridge.
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Friedan, D. and Windey, P. (1984) Supersymmetric derivation of the AtiyahSinger index theorem and the chiral anomaly. Nuc. Phys. B 235 395-416
Ikeda, N. and Watanabe, S. (1981) Stochastic differential equations and diffusion processes. North-Holland, Amsterdam Kostant, B. (1977) Graded manifolds, graded Lie theory and prequantization. Differential geometric methods in mathematical physics, Lecture notes in mathematics 570 177-306, Springer
Martin, J. (1959) The Feynman principle for a Fermi system. Proc. Roy. Soc. A 251 543-549 Rogers, A. (1980) A global theory of supermanifolds. Journ. Math. Phys. 21 1352-1365 Rogers, A. (1985) Integration and global aspects of supermanifolds, in Topological properties and global structure of spacetime, eds Bergmann, P. and De Sabbata, V, Plenum, New York Rogers, A. (1986) Graded manifolds, supermanifolds and infinite- dimensional Grassmann algebras. Comm.Math. Phys. 105 375-384
Rogers, A. (1987a) Supersymmetric path integration. Phys. Lett. B 193 48-54
Rogers, A. (1987b) Fermionic path integration and Grassmann Brownian motion. Comm. Math. Phys. 113 353-368 Rogers, A. (1990) Stochastic calculus in superspace I: supersymmetric Hamiltonians, King's College London preprint Rogers, A. (in preparation) Stochastic Calculus in Superspace II: spin bundles, supermanifolds and the index theorem
ON LONG EXCURSIONS OF BROWNIAN MOTION AMONG POISSONIAN OBSTACLES Alain-Sol Sznitman Courant Institute of Mathematical Sciences New York University 251 Mercer St., New York, NY 10012
0. Introduction Consider random obstacles on Rd, d > 1, given by means of closed balls of fixed radius a > 0, centered at the points of a Poisson cloud, with constant intensity v > 0, and law F(dw). Let Z. be an independent Brownian motion starting from the origin, T stand for its entrance time in the obstacles, and Po for the standard Wiener measure, Donsker-Varadhan [2] showed that: (0.1)
lim t-d/(d+2) log 1P ®P° [T > t] = -c(d, v) , too
(0.2)
c(d, v) = (vwd)2/(d+2)
with
d2 2 ,\d/(d+2) (±..) (_iL)
Here wd, Ad stand respectively for the volume of the unit ball in Rd, and the principal Dirichlet eigenvalue of -!A A in the unit ball of Rd. The constant c(d, v) comes in fact from the minimization problem: (0.3)
c(d, v) = in ff {vIUj + A(U)}
where U runs over the class of bounded open sets in Rd, with negligible boundary. In (0.3), JUG and A(U) denote respectively the volume of U and the principal Dirichlet eigenvalue of -1A in U. The optimal choice of U corresponds to a ball of radius Ad
R0
) 1/(d+2)
(2d vwd
The lower bound part of (0.1) can in fact be obtained by assuming that Z. does not leave until time t the ball centered at the origin of radius Rot'/ (d+2),
and that no obstacle falls in this ball. Alfred P. Sloan Fellow. Research partially supported by NSF grant DMS 8903858. Pres. addr.: Dept. Math., ETH-Zentrum, CH-8092 Zurich
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In this paper, we are mainly concerned with the large t behavior of the probability that Brownian motion surviving until time t, performs a "large excursion" at distance of order td/(d+2) from the origin. We show that in dimension d > 2, for x > 0: (0.5) -(c(d,v) + k(d,v,a)x) < F IM-
log P ®Po[suP IZu I > xtd/(d+2), T > t] u t] t-00 = -c(1, v - IhI) , when d = 1 = -c(d, v) , when d>2.
,
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So unlike the case of dimension 1, when d > 2, the result does not depend on h, provided 0 < JhJ < Let us give some indications on how the estimates (0.5), in the d > 2 situation, are derived. The lower bound is the easier part. One considers a long cylindrical tube of radius r and approximate length x td/(d+2), with a sphere of radius Rote/(d+2) attached to one end (Ro is defined in (0.4)). We now require that no point of the cloud falls in a neighborhood of size a of the union of the cylinder and the ball. We let Brownian motion start near the "free" end of the cylinder rush to the other end in a time of order td/(d+2), and then rest After some optimization this until time t in the ball of radius yields the lower bound part of (0.5). It is worth mentioning the following point. A variation of the previous argument, with a cylinder of sritaller size ta, a E (0, d/(d + 2)) shows that the basic asymptotic result (0.1) remains unchanged if we replace the event IT > t} by IT > t}fl{1ZtI > xta}. This explains why it is delicate to obtain a confinement property of surviving Brownian motion in scale So far it is known to hold in the two dimensional case (see [1], [7]). The upper bound part -)f (0.5) is more difficult. The crucial reduction step is derived in section II. It shows that to prove the upper bound, one can restrict the analysis to the following heuristic situation. The process vwd_lad-i.
Rots/(d+2).
tl/(d+2).
does not exit before time t a large box T centered at the origin, of size - const. This T is chopped in subcubes of size Ltl/(d+2) A finite number of these subcubes constitute "clearings" which can be used by the process as "resting places", whereas the other cubes, "the forest", t(d+1)/(d+2).
constitute to a certain degree, a hostile environment for the process, which is killed there at a given rate. Adjusting parameters, one can in fact assume that the process does not perform too many excursions (< lltd/(d+2)), in the forest, outside a neighborhood of size tl/(d+2) of the clearings, and does not spend too much time (< q t) in these excursions. The notion of "clearing" we use is based on the technique of "enlargement of obstacles" and the notion of "good obstacles", i.e. well surrounded obstacles, developed in [5], [6], [7]. The reduction step allows one to separate the clearings which have a small size - t1/(d+2), where the process
spends most of its time, from the forest, into which at some point, the process makes an excursion of size td/(d+2). In this division, the clearings account for the -c(d, v) term in the upper bound part of (0.5), whereas the excursions in the forest yield the -vwd_lad-lx term.
1. Lower bounds We consider random obstacles on Rd, d > 1, which are translates of
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B(0, a) the closed ball of radius a > 0, centered at the origin, at the points of a Poisson point measure N(w, ) with constant intensity v > 0, and law P(dw). We are also given an independent canonical Brownian motion Z.(w) on C(R+, Rd), and Po(dw) stands for Wiener measure. The entrance time of Z, in the obstacles will be denoted by T. Our aim in this section is to derive asymptotic lower bounds as t goes to infinity for the probability that Brownian motion has survived, i.e. has not entered the obstacles up to time t, and is located at a large distance from the origin. It will be convenient to introduce for t > 0:
(1.1) yt(dy)=P®Po[Zt Edy,T>t]=Po[Z'Edy,exp{-vIWfl}} where Wt = Uo 2, is uniquely attained at the solution ro > 0 of the equation: infr>o(vwd_1(a+r)d-1
(d - 1)vwd_I(a + ro)d-2ro =
(1.2)
2ad_1 .
We now want to prove:
Theorem 1.1. Suppose d > 2, and x > 0, (i)
if a
1, as will be clear from the proof, one obtains the same result as for Brownian motion in the absence of obstacles, namely: Jim t-(2a-1) log(µt(t"x, oo) ) _ t-.0
x2
2
Proof: Pick ,C3 E (0, a A d+2+2 ). For t >_ 1, we consider U the open subset of
Fed, which is the union of the cylinder C = (-ta, O+xt") x Bd_1(r) (Bd_1(r) stands for the open d- 1 dimensional ball of radius r centered at the origin), and of the d-dimensional ball B centered at the point (xta, 0, ... , 0) with radius Rot1/(d+2). Let us recall here that Ro defined in (0.4) is the radius of the ball such that vIB(Ro) I + X(B(Ro)) = c(d, v)
.
If U° denotes the a-neighborhood of U, we see that the volume of Ua is smaller than (1.3)
vtde_, Wd(ROt1/(d+2) + a)d +Wd-1(xt" + 2tfl + 2a)(a +
r)d-1
Consequently, as t goes to infinity, (1.4)
vt
-WdRot/(+), a < d+2 (WdRo +Wd-1(a + r)d-l x) td/(d+2) ,
-Wd_1(a+r)d-lxt", a>
a=
d
d+2'
d
d+2
Let us introduce a function s = s(t) < t, which we will specify later on, as well as the cylinder Cx C C: (1.5)
Cx = (xt" , xt" + 1) x Bd_1(r)
.
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Sznitman: Brownian motion among Poissonian obstacles
If 0u, u > 0 denotes the canonical shift on C(R+, Rd), and Tc the exit time of Z. from the cylinder C, we have: (1.6) µt(tax,oo) = Po[Zi > xta , exp{-vjWW I }] > exp{-vvt} Po [TC > s(t) , Z9(t) E C,x o 9, > Ze , H(Rot1i(a+2) _b) 0 09 > t - sI
,
where for any c > 0,
Hc=inf{u>O,IZu-Zoo>c},
(1.7)
and b = 1 + 2r is an upper bound on the diameter of Cx. It now follows that (1.8) log pt(tax, oo) > - vvt + log(Po [Tc > s , Z. E Cx] ) + log(Po [HRotl/(a+2) _b > t - s , Zt_8 > 0])
The second term in the right member of (1.8) equals (1.9)
log(P' -1 [Hr > s]) + log(Po [T(-tn,tfl+xt-) > s , Z, E (xta , xta + 1) ] ) Here, the superscript (d - 1) or 1 refers to the d - 1 or 1 dimensional Wiener measure. In view of (1.8), (1.9), we now have a lower bound of log µt(tax, oo) involving four terms. We will now specify our choice of the function s(t), and then study the asymptotic behavior of each term for large t. We pick (1.10) s(t) = (pta) A t
=pt,
,
when a < 1
,
with p a positive constant,
when a=1, with the constant 0 (t - s)/(Rots/(d+2) - b)2] t-00 Ad (1.12)
R0
if a 1, the method of images allows us to see that
P[T(-rata+xt«) > s , Z8 E I] > P [Ze E I] - P120 [Zs E I] - P2xt°'+2tl [Za E I] (xta + 1)2 (xta + 20)2 } exp{> (2irs)-1/2 ( exp{2s
- exp{-
2s
(xta + 20 - 1)2 2s
It then follows that 2
(1.13)
limt-.t-a
log P0, [T(-rata+xt.) > S , Z; E I] > -2
P
Collecting the asymptotic behavior of each term of the lower bound (1.8), (1 c
we see that when a
-vwdRo -
Ad
= -c(d, v)
0
Our claim i) follows thanks to the natural upper bound (0.1). When a = d/(d + 2), the left member of (1.14) is bigger than or equal to: 2
-c(d, v) - v wd_1(a + r)d-1 x - Ad-1 P -
2p
Optimizing in p and r, we find p = xr/ 2Ad_1, and r = r0 (given by (1.2)). Our claim ii) follows. When a > (1.15)
When
d+2
, we now have
limt-.ot-a log(Pt(tax, 00)) ? -vwd-1(a + r)d-lx - Ad-1
d+2
x r2 2p
< a < 1, optimizing over p, r, iii) follows. When a = 1, we
have the additional constraint p < 1, which leads to picking p = when x
t , Zt > xt1/3]) > -
2(x V to)2
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This and (1.16) now proves the lower bound part of ii). For the upper bound
part, it is enough to assume that x > to in ii). Considering the points of the cloud falling immediately at the left and at the right of 0, we see that: (1.17)
P ®Po [Zt > xt1/3 , T > t] e-v(u+v)tl/3 Po [T(_u,v) > t1/3
< J >0 , v>x du dv v212/3 Now
Po [T(-u,,,) > t1/3] < sup P., [T(o,I) > (0,1)
dx Pz [T(o,l) >
(27r)-1
(u + v)2
]
11/3
JoI
