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London Mathematical Society Lecture Note Series. 167
Stochastic Analysis Proceedings of the Durham Symposium on Stochastic Analysis, 1990 Edited by M.T. Barlow Trinity College, Cambridge and
N.H. Bingham Royal Holloway and Bedford New College, University of London
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CONTENTS Preface
List of participants An evolution equation for the intersection local times of superprocesses R. J. Adler and M. Lewin
vii
viii 1
The Continuum random tree II: an overview D. Aldous
23
Harmonic morphisms and the resurrection of Markov processes P. J. Fitzsimmons
71
Statistics of local time and excursions for the Ornstein-Uhlenbeck process J. Hawkes and A. Truman
91
LP-Chen forms on loop spaces J. D. S. Jones and R. Leandre
103
Convex geometry and nonconfluent I'-martingales I: tightness and strict convexity W. S. Kendall
163
Some caricatures of multiple contact diffusion-limited aggregation and the i7-model H. Kesten
179
Limits on random measures and stochastic difference equations related to mixing array of random variables H. Kunita
229
Characterizing the weak convergence of stochastic integrals T. G. Kurtz and P. Protter
255
Stochastic differential equations involving positive noise T. Lindstrom, B. Oksendal and J. Uboe
261
Feeling the shape of a manifold with Brownian motion the last word in 1990 M. A. Pinsky
305
Decomposition of Dirichlet processes on Hilbert space M. Rockner and T.-S. Zhang
321
A supersymmetric Feynman-Kac formula A. Rogers
333
On long excursions of Brownian motion among Poissonian obstacles A.-S. Sznitman
353
PREFACE This volume contains the proceedings of the Durham Symposium on
Stochastic Analysis, held at the University of Durham 11-21 July 1990, under the auspices of the London Mathematical Society. The core of the Symposium consisted of courses of lectures by six keynote speakers (Aldous, Dawson, Kesten, Meyer, Sznitman and Varadhan), three of which appear here in written form. In addition, there were twenty-six talks by invited speakers; the written versions of eleven of these make up the remainder of the volume. All the papers in the volume have been refereed.
It is a pleasure to thank here all those individuals and institutions who contributed to the success of the Symposium, and to these Proceedings. We thank the London Mathematical Society for the invitation to organize the meeting, and the Science and Engineering Research Council for financial
support under grant GR/F18459. We are grateful for the University of Durham for use of its facilities in Gray College and the Department of Mathematics, to our local organizers, John Bolton and Lyndon Woodward, and to Mrs Susan Nesbitt for her devoted secretarial help. Our main debt is to the speakers and participants at the Symposium, and to the contributors and referees for the Proceedings, and we thank them all. Last but by no means least, we thank David Williams, the organizer of the 1980 Durham Symposium on Stochastic Integrals, for much valuable advice during the planning of this meeting. Martin Barlow Nick Bingham June 1991
List of participants RJ Adler (Technion) DJ Aldous (Berkeley) MT Barlow (Cambridge) DJ Balding (Queen Mary Westfield) FG Ball (Nottingham) JA Bather (Sussex) JD Biggins (Sheffield) MS Bingham (Hull) NH Bingham (Royal Holloway) K Burdzy (Seattle) G Burstein (Imperial EA Carlen (MIT) TK Carne (Cambridge) T Chan (Cambridge and Heriot-Watt) JMC Clark (Imperial) NJ Cutland (Hull) IM Davies (Swansea) DA Dawson (Carleton ) RA Doney (Manchester) EB Dynkin (Cornell) DA Edwards (Oxford) RGE Elliott (Alberta) KD Elworthy (Warwick) M Emery (Strasbourg) AM Etheridge (Oxford) PJ Fitzsimmons (UC San Diego) L Foldes (London School of Economics CM Goldie (Queen Mary and Westfield) I Goldsheid (Swansea) PE Greenwood (University of British Columbia) DR Grey (Sheffield) B Hambly (Cambridge) J Hawkes (Swansea) RL Hudson (Nottingham) P Hunt (Cambridge) SD Jacka (Warwick) M Jacobsen (Imperial) J Jacod (Paris VI) DG Kendall (Cambridge ) WS Kendall (Warwick)
J Kennedy (Cambridge) H Kesten (Cornell) PE Kopp (Hull) H Kunita (Kyushu) GF Lawler (Duke) H-L Le (Cambridge) R Leandre (Strasbourg )
JT Lewis (IAS, Dublin) TM Liggett (UC Los Angeles) M Lindsay (Kings, London) T Lindstrom (Oslo) TJ Lyons (Edinburgh) Z Ma (Academia Sinica) D Mannion (Royal Holloway) P March (Ohio State) P McGill (UC Irvine) JM McNamara (Bristol) PA Meyer (Strasbourg) JR Norris (Cambridge) B Oksendal (Oslo) Papangelou (Manchester) E Pardoux (Marseille) EA Perkins (University of British Columbia) M Pinsky (Northwestern)
S Pitts (University College London)
P Protter (Purdue ) O Raimond (Paris VI) GO Roberts (Nottingham) M Rockner (Edinburgh and Bonn) A Rogers (Kings, London ) A-S Sznitman (Courant and Zurich) SJ Taylor (Virginia) B Toth (Heriot-Watt) A Truman (Swansea) SRS Varadhan (Courant) JB Walsh (University of British Columbia) D Williams (Cambridge) M Yor (Paris VI) I Ziedins (Heriot Watt)
An Evolution Equation for the Intersection Local Times of Superprocesses' ROBERT J. ADLER and MARICA LEWIN Faculty of Industrial Engineering and Management Technion-Israel Institute of Technology Haifa 32000, Israel
1
Introduction
The primary aim of this paper is to establish evolution equations for the intersection local time (ILT) of the super Brownian motion and certain super stable processes. We shall proceed by carefully defining the requisite concepts and giving all of our main results in the Introduction, while leaving the proofs for later sections. The Introduction itself is divided into four sections, which treat, in turn, the definition of the superprocesses that will interest us, the definition of ILT and some previous results, our main result - a Tanaka-like evolution equation for ILT - and an Ito formula for measure-valued processes along with a description of how to use it to derive the evolution equation. Some technical lemmas make up Section 2 of the paper, while Section 3 is devoted to proofs.
In order to conserve space, we shall motivate neither the study of superprocesses per se - other than to note that they arise as infinite density limits of infinitely rapidly branching stochastic processes - nor the study of ILT other than to note that this seems to be important for the introduction of an intrinsic dependence structure for the spatial part of a superprocess. Good motivational and background material on superprocesses can be found in Dawson (1978, 1986), Dawson, Iscoe and Perkins (1989), Ethier and Kurtz (1986), Roelly-Coppoletta (1986), Walsh (1986) and Watanabe (1968), as well as other papers in this volume. Material on ILT can be found in Adler, Feldman and Lewin (1991), Adler and Lewin (1991), Adler and Rosen (1991), Dynkin (1988) and Perkins (1988). 'Research supported in part by US-Israel Binational Science Foundation (89-298), Air Force Office of Scientific Research (AFOSR 89-0261) and the Israel Academy of Sciences (702-90).
Adler & Lewin: Intersection local times of superprocesses
2
Super Brownian Motion and Super Stable Processes. We
(a)
require some notation.
M = M(Rd) = {µ : µ is a Radon measure on Rd}.
Me =
M, (Rd)
= {p : p E M, f3td(1 + IIxII)-9µ(dx) < oo}.
C0(Rd)
= If : Rd
Co
=
CK
= Ch (Jtd) = If: Rd
R
,
f continuous, limll=Il-
f (x) = 0}.
R, f continuous with compact support}.
The Sobolev space of functions whose k-th derivatives are in £ is denoted by Wk,n. The d-dimensional Laplacian is denoted by A, and the fractional Laplacian by Da = , a E (0, 2). (c.f. Yosida (1965).) With some
abuse of notation, we shall let 02 - A. The domain of an operator A is denoted by D(A), Sd is the Schwartz space of rapidly decreasing functions on Rd, and Sd is the space of tempered distributions on Rd. (Q, F, Ft, P) is a filtered probability space.
The super Brownian motion, starting at µ E Mq, is a M9-valued, Ftadapted, strong Markov process with Xo =,u and satisfying
Xt) = Xt(v)
t
= µ(4O)+Zt(,v)+ f X8(AV) ds,
(1.1)
for every cp E D(0) fl Sd, where Zt is a continuous Ft martingale measure with increasing process [Z(4O)]t =
Jot
Xs(S 2)ds.
The super (symmetric) stable process is defined via the same recipe, but with
Da replacing A and Z" replacing Z throughout.
For the remainder of this paper, we shall take Xo = µ = Lebesgue measure, so we shall implicitly assume that q > d throughout.
(b)
Intersection Local Time (ILT). At a heuristic level, the (self)
intersection local time of a measure-valued process is a set indexed functional of the form
L(B) = fB dsdt
130 X Nd
6(x - y) X,(dx)Xt(dy),
(1.3)
Adler & Lewin: Intersection local times of superprocesses
3
where B is a finite rectangle in [0, oo) x [0, oo) and b is the Dirac delta function. A more precise definition of L(B) is obtained by replacing the delta function by an approximate delta function fE E Sd, and then taking an £2 limit as e 0. In particular, let U111>0 be a collection of positive C°° functions with f ff(x)dx = 1 for all e, such that fE -> 6 as e -> 0, where convergence is in the sense of distributions. Replace (1.3) by LE(B)
=
fB dsdt)RdXRd
f6(x - y)X3(dx)Xt(dy).
(1.4)
There are two qualitatively different cases that arise in this formulation. The first, and by far the simpler case, arises when the set B does not intersect
the diagonal D = {(t,t) : t > 0}. This case has been considered in detail in Dynkin (1988) to whom we refer the reader for details, noting here merely the fact that Dynkin has established the existence of L(B) as anG2 limit of L,(B) for all dimensions d < 7, and that he treats a wider class of superprocesses than that considered in this paper.
The more difficult case, in which B fl D # 0, requires a renormalization argument, since any attempt at a direct approach to (1.3) as a limit of LE(B) yields a plethora of infinities. In order to state what is known in this case, we adopt the notation (gyp, Xt) _ (cp(x), Xt(dx)) = )Rd p(x)Xt(dx) ('
,
X, x Xt) = (&(x, y), X3(dx)Xt(dy)) = )RdXRd O(x, y) X8(dx)Xt(dy)
Then Rosen (1990) has proven the following result, which, contrary to the assumption made above, holds only if the initial measure y = X0 is finite.
Theorem 1.1 (Rosen (1990)) . Let j,(T) be the approximate, renormalized, intersection local time defined by
y,(T) = f T dt Itt ds ((ff(x - y), X,(dx) Xt(dy)) -E{(f6(x - y), X3(dx)Xt(dy))I.F',}),
(1.5)
where the particular sequence If, = E d f f(x)dx = 1 is chosen for the approximate 6-functions. If Xt f is a super Brownian motion and d = 4 or 5, then 5E(T) converges in G2, to a limit independent of f, as e --40. G2 convergence also holds if Xt is a super
stable process of index a and d/3 < a < d/2. The limit process is denoted by y(T) and is called Rosen's ILT process for X.
4
Adler & Lewin: Intersection local times of superprocesses
Rosen's results also describe what happens when the conditions on d and a are not satisfied, and cover superprocesses defined over smooth elliptic diffusions.
It is worthwhile to note at this point that, as we shall see later, the renormalization in (1.5) obtained from the conditional expectation is not unique. Rosen's choice of renormalization arises naturally from his style of proof, which is heavily based on moment arguments. Our approach, which is stochastic analysis based, will yield a slightly different renormalization.
Note also that if d < 3 in the Brownian case, or a > d/2 in the stable case, the renormalization is not necessary, as both terms in (1.5) have finite second moment for all e > 0. Furthermore, in both of these cases, the super process has a well-defined local time, which can be used to both define the ILT and to develop an evolution equation for it. (cf. the discussion in Adler and Rosen (1991) regarding a similar situation for the Brownian and stable density processes.)
The primary aim of this paper is to go beyond the existence results of Theorem 1.1, and derive an evolution equation describing the spatial (via a parameter to be introduced below) and temporal development of -y(T). (c) An Evolution Equation Let pt and pt , a E (0, 2) be, respectively, the transition densities of Brownian and symmetric stable processes. Thus, with a slight abuse of notation:
pt (x, y) = Pj(x, y) = Pt(x - y) 1
((4irt _d 2
e-11x-Y1j2/4t
P101 (X) Y) = A, (X - y)
=
12
The corresponding Green's functions, for a E (0, 2] and A > 0, are defined by G« (x
- y) =
G« (x, y) = J
e-At p (x) y) dt
(1.6)
where we allow ourselves the luxury of writing Ga as a function of either one or two parameters at will. Note that, depending on the values of a and d, G° may be identically infinite. For all a and d, however, Ga(x, y) is finite if A > 0 and x # y. Furthermore, GA E Gl for all A > 0. We shall list a number of properties of Ga in the following section. The property of central importance to us is the fact that the distributional equation
(-A,, +A)u = S,
(1.7)
Adler & Lewin: Intersection local times of superprocesses
5
where b is the Dirac delta function, is solved by u = G. That is, for every test function cp E Sd,
I
((-0« + A)G")(x)tp(x)dx = to(0)
The fact that GA E £ for A > 0 implies that GA can also be treated as of CK functions such an Sd distribution. Thus, there exists a family that G, -+ Ga as e --> 0, in S. Since 0« is a continuous operator on Sd (cf. Hormander (1985), page 70) it follows that G° `'A :_ (-A« + A) GE -46
(1.8)
as e -> 0, where convergence is again in Sd. Thus it is not unreasonable to attempt to define a new approximate, renormalized ILT by setting, for every
'pESd, T
7E (T,
J0
Jt
dt
ds(cp(x)G"')(x - y), XS(dx) Xt(dy))
- jT(p(x)GE(x - y), Xt(dx)Xt(dy))dt.
(1.9)
There are a number of differences between the approximate ILT, -y (T, cp),
and the approximate ILT y,(T), of Rosen. Firstly, the addition of the test function tp gives information on the spatial dispersion of self-intersections that is not available otherwise. (The addition of this parameter is also necessitated by the fact that we work with an infinite initial measure.) Secondly, yE is not so much an ILT as a family of ILT's indexed by the parameter A. This emphasizes the non-uniqueness of the renormalization. The main difference, however, lies in the structure of the renormalization. As the proofs below will show, the renormalizing second term of (1.9) arises naturally from stochastic analysis considerations. The fact that it involves the "doubling" of the measure X - i.e. it involves X3 x Xt only for t = s - is highly reminiscent of the renormalized, self-intersection local time for Rd valued processes for which the renormalization involves the removal of "local double points" of the process. For this reason, we find our renormalization esthetically more appealing than that of (1.5).
Theorem 1.2 Let Xt be a super Brownian motion or super stable process, and let yE (T, cp) be its approximate renormalized intersection local time as defined by (1.9). If d = 4 or 5 in the Brownian case, or d/3 < a < d/2 in the
6
Adler & Lewin: Intersection local times of superprocesses
stable case, then for all A > 0, all T E (0, oo) and all cp E Sd, -t (T, ep) converges in £2 to a finite limit y>(T,'p) as e -> 0 which we call a renormalized ILT of Xt, and which is independent of the approximating sequence {GE}. Furthermore, -ya has the following representation in terms of the process Xt and the martingale measure Zt: 'YA(T, tP) =
IT T
0
fIT{jt(G(x - y)'(x),
Xt(dx)) ds} Z(dt, dy)
$
ft
+A
ds (G-\ (x - y)p(x), Xt(dx) Xt(dy))
dt
- IT (GA(x - y)cp(x), Xt(dx) XT(dy)) dt.
(1.10)
(d) About the Proof. As is the case with most evolution equations of the above kind, the derivation hinges on finding an appropriate Ito formula and applying it carefully. The following It-5 formula, which is actually slightly more general than we require, and is modelled on a similar result of Dawson (1978), will be proven in Section 3. Lemma 1.3 (It-5 formula). Let Xt be a Brownian or stable superprocess. Fix k > 1, ' E C2(R+ x Rk), and assume ' and its first and second order derivatives satisfy a polynomial growth condition at infinity. Let cpa, ... , cpk belong to the space W2 P n W2,1 for some p > 2. Let y) = (cpr, ... ,'k), and write (p, Xt) for the vector (('p1i Xt),... , (cpk, Xt)). Then, for all t > 0, t
`I'(t, (01 Xt)) = Vol
(0,
Xo)) + f Tt(s, (0, X,)) ds
t k
+Wxc(s, (0, Xs)) . f
X3) ds
0 i=1
1
+
2
f
t
+ It ft f
k
k
EE `Pxixj (s, (T, Xs)) . (ojcoj, Xs) ds i=1 j=1
x,(S, (0) Xs))coi(x) Z(ds, dx) i=1 E
where
'Pt = OW(t, x)/at, = aXF(t, x)/0xi, Txixj = a2XF(t, 'Px;
x)/axiax,j.
(1.11)
Adler & Lewin: Intersection local times of superprocesses
7
As is usual, the Ito formula also holds if ' is a non-anticipative functional of the process Xt. We shall be interested in the particular functional 41(t, x) = x
(1.12)
J0 t(V,, XS) ds,
where 0 E W2,P n w1,1 and x E Rd. Note that W(0, x)
0,
W (t,x) = x(O,Xt), t 'YX(t, x) = f (', Xs) ds, 'XX(t, x)
0.
Apply (1.11) with this choice of IF to obtain
Lemma 1.4 Let co, 0 E W2 'P n W2'1 for some p > 2. Then for every finite
T>0, pT
0=
Xt(dx) XT(dy)) dt
I dt f 0
0
IT
+
t
ds (b(x)oa(P(y), X3(dx) Xt(dy))
(0(x)cP(y), Xt(dx) XT(dy)) dt IT
+
fed f t ds (&(x)w(y), X3(dx)) Z(dt, dy).
(1.13)
This lemma is of importance in so far as it leads us to the following Lemma 1.5, which almost implies the central Theorem 1.2. Lemma 1.5, itself, follows in a reasonably straightforward fashion from Lemma 1.4 by first extending the latter to functions of the form 4)(x, y) = E^_1 (pi(x)z(i;(y) and then to general (D(x, Y) E C2(Rd X ltd) of compact support.
Lemma 1.5 Let 1(x, y) E C2(Rd X Rd) have compact support. Then (1.13) continues to hold if b(x)v(y) is replaced, throughout, by 4 (x, y), and O(x) y), where we use may) to denote A,,, operating on the xL cp(y) by second variable of D (x, y). That is, after reordering
I
T dt
0
+
J0
T
J0
=
y), X3(dx) Xt(dy))
t ds
IT (,I)
y), Xt(dx) Xt(dy)) dt (x, y), Xt(dx) XT(dy)) dt
_ fT r 0 Jd
/'t JO
ds
y), X3(dx)) Z(dt, dy)
.
(1.14)
8
Adler & Lewin: Intersection local times of superprocesses
To derive the evolution equation (1.10) for the ILT, y(T, cp), "all" that has to be done is to replace the function (x, y) of (1.14) by Ga(x - y)cp(x), and to note the relationship (1.7) between Ga, A., and the delta function b as well as the definition of the renormalized ILT as the limit of the 7,(T, V) of (1.9). Unfortunately, however, this simple recipe requires justification at a large number of stages. Hence the long proof of Section 3.
Acknowledgements: Our path, through the calculations of the following two sections, was made substantially easier by being able to obtain guidance from the Master of the Moment, Jay Rosen. Steve Krone pointed out some minor errors in an earlier version of the paper.
2
On Evaluating Moments
An important component in virtually all the proofs of the following section will be the evaluation of moments of the kind E 11;` 1(cp;, Xt; ), when Xt is a Brownian or stable super process. An algorithm for calculating these, based exclusively on the transition densities pt (x, y) and pt (x, y) of Section 1(c), has been given by Dynkin (1988)2.
E (11(vi, Xti)) i=1
Dk
f
dyv flPf(t)-s;(t) (Yf(1) -Yi(1)) vEV-
£EL k
x rJ dsvdyv 11 cpi(zi)dzi vE o
i=1
where pt (x) =- 0 if t < 0, and Dk is the set of directed binary graphs with k exits marked 1, 2, ... , k. The convention that pt (x) - 0 if t < 0 is important, and will be used throughout the remainder of this paper without further comment. Given such a graph, L is the set of directed links, and if the link f E L goes from vertex v to vertex w, we write v = i(t), w = f (.f). To each vertex v we associate two variables. \ Rd (Sv, yv) E R+ X
,
ZDynkin's formulae were acually proven for finite initial measures, and any positive, measurable rp. That the formulae also hold for a Lebesgue initial measure, and
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)
<(t
3'Z3)
(t 4,Z 4)
(t pZ1)
f dxi f dsi f dyi Ps (xi, yi) f ds2 f dye P 2-s, (yi, y2) f ds3 f dy3 Ps -32 (y2, y3) 4
X fPt,-31(Y"ZOK2-32(y2,z2)Pt3-s3(y3, z3)P"-s3(y3, z4) II V(zi)dzi i=1
Similar considerations apply in establishing the graphs and their contributions required for calculating third order moments. Since the pattern should, by now, be clear, we shall merely list the graphs, leaving the calculation of their contributions to the reader, and the discussion of the following section.
The three third order graphs
3 (a)
Proofs Proof of Lemma (1.3) - The Ito Formula. The Ito formula
Adler & Lewin: Intersection local times of superprocesses
13
(1.11) follows in a reasonably straightforward fashion from a standard Ito formula for continuous semi-martingales, and so we shall only sketch the proof, highlighting the technical dificulties. We shall also restrict ourselves to the one-dimensional case (k = 1 in (1.11)) which is all we really need in this paper.
Fix a test function w E W2,r fl W2'1, p > 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<1 be an jjcpjj. = sup,, Icp(x)I. Thus, let appropriate approximating sequence. We shall proceed by showing that each
of the four terms in (3.4) is Cauchy in G2(P), and that the limit variable is the corresponding expression with 4N replaced by 4). We shall give details only for the first and last terms. The first three are all similar, but the last, being a stochastic integral, is somewhat different. (A)
Because of the form of
The First Term:
as a sum, it is clear
that £2(P) Cauchy convergence of fo ("DN(x, y), Xt(dx) XT(dy)) dt will result from (3.5) y), Xt(dx) XT(dy)) I2 G C C. II'1)NIIq EI
for some q > 0, where C is a constant that may depend on T and
In we expect
particular, if D C Rd X Rd is the union of the supports of the C to depend on D. (D, of course, can be assumed compact.) The left-hand side of (3.5) is equivalent to l2
(N
EI 1: (
NN /'
"/,
= E E E {(ak, Xt) (tOj, Xt)(?k, XT)(Wj, XT)} .
(3.6)
j=1 k=1
Each expectation here can be treated via the graph formulae of Section 2. Since these are fourth order moments, there are six different graphs to consider. While the requisite calculations are not difficult, they are tedious, so we shall look, in detail, at only two cases. Consider the case arising from Graph 1, the simplest of the six. By the formula following the graph, the corresponding contribution to (3.6) is
NN
Ef j=1 k=1
(
d
/'
/'
Vk(z1)Pj(z2)4'k(z3)'l'j(Z4) 4
x pi(xl, zl)pt (x2, z2)pT(x3, z3)pT(x4, z4) jj dxidzi. i=1
Note that the p" are symmetric, and do the xi integrals. The resulting expression is
NN
E If j=1 k=1 f3Z4d
z4) dzldz2dz3dz4.
16
Adler & Lewin: Intersection local times of superprocesses
Since the ''N all have support within a common compact set D, this last integral is bounded above by IDl4ll4kNlI'.0, where IDS denotes the Lebesgue measure of D.
As a second example, consider the cases corresponding to the evaluation of the summands of (3.6) via Graph 2. There are really two different subcases here, depending on whether the two disjoint subgraphs there end with {(t,cpk),(t,Wj)} and {(T,1/'k),(T,'j)} or with {(t,cpk),(T,ij)} and {(t,Vj),(T,Ok)}. Consider the first of these. Their contribution to (3.6) is of the form rN N LS E I
dxldx2 J dy1dy2 J dslds2 ps (xl, yl)p 2(x2, y2)
x
IPi .,, (y1,z1)Pt'-.,, (y1,z2)PT-12(y2,z3)pT-12(y2,z4)
j=1 k=1
74
x Wj(zl)Vk(z2)4'j(z3)4'k(z4) 11 dzi. i-1
As before, note the symmetry pt (u, v) = p (v, u), and thus integrate out x1 and x2. Perform the yl, y2 integrals using the Chapman-Kolmogorov formula to obtain that (3.8) is equivalent to
E f dslds2 J P2(t-s,)(zl, z2)P2(T-sp)(z3, z4) j=1 k=1 x
pj(zl)Wk(z2)0j(z3)0k(z4)dzldz2dz3dz4
= f dslds2 f P2(t-s,)(Zl, z2)P2(T_32)(z3, z4) X
-I)N(zl, z3)'I'N(z2, z4)dzldz2dz3dz4.
Applying the same argument as before, and noting that $1, s2 < T, we obtain that (3.9) is bounded above by
The remaining graphs are treated similarly, as are the other three nonstochastic integrals in (3.4). It remains to show, however, that the Cauchy, G2(P), convergence is to the correct limit. In the current case, however, this is easy. Since the convergence of the 4 N to 1 is in the supremum norm, convergence is also uniform. For each
s, t > 0, Xt x X, is a.s. a finite measure on D, and so (IN, X, X Xt) 4 (4), X, x Xt). We have just shown that (IN, X, X Xt) is G2(P) convergent. Since the £2(P) and a.s. limit must agree, we are done. (B) The Stochastic Integral Term: We must now apply a similar argument to the last term of (3.4). We start with Cauchy convergence, for which we
Adler & Lewin: Intersection local times of superprocesses
17
take the 4)N as above. We shall show that (fT
E
0
ff 82d
2
t
y), Xs(dx)) Z(dt, dy)< ) C. II N00
ds
(3.10)
II2
0
where C = C(T, D). Since Z is a martingale measure with associated increasing process [(cp, Z.)]T = fo (p2, Xt)dt, (cf. Walsh (1986)), the left-hand side of (3.10) is equivalent to
E I f (ft(N(x, y), Xs(dx))ds)
X(dy) dt.
(3.11)
0 SNote
the form of 4)N, and expand the square to obtain that this is equivalent to
f
N N It ds' 1: 1: E {
T
dt
It
0
ds
0
0
Xs)
Xsi)
Yak, Xt) }
(3.12)
.
i=1 j=1
Evaluation of each summand here hinges on the three graphs at the end of Section 2. Consider, for example, the third and most complex of these. The corresponding contributions to (3.12) are of the form N N T t3 0t3 IT dt3 dtl f dt2 E > 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)
s(S1,y1)
(O,X1)0
(t 2, d' k (z 2 ))
AX 2 ) 0(t 3 d'j (z3 )) Then sum over j, k = 1, ... , N to obtain (3.19) for this D, following the style of argument of the previous proof.)
In order to simplify (3.19), use the symmetry of p" to integrate out xl and x2. Then use the Chapman-Kolmogorov equation to integrate out yl. This leaves T
T
I00 dsl f 0dt3
T
T
f dtl f1 3
8]
dt2 f3d dzldz2dz3 3
pt,+t2-23, (zi, z2)D(zl - z2)D(zl - z3)W2(zl) ,
X
(3.20)
where the lower bound of sl on tl and t2, which was implicit in (3.19), now needs to be noted explicitly. Furthermore, by ignoring a factor of 2, we can assume that tl < t2.
Noting now that D(z) = G,(z) - G,,(z), that the G, are Gl approximations to Ga, and that since ca < 2d we have GA E Gl for all A > 0, we can integrate out z3 in (3.20) to obtain 01
T
T
I00 dsl f 0dt3
T 3f,
T
dtl f dt2 f 2d dzldz2 pt,+t2-23,(zl - z2)D(zi - z2)
fT
fT
dsl 0
fT dt3 0
fT
dt2 f dzldz2 (tl + t2 -
dtl
3,
t2
2s1)-d/"
2d
X Pi W1 + t2 - 2sl)-11"(zl - z2)) D(zl - z2)V2(Zl) , where the second line follows from the scaling relationship
pt (x) = t-dj"pi (t-ll"x).
(3.21)
Adler & Lewin: Intersection local times of superprocesses
21
Now use the fact that stable and Gaussian densities are uniformly bounded to bound (3.21) by a constant multiple of T
T
T
f dsl f d13 f dtl f
1
1
dt2 (tl+t2-2s1)-d/cr x (3.22)
Using the finite support of cp and the fact that Ga E Gl, we have that the integral over zl and z2 is finite, and, by dominated convergence, can be made arbitrarily small by taking e, e' 10. Thus, to show that the contribution of the graph under consideration to the left-hand side of (3.18) is asymptotically
zero, we need only establish the finiteness of the time integrals in (3.22). In fact, there are only three time variables - s1, t1, t2 - of interest. The integration is elementary, and the integral is easily seen to converge if d < 3a.
To conclude, we consider a typical term corresponding to the third type of graph, viz. dx
f
T
dsl fRd dyl ps (x, yl)
Js
dtl
-s, (yl, zi)
Is
T
x s
ds2 Id dy2 psz-si (yl, y2)
dt2 fed dz2 s2
f, dt3
dz3
x pie-12 (y2, z2)pia-s2(y2, z3)D(zl - z2)D(zl - zs)
As before, integrate out x, and then use the Chapman-Kolmogorov equation to integrate out yl. The remaining integrand is K+s2-2s, (z1, y2)pi2-s2(y2, z2)pis-ss(y2, z3)D(zl -z2)D(zl -z3)cp2(zl) . (3.24)
Use the scaling relationship as before, but only on the first transition probability, and then integrate out Y2, again applying Chapman-Kolmogorov. Up to a constant multiple, the integrand becomes
(t2 + t3-2S2 rdlcv, + S2-2s, )-dl'D(zl- z2)D(zl -
(3.25)
Arguing as before completes the calculation as long as d < 3a, in order to do the time integrals, and if a < 2d, which is used to ensure that GA EL 2. This completes the proof.
References [1] Adler, R.J., Feldman, R.E. and Lewin, M. (1991), Intersection local times for infinite systems of planar Brownian motions and the Brownian density process, Annals of Probability, 19, in press.
22
Adler & Lewin: Intersection local times of superprocesses
[2] Adler, R.J. and Lewin, M. (1991), Local time and Tanaka formulae for super Brownian motion and super stable processes, Stochastic Processes and Their Applications, to appear. [3] Adler, R.J. and Rosen, J.S. (1991), Intersection local times of all orders for Brownian and stable density processes-construction, renormalization and limit laws, Ann. Probability, to appear.
[4] Dawson, D. (1978), Geostochastic calculus, Canadian J. Statistics, 6, 143-168.
[5] Dawson, D. (1986), Measure-valued processes: construction, qualitative behaviour and stochastic geometry, Proc. Workshop on Spatial Stochastic Models, Lecture Notes in Math., 1212, 69-93, Springer-Verlag.
[6] Dawson, D., Iscoe, I. and Perkins, E.A. (1989), Super Brownian motion: path properties and hitting probabilities, Prob. Theory and Related Topics, 83, 135-205. [7] Dynkin, E.B. (1988), Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times, Asterisque, 157-158, 147-171. [8] Ethier, S. and Kurtz, T. (1986), Markov Processes, Wiley, New York.
[9] Hormander, L. (1985), The Analysis of Linear Partial Differential Operators III, Springer-Verlag, Berlin. [10] Karatzas, I. and Shreve, S.E. (1988), Brownian Motion and Stochastic Calculus, Springer-Verlag, New York. [11] Perkins, E.A. (1988), Polar sets and multiple points for super Brownian motion, Trans. Amer. Math. Soc., 305, 743-795.
[12] Roelly-Coppoletta, S. (1986), A criterion of convergence of measurevalued processes: application to measure-branching processes, Stochastics, 17, 43-65.
[13] Rosen, J.S. (1990), Renormalization and limit theorems for selfintersections of super processes, preprint. [14] Walsh, J. (1986), An introduction to stochastic partial differential equations, Lecture Notes in Math., 1180, 265-437, Springer-Verlag. [15] Watanabe, S. (1968), A limit theorem for branching processes and continuous state branching processes, J. Math. Kyoto Univ., 8, 141-167. [16] Yoshida, K.Y. (1980), Functional Analysis, Springer-Verlag, Berlin.
The Continuum Random Tree II: An Overview David Aldous* University of California, Berkeley
1
INTRODUCTION
Many different models of random trees have arisen in a variety of applied setting, and there is a large but scattered literature on exact and asymptotic results for particular models. For several years I have been interested in what kinds of "general theory" (as opposed to ad hoc analysis of particular models) might be useful in studying asymptotics of random trees. In this paper, aimed at theoretical probabilists, I discuss aspects of this incipient general theory which are most closely related to topics of current interest in theoretical stochastic processes. No prior knowledge of this subject is assumed: the paper is intended as an introduction and survey. To give the really big picture in a paragraph, consider a tree on n vertices. View the vertices as points in abstract (rather than d-dimensional) space, but let the edges have length (= 1, as a default) so that there is metric structure: the distance between two vertices is the length of the path between them. Consider the average distance between pairs of vertices. As n -> oo this average distance could stay bounded or could grow as order n, but almost all natural random trees fall into one of two categories. In the first (and larger) category, the average distance grows as order log n. This category includes supercritical branching processes, and most "Markovian growth" models such as those occurring in the analysis of algorithms. This paper is concerned with the second category, in which the average distance grows as order n1/2. This occurs with Galton-Watson branching processes conditioned on total popu-
lation size = n (in brief, CBP(n)). At first sight that seems an unnatural model, but it turns out to coincide (see section 2.1) with various combinatorial models, and is similar to more general models of critical branching processes conditioned to be large (in any reasonable way). The fundamental *Research supported by N.S.F. Grants MCS87-01426 and MCS 90-01710
24
Aldous: The continuum random tree II
fact is that, by scaling edges to have length n-1/2, these random trees converge in distribution as n -+ oo to a limit we call the CCRT (for compact continuum random tree). This was treated explicitly in Aldous [2] in a special case and in Aldous [3] in the natural general case, though (as we shall see) many related results are implicit in recent literature. Thus asymptotic distributions for these models of discrete random trees can be obtained immediately from distributions associated with the limit tree. The limit tree is closely connected with Brownian excursion. In fact two different 1-parameter processes associated with the tree - the search depth process and the height profile process - are intimately connected with Brownian excursion (sections 2.4 and 3.2). Section 2 is a chatty account of 4 different ways of looking at the CCRT. In section 3 I take natural distributional questions about CBP(n) asymptotics (with known or unknown answers), which can be expressed in terms of the CCRT and see what can be said about the limit distributions, using the Brownian excursion representation in particular. Nothing I say is essentially new: I use the word "novel" (intended to be weaker than "new") to refer to results about CBP(n) asymptotics obtainable from known Brownian excursion results (e.g. Corollaries 3 and 6, and Proposition 12) and vice versa (e.g. (41) as a fact about Brownian excursion). One could conversely pick haphazardly some facts about- Brownian excursion and apply them to random trees, but that somehow seems less interesting. Scaling the edges of CBP(n) to have length n-° (0 < a < 1/2) gives (section 2.5) another limit tree I call the SSCRT (self-similar continuum random tree). Further, the same limit tree is obtained whether we root at the progenitor or whether we re-root at a uniform random individual in the population. This limit tree - which relates to the 3-dimensional Bessel process BES(3) in the same way that the CCRT relates to Brownian excursion - is less natural from the combinatorial viewpoint. But being more tractable (from the selfsimilarity inherited from BES(3)) it is useful in the theoretical stochastic process investigations below. Sections 5 and 6 are speculative. There has been recent theoretical interest in existence, uniqueness and properties of "Brownian motion" whose state space is some deterministic fractal set in d dimensions, the set typically constructed by some recursive procedure giving strong regularity properties. Our limit trees are "dimension 2" (inherited from Brownian sample paths), and it is intuitively clear that "Brownian motion" can be defined with these trees as its state space. Unlike other exotic state spaces, we can actually do some simple distributional calculations with these Brownian motions, and the purpose of section 5 is to present these back-of-an-envelope calculations. To develop
Aldous: The continuum random tree II
25
rigorously a theory of Brownian motion on general continuum trees would be an interesting project, and some thoughts are presented in section 5.2.
Section 6 is a quixotic venture into superprocesses. It is trivial to construct Markov processes indexed by a continuum tree. Making the index set the particular CCRT or SSCRT gives variants of the usual superprocess. This is the idea developed by theoreticians under the name "historical process",
but the theoretical literature makes this appear a deep and sophisticated object. I assert one should start from scratch and regard a superprocess as a tree-indexed process rather than as a measure-valued process. My purpose is to indicate (section 6.1) how this leads to insights which seem simpler or different from those obtained in the traditional approach. Acknowledgements. I thank Jim Pitman for much help with and information about Brownian excursion and BES(3), Steve Evans for help with superprocesses, and Martin Barlow for discussions on diffusions on fractals.
2 THE BIG PICTURE The first four subsections elaborate on the following four fundamental facts. Conditioned Galton-Watson branching processes correspond to a natural and well-studied class of combinatorial models of random trees.
One particular model can be constructed from simple random walk conditioned on first return to 0 at time 2n, and so its asymptotics can be expressed in terms of Brownian excursion.
Another particular model can be constructed from a direct (i.e. not involving conditioning) algorithm, and by taking limits one gets a direct algorithm for global construction of a limit tree. By considering asymptotics of subtrees spanned by a fixed number of randomly chosen vertices, one sees that the limit random tree must be the same (up to a scale factor) for all models in the class. Foundational work giving rigorous definitions and proofs concerning existence
of "continuum trees" (without any specific probability model present) and abstract convergence results is in Aldous [3], and it is not worth repeating such "general abstract nonsense" here. 2.1 Let
CBP(n) and Combinatorial Models > 0 be integer-valued and satisfy
E=1
26
Alrlous: The continuum random tree II
0
(1)
Such a is d-lattice, for some d > 1. We want to allow d > 1 for natural combinatorial examples (e.g. binary trees). Associate with the distribution defined by
+ 1)P( = i + 1), i > 0
(2)
and note that
Consider the simple Galton-Watson branching process with offspring distribution , starting with 1 individual in generation 0. Write T for the "family tree" of this branching process. Let T have the distribution of T conditioned on the total population size DTI = n. This CBP(n) (for "conditioned branching process") distribution is our object of study.
Tangential remarks. 1. If i.e. for some (c, 0)
and y come from the same exponential family,
i>0
then the conditioned branching processes constructed from and from y are identical. Thus we lose no generality by considering only critical branching processes. The chance that the total population size is exactly n decreases exponentially fast for sub- and super-critical branching processes, but only polynomially fast in the critical case: in this sense the critical case is most natural as a model for n-trees.
2. In the language of freshman statistics, if e is "number of daughters of a randomly-picked mother", then at (2) is "number of sisters of a randomlypicked girl". The two distributions are identical iff they are the Poisson(1) distribution. I use "Galton-Watson process" to mean the family tree of the process. Old-fashioned textbooks use it to mean the process of population sizes in 3.
successive generations, which I call the "height profile" of the Galton-Watson process.
Simply generated trees. Results about CBP(n) appear in the combinatorial literature under this name (introduced by Meir and Moon 135], apparently unaware of the branching process connection). Though the identification has subsequently become well known, there seems no convenient "translation guide" in existence, so I give one here.
Aldous: The continuum random tree II
27
A "rooted" tree simply has one vertex distinguished and called the root: imagine a family tree of descendants of a single progenitor, the root. We consider only rooted trees. Such a tree is called ordered if we distinguish birth order: if an individual (vertex) has 3 offspring then these are distinguished as "first", "second" and "third". Consider the family tree T of the unconditioned
Galton-Watson branching process with offspring . Write pi = P( = i). Then the distribution of T on rooted ordered trees t is
P(T = t) i>O
w(t) say
(3)
where d(v, t) is the out-degree (number of children) of vertex v in t, and Di(t) is the number of vertices in t with out-degree i. Thus the distribution of the CBP(n) tree Tn is specified by
P(Tn = t) is proportional to w(t) on It : Itl = n}
(4)
where Itl denotes the number of vertices in t.
One can get to (4) without explicitly mentioning Galton-Watson processes. Let (ci; i > 0) be non-negative constants with co = 1, and let 0(y) _
ciyi
be the associated generating function. Let ca(t) be some collection of nonnegative "weights" for trees. Define
yn = E W(t) t:Itl rn
and let Y(x) = En ynxn be the associated generating function. Then it is easy to see the following are equivalent.
Y(x)
x¢(Y(x)) w(t) _ cDi(t)
(5) (6)
i>o
A combinatorial definition of "simply generated tree" is "a family of weights satisfying (5), or equivalently (6)". So a random simply generated tree Tn is defined as
P(Tn = t) is proportional to ca(t) on it : Itl = n}.
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Aldous: The continuum random tree II
To see why this is really the same as the CBP(n) model, note that for any T with O(T) < oo we can define a probability distribution (7)
P(S = i) = pi = ciri/o(T), i > 0. Choose the T which makes E = 1. For w(t) defined at (3), we see w(t) = w(t) Titl-1/0ItI(,r)
Thus on It : Itl = n} w is proportional to Cv, and so the two models for T. are identical.
Elementary calculations from (7) show that the condition "E = 1" specifying T is the condition ,ro,(T)
= 40
and that the variance v2 = var(y) is U2 = 720"(T)/Y'(T).
(8)
The right-side expression appears in combinatorial papers without mention of its simple interpretation as "offspring variance".
Examples. The idea of all the combinatorial examples is that all n-vertex trees of a certain type should be equally likely. One aspect of "type" is that we can place restrictions on out-degrees. Another aspect is that sometimes we want to distinguish birth-order (ordered trees) and sometimes we don't. In the set-up above, ordered trees become the case ci = 1 if i is an allowed out-degree, = 0 if not and unordered trees become the case
ci = 1/i! if i is an allowed out-degree, = 0 if not Various offspring distributions pi = P( = i) are recorded below as a handy reference: the values of o are needed to connect our results with those in the combinatorial literature on special models. To reiterate the point: the uniform distribution on the following "types of n-vertex tree" coincides with the CBP(n) description with the stated offspring distribution. ordered (= planar) trees. Unrestricted degree: shifted geometric distribution pi = 2 Strict binary (0 or 2 offspring): po = p2 = 1/2; Q2 = 1.
i >_ 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
and the rescaled search depth process
Xt = n-112X([2nt]), 0 < t < 1.
(11)
Conventions about rescaling constants are awkward - e.g. one might want to rescale by (2n) -1/2 in (11) - but my conventions are chosen to make rescaled edge-lengths = n-1/2 consistently.
Returning to the unscaled process X(i), set X(0) = X(2n) = -1. For any model, the process X(i) has steps ±1 and first returns to the starting level after step 2n. The simplest model for such a random process would be "simple symmetric random walk, conditioned on first return to starting level at time 2n". The key fact is that this describes the depth search process in one particular model of random trees: the combinatorial model of "uniform ordered trees", which is the CBP(n) model with shifted geometric (1/2) offspring distribution. Various forms of this fact have been known to combinatorialists for a long time But its significance for probabilistic asymptotics was overlooked until recently (I learned it from Durrett et al [17], who attribute it to Harris). It is intuitively obvious (and true [16] - see also [13] and [9] p. 104 for references and history) that conditioned random walk rescales to Brownian excursion, and so (for this special model of random trees) the rescaled search depth process converges to Brownian excursion. It is equally intuitively obvious from (9) that the
rescaled height profile process converges to the total occupation density of Brownian excursion.
On a finite tree, the search depth process determines the ordered tree in a simple way: each +1 step draws a new edge, and each -1 step retraces an existing edge toward the root. So it is intuitively clear that, for the special "uniform ordered n-tree" model, there is a limit tree whose realizations can be constructed from realizations of sample paths f (t) of Brownian excursion.
In non-standard terms, an infinitesimal positive increment of f draws an infinitesimal new edge, and an infinitesimal negative increment of f retraces
an existing edge toward the root. In standard terms, given 0 < t1 < t2 <
... < tk < 1, let si = mint;
Aldous: The continuum random tree II
31
distance f (t;) -s1 toward the root, then make a new edge of length f (t;+i) -si (t;) and label its endpoint "t;+,". The shapes of these trees are consistent as varies, and define a "continuum tree" with vertices labelled by 0 < t < 1.
T t2
t3. root
t5
tl
t2
t3
t4
t5
L
figure 1 A rigorous treatment of constructing continuum trees from continuous functions is given in [3], Theorem 13: it turns out that distributional properties of Brownian excursion are irrelevant, and that any continuous function fo with certain qualitative properties (e.g. local minima are dense) can be used. Later we shall use (So, ,ao) to denote the continuum tree constructed from such a fo. Regard So as the vertex-set, labelled by 0 < t < 1, and regard µo as the "uniform probability distribution" on So induced from Lebesgue measure. Write (S, µ) for the particular continuum random tree ("the CCRT") constructed from (2x) Brownian excursion.
32
Aldous: The continuum random tree II
2.3 The Limit Trees: Global Constructions Another special case of CBP(n) is where the offspring distribution is Poisson(1). Combinatorially, this is the uniform random unordered labelled tree. Many algorithms for simulating this random tree are known: the following was discovered in Aldous [5].
Algorithm 1 Fix n > 2. Take a root vertex 1. For 2 < i < n connect vertex i to vertex V, = min(U;, i -1), where U2,.. ., U are independent and uniform on 1, . . . , n. Randomly permute the labels.
The advantage of this particular algorithm is that the n --+ oo limit behavior is intuitively easy to see. It is proved in Aldous [2] that the first process (the CCRT) described informally below is the limit when edges are rescaled to length n-1/2, and the second process (the SSCRT) is the limit when edges
are rescaled to length n-a, 0 < a < 1/2, or more generally to length 1/a(n) where a(n) --+ oo,a(n) = o(n1/2).
The compact continuum random tree (S, µ). Take a half-line [0, oo), and cut-and-paste as follows. Let C1, C2, ... be the times of a non-homogeneous Poisson process of rate r(t) = t. Cut the halfline into intervals [C1, C;+1) . Start with the line segment [0, C1), and make 0 the root. Grow a tree inductively by adding [C;, C;+1) as a branch connected to a random point J;, chosen uniformly over the existing tree. The process is the closure of the union of all branches.
The self-similar continuum random tree (R, v). Start at time 0 with an infinite continuous line [0, oo), and make 0 the root.
At time 0 < t < oo there is a tree composed of the original line and of finite line segments connected with each other; only a finite number of such segments connecting with each finite interval of the original line. The process grows according to the rules (i) in each time increment (t, t + dt), in each segment (x, x + dx) of the tree constructed at time t, there is chance dt dx of a "birth"; (ii) if a birth occurs at time t and place x, then a new branch with random exponential(rate t) length is instantly attached at x. The process is the closure of the tree at time infinity. In these limit processes, regard S and R as random sets, indicating the spatial position of the limit "continuum tree". Then p and v are random measures supported by S and R, representing how the vertices are spread over the tree.
Aldous: The continuum random tree II
33
In other words, with the tree T constructed by Algorithm 1 we associate the
empirical distribution µ of the vertices: p, puts mass 1/n on each vertex. As space-rescaled T converges to S, so does p,, (with the induced spacerescaling) converge to p. Similarly, when edge-lengths of T. are rescaled to length 1/a(n) to get the limit R, let v be the measure putting mass 1/a2(n) on each vertex: then v,,, with the induced space-rescaling, converges to v. As the notation suggests, we shall see below that the CCRT (S, p) constructed above is the same as that constructed from (2x) Brownian excursion in the previous section. 2.4
The Convergence Result for CBP(n)
The results in the previous two sections depended on exact combinatorial relations for finite n, in the two special cases. A natural first step in seeking to generalize is to consider the general CBP(n) model. Neither of the previous methods works: there is in general no constructive algorithm like Algorithm 1 known, and while any tree can be coded as a walk with steps ±1 as in section 2.2, the process obtained from general CBP(n) does not have any standard dependence structure which makes convergence to Brownian excursion look easy to prove. But using different techniques (outlined below), an abstract result "rescaled general CBP(n) converges to the CCRT" is proved in Aldous [3] Theorem 23. Without setting up the precise statement of the abstract result, let us state the concrete consequence (which actually turns out to be equivalent to the a priori stronger abstract result - c.f. [3]) for the rescaled search depth process X" at (11).
Theorem 2 For CBP(n), as n - oo
(Xz;0
An immediate corollary is the result for the rescaled cumulative height profile process H" at (10).
Corollary 3 For CBP(n), as n --> oo (H,,; s > 0) -+ (Ho,/2i s > 0)
(12)
34
Aldous: The continuum random tree II
where
H. =
f
0
1
l(w,<8)dt.
To use an old-fashioned term, the abstract result behind Theorem 2 is an invariance principle: the distribution of the limit tree S doesn't depend on the offspring distribution , except through the s.d. a as a scale factor. (This may be thought surprising - one's first guess might be that a would affect the shape of the tree). As with the classical invariance principle (convergence of i.i.d. partial sums to Brownian motion) one might expect the result to be true for much more general models, and we discuss this briefly in section 4.
A final ingredient of the big picture is an "intrinsically tree-ish" description of the CCRT S. To give this, we need to introduce a different species of tree t. Let t have k labelled leaves, a root with degree 1, and binary branchpoints (and hence 2k - 1 edges). Let the edge-lengths be positive reals, and regard the tree as unordered. Such a tree t can be specified by its topological shape t*, say, and by the 2k - 1 edge-lengths (ii). Define R(k) to be a random tree of this type with density 2k-1
f (t*, 11, ... , 12k-1) = s exp(-s2/2), s =
1,.
(13)
i-1
In other words, the edge-lengths are independent of the shape of the tree, which is uniform on all shapes; moreover the edge-lengths are exchangeable (and hence we didn't need to specify exactly which edge was edge i). These random trees satisfy the natural consistency condition in k. It turns out that the distribution of (S, p) is specified by the fact that the subtree R(k) spanned by k "uniform" (i.e. chosen according to it) random vertices has density (13). More generally, just as ordinary stochastic processes can be specified via consistent families of f.d.d.'s, so ([3] Theorem 3) a random continuum tree can be specified by "random f.d.d.'s", the subtrees spanned by randomly-chosen vertices. The point is that in general there is no "canonical" way of labelling
vertices of continuum trees, so random f.d.d.'s are a natural substitute for ordinary f.d.d.'s. The proof in [3] that rescaled CBP(n) converges is based upon convergence of random f.d.d.'s. Fix k, choose at random k vertices from CBP(n), consider the subtree spanned by these k vertices and the root, rescale and let n oo. Using classical asymptotics for sizes of critical BPs it can be shown that the
limit tree is (a scale factor a-1 times) R(k). In [3] we develop such "exchangeability and weak convergence" techniques as a hopefully useful way of
Aldous: The continuum random tree II
35
establishing convergence of more general models to more general limit continuum random trees. In principle one could seek to prove Theorem 2 directly, by first proving convergence of finite dimensional distributions (X', ... , X n ). In a special model of binary trees this was recently carried out by Gutjahr and
Pflug [26], based on exact combinatorial formulas, but the general CBP(n) model seems less tractable. Direct approaches to Corollary 3 are easier, but not powerful enough to establish the Theorem. We now tie this up with the special constructions of S in sections 2.2 and 2.3.
The construction in section 2.2 of a tree from a function f and points (ti), applied to a sample path of 2W and to k uniform random points, plainly must give a tree isometric to R(k). The connection with the global construction in section 2.3 is more surprising. Let R(k) be the subtree obtained from the first k branches [C3_1, C;], i < k in the global construction. Then a direct computation ([3] section 4.3) shows that R(k) has distribution (13):
R(k) I R(k)
(14)
In other words, t(k) is isometric to the subtree R(k) of S spanned by k randomly chosen vertices of S.
It is intuitively clear that the natural "local" result associated with Corollary 3 should be true. Write H(j) for the number of vertices at height j from the root, and define the rescaled height profile process h; = n-1/2H([nl12s])
Conjecture 4 For CBP(n), as n -+ oo,
(h8;s > 0) - (ZlO8/2;s > 0) where
_ 1°
d
r
ds Jo 1(w<<,)dt
is the total occupation density of Brownian excursion W.
The "weak convergence" methods of [3] are too weak to be used here. In some special cases there are exact combinatorial expressions for means and moments of H(j), and in these special cases one could no doubt establish Conjecture 4, but the general case seems to require delicate analytical asymptotics. Distributional properties are discussed in section 3.2.
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Aldous: The continuum random tree II
Remark: local time convention. Above I use total occupation density for Brownian excursion, and later I use total occupation density for BES(3). These are of course "local times as space-indexed processes", up to normalization conventions. Occasionally I use "local time at a point" as a time-indexed process, still using the occupation density normalization.
Technical note. Using the construction of S from Brownian excursion, we get a random measure p on S induced from Lebesgue measure on [0, 1]: this is the same measure y which occurs as the limit empirical distribution of vertices
(section 2.3). The same applies to (R, v), the SSCRT: in the construction below from 2-sided BES(3), v is the measure induced from Lebesgue measure on the line.
a
figure 2
b --------
Aldous: The continuum random tree II
37
2.5 The Self-Similar Continuum Random Tree Sketched above is the SSCRT (1Z, v) given by the global construction in section 2.3. (The "baseline" is drawn horizontally.)
Recall that standard BES(3) is the process distributed as the radial part of 3-dimensional standard Brownian motion started at 0. We shall be concerned with 2-sided standard BES(3) B = (B37 -oo < s < oo). Here 2-sided means that (Bt; t > 0) and (B-t; t > 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
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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
= a-1(n)X*([2a2(n)s]), -oo < s < oo.
(16)
Theorem 5 For CBP(n), as n -+ 00 (X; ; -oo < s < 00)
(2v-1B87 -oo < s < oo)
where B is 2-sided standard BES(3).
Corollary 6 For CBP(n), as n -> 00 (k,,,; s > 0)
- (Qo8/2; s > 0)
where
(17)
00
Q8 = f
1(B,<8)dt.
Here is the analog of Conjecture 4 for the (local) height profile process.
Conjecture 7 For CBP(n), as n -> 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
Wu + Wu+,-1 - 2
Wt, 0<s<1-u
inf
u+e-1
Wt, 1 - u < s < 1.
Then (20) becomes:
W I WU, where U is uniform on [0, 1].
(21)
This is much less helpful than (20). I find it conceptually helpful to think of trees as purely spatial objects, without any notion of "time" involved. The
Aldous: The continuum random tree II
41
point is that here are many different ways to associate "time" with a tree: the "intrinsic" time mentioned below if different from the notions of time induced by the Brownian excursion construction in section 2.2 and different again from the notions of time in the global constructions in section 2.3. Further, in section 5 we will consider trees as range spaces for random processes, in which setting having a notion of "time" attached to the tree itself is really confusing.
Having said this, recall that in a discrete-time branching process such as CBP(n) we would normally think of the vertices at distance d from the root as "the individuals alive at time d", since we are drawing the family tree with edges of unit length. Analogously, in a continuum tree we may consider
"time" to be "distance from the root" - I call this intrinsic time. Loosely, we may think of the CCRT as a family tree for individuals with infinitesimal lifetimes, the vertices at distance t from the root representing the individuals alive at time t. Thus the processes (It) and (qt) in the previous sections represent population sizes at time t in the CCRT and SSCRT. But the interesting symmetries of our trees, such as (20), involve changes in direction of intrinsic time, and this is why it helps to think of the trees as purely spatial objects.
As illustration, consider the interpretation of the SSCRT as an ancestor process. In section 2.5 the SSCRT was presented as a limit of rescaled CBP(n) as seen from the progenitor. Here the direction of time is indicated in the top diagram in figure 3. But as at (19) and (20) we can look at CBP(n) from the standpoint of a uniform random individual. Then rescaling as in section 2.5 gives the same limit SSCRT. Here the interpretation of the baseline is as the ancestral line back from the random individual V towards the progenitor, and a bush branching off the baseline at b indicates relatives of V whose last common ancestor with V was at (rescaled) time b in the past. See the middle diagram in figure 3. (Incidently, the bottom diagram arises in a context discussed in section 6.) Relations between the limits. There are several relations amongst these processes.
7Z is the "large-scale" limit of T,,,, (the discrete infinite tree), and the "small-scale" local (i..e. around the root) limit of S ([2] Theorem 11). The 1.
latter fact is a translation of the fact that BES(3) is the rescaled limit of Brownian excursion near 0. 2. R can be obtained by attaching to the baseline a o- finite process of (mostly
Aldous: The continuum random tree II
42
t=1.0
-
t=o
t=0.25
t=0
t=-0.25
Figure 3
Aldous: The continuum random tree II
43
small) rescaled copies of S ([2] section 6). In figure 2, the "bush" attached at a arises from the excursion of 2B above a, drawn over the dashed line. This
translates to a "last-exit" decomposition of BES(3) into excursions above levels b ending at the last exit time from b. See section 3.5 for applications. 3. The fact that Brownian excursion is "BES(3) bridge" is suggestive, but I see no solid interpretation in terms of our trees. 2.8 Discussion The preceding sections contain my subjective view of "the big picture". But there is much more one could say about related matters. 1. In classical applied probability, there is a branching process description of the total number of customers served in a busy period of a M/G/1 queue. For a critical M/MI1 queue, this gives a correspondence between the continuous time simple symmetric random walk (number of arrivals - number of departures) and the shifted geometric (1/2) Galton-Watson branching process, and this is exactly the correspondence of section 2.2 translated into continuous time.
2. There is recent theoretical literature on trees associated with Browniantype processes. Neveu and Pitman, whose work is summarized in [37], discuss trees associated with upcrossings of size h in Brownian excursion conditioned to reach height h (instead of conditioning on duration). The trees they obtain
are the family trees of continuous-time critical branching processes where individual lifetimes have exponential(2/h) distribution and are followed either
by death (probability 1/2) or by a split into 2 new individuals (probability 1/2). Their construction resembles that in section 2.2 in that branchpoints correspond to local minima. But fundamental to my set-up is the idea of trees as having distances between vertices, and one really needs to draw the trees as in figure 1 to make this work. Conversely, Waymire et al. ([25],[9] p. 284) start with the continuoustime binary branching process above and show that, conditioning on total population size = n and letting n --+ oo, the time to extinction rescales to 3.
the maximum of Brownian excursion.
4. For the reader interested in pursuing distributional properties of Brownian excursion, relevant papers include Chung [12]; Knight [32]; Salminen [44]; Imhof [27]; Biane [10]; Biane and Yor [11].
44
Aldous: The continuum random tree II
5. As a fanciful analogy, there are two ways to paint a picture on a piece of paper. You can divide the paper into small pixels and paint each in turn; or you can start with broad brush strokes in the middle and then fill in medium and smaller size details. The latter is analogous to the global construction of the CCRT in section 2.3; the former is analogous to its construction from Brownian excursion, where the sample path of the excursion "traces the outline of the tree". 6. Obviously we could replace the CBP(n) model with the model of critical Galton-Watson branching processes conditioned to have height (i.e. number of generations before extinction) greater than h. Then a rescaled h --+ 00 limit is the variant of the CCRT obtained as in section 2.2 from Brownian excursion conditioned to reach height 1 at least. This seems less natural from the viewpoint of discrete random tree models. I do not know if this limit has a global construction like those of section 2.3, or a simple description of random subtrees like (13). 3
DISTRIBUTIONAL PROPERTIES
Obviously branching processes are very amenable to study via generating function methods. Various questions about CBP(n) have been studied by combinatorialists (and some probabilists) using exact formulae in special cases and generating function asymptotics for the general case. Kolchin [33] provides a useful summary of the extensive Russian work in this area. We shall see how well the "weak convergence, continuum trees and Brownian excursion" approach does on these questions.
Height Write G for the height of CBP(n), i.e. the number of generations before 3.1
extinction. Since G. is the maximum of the search depth process, an obvious corollary of Theorem 2 is
Corollary 8 For CBP(n), as n --> 00
n-1/2G -
2Q-1W*
where W* = supo
Expressions for the mean and the distribution of W* are well known in the stochastic processes literature, e.g. Kennedy [29] or [9] p. 85:
EW* = 5/2
(22)
Aldous: The continuum random tree II
45
00
P(W* < x) = 1 - 2 E(4x2k2 - 1) exp(-2x2k2).
(23)
k=1
It is undoubtedly true that all moments converge in Corollary 8, but I did not keep track of moments in [3] so this does not rigorously follow from our approach. The result for means n-1/2EGn -+
2ir
-1
(24)
was established via generating function asymptotics by Flajolet and Odlyzko [22], generalizing various special cases known earlier. The general limit distribution result of Corollary 8 is Theorem 2.4.3 of Kolchin [33]. Special cases have been known for a long time: Renyi and Szekeres [42] studied the "uni-
form random unordered labelled tree" and obtained an expression for the limit distribution which (using Corollary 8) becomes the expression
P(W* < x)
= 21/27r5/2x-3
E00 k2 exp(-k2rr2/2x2).
(25)
k=1
So the right sides of (25) and (23) must be equal. The special case of "uniform random ordered trees" (where of course the result is immediate from the ideas of section 2.2) has also been studied - see Takacs [49] for a recent treatment and references.
Remark. Here and elsewhere, combinatorial arguments typically give local limit theorems, which are stronger than the convergence in distribution obtained by our methods. 3.2 Height Profile Corollary 3 and Conjecture 4 provide a connection between occupation density (1,; s > 0) of Brownian excursion Wt and asymptotic height profiles of CBP(n). In this section we look at the explicit formulas available in the literature. Working directly with Brownian excursion, Knight [32] Theorem 2.3 gives the following expression for the marginal density of l,.
f
- t)
7r2 (.2s2 1
1
fte(y) = 23/2Ir5/2s-3
J fW' (
)f (t, y)dt
where fw* is the density of W* (i.e. the derivative of (23)), and where
f(t,y) = -
(27rt)-1/2 00 1 d'-1 2s
E i=O
z!dyi-1(Y
d2
exp(-(2t-1s2(y+i)2)))
46
Aldous: The continuum random tree II
Convergence of 1-dimensional distributions in the setting of Conjecture 4 follow from classical asymptotics for generation sizes and extinction times in critical branching processes: see Theorem 2.5.6 of Kolchin [33] or Kennedy [30] Theorem 3. This approach leads to the following indirect expression for the density. S
J'(1 - t)-3/2 exp()92,(y/2, t) dt f'. (Y) = 4 8(1 - t) where g,(y,t) is the density whose joint characteristic function 0,(01i82) is given by
1/0'(01,82) =
sinh(s -2i 2) s
-22 2
- i91(
sinh(s -192/2) )2. S
-iB2/2
Finally, by combinatorial analysis of the uniform random ordered tree, Takacs [47] obtains the formula
fi.(y) = 200 E E k 3
)e-(v+2aj)2/2
(
_yl)
Hk+2(y + 2s))
j=1 k=1
where Hk are the Hermite polynomials.
There are simpler formulas for moments, e.g. for means
El, = 4s exp(-2s2) but these are best though of as facts about the distribution of heights of random vertices of S, as in section 3.3. Instead of emphasizing exact formulas (about which I have nothing new to say), let me emphasize some symmetry properties. In terms of CBP(n) with height Gn, there is no offspring distribution for which the height profile process exactly satisfies
(H(j); 0 < j < G,) I (H(Gn - j); 0 < j <
(26)
So from the branching process viewpoint there is no reason to suspect that the occupation time process (1,) has the height-reversal symmetry property
(l,; 0 < s< W*)
d
(lw._8; 0 < s< W*).
(27)
But this is indeed true. Then Corollary 3 gives a sense in which (26) is always asymptotically true as n -> oo. The symmetry (27) and a related identity sup 8
1,
d
2W*.
(28)
Aldous: The continuum random tree II
47
have some relevance to interesting questions about CBP(n). Here is one example: others are in the next section. Odlyzko and Wilf [38] were interested in the maximal height profile
Hn = maxH(j) 7
for CBP(n). This is difficult to analyze by combinatorial methods, and required a lot of work to get a 0(n1/2 log n) upper bound for EH.*. In view of (28), Conjecture 4 would imply
n- 1/2H.
d
and suggest the result for means
n-1/2EHn -, a x/2. Finally, one could consider the sum >J i j H(j) of heights of all n vertices of CBP(n). Corollary 3 implies
Corollary 9
n-3/2 > jH (j) - 2Q-1I
i
where I =
f1
J0
W,ds.
Darling [14] gives an expression for the Laplace transform of I. Takacs [48] gives a combinatorial proof of a special case of Corollary 9 and gives a complicated expression for the distribution of I in terms of infinite sums and special functions.
Heuristics for (27) and (28). These results are a small part of a big picture discussed in detail by Biane and Yor [11]. From my viewpoint they are anomalous because they do not seem to follow from any symmetry property of the continuum tree S. Here are heuristics in terms of branching process asymptotics. Let U be the diffusion on state space [0, oo) with drift rate µ(x) = 0 and variance rate o2(x) = x. This is the continuous limit of the generation size process in a (unconditioned) critical Galton-Watson branching process. More exactly, the limit where the initial population is uon1/2, the offspring
48
Aldous: The continuum random tree II
variance is 1, the population size is divided by n1/2 and the inter-generation time is n-1/2. Thus the limit process in Conjecture 4 (with o, = 1) ought to be the conditioned diffusion
h(Ut;O
(29)
where T = inf{t > 0 : Ut = 0}. Thus we are conditioning on U having an excursion from 0 of area 1.
With this description of l* the "height-reversal symmetry" property (27) becomes intuitively obvious: a 1-dimensional diffusion is reversible, and conditioning on a reversible event preserves reversibility. From Corollary 3 we have
(1g1s>0)
(2l/2is>0)
(30)
where 1 is the occupation density of Brownian excursion W. But there is another way of looking at 1*. Being a drift-free diffusion, Ut is a time-change of standard Brownian motion /3(t). With this time-change representation, a miracle occurs: the conditioning in (29) becomes conditioning /3 to have an excursion of duration 1, i.e. to be Brownian excursion W. Precisely, we get
(h; t > 0)
d
(WL_1(t); t > 0) , where L(u) =
J0
u 1/W, ds.
(31)
Putting together (31) and (30) gives a result of Jeulin (really a conditional form of the classic Ray-Knight description of local time for Brownian motion - see Bianne [10] Theorem 3) saying that the occupation density for Brownian excursion is a random time change of another Brownian excursion. And this relation gives (28).
3.3 Heights of Specified Vertices Asking about asymptotics of heights h,,(v) of particular vertices v in CBP(n) doesn't quite make sense: one has to specify how the vertex is chosen. Obviously Theorem 2 gives one case. Fix s and let v be the [2ns]'th vertex visited in the depth search process: then n-1/2hn(vn)
2a-1W,.
The limit marginal density of Brownian excursion is given by the formula (Ito-McKean [28] section 2.9 (3a)) .fw.(x) = 21/27r-1/2s-3/2(1 -
s)-3 2x2 exp(-x2/(2s(1
- s)))
(32)
Aldous: The continuum random tree II
49
While the limit result (for general CBP(n)) is novel, this way of picking vertices is not particularly interesting from the viewpoint of discrete random trees. Instead, let us consider h(V), where V is a random vertex of S chosen
according to the "uniform" measure p, and h denotes height. So h(V) 2WU, where U is uniform on [0, 1]. As explained below, this has density
fh(v)(x) = xexp(-x2/2)
(33)
and so
Eh(V) =
x/2.
(34)
So Theorem 2 implies the result for uniform vertices V of general CBP(n):
n-1/2hn(V) - o 1h(V)
(35)
and suggests the result for all moments, in particular for means n-1/2Ehn(V) -+ U-1 2/7r.
(36)
These limit results (35,36) for general CBP(n) were proved by generating function methods by Meir and Moon[35] Theorems 4.5 and 4.6 (in special cases,
exact formulas are available). In fact they proved the local limit theorem corresponding to convergence of expectations of 1-dimensional distributions in Conjecture 4:
Eh;
= Q2s exp(-o2s2/2).
Note that implicit in (34) and (22) is the fact that the mean height of S is exactly twice the mean height of a random vertex of S:
Eh(V*) = 2Eh(V) =
27r
(37)
where V* denotes a vertex at maximal height. An explanation of "exactly twice" comes from the stronger fact E(h(V)Ih(V*)) = Zh(V*).
(38)
This follows from the height-reversal symmetry property (27) of the limit height profile process 1*, because s -> 1* is the conditional density of h(V) given S.
There are several ways to understand (33), of which integrating (32) over 0 < s < 1 is the least useful. The most elegant is to use the fact (14) that
Aldous: The continuum random tree II
50
the subtree R(k) of S spanned by k uniform random vertices is distributed as the tree produced by the first k branches in the global construction of section 2.3.
So h(V1) is distributed as the first cut-point in the global construc-
tion, which obviously has density (33). Properties of the joint distribution of (h(V1), ... , h(Vk)) can in principle by obtained from the explicit distribution (13) of the subtree. For example when k = 2, a tree with leaves at heights y1i Y2 has edges of lengths x, y1 - x, Y2 - x (for some x < y1 A y2), and so we can use (13) to see that (h(V1), h(V2)) has joint density f(yl,y2) =
//
yl Ay2
(yl + y2 - x)exp(-(y1 + Y2 - x)2/2) dx.
JO
Note that f (s, s) = El,*2, for the limit height profile 1; (i.e. for Brownian excursion occupation density, up to factors of two (30)). This provides some alternative explanations for formulas in Chung [12] section 6.
3.4 Diameter of the Compact Continuum Tree The diameter On of CBP(n) is the maximal distance between a pair of vertices. The abstract result behind Theorem 2 (or Theorem 2 itself) implies n-1/2/.
-> Q-10
(39)
where 0 is the diameter of the CCRT S. Using the representation of S in terms of Brownian excursion W,
0=2
sup o
(Wtl + Wt, - 2 inf Wt). tl
(40)
Szekeres [46] gave a generating function proof of the existence of a limit in (39) for the special case "uniform random unordered labelled trees". From his result we obtain the following expression for the density function f& (x). 3/ 00
E{
64 4(4b,'
x - 36b,311 x +
75bm
2x f&(x) =
x - 30b,,,,) +
(41)
8 (4b,3,1
x - 10bm x)} exp(-bm,x)
m=1
where b,,,,x
(87rm/x)2.
From this Szekeres computes
E0=3 2ir3EG where G is the height of the CCRT S.
(42)
Aldous: The continuum random tree II
51
As facts about Brownian excursion, (41) and (42) are novel. It is an open problem to establish them directly from (40). Incidently, the argument for (41) is similar to the argument giving (25) for the limit height; so it is likely that (41) has an equivalent expression resembling (23) in format. I want to present an informal "argument by symmetry" which explains the simple relation (42) between mean diameter and mean height. Given the continuum random tree S, choose a point V uniformly in the tree (according to the measure µ) and let G* be the height of the tree rooted at V. Then d G* G by re-rooting symmetry (20). I shall argue informally
E(G*IA) = 3A/4
(43)
which obviously implies (42).
As a preliminary, although µ(S) = 1 by definition, we can think of "S conditioned on µ(S) = c" as the limit of CBP(cn) with o = 1 under the n-1/2 rescaling.
It is clear that S has a unique "center" v, that is a point such that S can be regarded as the union of two trees S1, S2 rooted at v and each having height A/2. These trees have random sizes (µ(S1), µ(S2)) = (Al, A2), say, where Al + A2 = 1. The key fact is conditional on (A, A1), S, and S2 are independent and distributed as S conditioned on having
height = 0/2 and size = Al (resp. A2) One sees this informally by considering the "uniform random unordered labelled tree" model with even diameter, where a "center" exists. In section 3.2 we discussed the height profile process 1; for S in terms of excursions of the diffusion U. Writing 11* for the height profile process of Si we get in the notation of (29)
conditional on (A, A1), (1;*; 0 < s < 0/2)
d
(U8; 0 < s < A/2)IT = 0/2, f 218*ds = A1). 0
But the conditioning preserves the time-reversibility of the excursions of U. Thus, conditional on (A, A,),
f
0
O/2
sll*ds=0/4
52
Aldous: The continuum random tree II
(c.f. the argument below (38)). This expression gives the conditional mean distance from the center to a point Vl chosen uniformly in Si. Clearly the height of S rooted at Vl is this distance plus 0/2. Applying the same result for S2 gives (43).
3.5 Processes Associated with the SSCRT We now turn to the SSCRT 7Z constructed globally in section 2.3, or from 2-sided BES(3) B. in section 2.5. In section 2.5 we discussed the height profile process (q,): here I shall discuss some distributions of other processes defined
in terms of the tree R. For b > 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 b This can be obtained from the description of Fb in terms of BES(3): Fb =
sup 2B,. T;
For by the hitting probability formula for BES(3)
P( sup 2B, < a) = P(infa>0B, > b/2 I Bo = a/2) = 1 - b/a. 0<s
Last common ancestors. Any two vertices of R have a last common ancestor, the point at which the paths from the root to the vertices diverge. Questions like the following are natural in terms of the interpretation of the SSCRT as a limit of critical Galton-Watson branching processes conditioned to survive
Aldous: The continuum random tree II
54
forever. Define C6 to be the distance from the root to the last common
ancestor of all points at distance b from the root (this last common ancestor must be on the baseline). And define Gb to be the distance from the root to the last common ancestor of two randomly-chosen points V1, V2 at distance b from the root (chosen according to conditioned v, the measure with total
mass q,) - this last common ancestor need not be on the baseline. These processes inherit the 1-self-similarity property d
C6
bC1, Gb
d
bG,
as does F6 above.
Proposition 12 (a) P(C1 > c) = (1 - c)2, 0 < c < 1. (b) P(Gi > 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)
if T(1) < 0 < T(2) if not.
Routine but tedious calculations with BES(3) lead to (b).
4 DIFFERENT MODELS FOR RANDOM TREES. Notwithstanding the open questions mentioned in sections 2 and 3, I regard the story of the invariance principle for CBP(n) as now well-understood. From the viewpoint of asymptotics for discrete trees, there are two natural research directions. Similar limits for more general models. I am willing to make a bold conjecture:
Aldous: The continuum random tree II
55
In any reasonable model for random n-trees where the diameter is O(n1/2), the rescaled trees converge to limit processes which coincide with, or can be simply derived from, the limit trees discussed here.
Here is an example of a different model, IMST(n). Start with n isolated vertices. Repeatedly, choose a pair of vertices uniformly and join them by an edge, provided they are not already in the same component. Ultimately a tree is obtained. The discrete infinite tree limit (c.f. section 2.6) for this model was obtained in Aldous [4], This limit is different from the CBP(n) limit, but rescaling the discrete limit tree into a continuum tree gives exactly the SSCRT. This is strong evidence - but nowhere near a proof - that n-1/2 rescaling of IMST(n) gives the CCRT. The "exchangeability" formalizations in [3] are designed to help with examples like this. Loosely, all we need is an argument that
P(D" > xn1/2) -> exp(_x2/2)
(46)
where D is the distance in IMST(n) between two prespecified vertices, and then the techniques of [3] could bootstrap the argument into a proof of convergence to the CCRT. Unfortunately (46) seems difficult. Here are some rather different models where we expect the CCRT limit.
1. Uniform random unordered unlabelled trees.
2. Uniform random spanning trees of expander graphs (e.g. hypercubes) - see Aldous [5]. 3. Steele's [45] "exponential family" of random n-trees, with a parameter determining the mean proportion of leaves. Another interesting application is to random mappings, a well-studied topic surveyed in Kolchin [33]. Here we choose uniformly at random one of the n" functions f : {1,. .. , n} --+ {1,. .. , n} and consider the graph with edges (i, f (i)). The component containing 1 consists of a cycle with attached trees: by representing the trees as in section 2.2, one can represent the entire mapping as a walk of length 2n. In ongoing work with Jim Pitman it is shown that these walks rescale to reflecting Brownian bridge.
Different limit continuum random trees. A much harder topic is the study of asymptotics for random trees whose definition involves the geometry of d-dimensional space. Two combinatorial examples are the uniform random
56
Aldous: The continuum random tree II
spanning trees of Zd studied in Pemantle [39], and the Euclidean minimum spanning trees on random points in Rd studied in Aldous and Steele [7]. And numerous examples such as directed animals and DLA appear in the physics literature. In many of these examples it is natural to conjecture the existence of continuum limit trees after rescaling, but I am not aware of any rigorous proofs.
5 BROWNIAN MOTION ON CONTINUUM TREES. 5.1 Generalities I want to discuss distributional properties of "standard Brownian motion" (Xt; t > 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<,<X,} for Brownian motion on S, i.e. the first time at which the sample path has hit every point of S. Consider the related cover-and-return time
C+ = inf{t > 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
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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
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[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.
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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.
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[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.
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[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.
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[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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Aldous: The continuum random tree II
[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
72
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((
by killing X at the exit time TG = r(G) = inf It > 0: Xt
XtG =
Xt, 0
t>TG.
G}. That is,
Fitzsimmons: Harmonic morphisms
73
An alternative description of U is provided by the harmonic measures HG(x, ) defined for open G C E by HG(x,f) = HGf(x) = Pz(f(XTG); TG < () = Px(f(XTG)) (Here and elsewhere we use the convention that a function f : E -+ IR is extended to E U {A} by setting f (A) = 0.) A function u: G --+ [0, oo] lies in
U(G) if and only if it is lower semi-continuous and Hwu < u on G for all open W with W C G. Moreover a bounded continuous function h : G -+ IR is harmonic on G if and only if Hwh = h on G for all W as before. The path-continuity of X is reflected in the fact that the measure HG(x, ) is carried by the boundary of G if x E G. Let
1=h+po
(1)
be the Riesz decomposition of the constant function 1 into harmonic and potential components. We assume without loss of generality that po = Uk,
(2)
where k is Borel measurable and 0 < k < 1. (This can always be arranged by substituting p + po for p, which amounts to performing a time-change on X, but has no effect on thev-harmonic space (E,U).)
Let (F, V) be a second T-harmonic space with 1 E V(F), and let q be a bounded continuous strict potential q. Let Y = (Yt,Q') denote the associated Hunt process. We use V to denote the potential kernel for Y, and we assume that the analog of (2) is valid:
1=h+ve,
(3)
where h is V-harmonic and 0 < 2 < 1. Recall that a continuous additive functional (CAF) of X is an (.Ft)-adapted real-valued process A = (At) such that Ao = 0, t --r At is continuous and increasing, and At+s(w) = At(w) + A9(6tw),
s, t > O, W E Q.
(4)
Fitzsimmons: Harmonic morphisms
74
That (4) holds with no exceptional w-set is a consequence of a "perfection" theorem found in the appendix of Getoor [G]. By the same theorem we can (and do) assume that any CAF is adapted to the filtration (.F't+), where F is the universal completion of o {Xe; 0 < s < t}.
Let cp: E -. F be continuous, let A = (At) be a CAF of X with inverse B(t) = inf{s : A8 > 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
n=0,1.....
Note that if XS _ = A then Xt = A for all t > (0 + ... + (n. Thus, at the time of the "nth death" we resuscitate if possible (i.e. if
E E);
otherwise the killing is final. It is intuitively clear that X is a Markov process whose lifetime is predictable. Therefore 1 is harmonic for X. Moreover, it
turns out that if cp is a harmonic morphism between X and Y, then it is also a harmonic morphism between X and Y (the resurrection of Y). Thus
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77
the main result of [CFO] applies to cp, X and Y, and Theorem 1.3 obtains after we unravel the connection between X, Y, X and Y.
The construction of X outlined above is a special case of a general resurrection procedure studied by Ikeda, Nagasawa & Watanabe [INW], Meyer [M], and others. The present case is simpler than the general case since X and k are related by a multiplicative functional. In fact, as the reader can easily prove from the above description, the semigroup (Pt) of X is the (minimal) solution of
Ptf(x) = Ptf(x) + JfO,
-f(y)rx(ds,dy),
(7)
]
where I'x(ds,dy) = Px(C E ds,XS_ E dy). In view of (6), (7) can be cast into a more useful form: Ut
Ptf(x) = Ptf(x) + Px
Pt_3f(X3)k(X,) ds)
which suggests that
Ptf(x) = Px(f(Xt) eXp(f t k(X,) ds))
(8)
0
In fact, in the next section we shall see that (8) is appropriate as a definition of the resurrected process X. We close this section with a bit of notation. Recall [BG, III] that a (positive) multiplicative functional (MF) of X is a [0, oo]-valued process M = (Mt)
adapted to (.Ft) such that Mt+, = Mt + M,o9t for all s, t > 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).
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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,(
oo,
otherwise.
(9)
In particular, if G C E is open, then {TG = S < oo}
{TG = Si < oo}.
Also, since U1 = p is bounded we have 00 > U(x,1G) = Px
1G(Xt)
(10
dt)
,
W
soPx(TG
= Pz(Ci = oo) = h(x).
Thus h isU-harmonic. Since 1 E U(E) and h < 1, we have u = 1-h E U(E). If (G,,) is a sequence of precompact open sets with union E, then clearly
Fitzsimmons: Harmonic morphisms
79
-rGn T C. Therefore
limHGnu(x) = limPx(C= < oo,TG, < C) = limPx(-rGn < (t < oo) n
n
n
= Px(TGn < )Y < 00, Vn) = 0, since (Y is totally inaccessible. The last assertion of the lemma now follows
immediately. 0 Let K = (Kt) denote the dual predictable projection of the additive functional t H 1 {S;
Vx E E.
Recalling (2) and Lemma 2.1 we see that
P3(KS) = Px(K,,,) = Uk(x),
Vx E E.
(10)
It follows from (10) and [BG, V(2.8)] that
Kt =
ft
J
k(X3) ds,
t > 0.
0
Consider now the Do leans equation:
dLt = Lt_ dJt,
t > 0,
L0 = 1.
(11)
As is well known, (11) has a unique (semimartingale) solution (Lt). Since J is a martingale additive functional of X, L is a positive local martingale multiplicative functional of X. In particular L is a supermartingale, so if T is any stopping time,
Px(LT) < 1,
Vx E E.
Moreover we have the explicit formula (cf. [S, A4]) Lt = exp(Kt) (1{t
(12)
We now introduce the resurrected process X = (X, L). That is, X is the strong Markov process on E with semigroup Pt f (x) =
PZ(f(Xt)Lt).
Fitzsimmons: Harmonic morphisms
80
The process X is realized as the coordinate process on the same path space SZ as X under the laws {Px, x E E} determined by
Px(d';T < () = Px(ib - LT;T < (),
(13)
o {X97 0 < s < t}, and 4D is where T is any (.Ft+)-stopping time, F to bounded and .PT+-measurable; cf. [S, §62]. (We write ( rather than ( on the left side of (3) only for emphasis; actually (w) = ((w) for all w E ft)
Using the fact that Lt = exp(Kt) > 0 if 0 < t < (, it is easy to check that X is a standard process. In fact, much more is true:
2.2 Proposition X is a Hunt process with continuous paths. The lifetime ofX is predictable, so the constants are X-harmonic. If G C E is a precompact open set such that x E--> Pz(TG) is bounded, then the harmonic space associated with the killed process XG is a '3-harmonic space. So as not to interrupt the main argument, we defer the proof of Proposition 2.2 to section 3.
The same construction yields a process Y, the resurrection of Y. In the sequel the overbar will be used to signal objects defined relative to X or Y. (We also use N to denote the closure of the set N, but no confusion should arise.)
We now prove that the harmonic morphism property of the mapping cp is preserved by the resurrection procedure. For this we require a lemma, which is stated in terms of X, but applies just as well to Y, X, or Y. In what follows we say that a set G C E is an exit set (for X) provided G is precompact and open, and x H Px(TG) is bounded. It is known [BG, p.240] that there is a countable cover of E consisting of exit sets.
2.3 Lemma Let u > 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 <
= e(Hwl - Px(T < )) = ePz(T = C) = EPZ(KT) < EP'(-r) < EPx(T).
Fitzsimmons: Harmonic morphisms
82
Thus, since Hwc(x) = c,
u(x) - Hiyu(x) > -EPz(r) and the proposition follows from the lemma. 0
Proof of Theorem 1.3. The implications and (iii)=;,(i) can be proved by straightforward modifications of the arguments used in [CFO]. The only new ingredient is the identity
Px(f(Xr(G)-); TG = b) = P'(f(XCi-); (i < TG) = Px
f
r(G)
f(Xt) dKt
0
which is an immediate consequence of the definition of K. Passing to the proof of (i).(ii), let us assume that cP is a harmonic morphism
between X and Y. We begin by noting that it suffices to prove that there is a CAF A of X such that Px (10
r [f°W(Xt) - (f' Vf)°,P(Xt)] dAt)
(14)
+Px (f Vf°W(Xt)dKt/ +HG(Vf°W)(x), 0
where G is any precompact open set, r = rG, and x E G. Indeed suppose (14) has been proved, and recall the Riesz decomposition 1 = h+V2 where h is Y-harmonic. Taking f = 2 in (14), and then discarding the XG-harmonic
terms, using the fact that 1 - Uk is X-harmonic, we obtain
P (f o
r
P I f r(QI)o (Xt) dAt) \o
dKt I
/
+ p*
on G,
(15)
where p* is the XG-potential part of hop. Now by Lemma 2.1 and (13) r(G)
h(y) = Q'(rc < (Y)) = QY exP - f o
Q(Yt) dt I
/
; Td < ((1') > 0,
so by the uniqueness theorem [BG, IV(2.13)] for potentials of CAF's, applied here to XG, it follows that t
Kt = fo £oW(X,) dA, + Ct, t < r, 0
Fitzsimmons: Harmonic morphisms
83
for some CAF C. Consequently [BG, V(2.8)] there is a Borel function v with 0 < v < 1 such that t
t
J0
2°cp(Xe) dA9 =
J0
v(X9) dKB,
t < r.
(16)
Substituting (16) into (14) and writing ir = 1 - v, we finally arrive at
(Jr V f (po(x)) = Px
f °(Xt) dAt/
+ Px (fT
on G,
dKt) +
which is formula (5) in condition (ii) of Theorem 1.3, as desired.
For later use we note that the above argument reveals that Px(AT(G)) is bounded above by q°cp independently of G.
Formula (14) will be proved in three steps. The first of these is the main argument and it amounts to proving a localized version of (14). In the final two steps the localization is lifted. Recall that G C E is an exit set for X if G is precompact and open, and x H Px(rG) is bounded.
1° Let G be an exit set for X, and assume that G C F is an exit set for Y such that G C W-1 (6). Let r (resp. f) denote the first exit time of X (resp. Y) from G (resp. G). Let v(x) = Qx(fo f (Yt) dt), where f is a bounded positive Borel function on F. We are going to show that there is a CAF A=A G of XG such that
(fr
vMx)) = Px
dAt (17)
for all x E G. In view of Proposition 2.4, So is a harmonic morphism between k G and YG;
because G and G are exit sets we can apply Theorem 1.1 of [CFO]: there is a (perfect) CAF A of XG such that
V *fMx)) = Px f f °cp(Xt) dAt + HG(V* 0
x E G.
(18)
Fitzsimmons: Harmonic morphisms
84
Here V* is the potential kernel for the process YG, and HG is the harmonic measure for ±. Similarly, we write V * for the potential kernel of Y, and if B is a CAF Uag(x) := P2 107, g(Xt) dBt,
Us9(x)
Px J T 9(xt) dBt. 0
The following variants of the resolvent equation are simple consequences of (13): (19)
UAg=UAg+U*(k'UAg)=UAg+U*(k'UAg);
(20)
(21)
Now replace f in (18) by 2 V*f and subtract the resulting identity from (18). Using (19)-(21) we obtain
V*f(W(x)) = UA(f°cp - (Q'
HG(V*f°W)(x)
= UA(f °cp - (e '
HG(V*f°P)(x)
+ [UK(UA(f°o - (f V*f)°cp) + HG(V*f°so))(x)] But the term in square brackets is evidently equal to UK(V* f °cp), so (17) follows. A variant of the above argument shows that if N is an open subset of G, then (17) remains valid (on N) if r = TG (resp. G) is replaced by TN
(resp. N). 2° We use the notation of 1°. In addition we assume that there is an open
set D C G such that G C cp-1(D). Suppose h E V(G) is bounded and Y-harmonic on G. Then on G we have h°cp = UK(h°cp) - UA((fh)°cp) + HG(h°cp)
(22)
To see this let N be an open set with N C N C G, so that cp(N) is a compact subset of G. Since h E V(6), there is a sequence (V * f,,) of Y6potentials with V * f,, T h; since h is Y-harmonic on 6 we can assume that
Fitzsimmons: Harmonic morphisms
85
fn=O on cp(N) for each n. By 1*, r(N)
V *fn(cp(x)) = Px J
V * fn°cp(Xt) dKt
0
(t.
Px
dAt
(JT(N)
Letting n --> 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
(p
on
1 = Px(LT )
=
Tn < C) + Pz(LT; Cp
TO
=Pz(LT; Tn <()=Px(Tn <(), because of (13). Moreover, by (12) Px(T < C) = Px(LT; T < C) = Px(eKT ; T < (i,T < C)
=Px(eKT;T 0 the a-potential kernel Ua = f0 e-«tPt dt is a strong Feller kernel.
3.2 Lemma For each a > 0, Ua f is continuous whenever f is a bounded Borel function on E. Proof. In view of the resolvent equation, it is enough to prove the assertion for one a > 0, say a = 2. But we can view 02 as the potential kernel of the process (X, exp(- ff (2 - k(X.,)) ds)). By the perturbation theorem (V.2.7) of [BH], the strong Feller property of U is inherited by U2. 0
Lemma 3.1 and the next result imply the assertions in the first sentence of Proposition 2.2.
Fitzsimmons: Harmonic morphisms
89
3.3 Lemma For a > 0 and G an open subset of E, let Ua(G) denote the class of a-excessive functions of VG. Then (E,L ') is a 13-harmonic space. In particular X is a Hunt process with continuous paths.
Proof. We first need to check that the cone of excessive functions of the process (X, e-at) satisfies the axioms (B1)-(B4) on p.57 of [BH]. Axioms (B1)-(B3) follow from the fact that X is a right process with a reference measure. Moreover (B4) follows from the perturbation argument used in proving the previous lemma in case a > 1. To prove (B4) for a E]0,1] we must show that for each bounded positive Borel function f on E there exists a continuous a-X-excessive function v > 0 such Ua f /v E Co(E). See [BH, II.4.6]. But U" f E lfa(E) C U2(E), so there exists a continuous v E U2(E) with v > 0 and U' f /v E Co(E). Define v = v" + (2 - a)Uav. Then v > and because of the resolvent equation v E ua(E). In view of Lemma 3.2
0 < Uaf/v < Uaf/v E Co(E), so (B4) follows. The Hunt property for X is now a consequence of the general theory of balayage spaces; see [BH, IV.7.6]. It is clear that X has continuous paths on [0,C[ because of (13) and the analogous property of X. The quasi-left-continuity of X and the predictability of (Lemma 3.1) imply that X. = A on { < oo}, so t H Xt is continuous on all of [0, oo[. It now follows that (E,W) is a q3-harmonic space; see [BH, III,§8]. 0 If G is an exit set for X, then the argument used in the proof of Lemma 3.3 is easily modified to show that the sheaf of positive hyperharmonic functions associated with the killed process XG forms a 93-harmonic space. This
proves the final assertion of Proposition 2.2. 0
References [BG] Blumenthal, R.M. and Getoor, R.K. (1968). Markov Processes and Potential Theory. New York: Academic Press.
[BH] Bliedtner, J. and Hansen, W. (1986). Potential Theory. Universitext, Berlin: Springer. [CC65] Constantinescu, C. and Cornea, A. (1965). `Compactifications of harmonic spaces', Nagoya Math. J. 25 1-57.
Fitzsimmons: Harmonic morphisms
90
[CC72] Constantinescu, C. and Cornea, A. (1972). Potential Theory on Harmonic Spaces. Berlin: Springer. [CFO] Csink, L., Fitzsimmons, P.J. and Oksendal, B. (1990). `A stochastic characterization of harmonic morphisms'. Math Annalen 287 1-18.
[G] Getoor, R.K. (1990). Excessive Measures. Birkhauser, Boston. [INW] Ikeda, N., Nagasawa, M. and Watanabe, S. (1966). `A construction
of Markov processes by piecing out'. Proc. Japan Acad. 42 370375. [M]
Meyer, P.-A. (1975). `Renaissance, recollements, melanges, ralentissement de processus de Markov'. Ann. Inst. Fourier Grenoble 25 465-497.
[S] Sharpe, M.J. (1988). General Theory of Markov Processes. New York: Academic Press.
Statistics of Local Time and Excursions for the Ornstein-Uhlenbeck Process J. HAWKES and A. TRUMAN Department of Mathematics and Computer Science University College of Swansea SWANSEA SA2 8PP
Wales, U.K.
ABSTRACT The Poisson-Levy excursion measures for the Ornstein-Uhlenbeck process are calculated explicitly by utilising some simple connections with the quantum mechanical harmonic oscillator.
1 INTRODUCTION In this paper we discuss the local time and excursion theory for the OrnsteinUhlenbeck process X (E R), satisfying the Langevin equation dX(t) = -X(t)dt + dB(t)
(1)
,
B being a BM(R) process, with K(B(t)B(s)) = min(s,t) , I being the expectation with respect to Wiener measure µ .
Nelson showed that the Ornstein-Uhlenbeck process is associated with the stochastic mechanics of the ground state of a quantum mechanical harmonic 2
oscillator with Hamiltonian H = 2-1( -a2 +x2) (Refs.(4), (5)). Here, utilizing dx
some results of Truman and Williams (Ref.(8)) and bringing together some quantum mechanical results for the harmonic oscillator Hamiltonian, we obtain detailed information for the statistics of the local time and excursions for this Ornstein-Uhlenbeck process. The local time at a upto time t, La(t) , is defined by
La(t) = lim h.Lo
h-1
h
Leb IS E (O,t) X(s)E (a - 2, a +
h2)
,
(2)
Hawkes & Uhlenbeck: Statistics of local time t or formally La(t) = S(X(s) - a) ds . Defining the corresponding subordinator 92
0
ya by
ya(t) = inf (s > 0 : La(s) > t)
t>0,
,
(3)
we shall show that for a = 0
{ e-a'70(t) } = exp
{ -t
j 00(1
-
e-)'s)
0
dv0(s) }
,
(4)
for each ? > 0 , where duo(s)
ds
3
_ 2 e s (l- a2s) 2 47E
The map s i-+ X(s) is p a.s. continuous so for each a e R ( s > 0 : X(s) >< a) is an open subset of the real line which can be decomposed into its component intervals - the excursion intervals. Define #o(s,t) by
#o(s,t) = the number of excursions away from 0
of duration s to (s + ds) upto the local time at 0 equals t . Then the last result is equivalent to
IP( #o(s,t) = N) = etdvo(s)
(tdvN! ,
for N = 0, 1, 2,..., (6)
v0 being given by Eq.(5). In this paper we establish more general results for excursions above or below any point a, namely: if a (s,t) denotes the number of excursions below a, respectively, then _ -tdv}(s) a )P(+a(s,t)=N)-e
where
(tdva (s))N
N!
,forN=0, 1,2,...,
(7)
Hawkes & Uhlenbeck: Statistics of local time ( ds
D + ± - I 2 Y) e Y2I'2d Y
J
dya(s)
2 so ±+a2
an
Y >
n
93 2
-an (
{ D_-+(± 2y) } 2 dy
f
y >
a"
(8)
D being Weber's parabolic cylinder function, a = an being the successive values of a for which
D_a (±'/2a) = 0
.
(9)
As we shall see, Eq.(5) is a special case of Eq.(8), corresponding to setting a = 0, with va = v a + va . The above va is called the Poisson-Levy excursion measure.
Defining L±(t) = Leb { s e (0,t) : X(s) >< 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 >
12
dy }
2
, (38) yf
a{
2 dy
respectively.
Proof Because VO(x) T o as I x I T -, the Hamiltonian H+ has pure point spectrum f 1-21-
a±n
1
H+ having corresponding eigenfunctions D_ n=0,1,2,... ,
(+'2x)
(before normalisation), respectively. This is a consequence of singular boundary
100
Hawkes & Uhlenbeck: Statistics of local time
value theory (see Chapter 10 in Ref.(3), in particular p.524). The resulting spectral theorem for H+ gives the reality of the spectrum of H+ and the above
expressions for va . If We state without proof here that va give the excursions above a, respectively below (see Ref.(8)).
For a = 0 evidently an = a n as expected. These a values can be found from the condition D-a(0) = 0. Using Eq.(17), a must satisfy:
r-1(2 + a) = 0
(39)
.. an = an = -1 - 2n , for n = 0, 1, 2,
.....
(40)
The corresponding eigenvalues of H+ are ?, where
X-1 = 2n+1, f o r n = 0, 1, 2, .....
(41)
?=
(42)
+(2n+1), for n = 0, 1,2...... 2 1
The corresponding eigenfunctions are H2n+i (x) e 2
x
2
,
H2n+1 being the odd
Hermite polynomials, which are zero at the origin, so satisfying the Dirichlet boundary condition. (Hn(x) = ex2(-
)n e
2 ) . It only remains to evaluate
the inner product in the numerator of Eq.(38) to confirm that Eq.(5) is a special case of Eq.(38) with a = 0. The key to this is the observation that for n = 1, 3,
5, ... f p
°°
2
n-1
e xHn(x) dx = dxn -1 e x
2
(43)
I
x=O 2
and r.h.s. can be evaluated from Maclaurin's series for e -x
Hawkes & Uhlenbeck: Statistics of local time
101
References (1)
M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970.
(2)
I.T. Gihman and A.V. Skorohod, Stochastic Differential Equations,
(3)
Springer, Berlin, 1972. E. Hille, Lectures on Ordinary Differential Equations, Addison Wesley,
(4) (5) (6) (7) (8)
(9)
Reading (Mass), 1969. E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, 1967. E. Nelson, Quantum Fluctuations, Princeton University Press, Princeton, 1984. M. Reed and B. Simon, Mathematical Physics, Volume 2, Fourier Analysis and Self-adjointness, Academic, New York, 1975.
B. Simon, Functional Integration and Quantum Physics, Academic, New York, 1979. A. Truman and D. Williams, A Generalised arc-sine law and Nelson's stochastic mechanics of one-dimensional time-homogeneous diffusions, in Diffusion Processes and Related Problems in Analysis, Birkhauser (to be published).
D. Williams and L.C.G. Rogers, Diffusions, Markov Processes and Martingales, Volume 2, 116 Calculus, Wiley, 1987.
LP-CHEN FORMS ON LOOP SPACES J.D.S. JONES AND R. LEANDRE
University of Warwick and Universite de Strasbourg
INTRODUCTION
Let M be a Riemannian manifold. Our aim is to study differential forms on the following infinite dimensional manifolds: (1) the path space PM consisting of paths w : [0, 1] -> M,
(2) the loop space LM consisting of paths w such that w(0) = w(1), (3) the based loop space LxM consisting of loops w such that w(0) _ w(1) = x where x is a chosen base point in M. One consequence of the fact that these manifolds are infinite dimensional is that there are infinite sequences a,, of forms with each a,, homogeneous of degree n. These infinite sequences are very important in the geometrical applications of loop spaces; for example they are essential in the theory of equivariant cohomology in infinite dimensions as is made quite clear in [25]. In [11] Chen describes the theory of "iterated integrals"; this is a method of constructing differential forms on these infinite dimensional manifolds. We will refer to forms constructed by this means as Chen forms. The purpose of this paper is to study some of the analytical properties of Chen forms; in particular to make estimates for suitable LP-norms and to consider
various decay conditions which one might put on the terms in an infinite sequence of the kind mentioned in the previous paragraph. The motivation for studying forms on the loop space comes from the relation between the S'-equivariant cohomology of the loop space of M and the Atiyah-Singer index theorem [1], [7]. Here the circle acts on the loop space by rotating loops. In [17] and [16] this link between index theory and loop spaces is studied using Chen forms (called iterated integrals in [17] and [16]) and it is shown how the cohomology and the equivariant cohomology of the loop space LM can be computed using Chen forms. The result is best expressed using Hochschild homology, for ordinary cohomology, and Connes's theory of cyclic homology, for equivariant cohomology. Thus there is a rather remarkable interaction between loop spaces, index theory, and cyclic homology. More information and further references can be found in [1], [7], and [17]. These ideas are summarised in [24].
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Jones & Leandre: LP-Chen forms on loop spaces
Because of this connection with index theory we are particularly interested in LM. However there is no advantage to be gained by restricting attention to this case alone. Indeed at several points it is simplest to argue first with the full path space PM and then deduce the corresponding result for the loop spaces LM and LxM. Thus we take care to develop the theory in all three cases. In [17] the loop space means the smooth loop space L°°M considered as an infinite dimensional manifold modelled on the Frechet space C°°(S', ]Rd)
where d is the dimension of M. Of course the natural measures coming from stochastic analysis do not make sense on the smooth loop space and so to do LP-estimates it is necessary to work with the full loop space LM of continuous loops. This provides us with our first problem: give a precise definition of what we mean by differential forms on LM and extend Chen forms to differential forms on LM. This, inevitably, requires stochastic analysis. It leads to the theory of stochastic Chen forms which is described in detail in §1. Stochastic Chen forms are also described in [16], where they are called stochastic iterated integrals. Our approach, in keeping with our aims, is based on very down-to-earth estimates and is very different from that of [16]. The essential estimates, on which our work is based, are mostly carried out in §4 and §5. In §2 and §3 we introduce natural LP-norms on the space of forms on LM and study the completions of the space of Chen forms in these LP-norms. These completed spaces are the spaces of LP-Chen forms. In forming these completions we complete the direct sum of the homogeneous terms, not just each of the homogeneous terms separately. So the spaces of LP-Chen forms
contain infinite sums E a,r where each of the terms a is homogeneous of degree n; we will refer to such sums as forms of infinite degree. The condition that such a sum has finite LP-norm imposes decay conditions on the terms a,,. We show that these spaces of LP-Chen forms have the minimum algebraic properties we might hope for: the intersection of all the LP-spaces is closed under exterior products of forms and interior products of forms with vector fields. One way to think of these constructions is to use the following analogy. If we take the Hilbert space H = L2 [0, 1] we can consider two, isomorphic, versions of the associated symmetric Fock space. The first, which is essentially algebraic, is defined as a completion of the sum of all the symmetric powers of H. Thus we allow infinite sums E ,Q,,, where /3, is in the n-th symmetric power of H, which satisfy certain decay conditions. The relation between these two versions of the symmetric Fock space is given by iterated integrals and Wiener chaos. In our context the algebraic space is constructed as follows. Let Q be the space of forms on M and Q the space of forms of strictly positive degree. Then we take C = E SZ 0 0' suitably completed. The link between the algebraic space and the space of forms on
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the loop space is provided by stochastic Chen forms.
The similarity with the theory of stochastic differential equations appears very naturally. In [9], the convergence, in LP, of the Picard series of a linear equation is studied; the same problem is studied in [35] for the general equation. In both cases the components of the equation are scalars. Exponential forms on loop spaces are, formally, solutions of a linear equation whose components are differential forms and they give forms of infinite
degree. Bismut's equivariant Chern character [7] is an example of such a form. We choose our LP-norms so that the path ordered exponential series is absolutely convergent in all the LP. These exponential forms, which are allowed to take their values in a matrix algebra, could be called the coherent forms on the loop space, in analogy with quantum physics. In the theory of cyclic cohomology, Connes's entire cyclic cohomology, see [13], [18], and [26], is designed for "infinite dimensional cyclic cocycles": this means we allow infinite sums E Wn, where each term is homogeneous of degree n, but impose decay conditions. We make the link between our spaces of LP-Chen forms and Connes's entire cyclic cohomology and so give a simple condition under which a form of infinite degree lies in the intersection of the spaces of LP-Chen forms. One of our main theorems is that if an infinite sum EWn E C satisfies an algebraic condition, then the corresponding stochastic Chen form on the loop space belongs to all the spaces of LP-Chen forms.
This condition shows that Bismut's equivariant Chern character, see [7] and [17], belongs to all the spaces of LP-Chen forms. The method of proving this is to use the construction of the equivariant Chern character given in [17] in terms of Chen forms and then to use estimates to show that this infinite sum satisfies the appropriate algebraic condition. The subject is a mixture of homological algebra, geometry of loop spaces, and stochastic analysis. Many of the constructions have a natural interpretation in terms of homological algebra, compare [17]. We have tried to point
out these links but, in order to keep the paper to a reasonable length, we have not tried to describe them in much detail. We would like D. Elworthy and M. Emery for many helpful discussions and comments on the contents of this paper. The first author would also like
to thank Ezra Getzler for many invaluable discussions on Chen's iterated integrals which have been a major influence on some of the ideas behind this paper.
§1 STOCHASTIC CHEN FORMS
Let M be a compact Riemannian manifold of dimension d and let PM, the path space of M, be the Banach manifold of all continuous functions w : [0, 1] -+ M. The loop space LM is the submanifold consisting of those
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w such that w(0) = w(1). The based loop space, LxM is the submanifold of loops w such that w(0) = w(1) = x where x E M is a chosen base point. Note that
LxMCLM CPM and each manifold has codimension d in the next. Let dPx,y be the usual Brownian bridge measure on the space of paths Px,yM which start at x and end at y; we use the notation Ex,y for expectations computed with respect to this measure. We use the measure pi (x, x) dx dPx
on LM, where pt(x, y) is the heat kernel of M. This is the unique measure of the form
f(x)dxdPx which is invariant under rotations of loops, see [7] and [15]. We use the notation EL for expectations computed using this measure on the loop space. We use the measure dP = pi (x, y)dx dy dPx,y
on PM, this is the equilibrium measure for Brownian paths, and E for the corresponding expectation. Let Tx = TTM be the tangent space to x E M and, given a path w in M, let
- Tw, be the (stochastic) parallel transport along to. This operator is a linear isometry and it is defined for all continuous paths w, except for a set of T9(w) : T
0
measure zero. Our discussion is based on two simple observations:
(1) The tangent space to the Banach manifold PM at a continuous path w is the space of continuous vector fields along to. (2) Parallel transport defines an isomorphism from vector fields along to to a space of paths in T,,,,,. We begin by defining suitable Hilbert spaces of paths in a d-dimensional
inner product space T. Let 7-1(T) be the space of paths X : [0, 1] --> 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.
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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 <
sE[o,1]
,
for n # 0.
InI
Now let w be a path in M and let e1,.. . , ed be an orthonormal basis for
Two. Fori=1,...,dandnEZwedefine en,i E x(Two),
en,i(s) = bn(S)ei,
en,i E 'Hw,
en,i(S) = T9(w)(en,i(S))
The en,i are an orthonormal basis for the Hilbert space 7l(Two) and the en,i an orthonormal basis for fw.
It is possible to obtain a basis for 'Hw and Tw by a similar, but more intricate, construction. However we shall avoid doing this and refer all our computations to the bases (1.3) for 7-lw and 'H(Two). The reason for doing this is that the subspace Tw is an "anticipating" function of w in the sense that it involves the holonomy around the loop w. In particular any natural construction of a basis for Tw will involve anticipating terms and lead to anticipating stochastic integrals. Choose an orthonormal basis el(x), ... , ed(x) for Tx such that each ei is a measurable function of x. Such a basis always exists as can be verified by a straightforward argument. Let en,;(w) be the orthonormal basis for 7{w defined by el(wo),...,ed(wo). Now if b is any section of the field of Hilbert spaces xw then we say that b is measurable if each of the functions w'-' (b(w),en,i(w))
is a measurable function on PM. It is straightforward to check that this notion of measurability does not depend on the choice of basis ei(x). The
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LP-norm of a measurable vector field is defined to be the LP-norm of the function jjb(w)jj on PM. If b is a section of the field of Hilbert spaces T,,, then b is measurable if the functions w -4 (b(w),e,,,i(w)) are measurable
functions on LM and the LP-norm of b is the LP-norm of the function w i-a jjb(w)jj defined on LM. The definition of measurability for sections of fy,, is identical. We shall refer to measurable sections of these fields of Hilbert spaces as measurable vector fields. In the case of the loop space LM it can be checked that these notions are invariant, in the natural sense, under the action of the circle given by rotation of loops. We shall now give the definition of measurable forms. The main reason for
this is to be explicit about our conventions for the constants which appear in the theory of differential forms. Often these conventions do not matter but since one of our main purposes is to keep track of decay conditions we need to be precise. We use the notation IJI for the number of elements in a finite set J. Let bi, i E I, be any set of measurable vector fields on PM indexed by a totally ordered set I. If J is a finite subset of I we write bj for the ordered I J I -tuple
given by the b3 where j E J. A measurable r-form a on PM is defined to be a continuous, multilinear, alternating function
defined for almost all paths w, with the property that given any r measurable vector fields b 1 , . .. , b,. the function W --> aw(bi (w), ... , br(W))
is a measurable function on PM. The definition of measurable forms on LM and LxM is identical. To define the exterior product, and also for later use, we recall the definition of shuffles. Let I = {l, ... , r}; then a (rl, ... , of I is a partition of I into subsets Il, ... , In with jIk = rk. Let (rl , ... , be the set of (rl, ... , r,,)-shuffles of I; there are r!
r1!...r,,! such shuffles. A shuffle defines a permutation of I by re-ordering I as
11U12U...UI,and sign(Ij, ... ,
is defined to be the sign of this permutation.
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Let o be a measurable r-form and r a measurable s-form on PM. The exterior product a A T is defined by the formula sign(I, J)Q(bj)T(bj).
(a A T)(bl, ... , br+9) _ (I,J)E4'(r,s)
The interior product of a measurable r-form a with a measurable vector field b is defined by ib(O')(bl,...,br-1) =Q(b,bl,...,br-1).
Let en,i be the orthonormal basis for f,,, defined by a measurable orthonormal basis el (x), ... , ed(x) for T. This basis is indexed by Z x {1, ... , d} ordered lexicographically. Let On,i(w) denote the dual basis of 1-1,*,,. Now let I= {(n1,il),...,(nr,ir)}
be a finite ordered subset of Z x { 1, ... , d} and define #1(w) by the formula
#1 = inl,i, /1 ... A 13n.,i.
(1.4)
With our conventions for exterior products /31 is the unique r-form such
that (3I (enl ,i, , ... , en.,i. )w = 1
and (31 vanishes on any r-tuple of the en,i indexed by a set other than I. The ,QI are a basis, over the space of measurable functions on PM, for the space of measurable forms on PM. Now we introduce Chen forms as a means of constructing differential forms on PM. The original definition is due to Chen [11], and the general theory of Chen forms on the smooth loop space is extensively studied in [17]. Here we study stochastic Chen forms in the Stratonovitch sense. Let S2 be the space of all smooth forms on M and let Sl be the subspace of all forms
of degree r with r > 1. Let w : [0, 1] -> S1®n,
S H W1,9 ®... ®wn,s
be a map which is piecewise constant. We suppose that each wi is homogeneous and define ri = deg wi - 1.
Let On be the simplex
A" = {(sl, ... , S.) E R': 0 < sl < ... < sn < 1}.
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Now define (1.5)
o(w)=J wl(dw,-) A"'Aw,,(dw,-)9n. On
We must explain how o(w) is a measurable form on PM of degree
r=r1+"'+rnLet bl (w), ... , br(w) be measurable vector fields on PM, then (1.6)
o(w)(bl, ... , br)w =
sign(Ii, ... , I.)
wl(dw, bi, )91,w ...Wn(dw, bIn )en,w.
(rl> ,'rn We must make sense of the stochastic integral in (1.6). The first step is to express the b3 in terms of the basis en,i bo(w) n, z
where the Anj;(w) are measurable functions of w. Now (1.6) becomes (1.8)
o(w)(bl , ... , br) _
a(1)
i1
... , >
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}.
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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.
1
Recall that if w, is a form on M which depends on s Ik"IIk,oo = Sup (IIw9IIc + Ilowe1I.
+... + IlokwsIIo)l .
8
Let N be the frame bundle of M and let ir : N -+ M be the projection. Let w,(u), s >_ 0, u E N, be the horizontal lift of the diffusion w,(x), s > 0,
where x = 7r(u). The diffusion w,(u) is hypoelliptic if it satisfies the following condition: There is a sub-bundle P of the frame bundle N with structure group G C O(d) such that the diffusion to,(u) takes place in P and the components of the vector fields over P which occur in the stochastic differential equation of w,(u) satisfy the strong Hormander hypothesis, [30]. We refer the reader to [30] for a more detailed discussion of the hypoelliptic condition. If the dimension of G is zero, for example if M = Rd, there is no extra condition and the diffusion is hypoelliptic. If, at each point in M, the components of the curvature of M span the Lie algebra of G, then once more the diffusion is hypoelliptic. Furthermore the diffusion w,(u) is hypoelliptic for a generic metric on M.
Lemma 1.10. (1) For all x E M we have the estimate (1.11)
E',' [ I A(f; w)I P ]1/P <
C'(f )rn (p)"
IIw1II2d+3,oo ... Ilwnll2d+3,.-
where d is the dimension of M. (2) Suppose that for all u the diffusion w,(u) is hypoelliptic. Then, for all x E M, we have the estimate (1.12)
E'
[
I A(f;
w)IP ]1/P <
C(f)rn (p)"
(3) If each of the wi is a 1-form then (1.12) holds even if the diffusion w,(u) is not hypoelliptic.
This lemma is the most important technical ingredient in the proofs of our main results. It is proved in §4 and §5. Note that the constant C(p) occuring in (1.11) and (1.12) depends on the Riemannian manifold M.
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Furthermore the term C(f )r occuring in (1.11) and (1.12) can be replaced IlfrIl oo but it is more convenient, at several points in writing by 11f, II. the proof, to choose the constant C(f). If we take open Brownian motion, that is we consider the expectation E [ IA(f; w)I P ]"P without any conditioning, the estimate (1.12) is valid without the hypoellipticity hypothesis; this follows from the method of Schwartz [35]. It seems reasonable to conjecture that in fact (1.12) is valid in general without the hypoellipticity hypothesis. By combining Lemma (1.10) with (1.2) and (1.3) we immediately get the following estimate.
Lemma 1.13. For all p > 1 there is a constant C(p, w), depending only on p and Ilwill2d+3, 1 < i < n, such that (1.14) I
+ 1) ... (Inrl + 1)
where II - IILP is the LP-norm on PM. O The terms in the denominator are Ini I + 1 rather than Ini I simply to deal
with the possibility ni = 0. This is, of course, a very weak application of Lemma (1.10). To prove that the series in (1.8) is absolutely convergent almost everywhere we first prove the following lemma.
Lemma 1.15. Suppose that lan,;I2 E LP n,1
for all p. Then the series (1.8) is absolutely convergent almost everywhere.
Proof. It follows that
:I 1)i,(w)... ni,
am>it(w)I2 < 00
for almost all w and that
E [(I: .ni)i, ... Anrl it 11 / 12
< 00
where each of the sums is taken over
nl,...,nr) E Zr'
(i1,...,ir) E {1,...,d}r.
Now the Schwartz inequality shows that, almost always,
(1.16) E IA;,,) i1(w) ... A(r) it (w)l a(w)(en, i, , ... , &nr,=r )w
( Ian ,1,i1(w) ...
1/2 Anr),ir (w) I2
I
(E I?(w)(&,, ail 9 ... , 6nr,ir )+ I2)1/2
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Taking expectations and applying the Schwartz inequality again we get
(1.17) E [ E r/ I
I
L
\
IAni)il...A(r)i.I ja(w)(eni,ii),...,En,,i.)I] < (1)
(*) 12z111/2 it ... n.,i. /1 E
f( L
IQ(w)(E nl
2l 1/2 i l, ... , En.,i. )I/
The first expectation on the right hand side of (1.17) is finite and, from Lemma (1.13), the second is less than
(Y
C,
1)2)1/2
(In,I + 1)2 .. . (InrI +
Thus the right hand side of (1.17) is finite and this proves that the series in (1.8) is almost always absolutely convergent.
We now use a truncation procedure to show that, in general, we can reduce the proof of (1.8) to the case covered by Lemma (1.15). Since each of the bj occuring in (1.7) is a measurable section of the field of Hilbert spaces l,,,, it follows that An,i(w)IZ < 00 n,i
for almost all paths w. Define
ni(w) _
Ani (w) 0,
if IAnji(w)I2
so that E LP. n,i
Now let K -i oo to complete the proof of (1.8). We have now given a precise meaning to the formulas (1.5) and (1.6) and therefore we have defined
Q(w)W :I-lwX...Xilw->R. It is not difficult to get the estimates needed to prove that this function a(w)w is continuous. In §3 we carry out several such estimates; for example (3.11). It is clear that a(w)w is alternating and so we see that a(w) is indeed a measurable form on PM.
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We now explain how to "restrict" o ,(w) to the loop space LM to get a measurable form on LM. The point to take care of is that
o(w)w:f,u,x...xHw-*R is only defined for almost all w E P(M) and the loop space has measure zero in PM. So it could happen that u(w)w it is never defined for w E LM. To see that this is not the case we simply repeat the procedure used to define a(w). So we first note that the stochastic integrals o(w)(en, , i , ,
. , en-i. )w
are defined for almost all loops w. Then we express the general integral
o(w)(bl,...,br)w in terms of the previous integrals as in (1.7) and (1.8), where the An,i are now functions on LM. Finally we argue, as above, computing expectations using the measure on LM, that the series (1.8), where the An,i are functions on LM, converges for almost all w E LM. This shows that Q(w)W:xwx...x }{,m->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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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
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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,. <_ IIaIIq,a,
II aII LP, . < IIaIIL9,a .
Next we estimate the LP,''-norm of a wedge product.
Lemma 2.8. Let a, r be measurable forms on PM, LM, or L.,M, then: IIa A Tliw <_ IIaIIwIITIIw Ila A TIILp,
< C3(p)IIaIIL2n,a IITIIL2p,a.
Proof. We give the proof for PM, the other cases are almost identical. Pick a countable orthonormal basis b; for l,,, and let /3I be the corresponding orthonormal basis for alternating multilinear functions on f,,,. With respect to this basis we write
aw =
al,wflII
rw = > TJ,w/J J
I
so that
IlaArll.=I EaI,wrJ,w/3IANJII
:Iaj,,I ITJ,wl =
IIahIwIITIIw.
This proves the first inequality. To prove the second inequality we start from
(a A r)w = E aI,wrJ,w/3l A /3J and express the right hand side of this equation in terms of its homogeneous
components. Let K = I U J and then we get
(a A r)w = 1: 1: aI,wrJ,w1K K InJ=z
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Now we estimate the random (p, a)-norms: 2
IlaATIIp",w= E ai,wrJ,wl37 A /3J II
P, C"W 2
E aI,wTJ,w
_ 1: a(IKI,p)2
InJ=z
K
a(IKI,p)2 2'KI
E
a1'WrI,w
InJ=E To complete the proof of the second inequality we use (2.4) and the Holder inequality. K
Now we estimate the LP>"-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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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 <
Ian,il2
Ia(In,i)I2
(n,i)I it follows that
a(IIIep)2 1: a(In,i)2 )
IIZb(a)IIp,« <
I
(1: An,i n,i
(n,i) f i
Now we replace In,i by J to get 10,(j)12
a(IJI - 1, p)2
IIZb(a)IIp,a
J
E An,i
(n,i)EJ
n,i
(J_1P)2 IJI Ia(J)I2
An,i n,i
Now using (2.2) we conclude that Ilib(a)IIP,a <
Ci(p)IIbII22
II0'II2p,«
Finally the Cauchy-Schwartz inequality shows that IIZb(a)IILP,« < C1IIblIL2p IIaIIL2p,a.
Now suppose that a(w) is a homogeneous Chen form of degree r as in (1.6) and write out a(w), in terms of the basis Pi for forms given by (1.5), as
a(w) = > ai(w),Q1. We show that a(w) has finite Lp" -norm for all p by an argument very similar
to the proof that (1.9) is almost always absolutely convergent. In fact the Holder inequality for p > 2 shows that (2.12)
E
[(E Ial(w)I2) p/21
]2/p)1/2
(E E [ Iai(w)Ip
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Then, as in the previous section, if I = {(n1i i1), ... , (nr, ir)} we get the following inequality l
l
(In,I + 1)... (InrI + 1)
and so the right hand side of (2.12) is less than )1/2
C2
(In, I + 1)2 ... (InrI + 1)2
and, since this term is finite, this shows that a(w) has finite LP,a-norm. Now if o(9; w; y') is a general Chen form as in (1.18) a simple argument using Lemma (2.8) shows that o(9; w; cp) has finite LP,a-norm.
Definition 2.13. The spaces of LP,a-Chen forms CP,a(PM), Cp,a(LM), and Cp,a(LXM) on PM, LM, and LxM are defined to be the the closure of the spaces C(PM), C(LM), and C(LXM) in the LP,a-norm.
Note that in forming C,,,0, we complete the whole space C = ® Cr, not just its homogeneous subspaces Cr, so we allow certain infinite sums of homogeneous elements. In terms of homological algebra, compare [17], the spaces CP,a are completions of the various Chen normalised bar complexes. From now on we write CP,a for any of the spaces of LP'a-Chen forms and only specify which of the spaces PM, LM, or LxM we are working on if it is absolutely necessary. In particular any statement about CP,a is valid for PM, LM, and LxM unless specified otherwise. It follows that if q > p Cq,a C CP,a, and we define P
The following lemma follows from a standard diagonal argument.
Lemma 2.14. If a is in C,,,a there is a sequence a,, E C such that a,,, converges too in each of the
LP'a-norms.
Now we examine the wedge product.
Theorem 2.15. If v, r E C,,,,a then the exterior product o h 7 is also in C,,.,a.
Proof. Using (2.14) pick sequences o,, and r,, in C which converge to a and r, respectively, in all the Then Q A r E C and using Lemma (2.8) it follows that o,l A r,, converges in all the This proves the theorem. LP'a-norms.
LP'a-norms.
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Theorem 2.16. Suppose that o is in C,,,a and that b is a vector field which has finite LP-norm for all p; then ib(a) E C,,,,a.
Proof. Since o is in all the LP'a-spaces and b has finite LP-norm for all p (2.9) shows that ib(o) is in all the LP,''-spaces. It only remains to prove that that if a is a Chen form then so is ib(a). As in the proof of (2.8) we write a= b = E A,,,;e,,,;. o(I )OI,
Since each A,,,i is measurable, Lemma (1.23) shows that it is enough to prove that if a is a homogeneous Chen form then each i£ ; (a) is in CH,oo,a We prove this for the forms a(w) defined in (1.5). The more general case of forms o(9; w; (p) or a(9; w), defined in (1.18) and (1.19), follows using the fact that ib is a derivation and Theorem (2.15).
By using a partition of unity we can reduce to the case where there is an orthonormal basis e1(k)(x),... , ed(k)(x) for T,, which is smooth on the support of wk and we choose such bases. If X is a vector field defined on the support of wk then we write X3(k), where 1 < j < d, for the components of X defined using the basis ej(k). Define e,,,i(k) and e,,,i(k) as in (1.3); then it is enough to prove that the form
fm w1(dw _)9, n ... A wj (dw, e41 i(k), -)y; A ... A W,,,(dw, -)y,n is in C,,O,a for all j, k. We use the approximation of a loop by a piecewise geodesic path and the corresponding approximation for parallel transport [8].
Then, compare [8], the above form becomes the limit, as 1 -> oo, of a sequence of forms
(2.17) J w1(dw, -)y1 A ... A wi(dw, plen,i(k), -)s; n ... A
_)sm
m
where the function cpt(s, w) is of the following type: If Si E [11N, (1 + 1)/N], then
PI(Sj,w) ='bl,k(W1/N,W2/N,...,W(t-1)/N,wg,) where
0I,k : M' -+ R
is a suitable smooth function. By the Stone-Weierstrass theorem we can assume that each 0l,k is a sum of products of functions on M. The form (2.17) now becomes a sum of forms of the type J w1(dw, -)g1 A ... A wj (dw, _)e; ... A ... A w(dw, m
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and is thus a Chen form.
Now we come to our main results, motivated by [12], [18], and [26]. We will state the theorem for the loop space LM, but there are obvious analogues for the path space and the based loop space.
Theorem 2.18. Let 9n, and wi,n, where 1 < i < n, be differential forms on M and define wn = w1,n ® -0 Wn,n(1) Suppose that the power series
li=1 n
00
n
6
V
has infinite radius of convergence for all k < 2d + 3 where d is the dimension of M; then 00
E Q(en; wn) E Coo,a(LM). n=1
(2) Suppose that for all u the diffusion ia,(u) is hypoelliptic and that the power series n
n
e
00
n=1
i=1
V0
has infinite radius of convergence; then 00
1: o(9n;wn) E Coo,a(LM). n=1
Remarks.
(1) We believe that part (2) is true without the hypoelliptic assumption but we have not yet been able to prove it. (2) There is an exactly analogous theorem giving a condition which ensures that 00
E 01(en; wn; 4'n) E Coo,c (PM). n=1
The condition is given by multiplying the coefficient of zn in the above power series by IIVnllk,00 in part (1) and by jlylnljl,o in part (2). In this case however, part (2) is always valid since, as
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we remarked after the statement of Lemma (1.10), the basic estimate (1.12) is valid without any hypoellipticity hypothesis. The analogous condition for the based loop space is given by removing the term 9n. The proofs are obvious modifications of the proof we give below. (3)
The importance of this theorem is that it makes a link between Connes's entire cyclic theory and the spaces C,,,,,. More precisely, in terms of homological algebra, it shows the iterated integral map v, described in [17], maps Connes's entire cyclic complex, see [13], for the algebra 12 into the space of L°°'°`-Chen forms on LM.
Proof of 2.18. We must prove that IIQ(en;wn)IILP.a < 00 n
for all p. It is enough to deal with the case where On = 1 for all n and each of the forms wi,n is homogeneous; the general case follows easily from this case and Lemma (2.8). Set r;,. = degw;,n - 1 and rn = r1,n + + rn,n. In terms of the orthonormal basis ,3I for forms defined in (1.4) Q(wn) = 1: QI(wn)/3I
and we compute LP"'-norms using the expectation EL on the loop space described at the beginning of §1: IIQ(wn)IILP" = a(rn,p)EL [IIo(wn)IIP1"P (2.19)
= a(rn,p)EL < a(rn,p)
r (E I 1
`\
1 /p
I0'I(wn)I2)P/2] [I0'I(wn)Ip]2/p)
1/2.
(EEL
Using (1.6) we can express oI(wn) as a sum of at most rn! terms of the form A(f; w) = Jon wl
...wn (dw,T(frn))9n
where fl,... , frn are taken from the basis en,i for f-l(T,,,o) defined in (1.3). Now we can use part (1) of Lemma (1.10) to estimate the expectation of each of these integrals. We get E"2[IaI(wn)IP]1/P <
rn!C(p)n n!(Im1I+1)...(ImrnI+1)
IIwnIIk,oo
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Jones & Leandre: LP-Chen forms on loop spaces
where
I= IIwnllk,oo = IIw1,nhlk,00... IIWn,nhIk,oo
k>2d+3. From this it follows that p i(x,x)Ex,x [ITI(wn)IP]dx)
EL [I0'I(wn)IP]1/P =
1/p
(IM
rn! C(p)n IIWnhIk, n! (Iml I + 1) ... (Imr 1 + 1)
< and from (2.19) it follows that (2.20)
Il a(wn)II LP.a <
a(rn, p)rn!C(p)n
1
(Iml l + 1)2 ... (Imr(n) I + 1)2
n!
1 /2
Ilwn ll k,.'
The sum which appears in (2.20) is smaller than )rn
<2 ndh nd
2Tn
1>0 22
since rn < nd. Thus, using (2.3) to estimate rn! a(rn, p), we get (2.21)
Ila(wn)IILP, < : (
Ilwnllk,oo
and the first part of the theorem follows directly from (2.21). Part (2) of the theorem is proved in exactly the same way but using part (2) of Lemma (1.11) in place of part (1). O We can get the same kind of theorem if the w;,,, are matrices of forms on M. The statement is formally identical to Theorem (2.18) so we shall not repeat it in detail. This is particularly important for the study of the equivariant Chern character, [7] and [17]. We briefly recall the construction of the equivariant Chern character as a differential form on the smooth loop space L°°M using Chen forms, for more details see [17]. Let E be a Hermitian vector bundle over M equipped with a Hermitian connection VE. Given an embedding of E in the trivial bundle M x CN let p(x) be the orthogonal projection from CN to Ex and let
Vp=podop+p'odopl
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We choose the embedding of E in M X CN so that the connection Vp preserves the splitting of M X CN as E ® E1 and agrees with the given connection on E; this is possible by [32]. A simple calculation shows that the connection form of VP is
A = p(dp) + pl(dpl), that is VP = d + A, and the curvature of VP is F = dp A dp.
Define a differential form A on M x S' by A = A + F A dt.
Let w + A dt be a form on M x S' where w and
are forms on M.
Using such forms we can define a larger class of Chen forms as follows. Let ai = wi +- i A dt, 1 < i < n, be forms on M x S1 and let a = a, 0 ... 0 an; then define
&(a)=f This is a form on L°°M. Note that even if a is homogeneous this form on L°°(M) is not homogeneous. Thus we can construct the matrix valued form, of infinite degree, po(A®k). k
It is proved in [17] that the form
Ch(E, VE) =
trace (po(A®k))
on L°°(M) is Bismut's equivariant Chern character of (E, DE). It is not difficult to adapt the previous theory to show that Ch(E, VE) extends to a measurable form on LM and to use the matrix analogue of (2.18) to show that Ch(E, VE) lies in C,,,,,p. The important property of the L°°"-norm which is needed here is that the Picard series of the solution of the equation 3.16 in [7] is absolutely convergent. See [9] and [35] for similar problems when the components are scalars rather than forms. If, in the above notation, we define .F = A(dwt) + F'w, (dt),
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which is a sum of a 0-form and a 2-form on L'M with values in MN(C), then Ch(E, VE) is the time ordered exponential of F. We conclude this section by making a few remarks on the theory of time ordered exponentials of forms.
The time ordered exponential of the form w on LM is the solution at time 1 of the stochastic differential equation, in the Stratonovitch sense, dHt = Ht n w(dw, -)t,
(2.22)
Ho = 1.
We explain, briefly, why this equation makes sense. In terms of a basis /3j for measurable forms we write t
w(dw,, 10
Eat (w, 1))31,
Ht =
Ht(K)OK;
I
then
dHt =
dHt(K) A aK
H t A w(dwt, -) = E E Ht(I)dat(w, J)
,fix
(I,J)
K
where the sum over (I, J) is taken over those multi-indices (I, J) with
IuJ=K,
InJ=O.
The components Ht(K), for t < 1, are solutions of the system of linear equations
dHt(K) = > 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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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 <_ II/IIB11k111.
IIXrII.
This defines a Banach space norm II - II B on AT'(f* ). Note that the norms
II - II H and II - II B are inequivalent. Each occurs naturally, in different contexts, and so we develop the theory for both. We can use the norm II - II B to define a weighted Lp-norm on the space of measurable forms on PM, LM, and LxM. As in §2 we define the random (B, p, a)-norm of the homogeneous measurable r-forma by (3.1)
II0'IIB,p,o
w = a(r,p)IIQII B,w
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If o is not homogeneous then we define the random (B, p, a)-norm of o to be the sum of the corresponding norms of the homogeneous components of o. The LB,P,'-norm of o is defined by (3.2)
II aII LB.p
1/P
_
(E [II all e,p,«])
where the expectation is taken over PM, LM, or L,xM, according to the case, using the measures described at the beginning of §1. Once more there are infinite sums of homogeneous terms with finite LB,P,'-norm.
In this case the B in the notation for these norms is there to emphasise that we are dealing with the (p, a)-norms based on the Banach norm II - IIB. We will assume this and drop the B from the notation. So if o is a measurable form, then, for the rest of this section, II o II will mean 110`11B, IIoIIP,a will mean and II aII LP. will mean IIaI1 LB,p,.. IIaIIB,P,a,
If p < q, it follows from (2.1) that (3.3)
IIaIIP,a
IIaIIq,.,
IIailL- _< II0'IIL9,'.
The relation between the basic algebraic operations on forms and the LP'norm is given by the next two lemmas.
Lemma 3.4. Let o, r be measurable forms on PM, LM, or Lx M, then: IIo A TII LD,° < 2C3(p)II0'IIL2P,. IITIIL21,..
Proof. To begin with suppose that or is homogeneous of degree r and r is homogeneous of degree s and then
sign(I,J)a(bi)r(bj)
(o A T)(bi,...,br+9) _ (I,J)E4'(r,s)
There are (r+s)!/r! s! terms in this sum and each term has modulus smaller than
r+s IIaII 11TIIrjIIb=II. 1
It follows that 11o, n TII L- <
r + s. ' a(r + s, p)E [(Il all r! s!
IITIUPl1/P
[IITII2P]1/2P
<_
(rr ! A)! a(r+s,p)E
[IIa1I2p]1/2PE
< 2C3(p)a(r, 2p)a(s, 2p)E [ IICII2P ] = 2C3(p)IlchIL2p,.
1/2P
E
1 1 71 1 2 p]1 , 2P
IITIIL4,r .
We have used (2.4) to obtain the last inequality and the expectations are taken over PM, LM, or L1M, according to the case. A simple argument shows that this inequality is valid in the general, inhomogeneous, case.
Jones & Leandre: LP-Chen forms on loop spaces
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Lemma 3.5. Let b be a measurable vector field and a a measurable form on PM, LM, or LxM; then II2b(0')IILP.' <
Cl(p)IIblIL2P 11011L2P,..-
Proof. We can assume that o is homogeneous of degree r and then
l2b(a)(b1,...,br-1)I = 1o(b,b1,...,br-1)l r-1
<- IIall IIbII jI IIbill. i=1
This shows that we have the inequality Ilib(a)lI _< 110 '11
IIbII
Now it follows that II2b(0`)IILP," = a(r -1,p)(E[(llib(a)II)pl)1/P
C1(p) a(r,2p)(E[(Ilall Ilbll)P])"P )1'2P
<
C1(p) a(r, 2p) (E [(IIahl)2P]
=
C1(p) IIbllLZPIIa1ILZP,a.
)1'2p
(E [(Ilbll)Zp]
In this computation we have used (2.2) and the expectations are computed over PM, LM, or LxM according to the case. 0
We can get estimates for hall when a is a Chen form by the following procedure. Choose an orthonormal basis ei (x), ... , ed(x) for TTM such that each ei is a measurable function of x. Let b be a measurable section of the field of Hilbert spaces W,,,. Then we define components bj by the following formula: d
b(s,w) _
Ebi(s,w)T9(w)ej(wo).
j=1
For each w the functions b' (s, w) are in the Hilbert space 7l = ?1(R) defined in §1. It therefore follows that (3.6)
f Ib'(s, ' w)12 ds < IIbIl2,
f
1(dbj(s,w))2
ds
ds < IIbIl2
/J1
To simplify we will now work on the based loop space LxM and therefore we assume that b(0, w) = b(1, w) = 0. On the free loop space or the path
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132
space there are extra terms coming from the values of b(s, w) when s = 0, 1. These extra terms are not difficult to estimate. Now given measurable sections b1, ... , br of T,,, let us write the formula (1.6) in terms of the b,. To do this in a reasonably compact form requires extra notation. Let (Il , ... , In) be a (rl, ... , rn) shuffle of { 1, ... , r} where We write out Ik as
Ik = {2k,1,...,ik,rk}. Now let J = { j1, ... , jr} be a set of indices for the components of the b1 and define jk,1 =
.lk,rk =1r1+...+rk,
bIk = bik,1 'b
rk
.
Using this notation we get the following formula: Q(w)(bl,...,br)
(3.7)
e(Ij,...,In)A(J;I1.... I,,) (J;Ii,...,In)
where
A(J; Il, ...rln)w =
JOn
wl(dw,TeJ1)91,wbj,(s1,w)...wl(dw,TeJn)en,wbjn(sn,w)
The sum in (3.7) is taken over
IJI = r,
(Ii,
J.) E P(rl,...,rn).
Note that A(J; Il, ... , In) can be rewritten as (3.8)
J
n
I
3n
91
1=0
... Jt
wl(dw, rej1)t1d (bj1)t ...wn(dw,TeJn)9nd (brn)f n=O
1
n
1
Now let us compute (3.8) by interchanging the order of t; and s1 in the integral: we get
A(w; t; J)d (bJ)
(3.9) L1,...,)ELO,l] n
t1
d (b) to
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133
where t = (t1, ... , tn) and A(w; t; J),,, is given by
... J
i
(3.10)
1
n=tnn
81=t1
Using (3.6) we get the following bound J1
2
J(o,1]n
1/2
dbL2
db f1
ds
ds1 ... dst,
ds
lib=II
Si
and then the above formulas give the following bound on IIo(w)II: (3.11)
Ia(w)(b1,...,br)wI < r
fl IIb=II i=1
f
> (r;I1....,In)
IA(w; s; J)I2ds, ... dsn
[o,1]n
From (1.10) the LP-norm of each of the A(w; s; J) is bounded uniformly in s and it follows that the LP' norm of a(w) is finite. As in the previous section we can now define LP'a-Chen forms based on the Banach norm II - II B . The formal properties of these spaces are identical to those for the corresonding spaces based on the Hilbert norm II - IIH.
Definition 3.12. The spaces of LP'a-Chen forms CP,a(PM), Cp,a(LM), and CP,a(L,XM) on PM, LM, and L,xM are defined to be the the closure of the spaces C(PM), C(LM), and C(LXM) in the LP,a-norm.
Once more it is important to note that we are completing the space C, not just each of its homogeneous components so we allow certain infinite sums of homogeneous elements. It follows from (3.3) that
if q>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
134
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- <
P/21/P a(rn,
P)Ex,x
I A(wn; s; J)I2dsl ... dsn (JIO
But there are at most rn! shuffles so, from (2.3), it follows that <
Ila(wn)IILP
P/2 1/P
C1(P)r° supEz'x J
IA(Wn;s;J)I2ds1 ...dsn (Ito,I]n
Now using part (1) Lemma (1.10) and Holder's inequality we get
Ex'x
p/2 2/p
r
I A(Wn; s; J)I2ds1 ... dsn
J
< Ji(P)nIIW1,nIIk,....IIWn,n112 n!
where k > 2d + 3. Putting these inequalities together we get IIa(Wn)IILP,a <
h(P)nIIW1,n11k,oo...IIWnn1Ik00
which proves that the series E a(Wn) is absolutely convergent in all the LP,'. This proves part (1) of the theorem and part (2) follows by using part (2) of Lemma (1.10).
We get exactly analogous theorems for PM and LM and also if the w;,n are matrices of forms and it follows that the equivariant Chern character belongs to all the C,,.,« in just the same way as in §2. §4 THE SCHWARTZ LEMMA
We now give the proof of Lemma (1.10), the version of a lemma of Schwartz [35] appropriate to our situation. To begin with we recall the necessary terminology and the statement we need to prove. Fix x E M and let fl, . . . , fr [0, 1] - TxM be functions which are continuous and have bounded variation. As pointed out in §2 we need some hypotheses on the :
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136
fi since we are using Stratonovitch integrals. These hypotheses are adequate for our applications but it is presumably possible to get away with weaker conditions. We choose a constant C(f) such that II fi II o < C(f) for
1 < i < r. Let
w:[0,1]--+On
S"w1s®...0Wna
be a map which is piecewise constant and set
ri=degwi-1
r=rl+"'+rn
ji={rl+.+ri+1,...,r1+...+ri+ri+i} f = (fl,..M; as in §1, let (4.1)
A(f; w) =
j
w1
(dw, r(fi, ))s, ... wn (dw, r(fin )),n
Recall that if ws is a form on M which depends on s Ilwllk, = Sup (IIwsll. + IIVw9Iloo + ... + 9
Let N be the frame bundle of M and let it : N -> M be the projection. Let ivs(u), s > 0, u E N, be the horizontal lift of the diffusion ws(x), s > 0, where x = Tr(u). We can also consider the case where the diffusion ws(u) takes place on a sub-bundle of N.
The Schwartz Lemma. (1) For all x E M we have the estimate (4.2)
E x,x [ I A(f; w)I P ]1/P <
C(f)rn (p)n Ilwl
112d+3,oo ... Ilwn ll2d+3,.,
where d is the dimension of M. (2) Suppose that for all u the diffusion ws(u) is hypoelliptic. Then, for all x E M, we have the estimate (4.3)
E',x[IA(f;w)IP]1/P < C(f)rn(p)n
IIw,ll1
...llwnll,
.
(3) If each of the wi is a 1-form then (4.3) holds even if the diffusion vvs(u) is not hypoelliptic.
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137
The forms w; are assumed to be piecewise constant but it should be possible to generalise to the case where the w; are smooth in s, or to follow
the more general setting of the paper of Hu and Meyer [19]. As pointed out in §2 we can replace the constant C(f )r occuring in (4.2) and (4.3) by 11fl 11. but it is slightly easier, at several points in writing the proof, to choose the constant C(f). First we show how to reduce the general case to the special case where
p = 1. It is enough to consider the case where p = 2k. Then, as in the formula for the product of iterated integrals, compare [17, §3] or [21], we can express A(f; w)P in terms of sums of iterated integrals of length 2nk. By this procedure we get a sum of (2nk)!/(n!)2k terms of the type
J
2nk
1(dw)s, ...
where each term Oj(dw) is given by %.i(dw) =Wk(dw,T(JIk))
for some k with 1 -< k <- n, and exactly 2k of the Oj(dw) are equal to wk(dw, T (flk ))
Recall Stirling's formula,
n! ^
nne-n
27rn,
as n --> oo;
it follows that (2nk)! 2k
(2nk)2n147rnk < C(k)n n2nk( 2nn)2k
as n -+ oo
and this gives us an estimate on the number of terms. Combining this estimate of the number of terms with the estimate for each term
E-,- I J
L ,2nk
%,(dw),, ... W2nk(dw)s2nk
given by the Schwartz Lemma in the case p = 1, the case of general p follows from the case p = 1 and n even.
Define z'(s) to be the l-simplex
o'(s)=10<sl <...<si<s:s,ER); we continue to use the notation On for the simplex On(1). We use the notation
ij(dw) = wj(dw, r(fj)).
Jones & Leandre: LP-Chen forms on loop spaces
138
For any integer l < n define J(1, s) by
J(l,s) =
(4.4)
Jo`(s)
01(dw),, ... 01(dw),,
Our aim is to get an estimate for Ex,x[J(l,1)]. Our method is to do this recursively and this requires us to get estimates for the expectation of J(l, s) for s < 1. One reason for considering expectations of J(l, s) is that we will get bounds in terms of "explicit formulas involved with heat kernels" in the manner of [20]. First we prove (4.3). Because of the hypoellipticity hypothesis we can estimate the expectation of J(l, s) by working in the frame bundle N. Give N the metric constructed from the Riemannian metric on M and an invariant metric on the fibres, see [4, page 403] or [14]. Let X; be the canonical horizontal vector fields on N. Now consider the stochastic differential equation of the process zut(u) conditioned to end at v after time s:
dtvt(u, v) = AdB + Cdt,
(4.5)
In this equation
zvo(u, v) = u,
t < s.
d X;(w(u, v))dBi d
AdB =
=-1 d
C=
X;(zvt(u, v)) (X;(zut(u, v)), grad log As_t(tut(u, v), v)) s-1
where Pt is the heat kernel associated to tvt. The transition probability of z7vt(u, v) is
(4.7)
pt,9(u, r, v) =
Pt(u, r)Ps-t(r, v) P3(u,v)
We write Es," [J(l, s)] for the expectation of J(l, s) using the law of the process z`vt(u) conditioned to end at v after time s. Let it : N -* M be the projection. Then we have the following recursive formula: (4.8)
J(l, S) =
J
9
J(1 - 1, t)b,,t(Di(AdB + Odt)).
This is a Stratonovitch stochastic integral and the first step is to write it in terms of Ito integrals. Let SB' be the Ito differential of the flat Brownian motion B. Then we decompose J(l, s) into four terms (4.9)
J(1, S) = J1(l, s) + J2(l, s) + J3(l, s) + J4(1,8)-
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139
The first two terms are the natural ones:
J1(l, s) = f , J(l -1, t) bj,t(Dir(A6B)) 0
(4.10)
J2(l, s) = f J(1-1, t) bj,t(D1r(Cdt)). The term J3(1, s) arises from pairing J(1 - 1, t) with Oj,t(D7r(A5B)). It is given by a formula of the kind s
(4.11)
J3(1, S) = 10, 1(1
2, t)Pi (01-1t, l,t)dt
where P, (01-1,t, z'1,t) is wt(u, v) measurable and (4.12)
IP1(?G1-1,'1)I <
Here, since Ji = wj (-, r (fr;)) , d(j) is the degree of the form wj and II'jIIi,co = IIwjIII,.... . The last term J4(l,s) comes from the conversion of
the Stratonovitch differential element z,bj(Dtr(AdB)) into an Ito differential element. We get (4.13)
J4(1, S) =
J0
J(1 - 1, t)P2(z(il)dt
where once more P2(it) is wt(u,v) measurable. In this case we have IP2('t/)l)I < C(f)d11)II
(4.14)
,:II1,..
Now we show that, for almost all v,
E;,' [J1(1,s)] = 0.
(4.15)
If the local martingale t -+ Jl(1,t), t < s were a true martingale then, of course, (4.15) would follow. We will prove (4.15) for almost all v by the following argument using Lemma (5.45) from §5. It is enough to show that E;'"[IJ(I,tIP]] is bounded for all p and for t < s. Suppose, inductively, that
E;'"[IJ(1- 1, t,
]]
is bounded for all p and for t < s. We get a formula
analogous to (4.11) J(l, t) = J1(1, t) + J2(1, t) + J3(1, t) + J4(1, t)
where the J,(1,t) are given by integrals identical to those in (4.11) except
that s in the upper limit is replaced by t and t inside the integral is replaced by tt. The only problem is to show that if, for r < l and for all q, E;'" [ I J2(r, t)I9 ] is bounded for t < s then so is E;'" [J2(1, t)].
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Define
I'i(t, s) = (Si(wt(u, v)), grad log P9-t(ti t(u, v), v))
(4.16)
then, by induction we have: p 1IP
t
E v [(j Ifi(t',s)I J(1-1,t')I dt'1 < (4.17)
j
E[I(t',s)IJ(1- 1,t')dt'] 2p] 1/2p
< J tE9'v [i(t'
1/p
Eu,v [IJ(1-1,t')I2pdt']ll2pdt'
0
<
S C(s,P) E9 "v [IJ(1- 1,t')I2p]1/2pdt'
A 7S =1
where we have used the fact that Pt > 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 <_ K3II01-1 II1,.1101111,oo J 0
"
IN
[J(1- 2), t)] I
,9(u, r, v)drdt
IEe'v[J4(l,s)]I <_ K4
fNJ 9
I Et'r [J(1- 1), t)] I Pt,5(u, r, v)drdt.
Here the term 4 (u, r, v) occuring in the estimate for J2 is given by 4(u, r, v) = p9 (u, v) grad P,_t(r, v)
and the constants K2, K3, and
given by
h2 = CII 1II1,.C(f)d(1) K3 =
C2C(f)d(1)+d(1-1)
K4
= CC(f)d(1)IIO,II1,..
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141
Now let a = (al , ... , ar) be a sequence with each aj equal to 0 or 1 and at least 1- r terms equal to 0. We set r = IaI go,t(u, v) = Pt (u, v)
(4.19)
gl,t(u, v) _ 1: 1 CX;(u), grad P, (u, v) Using (4.15), (4.18), and induction on 1 it follows that, for almost all v, IE9,"
(4.20)
[J(1, s)] I < ClC(f)d(1)+...+d(l)II0IIII,oo
...
II,01111'.
l/2
I(a).
Here i(a) is given by the following integral:
I
ds * (s)
Ps1(UIV1)qa,,92-11 (yl,v2)...ga.,a-a,.(yr,y) dy
IN
P9(u, v)
where
ds = dsl
dsr,
dv = dvl
dvr.
This last formula is only true for almost all v but this is not a serious problem. Our goal is to estimate Ex>x[J(n,1)] and
Ex'x[J(n,1)] = J
(4.21)
Nz
P'(u,v)Ei'"[J(n,1)]dv
x, x
where Nx = 7r-1(x) is the fibre of N over x. Thus (4.18) gives Ex'x[J(n,1)] <-
CnC(f)nIIW1II1'....
J:I(a)
IIWnhI1,.
n/2
I4 (s)
dv N.
P, (u, yl )ga,,92-si (y1, v2) ... gar,1-9,.(vr, y)
J
PI (x, x)
dv
This estimate is valid for almost all x and by continuity it follows that it is true for all x. Now we must estimate the integral I(a).
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142
Lemma 4.22. There is a constant (C such that (4.23)
pat(u,v1)Qat,$2-at(v1,v2)...gar,1-s,(vr,v)dv
J
Cr(S2 - S1)-at/2(S3 - S2)-a2/2 ... (1 - Sr) -a,/2
Proof. Since N is compact we can use the short time expansion of [23] for the horizontal Laplacian of N. In terms of the associated sub-Riemannian distance d(u, v) and the volume of balls, we get, for t < 1, go,t(u,v) < (4.24)
V(
u f) C,
91,t (u, v)
C2d(u,v)2
exp
-C2d(u, v)2 t
eXp
fV(u, )
where V(x, s) = vol B(x, s). These estimates are true for all (u, v) and for some suitable constants C1, Cl', C2, CZ > 0. Using (4.24) we see that the integral in (4.23) is smaller than C3(s2 -
(4.25)
1
exp
s1)-at/2
(S3 - S2)-a2/2... (1 - Sr)-a'/2 IN
(_C4d(u,v1)2 81
V(u, V13--j)
1
V(v1 ,
S2 - S1)
exp
1
V(vr,
1 - Sr)
dv
(_C4Jviv22)
exp
S2 -SI C4d(vr, V) 2
1 - Sr
for some suitable constants C3 and C4. Let us apply the converse inequality of (4.24): (4.26)
Cl
V(u, f)
exp
(_CMv2) t
< pt(u, v)
and the doubling property of [23]: if p < 1, there is a constant K such that V(u, 2p) < KV(u, p)
(4.27)
for all u. We conclude that there is a constant po, independent of u, such that the expression in (4.25) is smaller than (4.28)
Cr(s2 - S1)-at/2(S3 -S2 ) -a2 /2 ... (1 - Sr)-a'/2
fNxxN
pos1(U, Vl)fPo(a2-at)(vl, v2) ... pPo(1-a2)(vr, v) dv.
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143
The proof of (4.23) is completed using the Chapman-Kolmogorov equation.
Now comparing the inequality (4.23) with the estimate of Ex,z[J(l,1)] given by (4.21) we see that we are left to estimate the integral
f f
(4.29)
ds1
ds2
41 (S2
- S1)132
dsr
r
(1 - Sr)Qr
where Ni = ai/2.
Lemma 4.30. If /.3 < 1 then (4.31)
1
10
smds
< mQ-1r(-3 + 1)
(1-S)ft -
Proof. We have 1
I
- s)mds
1 (1
SmdS
J0 (1 - S),6 - Jo
So
11 emlog(1-s)ds
- Jo
so
and since log(1 - s) < -s when 0 < s < 1 it follows that
f1 0
f1 e-msds < foo e-msds =
surds
(1 -S),6 -
0
SQ
s(i
0
mQ - r (1 - Q) 1
Lemma 4.32. If r > 2 then k
r
1
(2'' -
(4.33)
/33k)
(k/3k) j=0
k=1
Proof. Let us introduce the expression ds1 Ir(a, s) =
far(,,) S#1 (S2
ds2
dsr (3 - Sr)Qr
W- S]Q1
so
( = f Ir(fl,s)(1 (/ Ir+1(a) 1
0
ds
l - S)Qr+1
Using the change of variables si H st; we see that Ir()3, s) = svlr(/3),
v=ri=0
i
Pk+1
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144
Now, using Lemma (4.30), it follows that //
Ir+1()) = Ir(
)
I J
- S118.}, C Ir( )r(1 -
sv /1
Nr+l)LQnti-1
This proves (4.33) by induction. Now we compute the actual integrals we need to deduce (4.3) from (4.21).
Consider I(a/2) where each a= is zero or 1, ao = 0, at least n - r of the a; are zero and at most 2r - n of the a; are equal to 1. In this case Lemma (4.32) implies that r
I(a/2) < C(p)" II
(CPn
(ri)
k"k-1
k =1
C(p)"` r!
n!
1/2
((n_r)!) C(p)"
Vr! Now by using the binomial formula we get
C"
1
.,/r! (n - r)!
n!
and so I(a) < C(p)" n!
By combining this estimate for the integrals I(a) with (4.21), we complete the proof of (4.3). The proof in the case where all the forms w; are one-forms is similar but much simpler since parallel transport does not appear. This completes the proof of parts (2) and (3) of the Schwartz Lemma.
We now turn to the proof of part (1) of the Schwartz Lemma, that is (4.2). We estimate the quantity E2'x[J(n, 1)],
where J(n, 1) is as in (4.4), by a different procedure based on ideas in [29]. As a result we obtain (4.2) in the case p = 1. As we have already argued, this is enough to prove (4.2) in the general case. As in [29] we consider the auxilliary measure on M (4.34)
u.(f) = Ex [J(n,1)f(wl )]
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145
where the expectation is taken over the space of continuous paths wt which start at x. This measure has a density qn(y) given by (4.35)
qn(y) = Ex," [J(n,1)] pi (x, y)
Since pi (x, y) > 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
Y
In order to show (4.36) we will apply the Malliavin calculus. Since J(n, 1) is smooth in the Malliavin sense, and since the diffusion wt(x) is non-degenerate, we have an integration by parts formula µn [Y'f] = E [Ca'(w).f(w1)]
(4.37)
The expression Ca(w) is a polynomial in the derivatives of J(n, 1), of length < 21al, the derivatives of w1(x), and in the inverse of the Malliavin matrix
of w1(x). Here all the derivatives are in the Malliavin sense. Moreover Ca(w) is homogeneous of degree 1 in the derivative, in the Malliavin sense, of J(n, 1). We now take the Schwartz construction of the Brownian motion m
(4.38)
X;(wt(x))dB' + Xo(wt(x))dt,
dwt(x)
wo(x) = x
i=1
for some suitable smooth vector fields on M. As in the previous proof we convert the Stratonovitch integrals which appear in J(n, 1) into Ito integrals. This gives us an expression for J(n, 1) as a sum of at most 4n terms J(r, 1)
of length r where n/2 < r < n. Here a term of length r means there are exactly n - r contractions of terms of the form (SB', SB' ) and, therefore, there are exactly n -r terms in dt in the expression for J(r,1); this is exactly as in the previous proof. Recall the following formula from [21, page 107]: If F and G are Brownian functionals of the form 1
F = f usSBs, 0
f1
G=
J
0
hsds
146
Jones & Leandre: LP-Chen forms on loop spaces
then 1
r1
LF =
1
(Lu, - -u, I SB3
J \
2
o
LG =
/
J
Lh,ds
o
where L is the Ornstein-Uhlenbeck operator. These formulas lead to the following conclusion: By Meyer's inequality it is enough to apply L' with I aI < d+ 1 to each term J(r,1) to evaluate the contribution of the terms in the integration by parts formula which come from the derivatives. When we apply L' to J(r,1) we will get a sum of at most C(n)k"I integrals of length r with at least n - r terms in ds. Each integrand depends on the derivatives
of the forms w; up to order at most 2IaI + 1 < 2d + 3. Here the +1 arises from the conversion of the Stratonovitch differential into an Ito differential. The Ornstein-Uhlenbeck operator is an operator of order 2 in the Malliavin derivative. By using the chain rule it follows that the Malliavin deriv-
ative appears with order at most 21al < 2d + 2 for wt(x) and also for the parallel transport rt. These quantities are bounded in LP but not uniformly so it is not possible to immediately apply the Schwartz method where the components of the integrals are bounded. But, since V(F1F2) = V(F1)F2 +F10(F2), in each term there are at most 2d + 2 terms which are not bounded. This fact will allow us to adapt the Schwartz method to this situation. We will denote by J(l, t) an integral which contains a(l) < 2d + 2 unbounded terms. Let to be the number of appearances of the ds terms and l'' the number of appearances of bB terms so that 1 = to + lo. Let ,7(l) be the set of indices which occur in the forms appearing in the integral J(l, t). Suppose, by induction, that there exists a constant C(a(l), p) and two real numbers n(l, lo) > 0, k(l, lo) > 0 such that (4.40)
C(a(l), Ex,x [J(l, t)'] 1IP < p) L
J
-
11
tn(1,1a) Ilwi 112d+3
jEJ(i)
To estimate the expectation Ex,x [J(1 + 1,t)P]
"P
k(l, lo)
we use the procedure
given in [35, Lemma 1.1]. If
J(l + 1, t) = f J(l, s)UbBs, 0t
where U is bounded, we have a(l + 1) = a(l). By using the Burkholder
Jones & Leandre: LP-Chen forms on loop spaces
147
inequality for some k(1) (4.41)
E[I Jl + 1, t)I]
P/2
1/P
[(j
C(p)IIWk(1) II2d+3Ex2
J
1
P
IJlsI2ds) l 2n(1,10 )+1
< C(p) J
II wj II2d+3
jE9(t+1)
(a(I p)t k(1,1o) 2n(l, lo) + 1
If U is not bounded, we have a(l+1) = a(l)+1 and by the Cauchy-Schwartz inequality (4.42) 1
Ex x [l J(l + 1,t) I
P] /P
_< C(p)IIwk(1)II2d+3
(f
t
Ex'x [J2P(l, s)]
1/P ds)
1/2
0
2n(1,10)+1
< C(2p)
Ilwj 112d+3
iEJ(1+1)
k(1,1o)
V
2)(l, lo) + 1
If
J (1 + 1, t) =
f J(l, t)uds
where u is bounded then we have a(l + 1) = a(l). The ds can appear from a contraction or from Xo in (4.39). Let us consider the first case which is more complicated. For some integer k(l) P]1P
Ex'x [IJ(l + 1,t)I (4.43)
< C(p)IIwk(1)II2d+3IIwk(,)-1II2d+3 f t Ex'x [JP(l, s)]1/P ds 0
11 jE.T (1+1)
C (a(l),
,0+1
p)tn(t l
IIwjII2d+3k(1,1o)(n(1, 1o)+1)
If u is only bounded in LP we apply the Cauchy-Schwartz inequality to get (4.44)
Ex'x [I J(1 + 1, t)IP]
1/P
C(a(l), 2p)tn(t,lo)+1
< C(2p)
IIwjII2d+3
iEJ(1+1)
From these recursive relations we can deduce that a(1)
C(a(l),p)
11 C(2kp) t=o
k(l lo)(n(l, lo) + 1),
148
Jones & Leandre: LP-Chen forms on loop spaces
and since a(l) is bounded by 2d + 2 we have C(0(1), p) < C(p)t.
(4.45)
From (4.41)-(4.45) we deduce that
n(l,lo)=2+I2I By using a similar argument to that used in the final step of the proof of (4.2) we can show that
4-
t (1) t tn'_i 7
k(1,10) >
-1to +1
(4.46)
(1)1
=
lt(l0)!
Let us return to the problem of estimating Ey,x
1/P
[l i(r,1)IP]
In this case 3(r) is the full set of indices (1,.. . , n}. We have C(p) n
I /P
(4.47)
<
E [I.7(r,1)IPI
n!(n - r)!
112d+3,. ... Ilwl
Ilwtll2d+3,oo
since there are at least n - r terms in ds in these integrals. On the other hand the binomial formula implies that n
(4.48)
n!
(1 n-
)
r -
n
- Irn
This completes the proof of (4.2).
§5 THE BRIDGE OF A HYPOELLIPTIC DIFFUSION
The goal of this section is to study the bridge of a hypoelliptic diffusion over a compact manifold N; the example we wish to apply the theory to is the frame bundle N of the compact Riemannian manifold M. There are two cases to consider, the homogeneous case and the inhomogeneous case and we start with the homogeneous case.
Jones & Leandre: LP-Chen forms on loop spaces
149
Let X1,.. . , Xm be m vector fields on N. We make the following assumption on the Xi. (H1)
At each x E N, the Lie algebra spanned by the Xi is equal to Tx N.
The semi-group associated to the operator 1 E X, has a density pt(x, y) which is the density of the following stochastic differential equation in the Stratonovitch sense m
dwt = > Xi(wt)dB',
(5.1)
wo = x
i=1
The equation of the associated Brownian bridge wt(x, y) starting at x and ending at y after time 1 is m
(5.2)
dwt(x, y)
i-
Xi(wt(x, y))(dBi + ri(x, y)dt)
where m
(5.3)
ri(x, y) _
i- Xi(wt(x, Y)) (Xi(wt(x, y)), grad logpl-t(wt(x, y)))
Theorem 5.4. If the vector fields Xi satisfy H1 then, for all x, y and t E [0, 1], the Brownian bridge wt(x, y) is a semi-martingale.
Proof. It is enough to show that
JE [Ir(x,y)I] dt < oc The law of wt(x, y) has density Pt (x, z, y) =
z, y)
Pt (x
Pi(X)Y)
so, using (5.3), it is enough to show that 1
f o
IN
Pt (x, z) I (Xi(z), grad pi-t(z, y)) I dz
is finite. Using estimates of Jerison-Sanchez [23] and Kusuoka-Stroock [27] we know that there are constants C1, C2, C3, C4 such that (5.5)
Cl y)21 V(x ) exp - d(x, ) C2t C
< pt (x, y)
V(x ) --V( -d(x,Coty)2 ) C3
Jones & Leandre: LP-Chen forms on loop spaces
150
where t E [0,1], (x, y) E N X N, d(x, y)2 is the hypoelliptic distance associated to the operator, and V(x, f) is the volume of the hypoelliptic ball with centre x and radius f. We also know that there are constants C5 and C6 such that (5.6)
V7)
(X(x), gradpt(x, y))
exp
(_d(x)2)
We will use the doubling inequality: there is a constant C7 such that V(x, 2t) < C7V(x, t)
(5.7)
From (5.5), (5.6), and (5.7) we deduce that there exists an integer p such
that (5.8) Pt (x, z) I (Xi (z), grad pi _t (z, y)) C3C5
<
<
d(xC,
V(x,f)V(z, 1-t) 1-t eXp( K V(x,
)exp(_ d(z, y)2
ot
exp
2
z)2
C6(1 - t) )
y)2 (-d(x, z)2) exp I 2pd(z, C6(1 - t) )
for some constant K. Together (5.5) and (5.8) show that
J pt(x,z)I (Xt(z),gradpi-t(z,y)) Idz M
Kt
1-t
N
p2pt(x, z)p2P(i-t)(z, y)dz
< K'p2y (x, y)
1-t
From this it follows that
f IN pt(x, y) I (Xi(z), grad pi-t(z, y)) I dz 0
is finite using the fact that
Let us now consider the inhomogeneous case of the diffusion associated to
X0+1Xi.
Jones & Leandre: LP-Chen forms on loop spaces
151
In this case, in general, we do not have the Jerison-Sanchez estimates for pt(x) y) so we use another approach which follows computations of KusuokaStroock [28, part 2]. Let us now make the following assumption on the Xi:
(H2) At each x E N the space spanned by the Lie brackets of length > 2 constructed using Xi, i = 0, ... , m and Xj, j = 1, ... , m is equal to TIN. The solution of the Stratonovitch equation m
Xi(wt(x))dBi,
dwt(x) = Xo(wt(x))dt +
(5.10)
wo(x) = x
i=1
has, at time 1, a smooth density pi (XI y). Let us denote by PI (dy) the law of probability p1 (x, y)dy. Let us consider the equation of the Brownian bridge
arriving at y after time 1 m
(5.11)
Xi(wt(x, y))(dB' + rt(x, y)dt),
dwt(x, y) = Xo(wt(x, y))dt + i=1
wo(x, y) = x
where rt(x, y) is as in (5.3). We have the following theorem.
Theorem 5.12. For almost ally for P1(dy) the process wt(x, y) is a semimartingale.
This theorem says that wt(x) is still a semi-martingale for the filtration augmented by w1(x). Proof. Let us first give a stochastic interpretation of F(x, y) = (Xi(x), grad log pt(x, y))
Consider the measure p on N defined by µ(f) = 11V F(x, y)f (y)pt (x, y)dy
(5.13)
= (Xi(x),f gradpt(x,y).f(y)dy) _ (Xii(x), E [.f(wt(x))])
E {(df(w(x)), dwt(x)Xi(x))] By using an integration by parts formula we get (5.14)
E [(df(w(x)), dxt (x)Xi(x))] = E [t(x).f(wt(x))]
Jones & Leandre: LP-Chen forms on loop spaces
152
and this shows us that (Xi(x), gradlogpt(x, y)) I = (5.15)
I
wt(x) = y].
Now let us estimate the quantity Ewl(x)
[Ew.(x,y) [I
rt(x, y)I]
We see that (5.16) Ewi(x)
[Irt(x,y)I]] <
J
T)t(x, pz1)P1 NxN
z, y) E
x,
[If1-t(z)I : wl-t(z) = y] pi (x, y) dydz
Jpt(x, z)pl-t(z, y)E [If1-t(z)I : w1-t(z) = y] dydz
NxN
= f pt(x, z)dz f E [If1-t(z)I : wi-t(z) = y] p1-t(z, y) dy N
N
= IN pt(x, z)E [Ifi-t(z)I] dz where the last equality follows from the definition of the conditional expectation. So, to prove the theorem, it is enough to show that E [I f1-t(z)I ] <
for some constant C independent of z. We follow the method of [28, Chapter 2]. Let a = (a1i ... , a,,) be a sequence of n integers between 0 and n and define IIall to be the number of ai which are non-zero plus twice the number of ai which are zero. Now we put X(a,i) _ [Xi, Xa],
X(i) = Xi,
for i # 0,
and then define m
(5.17)
G,(t, x) = E E tlla11+1X(a,i)(x) 0 X(a,i)(x) i=1 11aII<1-1
Jones & Leandre: LP-Chen forms on loop spaces
By hypothesis there is an integer to such that
Glo(t,x)-1
153
exists and is
smooth on N. From the compactness of N there exist aa,# E C°°(N, R)
such that X(a.) =
(5.18)
Y,
aa,p(x)X(Q)(x)
0540,11011<10
for any a with Hall > lo. This follows since, for any vector field X(x), (5.19)
=E
X(x)
(X(x),G10(1,x)-1X(a)(x))X(a>(x)
a#0,11«11<_1o
Let cpt(w, x) be the stochastic flow associated to (5.10); then it follows from [27] that -1
eat 8x
(5.20)
X(a)(x) _
Ca,Q(t,x)X(p)(x)
where E[
(5.21)
x)1" ]1/n < C(p)(f)11#11-Il° 11.
I
Now let
Mt(x) = (Dwt(x), Dwt(x)) be the Malliavin matrix associated to wt(x). Classically 2
Mt(x)
=
ft °
i-o
(8x8)-1 X=(we(x)),-
ds
and, moreover, (5.22)
M, (X) = E
E ea,a(t,x)X(a) (x) 0 X(a) (x) Q00,11011<
where (5.23)
sup E [I ea,Q(t, x)Il']1/) < C(p)(f)IIQII-11("11, X
We get the following lemma, analogous to Theorem (2.8) of [28].
Jones & Leandre: LP-Chen forms on loop spaces
154
Lemma 5.24. Let )(t,x) = inf
1};
E
then there exist constants C, c > 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<<-t0-1
with Ri(s, x) _
(5.28)
r(Q)(s, x)X(Q)(x). hall
Now 2
ds
hall
=
f
2
1/K
ds llal1:51o-1
and so from Theorem A.6 in the appendix of [27] we deduce the following estimate: (5.29)
P inf
{dji/K
2
ds llall
< C" exp(-c"K" )
<
2K-t0-1/ a
Jones & Leandre: LP-Chen forms on loop spaces
155
where S = {E :
G(o(t, x)E) =1} .
On the other hand rQ(s, x) (
R= (s, x)) 1
,
X(a)(x)))
11011 <10
(5.30)
t-1o+1
<
K IIQII<_(o
sup s
x)
So we need only to find an upper bound for (0-1 /3
d
P 1: 1:
i=1 IIQII<(o
sup o<s
K
By the computations of the appendix of [27] this is smaller than
C" exp(-c"K°) From the previous lemma we deduce the following lemma which is analogous to Corollary (2.10) of [28].
Lemma 5.31. There is a constant C(p) such that for all x and t < 1 (5.32)
E [I (X(a)(x),Mt(x)-1X(Q)(x))I"I
1/p
< C(p)t- 11-1111,611 2
for a#0,/3#0,a
(X(a)(x),Mt(x)-'X(,3)(x))p < (X(a)(x),Mt(x)-'X(a.)(x))pl2 (X(Q)(x),Mt(x)-l
so it is enough to show (5.32) when a = Q. Let us write GI(, (t, x) = UUt,
where Ut is the transpose of U, and
Mt(x) = U
Ut.
0... Ad
X(Q)(x))p/2
156
Jones & Leandre: L-Chen forms on loop spaces
Then lemma (5.24) shows that
(A1...O)_'
belongs to all the LP uniformly in x and t. So it is enough to show that
E
(5.34)
[IU-'X(«)(x)Ipll/p
< tllC/2
1
However we have
UUt = E ta«0X(«)(x) ® X(«> (x) «#o,ll«II-
(5.35)
_
tll«Ilx(a)(x)x(t«)(x). E «#o,ll«u
But
U-IUUt(Ut)-l
tll«IIU-'x(«)(x)x(«)(x)(Ut)-1 «#O,II«IISto
(5.36)
_E
tII«II U-1x(«)(x) ® U-'X(«)(x)
«#o,II«II
and this shows that IU-lX(«)(x)I < Ct-II«II12. This proves (5.34) and so completes the proof of Lemma (5.31).
As a consequence of Lemma (5.31) we get the following result. We use the notation II - I1p,1 for the Sobolev (p, 1)-norm.
Lemma 5.37. For all x,t<1,a#0,/3#0 where llall :5 lo and 11011 :5 lo (5.38)
(X(.) (X), Mt(x)-'X(Q)(x)) Ilp,' < C(p)t
°z
11
1111110
Proof. Using (5.23) and computing derivatives we get (5.39)
D(X(«)(x),Mt(x)-'X(Q)(x)) =
EDelh,.2(t, x) (X(a)(x), Mt(x)-'X(11)(x)) (X(72)(x),
Mt(x)-1
X(Q)(x))
The sum in this formula is taken over the set of -ti. and y2 such that ys
0,
II7=II < 10.
Jones & Leandre: LP-Chen forms on loop spaces
157
We can improve (5.23) to the following estimate (5.40)
sup II ea,Q(t, x)II p,l < C(p,1)(/)IIQII+IIaII x
The lemma now follows from (5.40) and (5.22).
We can now compute explicitly the term 1(x) which appears in (5.14). Let us recall (5.13) and write, for j # 0 m
(5.41)
Xi(x) _
hi,.i(s) _
f
t
C ax9
(ax9)
/ -1 Xi(w9(x))hi,i(s)ds
Xi(w9(x)),Mt(x)-1X.i(x)
XiE [f(wt(x))] = E [(Df(wt(x)), h)] = E [f(wt(x))Sh] where Sh is the Skorohod integral of h [31]. Let us recall the following fact from [31]. If h is a process in the Cameron-Martin space, F a scalar Wiener functional (5.43)
S(hF) = F8h - (h, DF)
By using the fact that the Skorohod integral of a non-anticipative process is nothing more than the Ito integral and by using lemma (5.?) and (5.20) we get (5.44)
E[IShIP]1"P <
f
Lemma 5.45. The finite variational part of the semi-martingale wt(x, y), denoted by r
J
V, (x, y) ds
satisfies Ex.(x,Y) [IV,(x,y)IP]'1P <
1 C
s
Jones & Leandre: LP-Chen forms on loop spaces
158
Note that this is exactly the Lemma required to deduce the inequality
J
tEe,v [If'(t'
s)2pj11/2P
-1,t')I2pdt']1/2pdt'
E;,v
J
I8 C(s ,P) E,," [I j(1 -Ifs --t 1
- l,t')j2p] 1/2p dt'
used in the proof of (4.15). This gives a slight improvement on Theorem (5.12). Our technique will give the analogue of [28, (2.18)] in our situation. Proof of Lemma (5.45). We can in fact improve (5.16) (5.46)
E"(x) [E'' (=,y) [P1-t(x,y)p ( 1 - t)p]] NxN
pt(x, z)E [( 1 - t)e1-t(z) I xl-t(z) = y]p Pl-t(z, y)dy
INJN
E [( 1 - t)pi-t(z) I X1-t(z) = y] Pl-t(z, y)dy
= IN pt(x, z)dzE [( 1 - t)p ei-t(z)] < C So
we have shown that for almost all y w t(x,
where
y) = Martingale +
j
E_ (=,y) [IVs(x, y)I p] 1/' <
V3(x, y)ds
C 1 -- S
for all p.
Remark 5.47. Suppose that the Lie brackets of length 2 span TTN for all x and that the horizontal space over N is of constant dimension d'. We are of course in the situation of Theorem (5.12). In that case, it follows from the Nagel-Stein-Wainger estimates [33] that the volume of the hypoelliptic ball is bounded by
C(f)2d-d' < V(x, V t-) <
C'(\)2d-b = Ct( f)v
uniformly for x E M and t < 1. From a recent result of [36], we get for all e > 0 and for x,y E M and t < 1 C'(E)
V(x, ')
exp (_ d(x, y)2 \ 21(1 - e))
C(e)
(_ d(x, y)2
- Pr(x,y) < V(x, ) exp \
2t(1
+e))
Jones & Leandre: LP-Chen forms on loop spaces
159
From [28, Theorem (2.18)] we deduce, for any b < 1, the existence of C(b) such that I (X=(x),PI(x, y))I
- I (X=(x),ptb(x, z)) pt(1-6)(z, y)dz1 Cam)
C
(\IM ptb(x, z)pt(1-6)(z, y)2dz)
1/2
(5.48)
By substituting the above bounds for V(x, f) and pt(x, y) in the inequality (5.48) we get I (X1(x),Pt(x, Y)) I
d x z)2
C (b, e
d(z, y)2
( f)1+ /2 (IM exp (26t(1 +,F) ) exp ( (1 - 6)t(1 C(b,e)
C(b, e)
- (f)1+v
2 Pt(,_,)(z+z)(x, Y)
d(x, y)2
exp (-2()1+ b)
From this we deduce that for any e 2
I(Xi(x),pt(x,y))
2t(1,+)e)
exp (
and that 1
Em.(=,y) [IV,-t(x,y)IP ( 1 - t)P] < C(p).
1/2
dz)
160
Jones & Leandre: LP-Chen forms on loop spaces REFERENCES
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161
18. E. Getzler and A. Szenes, On the Chern character of a theta summable Fredholm module, Journal of Functional Analysis 89 (1989), 343-358. 19. Y.Z. Hu and P.A. Meyer, Chaos de Wiener ei integrale de Feynmann, Seminaire de Probabilites XXII (J. Azema, P.A. Meyer, and M. Yor, eds.), Lecture Notes in Mathematics, vol. 1321, Springer-Verlag, BerlinHeidelberg-New York, 1988, pp. 51-72. 20. Y.Z. Hu and P.A. Meyer, Sur les integrales multiples de Stratonovitch, Seminaire de Probabilites XXII (J. Azema, P.A. Meyer, and M. Yor, eds.), Lecture Notes in Mathematics, vol. 1321, Springer-Verlag, BerlinHeidelberg-New York, 1988, pp. 72-82. 21. Y.Z. Hu, Calculs formels sur les equations de Stratonovitch, Seminaire de Probabilites XXIV (J. Azema, P.A. Meyer, and M. Yor, eds.), Lecture Notes in Mathematics, vol. 1426, Springer-Verlag, Berlin-HeidelbergNew York, 1990, pp. 453-461. 22. N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North Holland, 1981. 23. D. Jerison and A. Sanchez, Estimates for the heat kernel for the sum of squares of vector fields, Indiana Univ. Math. Jour. 35 (1986), 835-859. 24. J.D.S. Jones, Cyclic homology in geometry and topology, Bull. Lond. Math. Soc (to appear). 25. J.D.S. Jones and S.B. Petrack, The fixed point theorem in equivariant cohomology, Transactions of the A.M.S. 322 (1990), 35-49. 26. D. Kastler, Introduction to entire cyclic cohomology, Stochastics, algebra, and analysis in classical and quantum mechanics (S. Albeverio, Ph. Blanchard, and D. Testard, eds.), Kluwer, Academic Press, Dordrecht-Boston-London, 1990, pp. 73-153. 27. S. Kusuoka and D. Stroock, Applications of the Malliavin calculus, Part II, J. Fac. Science University Tokyo 32 (1985), 1-76. 28. S. Kusuoka and D. Stroock, Applications of the Malliavin calculus, Part III, J. Fac. Science University Tokyo 34 (1987), 391-442. 29. R. Leandre and P.A. Meyer, Sur le developpement en chaos de Wiener d'une diffusion, Seminaire de Probabilites XXIII (J. Azema, P.A. Meyer, and M. Yor, eds.), Lecture Notes in Mathematics, vol. 1372, SpringerVerlag, Berlin-Heidelberg-New York, 1989, pp. 161-165. 30. P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proceedings of International Conference in Stochastic Differential Equations (K. Ito, eds.), Wiley, 1978, pp. 195-263. 31. E. Nualart and E. Pardoux, Stochastic calculus with anticipating integrands, Probability theory and related fields 78 (1988), 535-581. 32. M.S. Narasimhan and S. Ramanan, The existence of universal connections, American J. Math. 83 (1961), 563-572. 33. A. Nagel, E. Stein, and M. Wainger, Balls and metrics defined by vector fields, Acta Math. 155 (1985), 103-197.
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34. L. Schwartz, Construction directe dune diffusion sur une variete, Seminaire de Probabilites XIX (J. Azema, P.A. Meyer, and M. Yor, eds.), Lecture Notes in Mathematics, vol. 1123, Springer-Verlag, Berlin-Heidelberg-New York, 1984, pp. 91-113. 35. L. Schwartz, La convergence de la serie de Picard pour les equations differentielle stochastiques, Seminaire de Probabilites XXIII (J. Azema, P.A. Meyer, and M. Yor, eds.), Lecture Notes in Mathematics, vol. 1372, Springer-Verlag, Berlin-Heidelberg-New York, 1984, pp. 343-355. 36. N. Varopoulos, Small time Gaussian estimates of heat diffusion kernels H. The theory of large deviations, Journal of Functional Analysis 93 (1990), 1-34. MATHEMATICS INSTITUTE, UNIVERSITY OF WARWICK,
COVENTRY CV4 7AL, ENGLAND
DEPARTEMENT DE MATHEMATIQUES,
ULP, RUE DESCARTES, 67084 STRASBOURG,
FRANCE
Convex Geometry and Nonconfluent r-Martingales I: Tightness and Strict Convexity. Wilfrid S. Kendall. Department of Statistics, University of Warwick.
1. Introduction. In Kendall (1990) it is explained how three nonlinear Dirichlet problems are closely connected to a problem about the existence of a certain convex surrogate distance function. Here we consider an aspect of these relationships in a setting more general than the Riemannian case of Kendall (1990). The problems are as follows. Suppose M is a smooth manifold equipped with a connection IF and separately with a reference Riemannian structure (the connection need not be compatible with the metric!). Consider B a closed region in M. (In the sequel B is generally compact, but we prefer to state the following properties as applicable to a general region.)
(A): Does B have r-convex geometry? That is to say, does there exist a (product-connection) convex function Q : B x B
[0, 1] vanishing only on
the diagonal A = {(x, x) : x E B}? Here the "r-" in "r-convex" refers to the use of the connection r to build the product-connection, instead of the Levi-Civita connection supplied by the reference Riemannian metric. (In the rest of the paper the prefix "r-" is omitted; by "convex" we mean "r-convex" unless indicated otherwise.)
(B): Dirichlet problem for r-martingales lying in B. This problem requires one to find r-martingales X (under a given filtration) attaining a given terminal value X(oo). In the following the heading (B) refers specifically to whether the Dirichlet problem is well posed and has unique solutions. By "well-posed" we mean a non-uniform interpretation: if X', X2, ..., X'° are r-martingales converging in B as time tends to infinity, subject to the same
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Kendall: Convex geometry and nonconfluent r-martingales I
filtration, and such that X"(oo) -i X-(oo) in probability then
lim sup dist(X"(t),X°°(t))
n-.oo tE(O,oo)
=
0
using convergence in probability. We do not use well-posedness for (B) in the following, but it will play a part in later work. Note that the question of uniqueness is also described in the literature as the problem of nonconfluence of I'-martingales for B, explaining the title of this paper. (C): Dirichlet problem for I'-harmonic maps with image in B. This problem requires one to find 1'-harmonic map extensions F : 0 - B of the boundaryvalue data OF : 80 - B, where 0 is an open subset of a Riemannian manifold and 80 is its (potential-theoretic) boundary. By "well-posed" here we mean:
if Fl, F2, ..., F'° are I'-harmonic maps from 0 to B such that OF" - 8F°O almost surely in 0-harmonic measure then lim sup {dist(F"(m), F°°(m))
n-oo
:
m E K}
=
0
for each compact set IC C 0. Again in the following the heading (C) refers specifically to whether the Dirichlet problem is well-posed and has unique solutions.
(D): Dirichlet problem for 1'-geodesics lying in B. This problem requires one
to find I'-geodesics -y connecting given pairs of points in B. Again in the following the heading (D) refers specifically to whether the Dirichlet problem is well posed and has unique solutions.
Problem (D) can be viewed as a special case of problems (B) and (C). The whole setting generalizes Kendall (1990) because a general connection 1' is used rather than the Levi-Civita connection supplied by the reference Riemannian metric. The metric is used simply as a convenience; definitions and constructions are all essentially independent of the choice of metric.
Kendall: Convex geometry and nonconfluent r-martingales I
165
Figure: Three implications established in Kendall (1990) for a compact convex region B (in the Riemannian case; the case of general r is a straightforward extension), together with the Emery conjecture (dotted arrow). The figure illustrates some elementary and fundamental logical relationships between these Dirichlet problems in the case of compact and convex B. Suppose first that B is compact. Kendall (1990) treats the Riemannian case (I' is the Levi-Civita connection for the metric) and establishes implications corresponding to the three solid arrows in the figure. In particular convex
geometry (A) implies that if solutions exist for the problems (B) and (C) then they are unique and depend continuously on the data. Indeed, in the Riemannian case (I' the Levi-Civita connection) Kendall (1990) shows that a consequence of (A) is that solutions exist for (B) and (C). The major part of the work of the 1990 paper concerns existence arguments rather than unique-
ness; the three uniqueness implications are straightforward. They extend
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Kendall: Convex geometry and nonconfluent r-martingales I
directly to the non-Riemannian case (indeed the existence implications also extend, since a check shows that the argument of section 8 of Kendall, 1990, allows one to use a reference Riemannian metric unrelated to the connection). In particular convex geometry of B implies nonconfluence of F-martingales in B.
Emery, in a separate investigation into r-martingales (for example Emery, 1985; Emery and Meyer, 1989; Emery and Mokobodzki, preprint), has made the bold conjecture that in the Riemannian case if B is convex and compact (convexity required so that solutions to (D) exist) then uniqueness for (D) implies (A). In the more general non-Riemannian case considered here he has a counterexample which suggests his conjecture should be strengthened to require in addition that (D) be well-posed (Emery, personal communication). In the Riemannian case our best information to date is that the conjecture is verified by explicit calculation of the required convex function for the case of compact subsets of hemispheres and (a slight generalization of this case) for closed subsets of regular geodesic balls lying in general complete Riemannian manifolds (Kendall, 1990, 1991). The next step is to consider compact regions in two-dimensional paraboloids and to verify by explicit (and probably tedious) calculation whether (A) holds in those instances for which (D) holds.
Were some form of the conjecture to prove correct, we would be able to close
the circle of implications of the figure. This demonstrates the importance of Emery's conjecture - the extent to which it is true has profound implications for the nonlinear elliptic variational theory of harmonic maps! The figure suggests a plan for an indirect attack on Emery's conjecture: instead of attempting direct calculations one might try to reverse each in turn of the solid arrows of implication in the figure. In this approach the Emery conjecture is broken up into three possibly more manageable problems, listed as conjectures in section 4 below and each possessing an intrinsic interest.
The purpose of this note is to advertise this picture of Emery's conjecture and to make a small step towards fulfilling this indirect programme of attack.
We shall show below (theorem 3.2) that under a natural condition on the compact region B (that B supports a strictly convex function) the uniqueness
part of (B) implies (A). Thus if B is strictly-convex-supporting and compact then nonconfluence of r-martingales in B implies convex geometry of B. The property (B) can be viewed as defining a problem in stochastic control
involving r-martingales. Our method is to show that the convex geometry required by (A) is provided by the value of the stochastic control problem.
Kendall: Convex geometry and nonconfluent r-martingales I
167
Note that, in a discussion of set-valued barycentres, Emery and Mokobodzki (preprint) obtain results closely related to those given below. In particular it is possible to prove theorem 3.2 using their methods. Note also that Picard (1991) gives an alternative approach to existence aspects of the Dirichlet problem for r-martingales.
The following note was added just before this paper went to press: I believe I can now construct a counterexample to Emery's conjecture (uniqueness for (D) implies (A) for convex and compact B). Details will be published elsewhere.
The comments of an anonymous referee were helpful in improving the clarity of the exposition.
2. Weak Convergence and r-martingales. In this preliminary section we establish an elementary criterion for the weak convergence of r-martingales. Our approach is based on a tightness criterion for "Ito processes" given in Stroock and Varadhan (1979). An alternative approach to r-martingale tightness is implicit in the treatment of set-valued barycentres in Emery and Mokobodzki (preprint); this uses the Skorokhod representation of weak convergence and starts with Rebolledo's theorem on weak convergence of continuous Euclidean martingales (Rebolledo, 1976).
Let D be a closed region in a smooth manifold N, such that N is furnished with a connection r and a reference Riemannian metric as given in the in-
troduction. (Actually we have in mind D = B x B C N = M x M, with the product connection and metric.) Let w denote the canonical continuous sample path stochastic process in D, so that w
C([0,0o);D).
E
We endow C([0, oo); D) with the metric d(w, w')
=
2-r
sup
dist(w(t), W'(0)
)
This makes C([0, oo); D) into a Polish space. We shall consider the weak convergence of Borel probability measures on this Polish space C([0, oo); D);
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Kendall: Convex geometry and nonconfluent r-martingales I
if P " weakly converges to P in this sense then we write P " = P. Note that the topology for C([0, oo); D) (and hence the weak convergence topology for probability measures on C([0, oo); D)) does not depend on the reference Riemannian metric used for dist(w(t), w'(t)) - indeed the same topology and
weak convergence of probability measures result if dist is replaced by the Euclidean metric inherited from a smooth embedding of D into Euclidean space (a fact exploited in the proof of Theorem 2.5 below). For the notion of a r-martingale see Emery and Meyer (1989) or the introductory material in Kendall (1987, 1988, 1990). A r-martingale Z on V is a manifold-valued semimartingale whose IF-development is a Euclidean local martingale. Recall that the r-development M of Z is defined via the rparallel transport using the Stratonovich form of the equations of Cartan development and parallel transport:
= =
d5Z
dso
E d5M (1)
HE dsZ
where H= is the horizontal subspace sitting above S" of the tangent frame bundle T (GL(D)), given by and characterizing the connection r. It is convenient to have the following notation for exit times from geodesic balls.
DEFINITION 2.1. Exit times. For each u > 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-)]
<
Adt
(5)
where (=")T is the adjoint of E" and the u' form an fixed orthonormal basis field over all D. Then the sequence {P" : n > 1} of probability measures, formed by the laws of the Z", is tight.
Remark: It suffices that the initial laws {Z'(0) : n > 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])]
<
2 supf K2
Proof: From equation (6)
E [f(Z(oo))] - f(Z(0))
=
2E
[j°(Hessf(,),d[M,M})]
The lemma follows by applying inequality (8). o
Remark: Because of the Darling-Zheng F-martingale convergence theorem (Darling, 1983; Zheng, 1983), it follows that the strictly-convex-supporting property for a compact region B implies that all I -martingales in B converge at time oo.
THEOREM 3.2. Suppose the compact region B supports a strictly convex function. Suppose further that solutions to the I'-martingale problem for B are unique when they exist. Then V equals 1 only on the diagonal 0 and consequently B has convex geometry.
Kendall: Convex geometry and nonconfluent I'-martingales I
173
Proof. Suppose V(x,y) = 1. Let {(X4,Yn) : n > 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
= <
tr ((( 'n, `In) (& ('n, `pn)) . d[(Mn, Nn), (Mn, Nn)])
2 tr((n (& in) d[M', Mn]) + 2 tr((P" ® qn) d[Nn, Nn]).
Consequently by lemma 3.1, and using the notation of that lemma, E[Tn]
<
8 supf X2
and so for each h > 0 P [Tn > h]
<
8supf h-1 K2
Hence T" < oo with probability one, and so the limits Xn(oo) and Yn(oo) genuinely exist. A similar argument holds for the intrinsic time of the weak limit I'-martingale (X, Y), so that (X, Y) also converges at infinity. By the boundedness of B and a dominated convergence argument E [dist(X"(oo), Yn(oo)]
-+
0.
Since X' and Yn stop at time Tn we deduce from the upper bound on P [Tn > h] that E [dist(Xn(h), Y" (h)]
<
E [dist(X"(oo), Y"(oo)] +
8 sup f
(diam B) h-'
and so it follows
limsupE[dist(X'(h),Y'(h)] n-
<
8supf (diaamB)h-' K2
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Kendall: Convex geometry and nonconfluent r-martingales I
and so by dominated convergence again E [dist(X(h), Y(h)]
Taking the limit as h
8 sup f
(diam B) h''
.
co we see
E [dist(X(oo), Y(oo)]
=
0.
By uniqueness of the I'-martingale Dirichlet problem for B it now follows X, = YY for all t and hence in particular x = y as required. The theorem follows. c
4. Discussion. If the strictly-convex-supporting condition is removed in theorem 3.2 then one may still construct a V supremum-attaining sequence of I -martingales {(X",Y") : n >_ 1} with (X",Y") . (X,Y) and P[X"(oo) = Y"(oo)] 1, but control is lost over the convergence of (X"(oo), Y"(oo)). It is natural to conjecture that the use of well-posedness might recover some control:
CONJECTURE 4.1. (B) * (A). Let B be a compact region for which the l'martingale Dirichlet problem has unique and well posed solutions (when they exist). Then B has convex geometry.
Remark: This conjecture has now been verified. A proof will appear in Kendall (preprint).
Note that (B) * (A) is immediate if well-posedness holds uniformly. Of course strengthening (B) to require uniform well-posedness would make the (B) below still harder! (C) We sketch some further considerations which may be informative.
Suppose that B is a convex compact region. Set
z=
{(x, y) : V(x, y) = 1}
.
Kendall: Convex geometry and nonconfluent r-martingales I
175
Then A C 2 and 2 is a convex closed set in B x B, because if (x, y) lies on a geodesic between two members of Z then a simple pasting argument yields a V-supremum-attaining sequence of r-martingales for (x, y) built from the two Vsupremum-attaining sequences for the two members of 2 (it is at this point that we need convexity for B).
The geometry of Z is intriguing. Any point z E 2 \ A has running through
it a r-martingale Z staying entirely in Z, obtained as a weak limit of a supremum-attaining sequence constructed as in the proof of theorem 3.2. The special form of the intrinsic time measures of the limiting sequence, and the requirement for each of the members of the sequence to move from z to A,
allow one to argue that Z is non-trivial. But this means that at every point z of (OZ) \ A there must be at least one r-geodesic with contact of secondorder. For otherwise one can construct a convex smooth function f defined in a neighbourhood of z such that f (z) = 0 and f < 0 where it is defined on 2, and a contradiction follows from consideration of the submartingale properties of f (Z). So if there are non-trivial 2 then their boundaries have an interesting second-order geometry. For two-dimensional B x B it follows that (8Z) \ A is the union of images of r-geodesics connecting points in A. It follows from uniqueness for (B) (and hence for (D)) that 82 C A and hence 2 C A. Unfortunately dim(Bx B) = 2 means B is one-dimensional and hence geometrically trivial.
However the whole line of argument of this paper also applies to an analogous set of problems in which the special case of two dimensions is not so trivial. These are problems centering around whether there are no non-trivial r-martingales converging to deterministic limits: we indicate them in the following list.
(A'): Do all singleton point-sets of B occur as zero sets of nonnegative convex functions?
(B'): r-martingale Liouville property. No non-trivial r-martingales converge to deterministic limits, moreover a r-martingale with an almost deterministic limit is almost trivial (that is to say, is almost constant in time).
(C'): r-harmonic map Liouville property. No non-trivial r-harmonic maps
F : 0 - B have constant boundary-value data 8F : 80 - B, moreover a r-harmonic map with almost constant boundary data is almost trivial (that is to say, is almost a constant map).
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Kendall: Convex geometry and nonconfluent r-martingales I
(D'): r-geodesic Liouville property. There are no nontrivial r-geodesics with coincident endpoints. Moreover a r-geodesic with almost coincident endpoints is almost trivial (that is to say, is almost a constant geodesic).
(C') = (D') (for . (B') example (B') . (C') is described in Kendall, 1988 Theorem 6). We can argue much as in section 3 to show (B') * (A') under the additional condition that B is strictly-convex-supporting. From the considerations given above it can be shown that (B') * (A') without any further conditions when B is convex and two-dimensional. Emery conjectures (D') * (A') when B is convex and compact. As before it is straightforward to show (A')
Reverting to Dirichlet problems, for completeness we mention the two other conjectures raised by the figure.
CONJECTURE 4.2. (C) * (B). Let B be a compact region which is a codomain for a r-harmonic map Dirichlet problem with unique and well-posed solutions (when they exist). Then the same is true for the r-martingale Dirichlet problem for B.
CONJECTURE 4.3. (D) * (C). Let B be a compact convex region for which the r-geodesic Dirichlet problem has unique and well posed solutions. Then the same is true for the r-harmonic map Dirichlet problem for codomain B when its solutions exist.
Progress on the (C) * (B) conjecture awaits a study of the fundamental problem of approximations of r-martingales by images of Riemannian Brow-
nian motions under r-harmonic maps. The (D) * (C) conjecture appears almost as difficult as the Emery conjecture itself, and it may be better to try to show (D) * (B). For example, under (D) an inductive argument shows (B) for the restricted class of dyadic r-martingales built from iterated barycentres by Emery and Mokobodzki (preprint; see proof of Theoreme 2).
The three conjectures of this section would all be resolved affirmatively if the Emery conjecture were shown to be true. A major difficulty in making progress towards resolving the Emery conjecture is that direct checks on specific cases typically require non-obvious and tedious calculation. These three conjectures define a useful indirect programme for an attack on Emery's fun-
Kendall: Convex geometry and nonconfluent r-martingales I
177
damental question, of whether uniqueness and well-posedness for (D) implies (A). Moreover they have intrinsic interest, so that the indirect approach offers rewards even if the Emery conjecture proves false. Theorem 3.2 of this paper makes the first and simplest step of this indirect programme, by giving
a partial resolution of the (B) = (A) conjecture. (But see the note added at the end of the introduction, section 1).
References.
R.W.R.Darling (1982), "Martingales in Manifolds - Definition, Examples and Behaviour under Maps." Sdminaire de Probabilities XVII, Lecture Notes in Mathematics 921, Berlin, Springer, 217-236.
R.W.R.Darling (1983), "Convergence of Martingales on a Riemannian Manifold" Publ. R.LM.S Kyoto Univ. 19 753-763. M.Emery (1985), "Convergence des Martingales dans les Varietes." Colloque en 1'Honneur de Laurent Schwartz (Volume 2), Asterisque 132, 47-63.
M.Emery and P.A.Meyer (1989), Stochastic calculus in manifolds. Berlin, Springer.
M.Emery and G.Mokobodzki (preprint), "Sur le barycentre d'une probabilite dans une variete." Submitted to SIminaire de Probabilities XXV. M.Emery and W.A.Zheng (1984), "Fonctions Convexes et Semimartingales dans une Variete." Seminaire de Probabilities XVIII, Lecture Notes in Mathematics 1059, Berlin, Springer, 501-538.
W.S.Kendall (1987), "Stochastic Differential Geometry: An Introduction." Acta Applicandae Math. 9, 29-60. W.S.Kendall (1988), "Martingales on Manifolds and Harmonic Maps." Contemporary Mathematics 73, 121-157 (but see also the correction sheet available from the author).
178
Kendall: Convex geometry and nonconfluent r-martingales I
W.S.Kendall (1990), "Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence." Proceedings of the London Math. Soc. (3) 61, 371-406.
W.S.Kendall (1991), "Convexity and the hemisphere." To appear, Journal of the London Math. Soc.
W.S.Kendall (preprint), "Convex Geometry and Nonconfluent r-Martingales II: well-posedness and r-martingale convergence."
J.Picard (1991), "Martingales on Manifolds with Prescribed Limits." To appear Journal Funct. Analysis. R.Rebolledo (1976), "Convergence en loi des martingales continues." Acad. Sc. Paris 282, 483-485.
C. R.
D.Stroock and S.R.S.Varadhan (1979), Multidimensional Diffusion Processes. Berlin, Springer.
W.A.Zheng (1983), "Sur la Theoreme de Convergence des Martingales dans une Variete Riemanniene" Z. Warscheinlichkeitstheorie verw. Gebiete 63, 511-515.
Department of Statistics University of Warwick Coventry CV4 7AL, UK.
Some Caricatures of Multiple Contact Diffusion-Limited Aggregation and the q-model HARRY KESTEN*
Department of Mathematics, Cornell University, Ithaca, N.Y. 14853 Abstract. We consider some variants of DLA which shift the distribution of the place where a new particle is added in a very strong way to the points of maximal harmonic measure. As a consequence these variants can grow like "generalized plus signs", with the aggregate containing only points on the coordinate axes at all times.
1. Introduction and statement of results. We construct connected lattice sets A,,, n = 1, 2, . .., by two procedures, both of which are variants of common procedures in DLA (Diffusion Limited
Aggregation). The original DLA model was introduced by Witten and Sander [22]; see also [13] and [21, Sect. 6] for a general introduction to DLA. We only consider lattice models, so that A,, is a connected subset of V. We always take Al = {0} and A1, will contain exactly n sites. The site added to An to make A,,+1 is denoted by yn, so that An+1 = Aft U {yn}. yn is chosen from COAR, the boundary of An, which is the collection of sites adjacent to An, but not in A. To describe the distribution of yn we
introduce some notation. Let S,.., k > 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
Kesten: Multiple contact diffusion-limited aggregation
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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182
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
183
Kesten: Multiple contact diffusion-limited aggregation
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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Kesten: Multiple contact diffusion-limited aggregation
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.
Kesten: Multiple contact diffusion-limited aggregation
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
Kesten: Multiple contact diffusion-limited aggregation
186
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
<
if t1 , t2 are tips of A and k > K21 and also (2.5)
µ8A (/w)
µ83(t)
<
1-a
if t is a tip of A, but w e 8A is not a tip of A and k > K2 (w is otherwise arbitrary in OA). If d > 4, then (2.4) can be sharpened to (2.6)
Iµ8A/(12)
1
<
I1'Ik3-d
µ83(t1)
We break up the proof into several lemmas. In Lemmas 2-6 we take
d=3.
LEMMA 2. Let
Ik = {-jet : 0 < j < k}. For each fixed v E Z3 there exists a constant CI (v) > 0 such that (2.7)
Eslk (v) .., C1(v)(log k)-1/2,
k , oo.
Kesten: Multiple contact diffusion-limited aggregation
187
Moreover, for some constant K3 (independent of k, j) we have (2.8)
EsJk(-jet)
<
K3(logk)-1/2{log((j A(k - j))+2)}-i,2,
k>2, O<j 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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Kesten: Multiple contact diffusion-limited aggregation
(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.
Kesten: Multiple contact diffusion-limited aggregation
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<
e
k f2k n2
=
1
2 3
k
r2k n2
= o ((log n)-1)
1 - 2r2n2/E e 3
(summation by parts)
(by (2.14)).
For small e, the sum over k > n2/e is by (2.13) and (2.14) approximately 2 2
(log n)-1
Therefore,
(2.15)
P{Y > n} = P{Y < -n} ti 6 (log n)-1,
This replaces (2.27) in [7].
n -*oo.
Kesten: Multiple contact diffusion-limited aggregation
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},
Kesten: Multiple contact diffusion-limited aggregation
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{TP<-kITo=j} 00
G(j, £)P{Y < -(k + £)} e=1
j-1
cc
v(n) E v(e)6 [log(k+f+j-n)]-1 e=0
n=O
3/2 j-1
2 (3)
v(n)(log k)-
k ->oo.
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
Kesten: Multiple contact diffusion-limited aggregation
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 :e>
=eel}Ple,{SoP
Ik}
e=1
= P{Y<-(k- j)}+>P{Y= j+e}l:G(e,r)P{Y<-r-k}. 1=1
r=1
Kesten: Multiple contact diffusion-limited aggregation
193
The first term in the right hand side is O((logk)-1) for j < k/2, by (2.15). Also, by (2.17) and (2.15), for large k 00
00
£Ar
r=1
n=1
G(f, r)P{Y < -r - k} < r=1
v(f - n) v(r - n) (log(r + k))-1 Co
L
v(.t - n) 1: v(s) (log(s + n + k))-1
6
n=1
s=0
If we now use partial summation and (2.16) (as in the proof of Lemma 6 of [7]) we find that 00
G(e, r)P{Y < -r - k} r=1 00
£
< K4 E v($ - n)
(1 + log s)1/2(s + n + k)-1(log(s + n +
k))-2
s=0
n=1
K5 E v($ - n)(log(n + k))-1/2 n=1
2h
log(P+ 1)
}1/2
log k
Of course, we also have for all t > 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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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) <00, SR(7) = -eel + e2, SR(7 )+1 = SR(jk) =-eel} C-L
(log 2)-1/2
< K4Pie,{R(Ik) < oo, SR(rk) = -eel
for some
L < e < k}
< K4Pie1{S. hits negative first coordinate axis first in {-eel : e> L}}.
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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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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 <
Kl(logk)-3/2
< K2(log k)-1 EsA(t + w)
(2.32)
for k sufficiently large. In particular
EsA(t) - C3(log k)-X12,
(2.33)
k -+ oo.
Proof: Once we prove the first inequality in (2.32), (2.33) follows from (2.23) (take w = 0). It is also easy to obtain from (2.23) (use an estimate like (2.30)) that Esyk(e1 +iv) > 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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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
Thus (2.34) is immediate from (2.35). As a consequence of (2.34) we find with the help of (2.21) that
Pz{R(Ik) < oo} <
(2.36)
K61 -
[d(z, Ik) + 1]-1
Next, with T(.) as in (2.1), it follows that
ESIk(v) >
R(7k) and S. does not hit Ik after T(C(2k))}
R(Ik)} inf Pz{R(Ik) = oo} zEC(2k)
I'6 ) > Pv{T(C(2k)) < R(Ik)} (1 - logk
.
Thus, uniformly in v (2.37)
Pt,{T(C(2k)) < R(Ik)} < 2EsTk(v).
In particular
Pe,+w{T(C(kl2)) < R(Ik/2)} < Pe,±,;,{T(C(kl2)) < R(Ik/4)} < 2Esyk/9 (e1 + w) (2.38)
< K7(log k)-1/2 (by Lemma 2).
Now assume for the sake of argument that k(1,+) > k and that t is the tip (k(1, +) + 1)e1 of A and iv = w. Note that for large k, t + w lies inside the sphere t - el + C(k(1,+)/2), and A \ 70, +) lies outside this sphere. Therefore a random walk starting at t + w cannot hit A \ J before it hits t - el + C(k(1,+)/2). By a decomposition with respect to
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T(t - el + C(k(1, +)/2)) we find (2.39)
0 < Es7(1 +)(t + w) - EsA(t + w)
(since J C A)
= Pt+w{S. does not hit 7(1,+), but does hit A}
= Pt+w {T(t - e1 + C(k(1, +)/2)) < R(7(1, +)), but S hits \ 7(1, +) after R(t - e1 + C(k(1, +)/2))}
< Pt+w{T(t - e1 +C(k(1,+)/2)) < R(J(1,+))} max
_
{ 1 - EsA\7(1,+)(z)
}
z E (t-ei+C(k(1,+)/2))\ J(1,+)
A translation by k(1, +)e1 shows that the first factor in the last member of (2.39) is at most Pei+w {T(C(k(1, +)/2)) < R(Ik/2)} < K7(log k)-112.
(2.40)
As for the second factor,
1 - Es-X\?(, +)(z) < Pz IS. hits one of the arms J(1, -)
(2.41)
or J(i, f), i > 2}. But for z E t - e1 + C(k(1, +)/2))
d(z, J(1, -)) > 2 k(1, +) - 1 and d(z, J(i, f)) > 2 k(1, +) - 1 for i > 2. Thus, (2.36) and obvious symmetry considerations show that
1 -EsA\J(1,+)(z) < 2(2d- 1) K6(log k)-1,
(2.42)
uniformly for z E C(k(1,+)/2)). It follows from (2.39)-(2.41) that I EsT(1,+)(t + w) - Es -(t + w) < 2(2d - 1) K6K7(log
k)-3/2.
I
However, the same argument as used in (2.39)-(2.41) shows that for r
k(l,+) < Eslk+rel (t + w) -
w) < 2K6K7(log 0-3/2
k).
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Since
Eslk+rel (t + w) = Eslk (e1 + w), (2.32) follows when t = (k(1, +) + 1)e1 and w = w. For the other tips of A and for other w, (2.32) follows by symmetry.
Lemma 4 proves (2.4) for d = 3, because (2 .43)
PaA(y) _
where C(B) = capacity of B (see [19, Prop. 26.2], [7, equation (1.4)] ).
We leave it to the reader to check that for d = 3 and any fixed finite set F, even I EsAUF(t + w) - Es7,, (e1 + w) < K1(log
k)-3/2
< I1(log k)-1EsAUF(t + w),
(2.44)
where A, t and w are as in (2.32), but Ki may now depend on F as well. Basically the only change to be made in the proof is to replace A by A U F in (2.39)-(2.42) , and 7(1, -) by J(1, -) U F in the right hand side of (2.41). (2.44) allows us to generalize (2.4) to (2.45)
Po(AUF)(t2)
-1
< Kl(logk)-1
Pa(AUF)(t1)
with A, t1, t2 as in (2.4), F a fixed finite set.
It is also easy to see how to modify the proof of Lemma 4 for d> 4 to obtain (2.6). We now simply use the lower bound
E {number of visits by S. to Ik}
1
for u E Ik, and the asymptotic relation
9(x, y) - adIx -
y12-d
(cf. [10, Theorem 1.5.4]) to obtain (2.46)
Pz{R(Ik) < oo} < Ksk[d(z,Ik)+ 112-d.
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This replaces (2.36). We then have for
N := {-jel : j > 0} = negative first coordinate axis,
andv0N Eslk(v) -+
0
as
k - oo
and also
R(Ik/4)} , EsN(v) > 0 It is now easy to imitate the argument from (2.39) on (with (2.46) replacing (2.36)) to obtain (2.6).
The next lemma begins the proof of (2.5). Again we restrict ourselves to d = 3.
LEMMA 5. Let d = 3. There exists a constant 0 < a < 1 such that for any generalized plus sign A(k(i, E)) which satisfies (2.3) with k(1, +) > k and
anywoftheform w=±e1-je1,i=2,3,j>0fixed, onehas (2.47)
EsA(k(l, +)e1 + w) POA(k(1, +)e1 + w) l.POA((k(l, +) + 1)e1) = Es-((k(1, +) + 1)e1) < 1 - a for all sufficiently large k.
Proof: The equality in (2.47) is again obvious form (2.43). In addition, by (2.32), it suffices to prove Eslk (w) (2.48)
< 1 - a for large k.
Esyk(e1) -
Now write Jk for {-jet : -1 < j < k}. Note that Jk = Ik U {e1} and (2.49)
7k \7k = {2e1, el + e,, el - e el + e3, el - e3}.
For any w a decomposition with respect to the last exit from Jk g ives (2.50)
Esjk(w) = Esy k(w) 00
+ E EPw{Sn=v,R(Ik)>n}Esyk(v). vEJk\Ik n=1
Also by a shift of -e1 Es7k(v) = Eslk+l (v - e1),
201
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and just as in (2.24) Es7k(v - el) = Eslk+l(v - el)
+E
1
z }Esyk(z).
Pv_eI{R(Ik+l) < oo, SR(Ik+I)
ZEIk+1\Ik
J
Since
Ik+l \ Ik = {-(k + 2)e1, -(k + 1)el ± e2, -(k + 1)e1 ± e3}, and (by (2.7) and (2.8))
Esyk (z) < Kl (log k)-1/2,
z E Ik+1 \ 'k)
we have (2.51) EsJk(v) = Esyk+I(v - el) = Esyk(v - el) + o(log
Es-1,'(v-e1),
k)-1/2
k --- oo,
for any fixed v for which there is some j > 0 with (2.52)
P,, -,,IS. hits jet before R} > 0
(see (2.28) for R). Indeed if (2.52) holds for some j > 0, then it holds for all j > 1 and
Esyk(w - el) > Pw_ei IS hits jel before R} Esyk(je1), and we saw in (2.31) that for large enough j
Esy (jel) >-
Cj(logk)-112
for some Cj > 0. In particular (2.51) holds for all v E Jk \ Ik (see (2.49)). Now we first apply (2.50) with w = e1. Observe that by symmetry
PeI IS,, = el + u, R(Tk) > n} Es7 (el + u)
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has the same value for u = fee, ±e3.
We therefore obtain from
(2.50)-(2.52) (and EsTk(e1) = 0) that (2.53)
Es-yk(2e1)
Esyk(e1) 00
Pe, {Sn = 2el, R(Ik) > n} Es7k(2e1) n=1 00 Pe,IS.=el+e2, + 41:
R(Ik)>n}Es-yk(e1+e2)
n=1
E57 (el)>Pe,{Sn 0" = 2e1, R(7k) > n} n=1 00
+ 4Esyk(e2)
Pe, {Sn = el + e2, R(7k) > n}. n=1
Thus (2.48) for w = e2 (or -e2 or ±e3) will follow once we show
(2.54) > Pel{ Sn = 2e1, R(Tk) > n} n=1 00
Pet {Sn = el + e2, R(7k) > n} > 1 + b
+4 n=1
for some b > 0, independent of k. Simple counting of paths shows
Pe,{Sl = 2e1, R(jk) > 1} = Pe,{Sl = el + e2, R(7k) > 1} =
PeS3 = 2e1, R(Ik) > 3} =
6-1
9.6-3
Pe,{ S3 = el + e2, R(7k) > 3} = 7.6-3. Thus the left hand side of (2.54) exceeds
6-1+9.6 -3 + 4.6-1 + 28.6-3 = 217.6-3 = 1+6 -3
;
(2.54) holds for b = 6-3 and hence (2.48) holds for w = ±e2, ±e3.
It is now easy to obtain (2.48) for any w = +ei - jet, i = 2, 3, j > 0, by induction on j. Indeed, (2.50) and (2.51) for w = e2 - je1 show as above that Es-I,, (e2 - jet) - Esyk(e2 - (j + 1)e1)
P. {Sn = v, R(lk) > n}Eslk(v - e1).
+ n=1
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In particular lim sup
Esyk(e2 - (j + 1)el) Eslk(e2 - 3e1)
k-oo
< 1.
Thus if (2.48) holds for w = e2 - jet and some a > 0, then it holds for w = e2 - (j + 1)e1 and a decreased by an arbitrarily small amount. By induction we therefore have (2.48) for w = e2 - jet, j > 0 and then also
forw=±e;-je1,i=1,2, j>0.
Remark. It is easy to refine this proof to show that the left hand side of (2.48) has a limit as k -> oo, but this is not important for us. LEMMA 6. (2.5) holds.
Proof: Let A satisfy (2.3), and for the sake of argument assume again
that k < k(1, +) < 2k. Then forw=e2+te1,0
EsA(w) < EsJ(1,+)(w) < K3(log k)-112{log(e A (k(1, +) - e) +
2)}-1t2.
If we take L such that
Iii{log(L + 2) }-1/2 < 4 C3, then we find (see (2.33)) (2.55)
EsA(e2 + eel) <
Esq(t)
for L < Q < k(1, +) - L and t any tip of A. By obvious symmetry considerations the same argument works for any w of the form (2.56)
w = ±ej + eel ,
if
k < k(j, +) < 2k,
L < e < kt(t) - L, i # j or if k < k(j, -) < 2k and - k(j, -) + L < f < -L, i Thus for all such w we have paa(w)
pan(t)
<
1
2
j.
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204
if t is a tip of A and k > K4 for some K4 independent of w.
With L fixed as above we can now take care of the remaining w E 8A which are not tips by means of (2.47). Indeed pa-X(W)
< 1 -a
for any w of the form (2.57)
w = ±ei + fed if k < k(j, +) < 2k,
k(j, +) - L < f < k(j, +),
k < k(j, -) < 2k
and
i
j, or if
- k(j, -) < $ < -k(j, -) + L,
provided k > K5 for some K5 (which may depend on L, but we have fixed L now). This is immediate from (2.47) (plus (2.4) and symmetry considerations). Finally, if (2.58)
w = ±e i + Peg
0
with
and
k < k(j,+) < 2k,
i # j, or with
k < k(j, -) < 2k, -L < f < 0
and
i # j,
then we use the following symmetry argument. For the sake of argument take once again k < k(1, +) < 2k, w = e2 + 'e1, 0 < f < L. Then EsA(w) < Es-y(1,+)(w) Es-7(,,+)(e2 + £e1)
= Es7(1,+)(e2 + (k(1,+) - P)el) < (1 - a)Es7(1,+)((k(1,+)+ 1)e1)
fork>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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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
We next state the (partial) analogue of Proposition 1 for d = 2. PROPOSITION 7. Let d = 2 and F C Z2 a fixed finite set.
(a) Assume that A is a line segment (2.60)
{Pei : -k(i, -) < .£ < k(i, +)}
with k < k(i, -), k(i, +) < 2k for some k > 0 and i = 1 or 2. Then for some constants C1 > 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,
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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)
<1-a
for ti the tip with k(i, Ei) > 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.
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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).
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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) <
P { Uj contains at least m balls at time r(m) } 2
e-meEE {eee;}
j=2
where f, is again defined by (2.81), but EE denotes expectation when
pi = s-1 + Ei. E,,{eBli }
Again by [2, Ex. XI.2.d] we find
I- _ pl+pj L pi+pj pi
m ee
Pi
1
jpi+pj(1_e9)}
Since, by assumption pj/pl < 1 - 6, it is easy to see that for 6 < 1/2 one has e-b/2
Pi
<1-
pl + pj(1 - e672)
62
<
e_ 2/s.
8
Thus (2.79) holds.
Proof of Proposition S. We restrict ourselves to the representative cases of Procedure 2 with d > 4 and Procedure 1 with d = 3. For these cases the proof is based on Proposition 1. We note that in all our constructions A,,, contains exactly n vertices. Thus if A satisfies (2.3) for a certain k and S = S,, is as defined before (2.73), then (2.87)
(k - 1)ISI < n = 1 + E (k(i, E) - 1) < 2kjS1. (i,e)ES
Kesten: Multiple contact diffusion-limited aggregation
212
Thus k and n have to be of the same order and we may replace k by n in the right hand side of (2.4) and (2.6) for An. We first consider Procedure 2 and d > 4, which is the easiest case. Now by (2.5), if t is a tip of An and w E 8An is not a tip, then
n(n) < (1-a )n(n)paAn(t) < (1 - a)7 (n)Z0(17) ,
(2.88)
where Zn(y) is as in (1.2). Thus (2.73) is immediate from (1.1) and ,q(n) - Colog n with C0> 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.
Kesten: Multiple contact diffusion-limited aggregation
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 <
4
and all (i, E) E S},
f + 4 (r - e) f o r all e < r < n 2
and all (i, E) E S},
r = inf { r : Ar is not a generalized plus sign or k(i, e, r) > 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
-
(2.90)
P{yr = e(k(i, e, r) + 1)ej I .Fr}. r=1
Assume now that for some e, n > e and 0 < y <
(61SI)-1 (with the K.
of (2.74)) (2.91)
K2(log e)-1/2 <
and F(e, i1) and G(e, n) occur. 4
Then for e < r < r, e < r < o A n, Ar is still a generalized plus sign. Moreover, for (i, e) E S (2.92)
l k(i, e, r)
IZr(i,,,e)1 + I k(i
ISI
e e) - ' I
r-1
+ E Ply, = e(k(i, e, q) + 1)ei I-Tq} -
r lsl e
q=r
< 2e+4(r-e)+ije+4(r-e) _
+2r.
The first inequality here is immediate from (2.90), while for the second inequality we used that Zr (i, e,
(r - e) on G(e, r),
e+ 4
k(i, e, e) -
I
TS-1
< 71e
on
F(e, 77),
216
Kesten: Multiple contact diffusion-limited aggregation
and for q < c (2.93)
,
(i, e) E S,
P{yq = e(k(i, s, q) + 1)ei I .Fq} -
S
<
y 4
Since we assumed r < r, we also have
k(i, e, r) = 0 for (i, e)
S.
Thus F(r, 2r7) occurs. By virtue of (2.74) and (2.91) we see that (2.93) also
holds for j = r, or o-i > r + 1. Note that (2.93) automatically holds for q = f on F(f, 77) by this same argument. It follows that if (2.91) occurs, then
at > r A (n + 1).
(2.94)
For the remainder fix 0 < 17 <
(61SI)-1 and f such that K2(logf)-1/2
Now G(f) = n G(f, n), and also r > f on F(f, 77). Therefore
<4
n>C
P{G(f) occurs and r < oo I F(f, r7)} 00
< Y P{G(f, n) occurs and r= n+ 1 1 F(f, q) I n=1 00
< 1: P{r= n+1 1 F(f,r7)nG(f,n)n{r> n}}. n=1
However, we just saw that on (2.95)
F(f, r7) n G(f, n) n {r > n}
we also have F(n, 27). By (2.73) we therefore have on (2.95)
P{r= n+ 1 l.Pn} < Ia.4 ICln-3 < 2dK1n-2. Thus (2.96)
P{F(.f, 77) and G(f) occur, but r < oo} <
00
2dK1n-2
n =t
<
K3f'1.
We further note that from (2.94), on F(f, 77) n G(e) n {r = oo} also o = oo.
Thus outside a set of probability at most K3f-1 we have r = o,e = 00
Kesten: Multiple contact diffusion-limited aggregation
217
on F(e, 77) fl G(t). In addition, (2.92) shows that F(r, 377/4) occurs on
F(f, j) fl G(e) fl {-r = at = oo} for all r > 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
218
Kesten: Multiple contact diffusion-limited aggregation
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 <
(3.11)
(1-a(16))irj,
which implies for some ic = c(C0) > 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 <_ 2(1 + e0) n}
_n
(recall that Y1(n) + Y2(n) = (n - 1) + Y1(1) + Y2(1)). In particular, X,,, which is positive, is stochastically larger than the absolute value of a one dimensional simple symmetric random walk, {Un} say. Since for a simple random walk limsup IUnJ/ f = oo w.p.1 we also have limsup
(3.15)
n
= oo
w.p.1.
Step (ii). Let no be so large that (3.14) holds for n > 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
<
eA
e)(1 +
1] < - 21.
Since also
E{0?ITj_1} < 1, we have on (3.19) P{(3.21) occurs I .Fns }
1/2 E < P{-Vn > E4 n11 /2 + A 7 E{o, I j-1} 4 ni n,+1
for some n1
Kesten: Multiple contact diffusion-limited aggregation
223
The right hand side here is at most
exp (-
e2 A2
167
)'
by exponential bounds for martingales with bounded increments; for instance one may apply [15, pp. 154, 155] with c = 1, a = (eA/4)n1j/2, A = (eA/4)-y-1ni 1/2 < 1. Next consider a sample path for which (3.19) holds, but (3.21) fails. We then have from (3.20) that for n1 < n < n1(1 + y) nno
nno
E{Aj I Fj_1} >
2D(1 -
e)Aj-1/2
ni+1
n1+1
> 2D(1 - -)A(1 +7)-1/2 ni 1/2(n A 0' - n1). In particular, if o > 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 +
7
4
P -Vni(1+y)
{
< exp (-
4
7')A[(1 +7)n1]1/2 I ni J
Ant1/2 1/2
g '`ln1
-1/2
+ gA n1
E
E{O:i I.Pj_1} I F1}
n1+1
A2). 64
In the last step we again used [15, pp. 154, 155]. It follows from the above estimates that on (3.19).
P{a > 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)
Then v>aandforo- 2EO(2+2E0)-1n so that
E{On+1 I .Fn} > 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<,aI
+
ff
a1>
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 <
k) be elements of N. We denote by jk the least v field for which ,n, h < j <
.
k are measureble. We often write F,k by Fk and Fh = o(Uk h+l,k) The uniform mixing rate (0-mixing rate) of the process {; j E N} is a sequence {qk, k E N} of nonnegative numbers defined by
Ok = ssup sup{I P(FI E) - P(F)D; E E Fh, P(E) > 0, F E F+k,o,}.
(1.2)
Condition (e.2) The sequence Ion) satisfies El 1(supn on)!* < oo Let {¢ik} be the uniform mixing rate of the process
Then the inequal-
Kunita: Random measures and stochastic difference equations
232
ity ¢k < sup, Ok is easily verified for any k E N. Therefore Ek1 < co is satisfied. Then using the mixing inequality, we can show that the infinite sum 00 0k12
V(E1, E2) = 1:{E[XEI (Sl)XE2(Sj)] - ir(El)ir(E2)}
(1.3)
J=2
converges. Further if f, g E L2(ir), then V (f, g) = f fRmxRm f (A)g(A')V(da, da')
(1.4)
is well defined and satisfies
IV(f,g)I
00
'
g21r)1 .
(1.5)
j=1
Let {cn} be an increasing sequence of positive numbers diverging to infinity. The third condition is concerned with the vague convergence of measures {µn} on R'n - {0} defined by
pn(F) = nirn(cnF).
(1.6)
The sequence {cn} will be fixed throughout this paper.
Condition (e.3) The sequence {µn} converges vaguely to a Radon measure p on R'n - {0}, i.e.,
lim f
n-oo R._1ol
h(A)µn(da) = f
R__101
h(,A)p(da)
(1.7)
holds for any bounded continuous function h(A) such that h(a) = 0 holds on a certain neighborhood of 0. We will give the definition of the weak convergence rigorously. Let g(R'n) _ {Ek} (or g(R n - {0}) = IF;,I) be a ring of Borel sets of R'n (or R'n - {0})
such that Ek C R'n (or Fk C R'n-{0}) for any k and it generates the Borel field of R'n (or R' - {0}). We assume that any element E of g(Rn`)
(or F of G(R'n - {0})) satisfies irn(E) -> ir(E) (or pn(F) -+ p(F)). Let M(Rm) (or M(Rm - {0})) be the set of all additive set functions on g(Rm) (or g(Rm - {0})). Then we may define the law of (Bra, Nn) on the space D = D([0, oo); M(Rm) x M(R'n - {0})). (For the topology of the space D, see Kunita [7].) We denote it by Pn. If the sequence {Pn} converges weakly as n -+ oo, the sequence {(B', Nn) In is said to converge in law.
Theorem 1.1. Assume Conditions (t .1) - (e.3) for
j E N}, n =
.. Then the sequence {(B', Nn)}n converges in law. Let (B, N, P°°) be its limit law. Then the followings hold. 1, 2,
Kunita: Random measures and stochastic difference equations
233
(1) For any El,..., Eq of g(Rm), (B(t, Ei),..., B(t, Eq)) is a q dimensional Brownian motion with mean 0 and covariance
E[B(t, E;)B(t, F,)] = t(ir(E, n E;) +V(E;, E;) + V(E;, E,) - ir(E;)ir(E,)).
(1.8)
Further for any t, B(t, E) can be extended continuously to any E E 3(R'), which is a Gaussian random measure. (2) For any F of G(R'n - {0}),N(t, F) is a Poisson process with intensity
p(F). Further for any t, N(t, F) can be extended to any F of B(Rm - {0}), which is a Poisson random measure. (3) The processes B and N are mutually independent. Now we will represent points of R'n - {0} by polar coordinate (r, 0), where r E (0, oo) and 0 E S--1(m - 1 dimensional unit sphere with center 0). Then Rm - {0} is homeomorphic to (0, oo) x S'n-i and (0, oo] x S'n-i U {0} is a compactification of Rm. We can regard {oo} x S'n-i as the boundary of R'n and denote it by 8Rn'. We set km = R'n U 8Rn'. Let p(r), r > 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<_N
If(x)I 1 + IxI
+ E sup IDaf(x)I,
(2.1)
1
and II IIk = liMN-. II IIk,N. We denote by Cb, the set of all f E CA; such taht IIf lik < oo. We assume that f n(A) = f n(x, A) in equation (0.3) is a C22b*-
valued function which is continuous with respect to A or is a step function of A.
Condition (f.1) (1) The sequence { fn(A)}n satisfies sup Ilfn(A)II2,N/p(IAI) < cc n,a
for any N. There exists a function f (A) such that II fn(A)- f (A)II1,N converges to 0 uniformly on compact sets of Rn` for any N. Furthermore, the limit f (x, A) is integrable with respect to 7r for any x and satisfies
f.. f (x, A)ir(dA) = 0.
(2.2)
Kunita: Random measures and stochastic difference equations
236
(2) The sequence {bnr(x)}n defined by (0.6) is in,C'b , supra IIb,"ytII2,N < oo for any N and converges to bM(x) of C'e with respect to II 111,N for any N.
Condition (f.2) Set hn(A) = f"(A)/p(I AI). Each hn(A) is continuous and is extended to a continuous C -valued function hn(A) on k"'. Further there exists a continuous C'*-valued function h(A) such that converges to 0 uniformly on R"' for any N.
IIh"(A)-h(A)IIl,N
Condition (g.1) The sesquence {gn(x)} is in Cb and converges to g(x) with respect to II 111,N for any N.
of
By Condition (f.1), the sequence { fn(A)} satisfies IIJ n(A)II1,N <- CNP(I AI) with some positive constant CN. Then IIf(A)II1,N <- CNp(I a) holds. Set fn(A)
_
1
fn(CnA).
These functions satisfy I1fn(A)II1,N :5 CNP(CnI
fn(A) = P(c
I)
Vn_
(2.3)
AI)/vfn-
Further since
hn(cnA),
the sequence { fn(,)} converges uniformly to a continuous C' valued function f (A) = f (x, A) with respect to II II1,N seminorms by Condition (f.2). It satisfies
f(A) = i(IAI)h(oo,
IAI
if
JAI > 0.
(2.4)
Now the law of the 4-ple (Bra, Nn, Kn, on) is defined on D = D([0, oo) M(R'n) x .M(Rm - {0}) x M(R"') x Rd). We denote it by Pn. We show the weak convergence of {Pn} and characterize the limit (B) N, K.,
B'= n o(B(s, ), N(s, ), Ko(s, ) : s < t + e).
(2.5)
e>o
Let u(x, A), x E Rd, A E R'n be a measurable function, continuous in x such that u(x, ) E L2(ir) for any x. Let V )t an Rd-valued (Be) adapted cadlag process. Then the stochastic integral of u(1/i A) based on B(ds, dA) is defined by t
J R.
u(O,-, A)B(ds, dA)
lim0k=O E (JRm u( W
tk_, , A)B(tk, dA) -
k. u(, tk_1 A)B(tk-1, dA) 1 (2.6)
Kunita: Random measures and stochastic difference equations
237
. < t,t = t} and Ion = maxk tk - tk_l. It is a
where A = {0 = to <
continuous local martingale. Next let v(x, A) be another function with the same property as u(x, A) and let 71t be a cadlag process adapted to (Be). Then the stochastic integral fo f v(77,_, A)B(ds, dA) is well defined. We have
A)B(ds, dA), f Rm < ft Ju(),-,
= f0t (f +
t
Rm
Jv(r7,_, A)B(ds, dA) >
A)v(r7,_, A)ir(dA)) ds
f t (f JR'"XRm u(7',-, A)v(71,-, A')V(dA, dA')) ds
r (ffR"`xR"' u(-O,-, A')v(t1,-, A)V(dA, dA')) ds - Itt u(V,-, A)lr(dA)) ds It (IRm v(77,-, A)ir(dA)) ds. + Jo
(2.7)
(Jitm
Now set
N(t, F) = N(t, F) - tp(F).
(2.8)
We shall define stochastic integrals based on N. If u E L2(µ), the integral f u(A)N(t, dA) is well defined. It is a Levy process with mean 0 and covariance t f u(A)2p(dA). Let u(x, A) and v(x, A) be measurable functions continuous in x such that u(x, A) and v(x, A) belong to L2(1c) for any x. Then the stochastic integrals
u(V,-, A)N(ds, dA),
v(77,_,
A)N(ds, dA)
f t fR o f t fR o are well defined similarly to (2.6). These are local martingales and their bracket process is r
ft
<
JR"'-{o}
= jt Similarly
u('),-,
A) AT (ds, dA), f
(JRm_{o}
IR.-{o}
v(t1,-, A)N(ds, dA) >
A)v(i1,-, A)(dA)) ds.
(2.9)
we can define the integrals of the form r
IS,
u(0)Ko(i, d9),
Jo
fsu(O,-, 9)Ko(ds, d9).
The former is a Brownian motion with mean 0 and variance t f u(9)2v(d9). The latter is a continuous local martingale. We have < It fsm_t u("O,-, 9)Ko(ds, d9), f t f _, v(77,-, 9)Ko(ds, d9) >
= 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<1al
ftf
o
f
a)1V(ds' dA)
f (cp,_, A)N(ds, dA)
+ ft{g + b, + c}(,p,_)ds.
(2.11)
Here the function c(x) = (cl(x), ..., cd(x)) is defined by cs(x)
= f JR-xRm f
(x,
A) . Vfi(x, A')V(d\, dA'),
(2.12)
where f (x, A) = (f, (x, A),..., fd(x, A)) and a b denotes the inner product of two vectors. The case where the limit process cot becomes a diffusion process is studied in [7]. In the present context, it is stated as follows.
Theorem 2.2. Assume Conditions (e.1) - (e.3), (e.4)' and (f.1), (g.1) for equations (0.4). Then the sequence {(B"(t) cpt)}n converges in law. Let (B, co) be its limit law. Then cot = xo + f t JRm f ((p,_, A)B(ds, dA) +
f{g + bl + c}(co,_)ds.
(2.13)
Finally we will discuss the case where the coefficients f" and g" of equation (0.3) does not depend on n. Consider random difference equations
').i = ').i-1 +
=f (V1-l) ') + ng(Oi-1)'
(2.14)
where f (A) = f (x, A) is a continuous C2*-valued function satisfying (2.2), g is an element of C', and is a sequence of stationary processes satisfying
Kunita: Random measures and stochastic difference equations
239
Conditions (e.1) - (e.3). Let ijn be the solution of the above equation with the initial condition z/i0 = x0 and set apt = 1/i1ntj as before.
Corollary 1. Assume supa JJf (A)JJ1,N < oo for any N > 0. Then the sequence {(Bn(t), tOn)}n converges in law. The limit (B(t), ept) satisfies (2.13). We next consider the case where supA I I f (A) I I1,N = oo for some N.
Corollary 2. Assume that Conditions (f.4) and (e.5) are satisfied for p(r) = (1 + r)" and cn = nQ where a, Q > 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}(
(2.16)
0
(2) If = 1/2a, then the sequence {(Bn(t), Nn(t), Kn(t), apt)}n converges in law. The limit satisfies equation (2.11) with IAI°h(x, oo,
IA ).
(2.17)
The proof of Theorem 2.1 will be given at Section 4. In the next section we will formulate Theorem 2.1 in a different way and establish Theorem 3.1, which will be applied in Section 4 for the proof of Theorem 2.1. 3.
Martingale problem
Let cpt be the solution of equation (0.4). Its law Qn can be defined on the space D = D([0, oo); Rd). In this section we show the weak convergence of the sequence of the laws {Qn} assuming the same conditions as in Theorem 2.1. Our goal is to characterize the limit law Q°° as a solution of a certain martingale problem.
Kunita: Random measures and stochastic difference equations
240
We first define two integro-differential operators G(1) and G(2). Let M be a positive constant. For a C2-function F(x), set
f(1)F'(x) =
OxiOxj (x)
.a,7(x)_92F
2
tj
+ E{bM,i(x) + ci(x) + gi(x)} i
8- (x)
+ J <,a,<MIF(x + .f (x, A)) - F(x) -
.fi(x, A)
OF
(x)}µ(dA), (3.1)
where
aij(x) =
fi(x, A)f;(x, A)ir(dA) JI
iA)fj(x, ^,) + fJ(x, A)fi(x, ^t)}V (d^, d^t)
+
+
j.,,{i(x,
Sm-1
hi(x, oo, 8)hj(x, oo, B)v(d9)
(3.2)
JJJJJ
and for a Cb2 -function F, G(2)F(x)
*M
{F(x + f(x, A)) - F(x)}µ(dA)
(3.3)
and define L = G(') + G(2).
Theorem 3.1. Assume Conditions (e.1) - (e.5), (f.1), (f.2) and (g.1) for equations (0.4). Then the sequence {cpi } converges in law. Let (apt, Q°°) be its limit law. Then for any C2 function F, F(oot) - Jot C(')F('p,_)ds
- E {F(cp,) - F(cp,_)}ON(s, {a : I .I > M})
(3.5)
O<s
is a locally square integrable martingale. Further for any Cb function F, F(cpt) -10 CF(W,_)ds
(3.6)
is a square integrable martingale, where LN(s, F) = N(s, F) - N(s-, F).
Kunita: Random measures and stochastic difference equations
241
In the rest of this section, we give the proof of Theorem 3.1. We will apply a criterion of the weak convergence obtained by Fujiwara- Kunita [3]. It tells us that Theorem 3.1 is valid if the following Conditions (A.I)-(A.IV) are satisfied. For a positive constant M, we set fn'M(x) = E[f'(x, Sj'M)]1
fn'M(x, A) = fn'M(x, A) - fn'M(x).
Then by Condition (f.1) (2), f n'M -> 0 and f n'M(.,
A)
, f(.,A) (uniformly
on compact sets for the latter convergence) with respect to 11 II1,N seminorms.
Condition (A.I) (1) For every M > 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
[nt]
_
limo lim E[ sup I-E[ fi e(x, sI- n k=[ns]+1
CC )f, (x, k'C)I ns1n-oo
-(t - s){ JRm fi(x, A)ff(x, A) ir(dA) + fm-1 hi(x, oo, 9)hj(x, oo, 9)v(d9)1I1 = 0. S
l J
Kunita: Random measures and stochastic difference equations
242
(3) For any N > 0 and 0 < s < t, f
1
lim lim E I sup I -E [ e-'0 n-+oo L
xI<_i
n
[2
n,e
ne
n,e
n,e
>2 .f: (x, k' )f. (x, t ) I
l
ns1J
k=[ns]+1 t=k+1
-(t - S) f JRmXR' fi(x, A)fi(x, A')V(da, dA1)I ] = 0.
(4) For any N > 0 and 0 < s < t, [nt]
1
r
lim lim EI spl e-.0 n-.oo
f
111 Ix'I<_v n
[nt]
El k=[ns]+1 t=k+1 ftte(x,k'M)'Ofi e(x,i'M) `r [ns]I >2
-(t - s) .l .lRmxR m f (x, A) V fi(x, A')V(dA, da')Il
= 0.
1
(5) limn- gn(x) = g(x) uniformly on compact sets.
Condition (A.III) For any a with at < 1, M > 0, N > 0 and 0 < s < t, [ntl
nlim .°° E sup I
1
lim n- oo
E l is
3
[nt]
[nt]
EI k=[ns]+1 t=k+1
n)
N
E [D- fn,M(x, k'M)I'F[n,] IJ II
Ixl
f(x,
(3.8)
= 0,
n,M
]I
1
SkM)I2J
XI fnM(x,
tt
= 0.
Condition (A.IV) For any t > 0 and N > 0, lim lim sup P SUP II fn( k )II1,N > M = 0.
M-00 n-'oo
k<[nt]
Cn
In the remainder of this section, we will examine these conditions one by one.
Lemma 3.2. Condition (A.I) is satisfied. Proof can be carried out similarly as in the proof of Lemma 2.2 in [8 J. It is omitted.
Kunita: Random measures and stochastic difference equations
243
Lemma 3.3. Condition (A.II)(1) is satisfied. Proof. Using the mixing inequality,
E sup IE 1xj
E h(fn( x, Cn)/ /X(6,M]Cn 1N) s1J
I
LLLk[ns]+1
l
[nt]
-E r
L k=[ns]+1
1
fk l ] Cn)/X(6,M](IIi()IIl,N)] / Cn 22
nt
2
(on k-[ns])' E
Cn )
lxl
k=[ns]+1
sup h (f n (x,
( 2(o)') JR"'-{o} Is
)
P h(f n(x' A) Un(dA)
where C' is a positive constant independent of n. Therefore the above converges to 0 as n -* oo. On the other hand, we have by Conditions (e.3) and (f.2), nt]
NIE[
l-moosI-
[E
k=[ns]+1
h(fn(x,Cn)IX(6,M](Ilfn(Cn)II1,N/J /
-(t - s) fRm-{o] h(f(x, A)) X(6,M1(Iif(A)II1,N),(da)I = 0. Therefore Condition (A.II)(1) is satisfied.
Lemma 3.4. Condition (A.II)(2) is satisfied. f;'6
Proof. For the convenience of notations, we denote nent of f n'`) by f°'`. We have
E sup -
[nt]
(E[fjn'-(x,Gn'-)fjn'-(X1G ,e
I
(xl<_ N
k=[ns]+1
-E [fin'-(x, Sk")fj "(x, Sk f )]
l
)Ins]]
f IJ
/
[nt]
<
( the i-th compo-
ttn,e
2
E 2(onk-[ns] )' E [ sup f: (x ' Skcn ) f '
ffn,s
E
2
i
Sk
(x '
cn
)]
There exists a positive constant CN such that I,In(A)II1,N C CNP(CnIAI)//
holds for all n and A. Since Ilk'`/cnl < e, the last term is bounded by 2CN(P(CnF)/V/n-)ZEk1(ok)1t2
It converges to 0 as n --+ oo and a --> 0.
244
Kunita: Random measures and stochastic difference equations
Now define hn,e(x, A) = f n,e(x, A)/p(IAI). Then A) can be extended to a C' -valued function hn'e(A). The sequence {hn'e(x, A)}n converges to h(x, A) of Condition (£2) with respect to II II1,N uniformly in A for any fixed e. Note the equality 1
[nt]
n k=[ns]+1
/ tt // E r[fn'e(x Sk'e)fn'e(x, e'e)Jl
= [nt] n [ns]
In- h; '`(x,
A)vE (dA).
Then the right hand side converges to (t - S) Jfm h,(x, A)hi(x, A)ve(dA)
uniformly on compact sets with respect to x. Further as e - 0, the above converges to
(t - s)
fnm h,(x, A)h2(x, A)v(dA)
= (t - S) (JRm fi(x, A)f, (x, A)ir(dA)
+
J
h; (x, oo, 9)hj(x, oo, B)v(d9)) .
Before we proceed to the proof of (A.II)(3) and (4), we need a lemma.
Lemma 3.5. Let k < P. Then p(Iek'el) > c
n
or
gjI
,n,dI)>c]
=
0.
(3.10)
Proof. We will first prove
lim sup E[p(I,k'eI)p(Iek'eI); p(IGk'`I) > c] = 0.
c- 00 n
We have E[p(IGk'eI)p(Iek'el); p(ICk'eI) > c]
<
p(Iek'sI) > Cl
< E[p(Iek'ei)E[p(Ien'eI)2IT`-1]'; E[p(Iei'eI)2I.F 1]' > c,]
(3.11)
Kunita: Random measures and stochastic difference equations +c'E[P(I
k'`I)i
245
P(I&k-'`I) > c]
< E[P(Iek'sI)2I]'E[E[p(Iei'tj)2IFi i]i
E [P(I Sk `I
)'+ry
P(Ie `I) >
c],
where 0 < y < 1 and c' > 0. Note that is bounded in n and both {E[p(II)2.r7_i]} and {p(I4k' `I)1+7}n are uniformly integrable by n
Condition (e.5). Then the supremum of the above with rspect to n converges to 0 as c - oo and c' -* oo. Therefore (3.11) is established. Now for any S > 0, there exists c > 0 such that sup E[P(Ii n
`I)P(Ik'`I)i
(3.12)
P(I k'` I) > c] < S
by (3.11). We have sup E[P(Iei'`I)P(Iek'`I); P(Iek'`I) < c,P(Iei'`I) > n
< c sup E [p(Iei'` I)i n
P(I Ui'` I) > c ]
Cl
.
Since {p(I&i`I)}n is uniformly integrable, there exists c' > 0 such that c- sup E[P(I&i'`I)i P(I&i'`I) > c'] <S. n
(3.13)
Then (3.12) and (3.13) imply sup E[p(I&i'`I)P(IGk'`I)i n
P(I &t'`I) > c ] < 26.
Therefore (3.10) is satisfied. 0
Lemma 3.6. Condition (A.II),(3) is satisfied. Proof. Set Lk.t = .f i M(x, Sk M)f . M(x, St M) and Iik,t(x) = Lk,t(x) - E[Lk,t(x)]
Then we can prove similarly as in the proof of Theorem 4.2 in [3] that [nt]
[nt]
lim sup 1 E E (E[Lk,t(x)IFnsj] - Lk,t(x)) = 0. n-, 1-1:5N n 1k=[ns]+l t=k+l
(3.14)
246
Kunita: Random measures and stochastic difference equations
Hence it is sufficient to prove
lim sup 1 [E
n-oo
[E Lk,,(x) - (t - s)V(f;(x), fl(x)) = 0.
(3.15)
I=K-N n k=[ns]+11=k+1
By Lemma 3.5 for any 6' > 0 there exists a continuous function Ojx) such
that 0<0,(x)<1,+kjx)=1ifI xI
[nt]
k+kp
E EE
sup I txI-N n k=[ns]+11=k+1
ll
n,M)/ J I < 8'
[Lk,l(x)- 0-(G
(1
(3.16)
holds for all n. Then use this inequality instead of (58) in the proof of Lemma 2.3 in [8]. Then we obtain (3.15). The proof of (A.II)(4) can be carried out similarly. Condition (A.11)(5) is immediate from Condition (e.5) and (g.1).
Lemma 3.7. Conditions (A.III) and (A.IV) are satisfied. Proof. Note that [nut]
t
ll
E[Dafi'M(x,S,'M)I'F[na,]j] [nt]
L
{
1
rn
tt
'M(x,Sk'M)Z]
[ns]+1
Since j suplx1
ek)]
is a bounded sequence, the last term
}
n
`
of the above converges to 0 uniformly on compact sets as n -+ oo. This proves (3.8). The convergence (3.9) can be proved similarly. Now, since ll f n(,)ll1,N C CNp(cnl Al )/ f, we have n I\/
p\k<[Pllll
(Cn)iii,N
> M/ <µn(I
> M
J
The right hand side converges to p({a : 77(I AI) > M/CN}) if the set {...} is the continuity point of p. It converges to 0 as M -+ oo. Therefore Condition (A.IV) is satisfied.
Finally we remark that Condition (f.2) can be relaxed in Theroem 3.1 in the following way.
Condition (f.2)' (1) There exists Cb.-valued functions hn(A) = hn(x, A)
Kunita: Random measures and stochastic difference equations
247
(n E N) and h(A) = h(x, A) on R' such that hn(x, A) - fn(x, A) p(IAI) '
h(x '
)
__
f (x, A) p(lAI)
if
A E R-
and satisfies
urn sup 11 h; (x, A)h; (x, A)vn (dA) - f h;(x, A)hj(x, A)ve(dA)I
n-oo xl
(2) Define f n(x, A) by (2.3). Then j (x, A) = limn-,,. and A and satisfies
lim sup I
n-
Ixl
J
fn (X,
0.
A) exists for all x
h \fn(x, A))X(6'M] (IlT(AIIlN)PdA)
-f h(f(x,A))X(6,M](llf(A)lI1,N)µ(dA)I = 0 for any bounded continuous function h.
Theorem 3.1'. Assume (e.1) - (e.5), (f.1), (f.2)' and (g.1) for equations (0.4). Then the assertion of Theorem 3.1 is valid. Indeed, the proof of the theorem can be carried out without any essential change as in the proof of Theorem 3.1. 4.
Proofs of Theorems
Let us first observe that Theorems 1.1 and 2.1 can be reduced to Theorem 3.1 or Theorem 3.1'. Let El, .., Eq E G(Rm), F1, ..., Fr E G(Rm - {0}) be arbitrarily fixed sets. Set for any n = 1, 2,... Xn('j) = ,n (A)
=
XE,(A) - in(Eq)), XF1
) - nµn(F1), ..., XFr( ) -
1 /_Zn(F,)),
(4.1) (4.2)
and for y = (u, v), u E Rq, v E R' define Fn (y, A) and Gn(y, A) by Fn(y, A) = (X' (A), icn(A)), Mn(F'1), ..., ld"(F,)), G" (y) _ (0, ..., 0,
(4.3) (4.4)
and consider a stochastic difference equation on Rq': jpn
_ `Yk_1 +
1Y0 = (0, x).
k) + .Gn('I'k-1),
(4.5) (4.6)
248
Kunita: Random measures and stochastic difference equations
Set 4)t = W["t]. Then W is represented by t = (B"(t), N"(t)). Next let G1, ..., G, E G (P-') be arbitrary fixed sets. Set (n (A)
= (PaxGi (A)X[O,cn<](IAI) - JG P(I AI )µE (dA), ... , P(IAI)xo,(A)X[o,C, j(JAI) - IG. P(IAI)M (da))
(4.7)
and define for y = (u, v, w, x) E R9+r+s+d (4.8)
F"(y, A) = (X" (A), c" (A), ("(A), f"(x, A)), G" (y) = (0, ..., 0, µ"(F1), ..."an (F,), 0, ..., 0, g" (x)),
(4.9)
and consider a stochastic difference equation (4.5), (4.6) on R9+r+++d Set (bt q["t] Then 4 t is represented by I = (B"(t), N"(t), Ke (t), P'). There= fore we can apply Theorems 3.1 or 3.1' for the weak convergence of the above sequence {4Dt }", by checking Conditions (f.1),(f.2)' and (g.1) in these cases. Proof of Theorem 1.1. Define F" and G" by (4.3) and (4.4), and consider equations (4.5), (4.6). Conditions (e.4) and (e.5) are trivially satisfied by setting p(r) = 1. Further conditions (f.1),(f.2)'and (g.1) are satisfied for these F" and G" in place of f" and g". Indeed, the sequences {F"(y, A)} and {G"(y)} converge to F(y, A) = (X(A), 0) and G(y) = (0, µ(F1), ..., p(Fr)), respectively in the sense of Conditions (f.1) and (g.1), where X(A) is defined by (4.1) replacing .7r" by it . Further, H"(y, A) = F"(y, A)(p = 1) satisfies Condition (f.2). Let Q" be the law of (B" (t, E1), ..., B" (t, Eq), N"(t, Fl), ..., N"(t, Fr)) in the space D = D([0; oo), R9+r). Then the sequence {Qn} converges weakly by Theorem 3.1. Let Q°° be its limit law. We will apply Theorem 3.1 to the Cb-function F(u, v), where we assume q = r = 1 for simplicity of notation. Then the operator G of (3.4) is represented by \ 02 F
GF(u, v) = 2 (7r(E) + 2V(E, E) - 7r(E)2 l
(u, v) 9U22
+ fo
v)}µ(da).
(4.10)
Therefore,
F(B(t), R (t)) - f t CF(B(s-), 1V (s-))ds
(4.11)
is a martingale with respect to Q°°. The fact implies that (B(t), N(t)) is a Levy process and satisfies all assertions of Theorem 1.1. See Kunita [7]. Finally let P" be the law of (B"(t), N"(t)) on the space D([0, oo), M(Rm) x M(Rm-{0})). Then the sequence {P"} is tight, which follows from the tightness of {Q"} for any choice of {El, ..., Eq, F1, ..., Fr}, q, r = 1, 2, ... Let P°° be
Kunita: Random measures and stochastic difference equations
249
any weak limit. Then the law of (B(t, E1), ..., B(t, Eq), N(t, F1), ..., N(t, Fr)) with respect to P°° coincides with Q. Hence P°° is unique. Proof of Theorems 1.2 and 2.1. Consider equations (4.5), (4.6), where F" and G" are defined by (4.8) and (4.9). Then the sequences {F"(y, A)}" and {G"(y)}" converge to F(y, A) = (X(A), 0, ((A), .f (x, A)),
G(y) = (0, p(F), 0, 9(x)),
(4.12)
where X(') and ((A) are defined by (4.1) and (4.7) replacing ir" and vn by 7r and ve, and X[o,cne](IAI) by 1, resepectively. Hence Conditions (f.1) and (g.1) are satisfied. Set Fn (Y ' A)
H"(Y' A) =
(4.13)
XIAD
A) can be extended to a function A) on ft". Then Further the sequence- {H"(., A)} converges to H(., A) with respect to II II1,x seminorms. It is represented by I, ( Y'
A) =
if
A E R'" ,
(4. 14)
y , A) AIAD
H(y'00, ICI) _ (o, o, XG00,
)X[O,e](IAD,h(X,oo,
(I
I !
(4.15)
if A = (oo, A/IAI) E Mm. Therefore Condition (f.2)' is satisfied for the Fn(y, sequence {F"}. Further, the sequence A) = F"(y, c"A)/ f, n converges to F(y, A) _ (0, XF(A), 77(IAI)Xc (00, IAI) Xio,tl(IAI),1(x, A)),
(4.16)
if JAI>0. Let Q" be the law of (B"(t), N"(t), K' (t), et ) on the space Rq+f+a+a Then {Q"} converges weakly by Theorem 3.1'. Let Q°° be its limit law. We will apply Theorem 3.1 to the function F = F(u, v, w) assuming q = r = s = 1. The operator G of (3.4) is represented by z
GF(u, v, w) = 2 (lr(E) + 2V(E, E) -1r(E)2) +2 (Jc p(IAI)2ir(dA) -
au2
(J0PlAldA)2
/+2
f JcxG
2 p(IAI)p(IA'I)V(dA,
dA') + v(G) )
+(JcrlE p(I AI)ir(da) - ir(E) Jc
p(IAI)ir(dA)
aw
250
Kunita: Random measures and stochastic difference equations 92F
+ff
(xxa(A, A')p(IA'I) + XGxE(A, A')a(IAI)) v(dA, day)) auaw
+ J
IF(u, v + XF(A), w + FA-1
)
-F(u, v, w) - ?1(IAI)X(o,.)(IAI)xo oo, IaI) aw }P(da)
+1, M
{F(u,v+XF(A),w+n(IAI)X(o,e](IAI)XG(°°'
IAA))
-F(u, v, w) }P(dA).
(4.17)
Therefore
F(B(t), N(t), K. (t)) - f t CF(B(s-), N(s-), K,(s-))ds is a martingale by Theorem 3.1. The fact implies that (B(t), N(t), Ke(t)) is a Levy process and satisfies all assertions of Theorem 2.2. See [7]. Next we apply Theorem 3.1' to the function F = F(u, v, w, x). We will not represent the operator G explicitly since it is long. However if F = x, the theorem tells us that
E
Mt =- Ot - ft(bM + c +
{IAI > M})
(4.18)
,
is a locally square integrable martingale. Let Mt be its continuous part and Mt be its discontinuous part. Their bracket processes are computed from coefficients of a2/ax2 of the operator C. Indeed, we have
< Mt >= f
t
a,(
(4.19)
0
where al (x)
=
fRm .f (x, A)2ir(dA)
+211Rm X Rm .f (x, A)f (x, A')V (dA, dA') +f
sm-h(x, oo,
9)2V(d9).
(4.20)
Similarly the joint quadratic variation of Mt and B(t, E) is obtained from the coefficient of a2/auax of the operator C i.e., t
< Mt , B(t, E) >= f a2(p E)ds, 0
(4.21)
Kunita: Random measures and stochastic difference equations
251
where
a2(x, E) = JE f (x, A)ir(dA)
+ J fRm.R.IXE(A)J (x, A') + XE(A')f (x, A)IV (dA, dA').
(4.22)
We have further t
< Mt , K, (t, G) >= f a3(
(4.23)
0
where a3(x, G) = JGnRm P(I AI )f (x, A)ir(dA) +J JRmXRm{XGnRm(A)P(IAII)f(X
A,)`
+XGnRm(A')P(IA'I)f(x, A)} V (dA, dA')
+ IGnaRm h(x, oo, 9)v(d9).
(4.24)
We will next consider the bracket processes of discontinuous martingales. It can be computed from the part Jogal<.u {F(y + -'(y, A))
OF
- F(y) -
Pi(y, A)
(y) JP(dA)
(4.25)
of the operator G where y = (u, v, w, x). Set F = x2. Then the above is equal Then (Mt )2 fo (fo
-
A)I2µ(da).
< Mt > = f 0t
(10<1INKM
I f(3-, A)I2P(dA)) ds.
(4.26)
Next set F = vx. Then (4.25) is equal to fo= lot (10<1;kj<M
1 A)XF(A)lb(dA)111 1 ds.
(4.27)
0
Similarly we have
< Mt , Ke(t, G) >_< Mt, Kd (t, G) >
= lot (10
A)77(IAI)XG(A)X(O,e](IAI)/-t(dA)) ds.
(4.28)
252
Kunita: Random measures and stochastic difference equations
Now define Mt by
Mt =
ft fRm f (ps-, A)B(ds, dA) t
+ f0
fm-t h(cp,-, oo, 9)Ko(ds, d0) S
0t
+
f (cp,-, A)f 1(ds, dA).
0
(4.29)
Three terms of the right hand side are orthogonal local martingales. We can
compute < Mt > by (2.7), (2.9) and (2.10). Then we get < M >t=< M >t. We have by (4.21),
< Mt,
A)B(ds, dA) >
Jo JR'" f (GPs
t
r
=< Mt , Jo JRI
=
It
.)B(ds, d,A) >
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<1aI<M
0
_ It (f<1,\I<m f (Ws
A)zµ(dA) ds.
(4.33)
Consequently from (4.30), (4.32) and (4.33) t
< Mt, Mt >=< Mt,
+ < Mr,
JRI f (Vs-, )t)B(ds, dA) >
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
0
+E
{IAI > M}).
(4.35)
s
The functions bM and bi are related by bM(x) Therefore
ft 0
bi(x) +
bM(c03-)ds + f
Ii
t f0<1al<M
f (x, A)p(dA).
A)N(ds, dA)
0
ft
= +
Jot Jo
ti 0
f (t' -, A)N(ds, dA)
f (cp,_, A)N(ds, dA).
(4.36)
Substitute the above to the right hand side of (4.35) and let M tend to infinity. The last term tends to 0. Then we obtain the representation (2.11).
Finally the weak convergence of {P' } can be shown similarly as in the proof of Theroem 1.1.
254
Kunita: Random measures and stochastic difference equations
Bibliography [1] T. Fujiwara, On the jump-diffusion approximation of stochastic difference equations driven by a mixing sequence, J. Math. Soc. Japan, 42(1990), 353-376.
[2] T. Fujiwara, Limit theorems for random difference equations driven by mixing processes, preprint. [3]
T. Fujiwara and H. Kunita, Limit theorems for stochastic differencedifferential equations, Nagoya Math. J., submitted
[4] B.V. Gnedenko and A.N. Kolmogorov, Limit distributions for some of independent random variables, Addison-Wesley, 1954. [5]
J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes, Springer-Verlag, 1987.
[6] H. Kunita, Stochastic flows and stochastic differential equations, Cambridge Univ. Press, 1990.
[7] H. Kunita, Limits of random measures induced by an array of independent random variables, to appear in Constantin Caratheodory: An international tribute ed. T.M. Rassias. [8] H. Kunita, Central limit theorems on random measures and stochastic difference equations, to appear in Proc. Conf. Gaussian random fields ed. T. Hida.
[9] H.J. Kushner and Hai-Huang, On the weak convergence of a sequence of general stochastic difference equations, SIAM J. Appl. Math., 40(1981),528-541.
[10] J.D. Samur, On the invariance principle for stationary co -mixing triangular array with infinitely divisible limits, Prob. Th. and Rel. Fields, 75(1987), 245-259.
CHARACTERIZING THE WEAK CONVERGENCE OF STOCHASTIC INTEGRALS by
Thomas G. Kurtz Dept. of Math & Statistics University of Wisconsin-Madison Madison, WI 53706
Philip Protter Dept. of Math & Statistics Purdue University West Lafayette, IN 47907-1399
For n = 1, 2,..., let 'E" = (r, .F", (.Ft )t>o, P") be a filtered probability space, let H" be cadlag and adapted, and let X" be a cadlag semimartingale. A fundamental question is: Under what conditions does the convergence in distribution of (H", X") to (H, X) imply that X is a semimartingale and that fo H,"_ dX; converges in distribution to fo H,_ dX,? A slightly more general formulation would put conditions on the sequence X" alone such that the convergence above holds for all such sequences H". A sequence with this property will be called good. To be precise, let Mk"` denote the real-valued, k x m matrices, and let DE[0, oo) denote the space of cadlag, E-valued functions with Skorohod topology. Definition: For n = 1, 2,..., let X" be an Rk-valued, (.F2 )-semimartingale, and let the sequence (X")">1 converge in distribution in the Skorohod topol-
ogy to a process X. The sequence (X")">1 is said to be good if for any sequence (H")">1 of Mkm-valued, cadlag processes, H" ()t`) -adapted, such
that (H", X") converges in distribution in the Skorohod topology on DMkm xRm [0, oo) to a process (H, X), there exists a filtration (.Ft) such that
H is (.Ft)-adapted, X is an (.Ft)-semimartingale, and
rt
dXa = jH8_
dX,.
Jakubowski, Memin and Pages [1] give a sufficient condition for a sequence (X")">1 to be good called uniform tightness or UT. This condition uses the characterization of a semimartingale as a good integrator (see e.g., Protter [4]), and requires that it hold uniformly in n. On 'E", let xn
1 Supported in part by NSF grant # 2Supported in part by NSF grant #DMS-8805595
256
Kurtz & Protter: Weak convergence of stochastic integrals
denote the set of elementary predictable processes bounded by 1: that is,
?l" = {H": H" has the representation Ht = Ho 1{o)(t)+
p-1
=1
H; 1[t;,t,+1)(t),
with H; E .Fti', P E N, and 0 = to < t1 < ... < tp < oo, IHs < 1}. Definition: A sequence of semimartingales (X")">1, X" defined on E-En)
satisfies the condition UT if for each t > 0 the set { fo H; dX; , H" E H", n E N} is stochastically bounded. Theorem 1. (Jakubowski-Memin-Pages). If (H", X") on E'En converges in distribution to (H, X) in the Skorohod topology and if (X")">1 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
where Ox, = x(s) - x(s-). Let fo JdA,1 denote the total variation of the process A from 0 to t (w by w).
Theorem 2. (Kurtz-Protter). Let (H", X") on 8" converge in distribution to (H, X) on EE in the Skorohod topology on DMkm xRm [0, oo). Fix b > 0 (al-
lowing b = oo) and let X"'6 = X"-J6(X"). Then X"'6 is a semimartingale and let X"'6 = M"'6 + A"'6 be a decomposition of X n,6 into an (Ft)-local martingale and an adapted process of finite variation on compacts. Suppose
(*) For each a > 0, there exist stopping times T"'a such that oo. a) < « and supE{[M"'6,M"'6]tATn.- + JonT^
P(T",a <
"
Then there exists a filtration (.Ft) on 8 such that H is (,Ft)-adapted and X is an (.Ft)-semimartingale, and (H",X", f H, dX,) converges in distribution to (H, X, f H,_ dX,) in the Skorohod topology on DMkm xRm xRk [0, oo).
That is, (X")">1 is good. It is shown in Kurtz and Protter [2] and also in Memin and Slominski
Kurtz & Protter: Weak convergence of stochastic integrals
257
[3] that UT and (*) are equivalent sufficient conditions for the sequence (X") of (vector-valued) semimartingales to be good. The next theorem, which is the principal result of this note, shows that the sufficient conditions of Jakubowski-Memin-Pages and Kurtz-Protter are also necessary.
Theorem 3. Let X" be a sequence of vector-valued semimartingales on filtered probability spaces n If X" is a good sequence, then X" satisfies the condition UT and the condition (*) of Theorem 2.
Proof: Since UT and (*) are equivalent, it suffices to show that UT holds. We treat the case k = m = 1 for notational simplicity.
Suppose that (X")n>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.
Lindstrom et al: SDE involving positive noise
263
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
265
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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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)
<_ IIFnIIH-MP(Rn) ' IIGmIIH-
(Rm)
Therefore we can extend in a natural way the product given in (3.13) to work for more general F E (L2)-, G E (L2)- by defining lim
f FndB®n)o(E JGmdB®'")
00 n=0
(3.17)
m=0
JFnGmdB®m). E n,m=0 00
In view of (3.16) we see that FoG E (L2)- if (for example) 00
(3.18)
E k ' k! E IIFnIIHk=0
n+m=k
(Rn) ' IIG.IjH
m411
)
< oo.
Remarks 1) In the literature the symbol : FG : is often used for the Wick product F o G. For more details on the connection between :: and o see the remark following Theorem 5.1.
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2) Several motivations exist for this definition of multiplication of functional processes. We give some of them here:
a) First note that multiplication of a deterministic quantity with a random one reduces to the usual multiplication. Going one step further, one might say that the Wick product can be regarded as the product one would obtain if the two stochastic factors were independent. In general our quantities are of course dependent, but in a sense we ignore this when we form the Wick product. This point of view is related to the situation in thermodynamics for example, where technically the path of a single particle of course depends on the paths of the others, but the connection is so weak/chaotic that it is not unreasonable to assume that the effect of it averages to zero in time. The Wick product may be viewed as an axiomatic way of stating such a principle. Moreover, this special product has the technical advantage of preserving the martingale property. And - as shown in Lindstrom, Oksendal and Uboe (1991) - the product provides a new approach to the Ito calculus.
b) Wick multiplication can also be motivated from a renormalization point of view: Arguing heuristically we see that white noise Wt may be represented as an element of (L2)- as follows t+ot Wt
oimo Ot
f dBu =
f
t
R
bt(u)dBu,
where bt E H-1(R) is the Dirac measure at t. Therefore one should have, by Ito's formula, t+Ot
Wt ^, (1 f dBu)2 = (&)2 t
t+Ot V
f (f t
At
t
The additive renormalization of this is t+Ot V 2
dB f f dBu At)Ot - f f 6t(u)bt(v)dBudB as At --> 0 (
t
t
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272
which motivates the definition
J btdB o J btdB =
r
J
St ®btdB(92
in agreement with (3.15).
3) It should be pointed out that the argument above, as well as the notation (3.19)
JFndB0'
for Fn E 1Y'
(R")
R^
from the general representation (3.10) of elements in (L2)-, is just formal: (3.19) is not defined as an Ito integral. Nevertheless, (3.19) - slightly modified and generalized - makes sense as a distribution valued process and this is the starting point for the type of generalized white noise functional processes (functional processes, for short) which we now construct. §4. Functional processes
Let s E R and suppose F E H-'(R"; L2(Rn)) is an L2-valued H-'distribution, i.e. F is a linear map from H'(R") into L2(R") such that there exists K < oo with (4.1)
11 11L2
where < F, 0 >:= 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.
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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<0 if 0
has a derivative equal to
d t = J(1t(u)S(v) + 2f
lt®btdB®2 = 2(1 1tdB) o (f btdB)
=2BtoWt, which is in agreement with usual (not Ito) differentiation rules.
§5. The Hermite transform l We now show that to a given functional process X0 we can associate an (a.e. defined) function fl(XO) := X0 : Co -> C
Lindstrom et al: SDE involving positive noise
276
where C = {x + iy; x, y E R} denotes the set of complex numbers and C0 = {(zl, z2i , ); z2 E C, 3N with zi = 0 for Vi > N}. This connection will be fundamental for the rest of this paper.
Fix an orthonormal family {(k}k 1 in L2(R). Let
=
00
IF,(') (u)dBOn
X0 n=0
be a functional process, and keep 0 E S fixed. Using multi-index notation each function Fo (ul, )un) maybe (uniquely) written j,'0 n
a(n) ®a (ul, ..
(u1 ' ... un) _
un)
lal=n
for suitable constants
c(an)
=
c(an)(4)), where
a = (al,
,
an), jal = al +
...+a.. and S®a
_ t®al
®C®«2 ®... j, t tam
This gives the (unique) representation (5.2)
X4, _
00
can)
n=0 lal=n
J
(®adB®n = E ca a
where can)(.) E H-O°(R") for n>
J
C®adB®lal
E H-°°(R).
By symmetrization of S®a and use of the Ito isometry we see that with such a representation we have (5.3)
a!ca,
E[XO2] _ a
where
The iterated Ito integrals on the right of (5.2) can be computed using the Hermite polynomials hn defined by hn(x) = (-1)ne'r The result is the following:
.
dn(e-4)
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277
THEOREM 5.1 (Ito (1951), p. 162) m
I/
(5.4)
J
c®al ®(®a2 ®... ®(mn'mdB®frI = II ha, (xi) := ha(x)
i_i
where xi = f (i(u)dB,.;j = 1,2,...,x =
(xi,x2,...).
Remark. Using Theorem 5.1 we can now explain more precisely the relation between the multiplication* defined by (3.15) and the classical Wick product: If X is a real Gaussian random variable with mean 0 and variance E[X2] _ a2 then the k'th Wick power of X is defined by
Xk:=a hk(-) (See e.g. Simon (1974) p.11).
For example, if X = f f (t)dBt with f E L2(R) then R
Xk := IIf Ilkhk(I fII) where 11f 11 = Of IIL=
On the other hand, applying Theorem 5.1 to a = al = k, (1 = II f II -'f , we see that X°k = (11111 I fII 'fdB)= Iit
J
Thus :
Xk
:= X°k for such X.
By polarization we obtain, if Y = f g(t)dB with g E L2(R),
2:XY:=:(X+Y)2:-:X2:-: Y2:=2XoY. However, if we more generally consider two random variables U, V of the form
U = f fndB®n,V = 1 gmdB®m with In E L2(Rn),gm E L2(Rm), Rn
R-
then by the definition on p.12 in Simon (1974) we have
: UV := UV - E[UV]
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278
V. For example, if This is not necessarily the same as U oJfdB
=with
U=
f = X[o,t],
R
R2
then U = V*2 so by the above
UoV = Vo3 = hIfIl3h3(11)
V3 -311 f112V = Bf
- 3tBt,
while
UV := (Bt - t)Bt - E[(Bt - t)Bt] = B3 - tBt. Nevertheless, we have found the relation between o and :: so close that we feel that it is natural to use the name Wick product for o. The next representation of Hermite polynomials is well known:
0 (5.5)
hk(x) =
(x + iy)ke-iYzdy
2x x
k=1,2,...
-00
Define a measure A on the product o-algebra on RN by (5.6)
f
00
00
f(y)dA(y) _ J ...(
00
f (f
f(yi,...,yn)YI
d23r)e-Y2
_00 -00
-00
d2
'gYn dyn v'r2-7-r
if f : RN -> 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
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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
<00.
In particular, if z = (zi,
, zN, 0, 0,
) and Izk I < M for all k then in
order to check (5.9) it is enough to sum over those a such that aj = 0 for
Lindstrom et al: SDE involving positive noise
280
j > 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!
<
00
M2n
CN+n-11 J
n
n=0 [?iJ!
00
<
n M2n(N + n)N < oo.
n=0 [-9
We conclude that (5.8) converges absolutely for such z. Therefore, if we for
all N put z(N) :=
(x1,... ,zN,0,0,...)) if z = (z1,... ,zN,zN+1,...) E CN
and define JCO(z(N))
(5.10)
then the power series for variables z1,
converges uniformly on compacts in the
, ZN. Hence we have proved:
LEMMA 5.3 X) is an analytic function in (z1,
,
ZN) E CN for all N
and all From (5.7) we get the following connection between X0 and its f-transform: XO(w) = [f XO(x + zy)d\(y)]Z= f cdB
(5.11)
Since ±O(z) is not defined for all z, the integral above requires some expla-
nation. It is defined by
J
X4,(x + iy)dA(y) = 1l i m f
g ,)(x +
It follows from (5.7) that this limit exists (in L2(µ)) if x; = f C3dB.
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281
EXAMPLE 5.4 Let X0 = WW = f qdB be the white noise process. If ¢ is fixed, p = II0IIL2 # 0, we may regard p`o as the first function (1 in our orthonormal basis {(k}. Then Wo =
J
WO(z) = pzl
so
pC1dB
and (5.11) says that 00
J
¢dB=[J
sv2
d27r]==fcidB
)e
-00
Co
_ E(O,(k)(k
where
(0,(k) =
k=1
cb(t)Ck(t)dt
J it
This gives
WO = E(O, (k) k
f
Ck(u)dB.
and therefore (5.12)
l'T(z) _ E(4', (k)Zk k=1
Hence 00
f fVm(x + iy)dA(y)
(k) k
f
(xk + iyk)e-Ie v: d is
(k)xk,
-00
which, after the substitution xk = f CkdB gives
E(4', (k) f CkdB =
J((c&, (k)(k(u))dB,, =
f
O(u)dB,,.
The representation (5.11) is useful for explicit computations, but the main reason for the importance of the fl-transform is the following property:
THEOREM 5.5 Let X0, Yo be functional processes such that X0 o Yo is a functional process. Then (5.13)
l(Xo o YO) = l(Xo) l(Y4,)
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282
Remark. The product on the right hand side of (5.13) is the usual complex product in the complex variables zj but a tensor product in the coefficients, i.e.
Ca(4)z' e,6(0)z' = (C« ® ep)(q 0
')za+6
Proof. It suffices to prove this when a = (a1 i
X0 = f
C®adB®1a1,
= Ca(O)C,e(V
,
)za+,6
8 = (N1,
, Qk)
and
Y4, = f c®,sdB®IRI.
Then XO o Yo =
r;®a ®(®#dB®la+#l = r S®(a+R)dB®Ia+QI J
J
and therefore
(X0 o Yo)-(Z) = x` = za z0 = X0(t) Ye(t), ; as claimed. COROLLARY 5.6 Suppose Xk is a functional process for all k < n. Then f o r all ak E R, k = 1, n
, n, we have n
akX0k = [J k=0
k=0
akX0(x +iy)dA(y)]x=f CdB
As promised above we shall end this section by briefly explaining the relationship between the Hermite transform and the S-transform as defined in Hida-Kuo-Potthoff-Streit (1991); i.e.,
Sf(0b) = f f(w +ib)dp(w),
(5.14)
St
where f E L2(µ) and 0 E S. If f has Wiener-Ito decomposition
f =
E f,idB®n, it turns out that (5.15)
Sf(tb) =
E
fn(u1,...,26n)W®n(ul,...,un)du,
n R,n
and using this formula, we easily establish the following relationship between the two transforms.
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283
THEOREM 5.7 If 1l denotes the Hermite transform with respect to the orthonormal basis (1, (2, , then for any f E L2(µ) and any sequence z =
) E Co , we have
(-'I) z2, (5.16)
H(f)(z) = Sf (z1(1 + Z2 S2 + ...).
Proof. If we write f = E E ca f (®adB®n, then according to (5.15) n Ia'=n
S®a(ul, ... 7 un)(E zi(i)1n(U1, ... , 2dn)du,
Sf (E z1(1) = E E C. n Ian=n
g,n
and hence all we have to prove is that
&(u1, ... , un)(1:
(5.17)
zibi)®n(u1,...
,
un)du = za
J
Rn , ik be the indices where oil, oiz ai,, are different from zero, and let o run over all permutations of the set {1, 2, , n}. The integral in (5.17) can then be written as
Let i1 , i2i
nl
7
iiluo(1))...
p
J
zj(jn(un))du.
is orthonormal, there will for each permutation v be exactly one sequence j1, -,in such that the integral is nonzero; we Since the sequence (1, (2,
have to choose jl = o-1(1),
-J. = o,-1(n). With this choice of j's, the
integral takes the value za, and (5.17) follows.
Remark: In the theory of the 8-transform, the ray analytic functions z S f (zzb) play a fundamental role. The theorem above tells us that we may think of the Hermite transform as an extension of these ideas to several variables (in fact, to infinitely many).
§6. A functional calculus on functional processes We now use the Hermite transform to show that for a large class of functions f and functional processes X0 we can define a new functional process f oX0.
Lindstrom et al: SDE involving positive noise
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The key to such a functional calculus is Corollary 5.6, which can be stated as follows (under the given conditions): (6.1)
n(X#) =
[f p(XO)(x +iy)d)t(y)JX= f
cdB
for every (complex) polynomial p(x) = j akxk with real coefficients ak, k=0
where p indicates that Wick powers of X1 are used.
The idea is to extend (6.1) to a larger family of functions than the polynomials. The following result will be useful:
LEMMA 6.1 Let f (XI, x2i
E[f(J (1 dB,
f
) E L1(A). Then C2dB, ...)] = Jf(xi,x2,.
.)dA(x)
Proof. It suffices to prove this in the case when f only depends on finitely many variables, and so we may assume that f E CO '(R") for some n. Then f is the inverse Fourier transform of its Fourier transform f : f(x1,.. ,xn) = (27r)-'2 .
f
f(yl,...,yn)' ei(x,b)dy
Rn
Hence, by (2.1)
E[.f
(f S1dB, ... , J (ndB)] = (21r)-t . f .f (yl, ...
yn)E[ei<w,E
,
>]dy
Rn
(27r)-t f J(yl) ... , yn)e
s > I'j dy
Rn
e-i(x,Y)dx)e-2 Ivl'dy = (27r)-n
f Rn
_
(27r)-'i
f f (x1 i . an
, xn)
=
(27r)-n
f(Jf(xi...x) Rn Rn
A XI, ... , xn) . (J e-i(x,v)- 2IYI2dy)dx Rn
(27r) 2 e- s I nI2dx = Jf(x)d.X(x), as claimed.
Lindstrom et al: SDE involving positive noise
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Here we have used the well-known formula 00
1 e-fr
f et«t-prat = J = -00
COROLLARY 6.2 Let g : R -* R be measurable and such that g(X,) E L1(µ). Then
E[g(Xo)] = f g(f Xm(x + iy)dA(y))d.\(x). In particular,
E[IXj121: Jf
1±0 12 (X
+ iy)dA(x)dA(y)
DEFINITION 6.3 a) We let A2 = A2(A X A) be the set of all formal power series
/E
«
n=0 l«l=n
with real coefficients c« for all a = (al,
,
am), such that
f(N)(x) := f(x(N)) is analytic for each N (using the notation of (5.10)) and f E L2(A X A), i.e. IlliIL 2(axA)
:= Jf If(x + iy)I2dA(x)dA(y)
(6.3)
N Jf
If (N)(X
+ iy)I2dA(x)dA(y) < 00
b) Define G to be the set of all functional processes X0 such that Xk E
A2(AxA)forallk=1,2,3, Remark:
C is an algebra (under Wick multiplication).
To prove this it suffices to show that if X0 E 9, then X°2 E this follows from Theorem 5.4.
also. And
Lindstrom et al: SDE involving positive noise
286
Our first main result in this section is the following:
THEOREM 6.4 (Functional calculus) Let X0 E 9 and f : C -+ C be given and assume that there exist polynomials Pk : C --k C with real coefficients such that
Pk(X,6)-+f(XO)inL2(Ax A) ask ->oo. Then
f(Xm) := [f f(XO)(x+iy)dA(y)]x= f (dB defines a functional process.
Proof. For each k we have that Pk(XS) = [f Pk(Xc)(x + iY)dA(y)]x= f (dB is a functional process and, by Lemma 6.1, E[IPk(X.) - P:(Xm)I2] = E[I J(PkCt) - P,(.; )) 0
<
(f (dB + iy)dA(y)I2]
E[f IPk(Xc) - Pt(X.,)I2 o (f (dB + iy)dA(y)]
= IIPk(X#) - Pl(X,)II L2(axa) -> 0 as k,1 -+ oo.
We conclude that {Pk(Xo)} constitute a Cauchy sequence in L2(µ). If we write f F(n'k)dB®n then Pk(X#) _ n En!IIF(n,k)
E[IPk(X.) -Pi(X0)I2] =
_
112
n
so for all fixed n we see that {FFn'k))k 00, converges to a limit G. ), say, in L2(R). Hence
f G( )dB®n,
limPk(XO) _ n
which is a functional process.
287
Lindstrom et al: SDE involving positive noise
EXAMPLE 6.5 The exponential of white noise is defined by (6.6)
Exp(Wt) :_ E
n
:= [J exP(Wt)(x
f (dB
n=O
With 0 E S given we can choose (, = p-14' where p = 1101I L2 and this gives (see Ex. 5.3)
Wt(z)=pzi ELU(AxA) forallp
exp(Wt)(z) = exp(pzi) =
r o n=0
kr o
ni(Wt)n(z), n=0
the limit being taken in L2(A x A). So Theorem 6.4 applies to define (6.6) as a functional process.
EXAMPLE 6.6 The square of white noise is given by Y = W;2 = f 0 ®4'dB112, Rz
so, with 4' = pCl, I',(z) = p2zi and hence YO E 9. However, exp(Y>) = e"
2Z2
which belongs to L2(A x A) only for p <
1 2,
but for such 0 we see that Exp(WWo2) is well-defined.
The inverse f-transform
The f-transform assigns to each functional process X. a function X,. C0 __+ C. Conversely, starting with a function g : C0 -, C it is useful to be able to find a functional process X whose ?f-transform is g. Clearly the candidate for X is
X=
[I g(z)dA(y)]x= f SdB
but the question is for what g this makes sense and defines a functional process.
Lindstrom et al: SDE involving positive noise
288
such that
Suppose that we are given distributions
(6.8) co(-) E H-°°(R) and E H-°°(R") for each multiindex a with jal = n > 1, such that (6.9) (6.10) F, a!c«(0) < oo for all 0 E S. a
Then, as shown by the proof of Lemma 5.3, the function
L >2 n=0 lal=n
(6.11)
ca(qf)za
go(z) := 0"
is defined and analytic on Co , for each 0 E S.
DEFINITION 6.7 The set of all such functions go is denoted by P.
THEOREM 6.8 (Inverse f-transform) Let go E P. Then Xo :_
[I 9o(z)dA(y)]Z= f tdB
defines a functional process whose f{-transform is go.
We will use f-'(go) or g0 as notation for the inverse Hermite transform. (Strictly speaking this notation is slightly in conflict with the notation of (6.5), but this should not cause any difficulties).
Proof of Theorem 6.8. If go(z) = E E ca(4)za then by (5.5) n jal=n
J9c&(z)(Y)=>2 >2 ca(c&)ha(x), n jal=n where h.,(xl)ha2(x2)... han,(x.) if a = (al,...,a,n)
ha(x) = So by (5.4) we get
Xs
g4(z)dA(y)]z=f CdB = E E ca(4) f S®adB®n
n (al=n
Lindstrom et al: SDE involving positive noise
289
a!ca(¢) < oo, we conclude that X0 is a functional process.
Since E[X2] _ Cr
§7. Positive noise We now return to the concept of positive noise, mentioned in the introduction:
DEFINITION 7.1. A functional process Xt(w) =
00
n=0
J
Flnl (u)dBr(w)
is called a positive noise if (7.1)
Xo(w) > 0 a.s. for all ¢ E S with II0iIL2 sufficiently small.
EXAMPLE 7.2. Wt 2(w) =
is not a positive noise, since 00
V
J00-00 J 0(u,
WW2 (w) = 2
for all 0 E L2(R2),
so choosing e.g. %(u, v) = X[0,1] X [0,I] (U, V) we get v
1
W2(w) = 2 j(1 0
2
0
f 0
which is not positive a.s. However, we have the following:
EXAMPLE 7.3 The exponential of white noise,
= Nt := Exp(Wt)
00
E 11(f b®ndB®n) n=0
is a positive noise.
Bi (w) -1,
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Lindstr0m et al: SDE involving positive noise
To see this we use the computation from Example 6.5: 00
NO(w) _ [ f eXP(Pxi +iPyi) . e-4vi dyl
2 '-f
-00
]x
S,dB
00
= exp(J ¢dB) .
27 J e1vb- . v2dy, 00
which combined with (6.2) gives (7.2)
r Exp(W4,) = exp(f 4dB -
E S.
In particular, Exp(WW) > 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 )
Lindstrom et al: SDE involving positive noise
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
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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_(<)z
is positive definite,
y,((7.14) n)(iy)e-(o)2 is positive definite.
Therefore the product onyn(y) _ Choosing p =
is positive definite.
this gives that
(XoY);(n)(iy)e-iY2 is positive definite, so from Theorem 7.4 we have X o Y > 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)
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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
_ -2
J{(u + s)'O(v + s) + 0'(v + s)o(u + s)}ds, 0
so 00
<
8x
>_ - 2
J 0
ds
(O(u + s)O(v + s))ds = 2 0(u)O(v)
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297
Hence
aG ).=(t,t) = bt ® bt fort > 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
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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)<&
i
0
t+h
h-2n (h - max(ui) + min(u=))2du i
0
i
0
max(ui)-min(ui)
t+h
t+h
+
J 0
..
r
f 0
max(ui)-min(u; )t
h2n(t - max(u) + h)2du i
Lindstrom et al: SDE involving positive noise
301
If n = 1, we have by (8.13) t+h
t
(8.14)
J(G1)(u))2du = r h-2 h2du + J h-2(t + h - u,)du =t+ 3 t
0
For general n, we easily get t+h
t+h
r
h-2" (h)2dU < J(G(n)(u))2du <
0
0
max(ui)-min(u;)<4 i
ul
t+h
t+h
I ...
J0
h-2nh2du 0
max(u;)-min(u;)
This again gives the estimates t u1+4
(8.15)
(8.16)
J(G(u))2du >
J(G(u))2du <
u1+4
J J ...
h-2n
J
0 u1-4
u1-4
t+h u1+h
u1+h
r r
J0 u1-h J
2
4
du
1 )n (2h
2
... J h-2nh2du = (t + h)h (2 )n u1-h
2
h
If we substitute (8.14), (8.15) and (8.16) in (8.12), we get (8.11).
Remark. It is interesting to note that the estimate (8.11) resembles some properties expected in a real world experiment. Using a very small averaging window (molecule-size?) the world is looking very discontinuous with rapid, chaotic variations. On the other hand, with a very large averaging window (much larger than the object we study), the image get blurred and matter again looks less structured. Somewhere in the middle there should be a scale where we see matter clearly and more or less independent of the size of the averaging window. Comparing with (8.11) we see that l oE[(X(e) - t)2] --+ oo and rlim E[(Xt`) - t)2] __+ co
302
Lindstrom et al: SDE involving positive noise
corresponding to the first two properties.
On the other hand, if s2 << h << t, we have E[[Xte)
- t]2) N 62t
more or less independent of the size of h. If e.g. e = 0.1, t = 100 then the variance is essentially equal to 1 when h ranges from 0.1 to 10. Acknowledgements
We are grateful to S. Albeverio, P. Malliavin and J. Potthoff for useful communications.
This work is supported by VISTA, a research cooperation between The Norwegian Academy of Science and Letters and Den Norske Stats Oljeselskap A.S. (Statoil).
REFERENCES E. Dikow and U. Hornung (1987): A random boundary value problem modelling spatial variability in porous media flow. IMA Preprint Series d 309, April 1987, Minneapolis, Minnesota 55455, USA. G.B. Folland (1984): Real Analysis. Wiley.
I.M. Gelfand and N. Ya. Vilenkin (1964): Generalized Functions, Vol. 4: Applications of Harmonic Analysis. Academic Press (English translation). T. Hida (1980): Brownian Motion. Springer-Verlag.
T. Hida, H: H. Kuo, J. Potthoff and L. Streit (1991): White Noise Analysis (Forthcoming book).
K. Ito (1951): Multiple Wiener integral. J. Math. Soc. Japan 3 (1951), 157-169.
H.-H. Kuo (1983): Brownian functionals and applications. Acta Appl. Math. 1, 175-188.
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H.-H. Kuo and J. Potthoff (1990): Anticipating stochastic integrals and stochastic differential equations. In T. Hida, H.-H. Kuo, J. Potthoff and L. Streit (editors): White Noise Analysis-Mathematics and Applications. World Scientific, 256-273.
T. Lindstrom, B. Oksendal and J. Uboe (1991): Wick multiplication and Ito-Skorohod stochastic differential equations. To appear in S. Albeverio, J.E. Fenstad, H. Holden and T. Lindstrom (editors): Ideas and Methods in Mathematics and Physics. Proceedings of a conference dedicated to the memory of Raphael Hoegh-Krohn. Cambridge University Press.
D. Nualart and M. Zakai (1986): Generalized stochastic integrals and the Malliavin calculus. Prob. Th. Rel. Fields 73, 255-280. D. Nualart and M. Zakai (1988): Generalized multiple stochastic integrals and the representation of Wiener functionals. Stochastics 23, 311-330.
J. Potthoff (1987):
On positive generalized functionals. J. Funct. Anal.
74, 81-95.
B. Simon (1974):
The P(q)2 Euclidean (Quantum) Field Theory. Prince-
ton Univ. Press. J.B. Walsh (1986): An introduction to stochastic partial differential equations. In P.L. Hennequin (editor): Ecole d'Ete de Probabilites de SaintFlour XIV-1984, Springer LNM 1180, 265-437.
B. Oksendal (1990): A stochastic approach to moving boundary problems. To appear in M. Pinsky (editor): Diffusion Processes and Related Problems in Analysis. Birkhauser.
Dept. of Mathematics, University of Oslo Box 1053, Blindern, N-0316 Oslo 3 NORWAY
Feeling the Shape of a Manifold with Brownian Motion-The Last Word in 1990 Mark A. Pinsky Northwestern University
Dedicated to Henry McKean on the occasion of his sixtieth birthday
1. Introduction. In 1984 [P1] we introduced the theme of inverse problems for Brownian motion on Riemannian manifolds, in terms of the mean exit time from small geodesic balls. Since that time a number of works have appeared on related stochastic problems as well as on some classical, non-stochastic quantities
which may be treated by the same methods. Most recently H.R. Hughes [Hu] has shown that in six dimensions one cannot recover the Riemannian metric from the exit time distribution, thereby answering in a strong sense the main question posed in [P1]. The general area of "inverse spectral theory" was initiated by Mark Kac in his now famous paper [Ka] on the two-dimensional drumhead. In the intervening years a large literature has developed on inverse spectral problems in higher dimensional Euclidean space and differentiable manifolds; for recent surveys see ([Be],[Br], [Go]). In these approaches one is given the entire spectrum of eigenvalues, from which one asks various geometric questions. Our approach, by contrast, is able to obtain strong geometric information from the sole knowledge of the principal eigenvalue of a parametric family of geodesic balls (see section 6, below). It will be our goal here to give an account of the above mentioned developments, focusing on the following topics:
i) Exit time distribution of Brownian motion ii) Exit place distribution of Brownian motion iii) Joint distribution of exit time and exit place iv) Principal eigenvalue of the Laplacian.
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For various reasons we are not pretending a survey of all the recent activity;
for example we refer to the work of Stafford [St] which charts out some promising new areas for development.
2a. Notations and definitions. The setting is as follows: (M, g) is a p-dimensional Riemannian manifold, not necessarily complete. The Laplacian is defined locally by the coordinate formula
0 = (l/vrg) 8/8x; (Jg`ia/axj) with the usual notation for the metric tensor gij, its inverse gij and its determinant g. The Brownian motion process (Xt, Px) is the (local) diffusion generated by i A. The exit time from a ball of radius r about m E M is defined by
T,. = inf{t > 0 : dist(Xt,m) = r}. The homogeneous decomposition of the Laplacian in a normal coordinate chart ( x1 , ... , xi,) about m E M is written 00
A=A-2+EO, k=0
where 0-2 = p 1 a218x; is the standard Laplacian of RP and {Lk}k>o are second- order differential operators whose coefficients are homogeneous
polynomials in the normal coordinates (x1i... , xP) with the following additional properties: 2a i): Ok maps a homogeneous polynomial of degree n to a homogeneous polynomial of degree n + k. 2a ii): The coefficients of Lk depend on the curvature tensor and its derivatives of order < k.
Do, Ol and L2 were computed in [GP] and 03 was computed in [KO].
The curvature tensor is computed from the Taylor expansion in a normal coordinate chart [G] by the formula
gij(xl, ... , xP) = 6ij
-3
P
Ria,jb(m)xaxb + O(Ix13) a, b=1
llxl y 0). //
I
Pinsky: Feeling the shape of a manifold
307
The Ricci tensor is defined by the first contraction: P
pij(m) = ERiaja(m) a=1
and the scalar curvature is defined by the second contraction: r(m) _ EP 1 pii(m). In this notation, we can display the operator Do as P
P
Riajb(m)xaxba2/axiaxj - (2/3) > pia(m)xaa/axi.
Ao = (1/3)
i,a=1
i,a, j,b=1
In the following sections we'll discuss results on the probability law of the random variables T, and XT, for a general manifold. In order to put these in perspective, we first recall the familiar properties of Brownian motion of Euclidean space.
2b.
Brownian motion of Euclidean space and related asymp-
totics. In the Euclidean space (RP, go) we have the following well-known properties
for paths starting at the center m of a ball of radius r.
2b i): The exit time Tr has the Brownian scaling property: Tr/r2 - T1 in the sense of the probability law P,,,.,
2b ii): The exit place XT is uniformly distributed on the sphere: XT Leb(SP-1) in the sense of the probability law P,,,., 2b iii): The exit time and exit place are P,,, -independent random variables:
ba>0,0EC(RP) Em (e-aT. 1b(XT. )) = Em (e-aT.) Em (Y'(XT,. )),
2b iv): The principal eigenvalue of 0 with zero boundary conditions on the ball of radius r is inversely proportional to the radius: A1(Br) = z2/r2
where z2 is the square of the first positive zero of the Bessel function 'J(P-2) /2 .
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In the following sections we will describe results which show, for example that the property 2b i) is characteristic of RP if p < 6 , but fails to discriminate in case the dimension p > 6.
For any Riemannian manifold it is not difficult to show that the above properties take hold in the limit r 10, in the following sense:
2b i'): The scaled exit time Tr/r 2 converges in law when r 10 to a limit T which is the exit time of Euclidean Brownian motion from the unit ball in RP.
2b ii'): The exit place XT, converges in law when r 10 to the normalized Lebesgue measure Leb of the unit sphere in RP. 2b iii'): The random variables T,., XT are asymptotically independent: Va > 0,7b E C(M),
EAm(e-CTr b(XTr))
Em(e-aTr)Em,b(XTr)(1 +o(1)),
r 10.
These probabilistic statements reflect the locally Euclidean nature of a Riemannian manifold; the corresponding properties of Euclidean space take over in the limit of small radius. The asymptotic behavior of the principal eigenvalue is the statement
2b iv'): A1(Br)
z2/r2
(r j 0).
In the following sections we investigate to what extent these quantities may be used to detect the intrinsic geometry of the Riemannian manifold (M, g). It is emphasized that in the formulations of the theorems, all considerations are local; there is no need to be concerned with the cut-locus or other global issues.
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3. Exit time distribution of Brownian motion The first general result on the exit time was obtained in [GP], from which it follows that if p < 6 the mean exit time alone is sufficient to detect the geometry of Euclidean space or other model spaces (constant curvature or more general rank-one symmetric spaces).
Theorem 3.1. For any p-dimensional Riemannian manifold (Mg), the mean exit time has the asymptotic expansion Em(Tr) =
0)
where c1i c2 are constants which depend on the dimension p. Here J R12 (resp. IpJ2n) is the sum of squares of the components of the curvature tensor (resp. Ricci tensor).
Corollary 3.2. Suppose that p = 2 and `dm E M the mean exit time satisfies E,,m(Tr) = r2/2 + o(r4), r J. 0. Then (Mg) is locally isometric to the Euclidean plane. Indeed, the scalar curvature (=Gaussian curvature) is the only local geometric invariant of a two-dimensional Riemannian manifold. In higher dimensions, we may obtain converse results by writing the third term of the expansion in a more transparent fashion. This may be expressed
in terms of the Weyl conformal curvature tensor Ctjkt ([BGM], p. 83) through the identity IRI2
- IPI2 = ICI2 + p _ 2
IP2I
If this term is zero and p < 6 then both coefficients ICI and IpI must be zero and the curvature tensor is identically zero. These remarks serve to prove the following
Corollary 3.3. Suppose that p < 6 and that `dm E M the mean exit time satisfies E,,, (T,) = r2l p + o(rs) r J. 0. Then (M, g) is locally isometric to the Euclidean space RP.
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Corollary 3.4. Suppose that p < 6 and that `dm E M the exit time dislaw of Tlr2 doesn't tribution has the Brownian scaling property: the depend on r for r < r(m). Then (M, g) is locally isometric to the Euclidean space RP.
One might infer from the above discussion that, in order to prove suitable generalizations to dimensions p > 6, it would suffice to take additional terms in the asymptotic expansion of the mean exit time. This is false as was shown by the following example of H.R. Hughes [Hu]. This is the
product manifold M = S3 x H3 with the product metric; here S3 is the three-dimensional sphere of constant sectional curvature = +k2 and H3 is the three dimensional hyperbolic space of constant sectional curvature = -k2 where k is any positive number. The following result will be proved.
Proposition 3.5. [Hu]. For any r < ir/k the probability law of the exit time Tr of S3 x H3 coincides with the probability law of the exit time Tr of the six-dimensional Euclidean space R6. This example depends on the structure of the bi-radial Laplacian of S3 x H3.
This is the operator on R+ which results from the Laplacian restricted to functions which depend only on the combinations ri = xi + x2 + x3, and r2 = x4 + x5 + xs. It will be shown below that this operator is conjugate to the bi- radial Laplacian of R3 x R3 from which many interesting conclusions follow.
Suitable counterexamples may be obtained in any dimension p > 6 by taking
an additional flat factor in the form M = S3 x H3 X RP-6.
4. Exit Place Distribution of Brownian Motion. Brownian motion of Euclidean space cannot be characterized by the law of its exit place, even in dimension two (to see this, think of a brownian particle on the surface of the earth starting at the North Pole and hitting the Arctic Circle). More generally, on any space of constant sectional curvature, the exit place distribution from a geodesic sphere is the uniform distribution. Therefore, in the general case, all we can hope for is to use the exit place distribution of Brownian motion to study the variation of curvature. This may be measured exactly by the covariant derivatives of the curvature tensor over the manifold, for example. To detect the variation
Pinsky: Feeling the shape of a manifold
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of curvature in different directions, it is useful to examine the Ricci tensor as a first approximation, as we shall see below.
In order to study the distribution of the exit place from a ball of radius r, it is convenient to work on the unit sphere of Euclidean space by means of the exponential mapping exp,,,. This is the differentiable map
exp,n:RP -*M which sends 0 E RP to m E M and maps straight lines to geodesics emanating from m E M. If T is a continuous function on the sphere SP-1 we define the harmonic measure ym(r, de) on the sphere by the operator
Hr' (m) - E,n%P(r-1exp;1XT.) _
J p-1
`F(O)ftm(r, d9)
From property 2b ii') these measures converge to the uniform measure Leb(dO) when r 1 0. A two-term correction formula was discovered by Ming Liao in 1988.
Theorem 4.1. [ML]. For any p-dimensional Riemannian manifold, the harmonic measure operator has the asymptotic expansion HrW(m) =
J
[1 - (1/12)r2pm(0) - (1/24)r3ptn(9)]W(9)Leb(d9) + O(r4) p-11
where pO denotes the traceless Ricci tensor defined by p4m(9) = (pii SiiTrn/p)OiOj and po,n(9) = apij/axkOiOjOk - Ok[OT/axk]/(p + 2) where repeated indices imply a summation. In the case of two dimensions the traceless Ricci tensor is zero and we can infer that the uniform distribution of exit place implies constant curvature, as follows.
Corollary 4.2. Suppose that p = 2 and the exit place distribution satisfies Hr'P(m) = fs1 T(9)Leb(d9) + o(r3), r 10 Vm E M. Then g has constant curvature.
In the case of higher dimensions we have the following general result.
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Corollary 4.3. Suppose that for some m E M the exit place distribution satisfies HrW(m) = fsy_, %P(9)Leb(dO) + o(r2) r .. 0. Then pm = 0. In particular if the condition holds Vm E M, then pt - 0 , i.e. g is an Einstein metric.
Combining this with the results of the previous section, we see that the combined hypotheses of these two sections suffice in all dimensions:
Corollary 4.4. Suppose that `dm E M the exit distribution satisfies H,%F(m) = fsP-, T(9)Leb(d9) + o(r2) and that the mean exit time satisfies Em(Tr) = r2l p + o(rs ), r 10. Then (M, g) is locally isometric to the Euclidean space RP.
To see this last fact, note that the second hypothesis implies that Tm - 0 which, taken with the first hypothesis, shows that the entire Ricci tensor p;j(m) - 0. Using again the second hypothesis and the expansion in theorem 3.1 shows that the curvature tensor R;jkt(m) - 0, which was to be proved.
5. Joint distribution of exit time and exit place. On any space of constant sectional curvature it may be shown that the exit time and exit place from a ball are independent random variables, for Brownian paths starting at the center. This is easily seen by the separationof- variables solution associated with the Laplace operator of such a space. In general, one may inquire to what extent independence characterizes a general Riemannian manifold. For a recent discussion of independence in the Euclidean case, see [Pt]. The random variables (T,., XT) may be studied jointly through the Laplace transform which defines a family of measures µm (r, ) on SP-1 by Em (e-(TT
/r2
`F(r-lexpm1XTr )) =
J
`F(9)lum(r, d9) P-1
When r J. 0 property 2b iii') indicates the factorization into the product of the uniform measure with the Laplace transform Eo a-T in the Euclidean case. The following results have been proved:
Pinsky: Feeling the shape of a manifold
313
Proposition 5.1. ([ML], [KO], [P2]) Suppose that `dm E M the random variables T,., XT, are independent Vr < r(m). Then the scalar curvature is constant.
One might conjecture that independence implies stronger curvature conditions, such as the Einstein condition obtained in the previous section. The following example of H.R. Hughes shows that one may have independence without the Einstein condition.
Proposition 5.2. (Hu] For any r < irl k the exit time and exit place from the ball of radius r in the product manifold S3 x H3 are independent random variables.
6. Principal eigenvalue of the Laplacian. Quite apart from considerations of Brownian motion, one may study this classical quantity, which may be defined as a Rayleigh quotient
A(B) 1
r
fB, Idf12
inf (f:f o,f=0 on 8B,.) fB, I f
2.
The following three-term expansion was obtained in [KP].
Theorem 6.1. For any p-dimensional Riemannian manifold (M, g), the principal eigenvalue Al has the asymptotic expansion
) i(Br) = x2/r2 - Tm/6 + const. r2[IRIm
- IP12 + 6ZTm] + 0(r4),
r 10
where the constant only depends on the dimension p. From the principal eigenvalue one obtains exactly the same conclusions as for the mean exit time in section 3.
Corollary 6.2. Suppose that p = 2 and Vm E M the principal eigenvalue satisfies Ai(Br) = const./r2 + 0(1), r 10. Then (M,g) is locally isometric to the Euclidean plane.
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Pinsky: Feeling the shape of a manifold
Corollary 6.3. Suppose that p < 6 and that Vm E M the principal eigenvalue satisfies a1(Br) = const./r2 + o(r2 ), r i 0. Then (M, g) is locally isometric to the Euclidean space RP. As with the previous studies, a counter-example is available in dimension six:
Proposition 6.4. For any r < 7r/k the principal eigenvalue of the ball of radius r in S3 x H3 satisfies A1(Br) __ const/r2, where the constant is the the square of the first positive zero of the Bessel function J2
.
7. Computations on M = S3 x H3. In this section we give the proofs of the above statements concerning the six-
dimensional counter-example. The metric is defined as the product of the corresponding metrics in the respective spaces, which are most conveniently written in terms of geodesic polar coordinates (r1, 91), (r2, 92). In S3 we have
ds1 = dr1 +
sin 2kr1
d91,
the latter being the standard metric on the 2-sphere. In H3 we have ds2 = dr2 +
sinl22kr2
d92.
One may note that in the product space the geodesic distance is correctly given by r2 = r2 + r2, where we must respect the cut-locus by requiring that r < 7r/k to obtain a ball which is diffeomorphic to a standard ball of R6.
The full Laplacian of the product space is written AM f = 82/8r2 + (2k cot kr1)8/8r1 + (k2 csc2 kr1)Os2
+ a2/ar2 + (2 k coth kr2)a/8r2 + (k2 csch2 kr2)OS2
where the operators L 2 , resp. L 2 refer to the Laplace operators of the standard 2-spheres in the respective spaces. This formula follows from the radial decomposition of the Laplacian of a Riemannian manifold and the properties of product spaces(e.g. [He]). Let q(r1i r2) be defined by sin kr1 sinh kr2 q(r1,r2) =
r1
r2
Pinsky: Feeling the shape of a manifold
315
The Brownian motion may be constructed in terms of independent radial processes rt rt and the associated angular processes 9t, O. The following statement gives the required between the relation between the generators.
Proposition 7.1. For any twice-differentiable function f (rl, r2) on R2+ we have
AM
f(rl,r2)
q(rl, r2)
=
1
q(rl, r2)
AR.f(rl,r2)
where the last term refers to the Euclidean Laplacian acting on "bi-radial functions": AR6 f (rl, r2) = 92f /art2 + (2/rl)af/arl + 492f
/art + (2/r2)af /are
In probabilistic terms we have two independent Bessel processes whose generator is conjugate to the generator of a pair of independent radial diffusions on the sphere and hyperbolic spaces. The proof by explicit computation is
given at the end of the section. One may note that by taking f = q the formula implies that q is a harmonic function for the pair of Bessel processes. Probabilistically we have an h-transform by the function q. This formula can equivalently be paraphrased as the statement that 1/q(rl, r2) is AM-harmonic. This fact and Proposition 7.1 could be proved directly by Ito's formula applied to the pair of radial diffusions. At the end of this section we have included a classical, non- stochastic proof. In order to use this to study the exit time, we may characterize the Laplace
transform u = E.M(e-°TT) as the solution of of the equation AMu = au which assumes the value u = 1 on the boundary where r2+r2 = r2 [D]. From the rotational invariance of the respective components it follows that u only
depends on the radii (rl,r2). Therefore we may use the above structure of the bi-radial Laplacian to reduce to a problem on R. Defining a new function by v = qu we have V aq =au=AMU=AM(q) = -AR6v
Therefore v satisfies the Euclidean equation AR6 v = av in the Euclidean
ball with the boundary condition that v = q on the sphere ri + r2 = r2 The probabilistic representation in terms of Euclidean Brownian motion is
Pinsky: Feeling the shape of a manifold
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v(0, 0) = E(o 0) [q(rT , rT )e-aT l Invoking the independence property 2b iii) this factors and we obtain 1
E
oe-aT,
= u(0,0) = [1lq(0, 0)]E(o
(0'0)[e-aT,]E(o,o)[q(r4
,rT,
Taking a = 0 shows that the product of the q-terms equals 1 and we obtain the desired conclusion in the form E o o)e-aT, =
E(o,o)e-aT,
which proves Proposition 3.5.
To study the independence, we examine the joint Laplace transform ua =
E.(e-'T 'bi(X(Tr))
We must show that this factors into a function of a and an operator on 0. To prove independence, it suffices to prove the factorization property of the Laplace transform for ?p of the form & = 'Y'1(r1)02(r2)41(91)02(02), where 01i 02 are the angular components. We first do the case of 01 = 1, 02 = 1.
To do this we characterize ua as the solution of the equation AMu = au which assumes the value on the boundary of the ball, where r2 + r2 = r2 [D]. For 0 of the type 0 = 01(ri) / 2 (r2) the solution is a bi-radial function.
Defining v = qu as above, we see that v satisfies the Euclidean equation AR,6v = av with v = qb on the surface. To prove the required independence we write, as above u(0,0) = E o o)[Y'1(4 )Y'2(4) e-aT,]
= q(0 0) ' 0 0)[q(4 , rT.) 1(rT)Y'2(4.)e-aTr] 6
= q(0 0)E (0, 0)[q(rT,4 )Y'1(4,) '2(rT,)]E o
o)[e-aT,]
where we have used the independence property 2b iii) of Brownian motion on Euclidean space. Setting a = 0 shows that the product of the first two factors in the final expression reduces to the expectation on the manifold E o o)(01(r4r)02(rT )) which proves the independence of Tr from the pair 1
2
(rT,,rT,)
Pinsky: Feeling the shape of a manifold
317
To prove the independence in general, note that the angular components of the M-diffusion may be represented in terms of the respective radial processes r1, r2 and independent S2- valued brownian motions. Given r1, r2 these are conditionally independent, by the usual skew-product decomposition [McK]. The joint Laplace transform is written 0)Ly1(rTr)02(rTr)01(uTr)c2(eTr)e-a
F'
TrJ
= E(M0,0)1 E(o 0)[01(OTr)Ir1,r2]E o
0)[02(9T2
2
.)Ir1,r2]' 1(r4r)02(rTr)e-°Tr]
Starting at the center (0, 0) the conditional hitting laws of BTr, BTr are uniform on the respective spheres, so that the 01, 02 factors are constant, equal to the respective averages with respect to normalized Lebesgue measure.
Therefore the independence property property has been reduced to the independence for radial functions, which has been proved above. This proves Proposition 5.3. To study the first eigenvalue, we write the numerator of the Rayleigh characterization in the form
L,.
Idf12= L
Of/or)2+(Of/ar2r
+f (Ido,f12+Idezf12)dvol Br
which is greater than or equal to the term obtained by ignoring the angular derivatives. Therefore A1(Br.) is greater than or equal to the quantity obtained when we minimize only over bi-radial functions. But for bi-radial functions we may use the conjugacy proved in Proposition 7.1 to determine the minimum in terms of the Euclidean Laplacian, which yields the minimum as z2/r2, where z is the first zero of the Bessel function J2, hence the inequality .X1(Br.) > z2/r2. To show equality, it suffices to transplate the exact eigenfunction v(r1i r2) = J2(r)/r2 by letting f = v/q. Substituting this into the Rayleigh quotient, we obtain the inequality )1(B,-) < z2/r2. The proof is complete. Proof of Proposition 7.1. In the sphere S3 we have for any F = F(ri, r2) AssF(r1, r2) = 02F/Or2 + (2k cot kr1)OF/8r1 = (1/ sin krl)C12/ar2(sin kriF) + k2F
Pinsky: Feeling the shape of a manifold
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In the hyperbolic space H3 we have AH3F(r1i r2) = a2F/are + (2k coth kr2)OF/ar2
= (1/ sinh kr2)a2/ar2(sinh kr2F) - k2F Adding these cancels the zero-order term and we have
OMF(rl, r2) = (1/ sin kr1)a2/ar2(sin kr1F) + (1/ sinh kr2)a2/ar2(sinh kr2F) = (1/ sin kr1 sinh kr2)[a2/ar2 + a2/ar2](sin krl sinh kr2 F).
Writing F = f /q and noting the definition of q yields the relation
AM[f/q] = [1/ sinkrlsinhkr2][02/ar2 +a2/ar2)(rlr2f) On the other hand we may compute the six-dimensional Euclidean Laplacian on bi-radial functions as AR$ XR3f
(rl, r2) = a2f /ar2 + (2/rl)af
a2f /ar2 + (2/r2)af /ar2 _ (1/rl)[a2f /ar2 (rlf )] + (1/r2)[a2 f/ar2(r2f )] _ (1/rlr2)[a2/arl +a2/ar2](r1r2f). /arl +
Comparing the two forms obtained yields the desired result.
Acknowledgement We would like to thank Rick Durrett, Randy Hughes, Joseph B. Keller, Ming Liao and the referee for some helpful comments on a preliminary version of this paper.
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319
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[KP] L. Karp and M. Pinsky, First eigenvalue of a small geodesic ball in a Riemannian manifold, Bulletin des Sciences Mathematiques 111 (1987), 222- 239. [McK] H. P. McKean, Stochastic Integrals, Academic Press, 1969.
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Department of Mathematics Northwestern University Evanston Illinois 60208-2730
DECOMPOSITION OF DIRICHLET PROCESSES ON HILBERT SPACE
M. ROckner
Institut fur Angewandte Mathematik Universitat Bonn Wegelerstrasse 6 D-5300 Bonn 1 Germany
Zhang T-Sheng Department of Mathematics University of Edinburgh The King's Buildings Mayfield Road Edinburgh EH 9 3JZ Scotland
1. FRAMEWORK AND INTRODUCTION
The purpose of this note is to extend results in [3] on the decomposition of diffusion processes associated with Dirichlet forms on infinite dimensional
state space to the case where the martingale part is not necessarily a Brownian motion. Our main techniques are those developed in [3] and the stochastic integral on Banach space defined in [6]. We also prove a corresponding decomposition in a forward and a backward martingale in the spirit of [7], [8]. To state our results precisely we need some preparations. We start with describing our framework. Let E be a separable real Banach space. Denote by B(E) the Borel
a -field of E . Let p be a probability measure on (E,B(E)) . L2(E,p) is the corresponding real L2-space with the usual inner product ( logical dual of E is denoted by E' . Define the linear space
*
,
)A
.
The topo-
The second-named author has been supported by the British SERC.
Rockner & Zhang: Dirichlet processes on IIilbert space
322
(1.1)
1 Cb(E) = {u : E --4R
I
u = f(ll,...,lm) , for some m E IN
,
E E' { .
f E Cb(Rm)
For U E I Cb(E) , k E E we introduce a=
(1.2)
Ts-
u(z+sk) 1s=0
,
z E E.
We say that k E E is well-(p )admissible if there exists Qk E L2(E,p) such that (1.3)
f
du = - f u fk du for all u E 7 Cb(E)
,
(cf. [1] for more details on this notion.) In addition, we assume that there exists a separable real Hilbert space (H,< , >H) satisfying (1.4) H c E densely and continuously. Identifying H with its dual we obtain E' C H C E densely and continuously. Note that hence the dualization El < , >E between E' and E coincides with < , >H when restricted to E' xH . For convenience below we also sometimes write 1(z) for 1 E E' , z E E , instead of E, E . We denote for z E E and u E 2 Cb(E) by Vu(z) the element in H repre-
senting the bounded linear functional h - M (z) , h E H , and denote the Banach space of all bounded linear operators on H by L"(H) equipped
with the usual operator norm. Let A : E -' L'(H) be a continuous map such that For each z E E , A(z) is symmetric and c IdH < A(z) < c-1 IdH (in quadratic form sense). Here c is a positive constant (indep. of z). From now on we also assume that any k E E' (c H c E) is well-(p-)ad(1.5)
missible and E' C L2(E,p) . We introduce the following symmetric form E(u,v) , on L2(E,p)
(1.6)
,
u(dz) for u,v E F C(E)
E(u,v) = 2 f KA(z)Vu(z),Vv(z)) E
H
Under our assumptions, it follows from [3, Theorem 3.1] (see also [4]) that the symmetric form E(u,v) is closable on L2(E,,a) . If we denote the closure of E by (EA,D(EA)) , then by [10, Proposition 3.2] and [2, Theorem 2.7] (see also [11]) we know that there exists a diffusion process M=
Rockner & Zhang: Dirichlet processes on Hilbert space
323
[,,(7t)t>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,s
< , >H 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
n
f
0 i=m+1
(a_1(Xs)k,ei)2 ds H
n
.
2
(rl(Xs)k,e) - 0
Since
Pa.e.
and
m,n -+ w
as
H
i=m+1 2
°D
Y, (01-'(Xs)k,ei)H
c
<
Ijk11H
, by the dominated convergence theorem we
i=1
get that for all t > 0 , Ez<Wk,n_Wk,m,Wk,n_Wk,m>t-0
as m - m,n -im.
Hence by Doob's inequality, a square integrable continuous process (Wt)t>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 to
Fix z E E\S , then, for t > 0 by the Lemma, (2.10) and the dominated convergence theorem
Rockner & Zhang: Dirichlet processes on Hilbert space
t
329
2
EZ f
Hc()Qn -
o(Xs)II ds --4 0
as n -4 w .
2
0
Thus, JS (t) -' Mt in L2(SZ --4E,PZ) as n --4 m , consequently n \
E1Cej,J(n(t) )E -4 E' <ej,Mt>E 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
336
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)
340
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.
Rogers: Supersymmetric Feynman-Kac formula
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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
Sznitman: Brownian motion among Poissonian obstacles
354
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
< (c(d, v) + vwd_lad-1x) , where the constant k(d, v, a) is defined as:
2.\d_1/r)
k(d, v, a) = mion(vwd-1(a + r)d-l +
(0.6)
In dimension d = 1, the situation is somewhat different, for these excursions are not so large, and we find: lim t'1/3 log P ®Po [ sup IZu I > xtl /3 , T > t] too U
(0.7)
_ -(v(x V 2Ro) +
7r2
2(x V 2Ro)2)
These bounds suggest the presence of the large deviation property for the law of t-d/(d+2) supu t] . This can- checked in the case of dimension 1, which however is somewhat singular (see Remark 1.4). The bounds (0.5), (0.7) have the following consequence. Eisele and
Lang, motivated by the work of Grassberger and Procaccia [4], proved in [3] the existence of a change of regime in the long time behavior for the survival probability of Brownian motion with a constant drift h, evolving among Poissonian traps, as IhI varies. Stated in terms of Wiener measure, they showed the existence of a constant a(d, v, a) E [vwd_lad-1, k(d, v, a)] such that lim 1 log E ®Eo [eh-Z` , T > t] = 0 t-.oo t
,
when
IhI < a(d, v, a)
>0,
when
IhI >a(d,v,a)
The estimates (0.5) and (0.7) allow us to refine (0.8) for small IhI, that is IhI < vwd_lad-l when d > 2, and IhI < v, when d = 1. Indeed, in this case we show:
(0.9)
lira t-d/(d+2) log E (g Eo[eh'z T > t] t-00 = -c(1, v - IhI) , when d = 1 = -c(d, v) , when d>2.
,
Sznitman: Brownian motion among Poissonian obstacles
355
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
Sznitman: Brownian motion among Poissonian obstacles
356
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<8 2, and we set it equal to v when d = 1. The infimum in the expression defining k(d, v, a), when d > 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 <
(ii)
if a =
lim t-d/(d+2) log(µt(tax, oo)) _ -c(d, v)
d d+2 '
t-.oo
d
d
limt_oot-d/(d+2) log(µt(td/(d+2)x, oo) )
2
> -c(d, v) - k(d, v, a)x
(iii) if
d
limt_.t_a log(pt(t«x, oo))
> -k(d, v, a)x
(iv) j n ...t-1 log(µt(tx, oo) )
> -k(d, v, a)x ,
for x <
> - min(vwd_1(a + r>O
r)d-lx
2ad_1
+
Ad-1) r2
-x
2
2
otherwise.
Remark 1.2. 1) The argument we use in the proof is valid in the one dimensional case as well, however it does not yield the correct behavior in case ii), as Theorem 1.3 will show.
Sznitman: Brownian motion among Poissonian obstacles
357
both Eo[Z1 > &x, exp{-vjWW j}] and 2) As long as a < d+2 d Eo[exp{-vjWW}] behave as exp{-c(d, v)td/(d+2)(1 +o(1))}. This explains that the presence of a confinement property of surviving Brownian motion in scale tl/(d+2) (see [7] in the 2 dimensional case, or [1] for 2 dimensional random walks), involves more than principal logarithmic behavior for the probability of excursions of surviving Brownian motion. 3) Let us mention for the reader's convenience that when 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)
.
358
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
The asymptotic behavior of the first term vvt of the lower bound (1.8), (1.9) is provided in (1.4). For the second and fourth term using scaling and symmetry, we see that (1.11)
tli
m t-a log(PP -1 [Hr > s]) _ -Ad_lp/r2 , and
lim t-d/(d+2) log(2Po [Hl > (t - s)/(Rots/(d+2) - b)2] t-00 Ad (1.12)
R0
if a<1,
0
=-- (1-P) 0
ifa=1.
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Sznitman: Brownian motion among Poissonian obstacles
Let us now study the third term. If I stands for the interval (xta, 1 + xta ), for t > 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 <
(1.14)
d
limt_,.t-d/(d+2) log lpt(tax
,
oo) > -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 <
2
xr0
tad-1 2Id_1/ro, and p = 1 otherwise. Our claim iv) now follows. 0
In the one dimensional situation, the separation between scales tl/(d+2) and td/(d+2) does not exist any more. If one sets to = 2R0 = (7r2/v)1/3, one has in fact:
Sznitman: Brownian motion among Poissonian obstacles
360
Theorem 1.3. When d = 1, i)
a<
lim t-1/3log(µt(tax,0o)) = -c(l,v) _ -3(7fv)2/3
t-00
IT2
ii) turn t-1/3 log(µt(t1/3x, 00)) = -(V(x V to) +
iii) lim t_« log(µt(t«x, oo)) = -vx t
00
3
2(x V 6)2
) 1
, 2
3
iv) tl r t-1 log(µt(tx, oo)) = -(vx + 2)
Proof: Let us first treat the cases iii) and iv). One has the upper bound:
P®PJ[Zt>xt",T>t] xto']
,
which immediately yields the upper estimates in iii) and iv). As for the lower bound part one uses (1.15), which in the one dimensional situation boils down to 2
li mt-OOt-, log(pt(tax, oo)) > -vx - 2p
,
a> d+2
with the constraint p < 1 when a = 1. This immediately provides the lower bound part of iii) and iv). The next observation is that i), (a similar remark could in fact be applied in the context of Theorem 1.1), is a consequence of ii). So there now remains to prove ii). Let us begin with the lower bound. Pick /3 E (0,1/3), and consider the interval J = (-t'3,0 +(xVfo)t1/3). One has: (1.16)
log(µt(t1/3x, co)) > -v(2a + 20 + (x V £o)t1/3) + log(PJ [TJ > t , Zt > xt1/3])
Using an eigenfunction expansion for the last term, it is standard to argue that PJ [Tj > t , Zt > xt1 /3] is equivalent to the principal term (with obvious notations): Oi (0) exp{-
IT2
2I
(xVto)t" +to
JI2 t}
Oi (u) du Ix1/3
From the explicit behavior of Oi it is now easy to deduce: 7r22
limt-.t-1/3
log(PJ [Tj > t , Zt > xt1/3]) > -
2(x V to)2
Sznitman: Brownian motion among Poissonian obstacles
361
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
<
tl/3
.2
(u - v)2
1]
t1/3
- 1) } , ((u + v)2 where the last inequality follows from the spectral theorem. If we use this estimate in (1.17), Laplace method now yields: <
(2ir)-1
exp { -
log
2
µt(t1/3x,
CO)
2 <-
2
inf(vt + 2Q2) = vx + 2x2
This finishes the proof of ii) and of Theorem 1.3. 0 Remark 1.4. It should be mentioned that the proof above works wi th trivial modifications if we replace the quantity pt(t"x, oo) in the four cases of Theorem 1.3 by P x Po[sup, xt' , T > t]. It is not difficult to see that t-1/3 sup,
t], with rate function I(x) = vy+
property under P ®Po
22
-c(1, v),
with y = (2x) A (to V x), whereas t-1/31Ztl under the same2yconditional measure satisfies a large deviation property with rate function I'(x) = v(x V
to) +
2(x V X0)2
- 41, v).
o
H. Upper bound: a preliminary reduction step. We suppose now that d > 2. We want to derive a preliminary estimate which will help us in proving upper bounds on the probability of existence prior to time t of an excursion of surviving Brownian motion at a distance larger than const. This estimate will show that with no loss of generality we can assume that Brownian motion does not leave a suitable cube T of size const. t(d+1)/(d+2), that in this cube the obstacles form finitely td/(d+2).
many "clearings" of size - t1/(d+2). The estimate will also allow us to
Sznitman: Brownian motion among Poissonian obstacles
362
assume that the number of excursions of the process "in the forest", that is outside a neighborhood of size tl/(d+2) of the clearings, compared to td/(d+2),
as well as the fraction of time spent during these excursions are arbitrarily small. We will now make these notions precise. First, it will be convenient to adopt respectively tl/(d+2) and t2/(d+2) as new space and time units. With this new choice of units, we now study a standard Brownian motion until time s Lef td/(d+2), evolving among a Poisson cloud with intensity vs of points xi of Rd surrounded by closed balls of radius at-1/(d+2) = as-1/d
We now introduce a number L > 0, and an integer N > 1, and for s > 1 we define the cube T of Rd:
T = (- N[s]L , N[s]L)d
(2.1)
,
([s] : integer part of s) .
For each multiindex m E [-N[s], N[s] - 1]d, Cm will be the open subcube of T:
Cm = {zERd, miL
(2.2)
With no loss of generality we will assume that no point of the Poisson cloud has one of its coordinates which is an integer multiple of L.
We now introduce two numbers b > a, and b > 0. Following [5], we define a point xi E Cm of the Poisson cloud to be a good point if for all closed balls C = B(xi , where e 1!1':f s-1/d, 0 <.£ and 10t+1eb < L/2, (2.3)
CmnCn( U B(x;,bE))I>3 ICmnCI. x j EC,,,
We will denote by G(m) the set of good points of Cm. Notice that for r > 0, and z E Cm, using a homothety of ratio 1/3 at z, one has: ICm n B(z, r/3)I > 3-dICm n B(z, r) I It now follows from the covering Lemma 1.3 of [5], that for each Cm: (2.4)
1c,. n ( U B(xi, be) )I
5ICmI = SLd
xi¢G(m)
xjECm
In other words, the union of balls of radius be at bad points of Cm cover a small fraction of the volume of Cm.. We then chop identically each segment [kL, (k+1)L] in at most
[iS1/d} +1 intervals of length
s-1/d, except
Sznitman: Brownian motion among Poissonian obstacles
363
maybe for the "last one". This yields closed boxes of diameter less than
\d
be = bs/d, in number less than ([V'dsh/c] + 1 I , whose union is C,,,.. We then introduce a number r > 0, and denote/ / by Clr the event that there is a "clearing of size r in the cube C,,,", that is: (2.5)
Clm = (w, I Um(w)I >_ 2-d I B(O, r)I = 2-dwdrd }
,
if U,n(w) is the random open set of C,,,, obtained by taking the complement in C,,, of the closed boxes where a good point of the cloud falls. We also
C T, where there is a
denote by B(w) the union of all closed cubes clearing of size r: (2.6)
1B(w)(z) _
1Cm(z) -
We now pick a number 2 > 0. Let us mention that we will not need to let £ vary and we could very well pick f = 1, once and for all. We now define the successive excursions of the process Z.(w), at distance f from B(w):
Di=inf{v>0,
D1 , Z, E B} =HB0OD, +D1 1: Dn+1 = H(B1)- 0 6R.. + Rn
,
Rn+1 = R1 0 8R + R. ,
here Be denotes the open neighborhood of point at distance less than f of B(w) (which is empty if B is empty). We set N. to be the number of excursions completed by time s: (2.7)
{N,=k}={Rk<s0,
with the convention Ro = 0. Finally, we define L, to be the fraction of time spent up to time s in the excursions: (2.8)
L,= 1E(RiAs-Dins). s i>1
We can now state the main object of this section:
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Sznitman: Brownian motion among Poissonian obstacles
Theorem 2.1. For anyL>0,r1>0,2>0: (2.9)
limN.00 limr-o limno-00 limb.oo, 6-0 lim,_,. llog(PxPo[{T>s} n[{TT<s}U{JBI>noLd}U{N,>[,s]}U{L,>7/}]])=-oo.
Here TT denotes the entrance time in T'. Proof: Our claim (2.9) will follow if we show that: (2.10)
(2.11)
limN
moo
lim,-.
s
log (Po [TT < s])
limno-.°° 1imb_oo,6-.0 lims-
,
s
= -00 ,
logP[IBI > noLd] = -oo
and with the notation T A TT =!h (2.12)
limr.o limb-.oo, 6-.0 lima-oo 1 log sup Po If T > s l S
W
n{N,>[iis]}]=-oo, (2.13)
limr.o limb-.., 6-.o lim,.,,. -log sup Po [{T > s} s
W
n{N,<[77s]}n{L,>i7}]_ -oo Let us start with the proof of (2.10). Using scaling, we immediately find:
Po[TT<s]NL[s]] o
Vis-
where Po is the one dimensional Wiener measure. It follows that: 2
li m8-00
s log Po [TT <s] < - 2 L2
,
which yields (2.10). Let us now show (2.11). From the definition (2.3), we see that for any
point xi E Cm, the property that xi is a good point of C,n just depends on the restriction of the Poisson measure to C,n. From this observation, it follows that the events Cl,,, ("presence of a clearing of size r" in Cm), for m E [-N[s], N[s] -1]d are i.i.d. Let us now give an upper bound on P[Cl,n].
Sznitman: Brownian motion among Poissonian obstacles
365
If U,,,(w) denotes the complement of the boxes in Cm. where some point falls (recall that for Um(w), it is a good point instead), we have since each box has diameter less than be:
Um C Um U ( U B(xi, bs-1Id ) n Cm) . .i G(m) si ECm
Thanks to (2.4), this yields: (2.14)
IUmI
This shows that cam = {IUmI >
2-dIB(O,r)I} c {IUmI > 2-d9B(O,r)I
- 6L d}
and it follows that: P[Clm] < P[IUmI > 2-dIB(0, r)I - 6Ld] dsi/d + 1)d < 2( b
exp{-vs(2-dIB(0, r)I
- 6Ld)}
Now, there are at most (2N s)dno possible subsets of no elements in the set of subcubes Cm of T. As a consequence, P[IBI ? noLd]
< (2Ns)dno2((LI6)vde'id+1)dn,, exp{-novs(2-dIB(0,r)I
- 6Ld)}
so that lime.
s log P[IBI > noLd] < no (log 2(b V d + v6Ld - v2 -d I B(0, r)I)
From this we obtain (2.11) immediately. Let us now prove (2.12). We first introduce the constant cl defined as: (2.15)
cl= 12 CEC of
Po[Hc0,
where H3 = inf{v > 0, IZ,, -Zo I > 3}, He is the entrance time in C, and C is the class of compact subsets of W(0,2), such that ICI > 2-d(1-2 -d )I B(0, 2)[. cl is easily seen to be positive, for instance by observing that the transition density ql (0, ) at time 1 of Brownian motion kill ed outside B(0, 3) has a
366
Sznitman: Brownian motion among Poissonian obstacles
positive infimum on B(0, 2). The notion of clearing we introduced will be of special help to us thanks to
Lemma 2.2. For any b > a, 1 > S > 0, 0 < r < L/4, N > 1, there exists so > 0 such that for s > so: inf
(2.16)
w, zEB(w)°f1T
Pz[T < H4r] > C1
.
Proof: First define the positive number: (2.17) a(S, b, a) =
inf
Izl<1, CEC'
PP[HC < Hlo] x inf Pz[HB(o,a) < HB(0,3b)-] IzI
where C' is the class of compact subsets of B(O,1) with relative volume no less than S/6d. Then set m(S, b, a) to be the smallest integer such that:
m log(1 - a) < - log 2 .
(2.18)
Consider now z E Cm a point at distance smaller or equal to be = bs-1/d of a good point x= E Cm. Suppose 10'+1 be + be < L/2, we have: (2.19)
Pz[H1om+4e+b,,
with Hp = inf {u > 0
,
I Zu - x; I > p}. Since x; is a good point of Cm, we
know that: ICm n c n
(U
B(x;
,
bs-11d))
I > 3 ICm n cl ? s Icl
x, EC-
for all balls C = B(x;,10e+1eb), with 0 < 2 and 10l+leb < L/2. Now from the definition (2.17) of the constant a, using scaling and Markov property, we see that 1 Pz[H1m+lb, < T] < (1 - a)n, <
As a consequence, we see that for any z in T within distance be of some good point in some cube Cm, we have (2.20)
Pz[Hr > T] > Pz[H1Om+be+be > T] > 1 - (1 - a)f1 > 2
provided s is large enough so that (2.21)
10'+1 be + be < r < L/4
.
Sznitman: Brownian motion among Poissonian obstacles
367
n B(w)', we know that lCim(,,,) = 0, and IUmI < Now, when z E 2-djB(0, r)j. As a consequence, the intersection of B(z, 2r) with the union of boxes in C,,, containing some good point has a volume bigger than or equal to:
I - 2-d1 B(0,r)I > (1 - 2-d)2-dIB(0,2r)I , since r < L/4
JB(z,2r) fl
Consequently, using scaling and the definition (2.15) of cl, (2.22)
Pz[Hcm\um < H3r] > 2c1
,
for z ECfl B(w)`
.
Combining (2.22) and (2.20), we see that when s is large enough so that (2.21) holds, for z E T fl B(w)c, Pz[T < H4r] > 2 x 2c1 = c1. This yields our claim.
Lemma 2.2 has the following consequence: for b > a, 1 > S > 0, 0 < r < L/4, N > 1, s > so (so appearing in Lemma 2.2), and p > 0: sup
(2.23)
PP[HB < T] < (1 - c1)[P/4r]
w, zE(BP)'nT
Indeed, if z E (BP)c fl T, the process has to exit successively at least 12-1 4r balls of radius 4r before entering B, so that (2.23) follows from (2.16) and a repeated use of strong the Markov property. Let us now continue the proof of (2.12). We have: sup Po[{T> s} fl {N8 > [is]}] < supPo[R[,,8]
W
Observe now that for k > 0: Po [Rk+l < T] < Po [Rk < T, Dk+1 < T, HB 0 ODk}1 < T o ODk+1 ]
< Po[Rk < T] x
sup Pz[HB < T]
(B<)cnT
< ( sup Pz [HB < T]) k+1 (BL)cnT
using induction. It now follows from (2.23), that when s is large enough:
supPo[{T > s} fl {N8 > [is]}] < (1 U;
from which we obtain (2.12) immediately.
Sznitman: Brownian motion among Poissonian obstacles
368
Let us now prove (2.13). Pick .X0 small enough so that P0[exp{A0H1}] < 1 + 2
(2.24)
Define H° = 0, and H'+1 = H` + H4r o 9H:, the successive times when Z. travels at distance 4r. Then choose A < Ao/16r2 and s > s0 given by Lemma 2.2, for z E (Bt)c nT, we have: (2.25) E,z[exp{ARi AT}]
= PZ[T = 0] +
Ez[Hk < HB A T < Hk+1 , ea(HBnT)] k>O
< 1 + 1: EZ[H' < HB AT , aHk] EO[eaH9, ] k>O
From scaling, E0[exp{AH4r}] = E0[exp{A16r2H1}] < 1 + cl/2. Now for
k>1: EZ[Hk < HB A T , eXH'] < Ez[Hk-1 < HB A T , eAHk-1 (EZHk-1 [H4r < HB A T] + EZHk_1 [e- \H--] - 1)] Cl)
< Ez[Hk_1 < HB A T, eAHk-1] (1
-
where we use the fact that ZHk-1 E B(w)c n T, on the event Hk-1 < HB A T, together with (2.16). It now follows that for A < A°/16r2, s > s0, z E (Bt)° n T, and any w: Ez[exp{ARi A T}] < 1 + (1 +
(2.26)
2)
(1 k>O
2
2
Picking A = A0/16r2, we find (2.27)
)k = 2 +
supPO[{T>s}n{N, <[is]}n{L,>ij}] W
< supexp{-Arks} Po [exp {A[(R1 A T) o 0D1nT W
+(R1 AT)o0D2AT+...+(Ri AT)o9D(ne]nT]}1 Using now the strong Markov property, and the fact that (2.26) obviously holds when z T, we see that the expression in (2.27) for s > so, is smaller than: exp{-A s}(2+2/c1)'?9+1 from which it follows that:
lifls-c
s
log sup P0[{T>s}n{N,<[r)s]}n{L,>rl}] W
6r2
+ log (2 + 2/cl)
This immediately yields (2.13) and finishes the proof of Theorem 2.1.
Sznitman: Brownian motion among Poissonian obstacles
369
III. Upper bound We suppose here, as in Section 2, that d > 2. Our main purpose is to derive an upper bound for the probability that surviving Brownian motion This will provide a companion in time t travels at distance of order estimate to the lower bound of Theorem 1.1, ii). td/(d+2).
Theorem 3.1. (3.1) FMt_""t-d/(d+2) log p ®Po[T > t , sup IZu I > x td/(d+2)] u
< -(c(d, v) + uwd_1ad-lx)
Proof: Using the same scaling argument and same notations as in Section 2, we want to show that (3.2)
limsy00s-1 log P ®Po[T
xs(d-1)1d]
> s , sup IZu I > u<9
< -(c(d, v) +
vwd-lad-1x)
,
where we recall that s = td/(d+2) and P now denotes a Poisson point process of intensity vs, the points of the random cloud being surrounded by closed
balls of radius as-l/d, constituting the obstacles. Thanks to Theorem 2.1, we see that for a given 0 < i < 1, and L > 0, we can pick N(i1, L), r(77, L), r7o(r7, L), b(r7, L), b(i, L) so that the probability of the event which appears in
(2.9), decays at a faster exponential rate in s than -(c(d, v) + vwd_1ad-1x) So, (3.2) will follow if we show that (3.3)
limL-. limn-o 11m,-.0 log P ®Po [T > s I BI < noLd
,
N. < [11s]
,
L. < t sup IZ.I >
, x8(d-1)/d]
v<s
< -(c(d, v) + vwd-lad-lx) Observe that the condition L, < 77 < 1, forces B(w) not to be empty. The number of nonempty subsets M of [-N[s], N[s] - 1]d, with no more than no elements is smaller than (2Ns)dno x no. It is consequently enough to show that: (3.4)
limL_,oo lim,7_,o limn-oc
N. < [17s]
,
L, < 17
,
sup IZuI > U<3
sup
1
logP ® Po[T > s ,
C -(c(d, v) + vwd-lad-1x)
370
Sznitman: Brownian motion among Poissonian obstacles
where now N. denotes the number of excursions between B = UM C. and (BL)c, as in (2.7) and L9 is the fraction of time spent up to time s, coming back from these excursions as in (2.8). Observe that the restrictions of the Poisson cloud to B2t and (B2e)c, are independent. We denote by PB2t and P(B2<). their respective laws. Conditioning on the restriction to Bet of the cloud, we see that the probability under study in (3.4) equals: (3.5) PB2t ® Po[T > s , N. < [ris]
, L. < q, sup IZuI > X3(d-l)/d u<s
exp{-vsJWW
3_l
n (B21)cI}]
where T = TT A T, and T denotes now the entrance time in the obstacles coming from the cloud with law PB2t. It will now be convenient to derive:
Lemma 3.2. Let 0 E Q[0, TI, Rd), ¢(0) = 0. For p > 0,
(3.6) IW(q)n(B2c)cI >
wd_lPd-l
sup ICI-no(L+4e+2p)J) -nowdpd . [0,T]
Proof: With no loss of generality, possibly reducing the interval [0, T], we assume that Ic(T)I = max[o,T] 101. Define for m E M, Cm 3 to be the closed cube with same center and parallel to Cm of side length L + V + 2p, and set U rim 3 B2c+a
.
M
We then define sl = inf{u E [0, T] , 0(u) E B} (Si = oo, if the previous set is empty). Then there is one or maybe several m E M such that q5(sl) E Cm. Define tl to be the supremum of the v E [sl,T] where r¢(u) belongs to the union o f these Cm. Continue then by defining s2 = inf{u > t1, 4(u) E B}, and construct a finite sequence, 0 < sl < tl < s2 t2 < < sk < tk < T, with 0 < k < no, such that for u V Ul
IWp(t')I -no(
(L+4Q+2p)Vdwd-lpd-l+wdPd)
>Wd-1Pd-l(IO(T)I -no(L+4Q+2p)Vrd-)-nowdpd which proves our claim (3.6) since IO(T)I = Iq(T)I = max[o,T[ 101.
11
,
Sznitman: Brownian motion among Poissonian obstacles
371
If we now apply Lemma 3.2 to (3.5), we find that the expression in (3.5) is smaller than: PB2c ®Po [T > s , N, < [77s] , L, < r7]
(x - 2nov(L + 4t +
exp{-vswd_lad-1
2as-'Id) 8-(d-1)/d) - 2nowdad}
So our claim (3.4) will follow if we show that: (3.7) limL-,oo lim,l_,o lime-0
sup
1109 PB24 (D Po [T > s,
1<J.MJ<no(,),L) S
N, < [r7s], L, < r7] < -c(d, v) .
Let us introduce the torus of size L, TL = (R/LZ)d, and denote by proj'r, the canonical projection on TL. We have:
IW;'-'/d n B2`I >
IprojT.L(W;'-'/d n B21)1,
where we use I I to denote the usual volume on TL as well. As soon as as-1ld < Q, the last quantity is bigger than ¢e-1/d
(ta-1/d
0.8-1/d
IProJTi(W[o,D1) U W[Ri,Dz) ... U W[RN,6ADN,))I
For as-11d < .e, we find
PBz<®Po[T>s, N.<[r7s], L,
e-11d
n B211} , N. < [77s]
,
L. < q
,
TT > s]
Po[exp{-vlprojT.L(GV[0 D0d U W[R1,Dz) U ... U W[RN.,enDN.))I }
e
N,<[17s], L,s].
(3.8)
Let us now denote by PL the law of the random point process obtained by picking a Poisson configuration of points with intensity vs in (0, L)' and extending it to the whole of Rd by periodicity. As before let us define the obstacles as closed balls of radius as-11d centered at each point of the periodic configuration. If T denotes as before the entrance time in the obstacles, we see that (3.8) equals PL
®Po[TT>s, T>D1, ToOR1>D1oOR1, ..., T o 9RN, > (s - RN,) A D1 o 0RN, , N. < [17s], L. < rl]
Denote by AL(w) the principal Dirichlet eigenvalue for the generator of the self adjoint trace class semigroup on L2(TL), corresponding to Brownian
Sznitman: Brownian motion among Poissonian obstacles
372
motion on TL killed on the obstacles on TL, projection of the periodic obstacles on Rd. Observe that on the event whic h appears in (3.9), T A D1 + T A D1 0 ORI +
+ T A D1 0 9RN, > (1-77)s
It follows that for M > 0, and p > 0, (3.9) is smaller than: PL®Po [TT > s, exp{(AL A M - p)+(T A D1 + T A D1 0 ORS + .. .
+T AD1 0ORN. - (1 -q)s)} , N, < [17s]] < pL [exp{-(JCL A M - p)+(1 - 71)s} Po [exp{(AL A M - p)+
T A D1 0 9R; 1(R; < oo)}] ] 0
< PL [exp{-(JCL A M - p)+(1 - 77)s}
(sup PZ[exp{(AL A M - p)+T}])ns+1 ] ZERd
If now PL denotes the law of Brownian motion on TL starting from z, using obvious notations, we have sup Px [exp{(AL A M - p)+T}] = sup Pz [exp{(AL A M - p)+TL}] Rd
TL
This last quantity, by inequality (1.22) of [5] is smaller than eM(1 + CL + cL v ), where CL stands for the supremum of the transition density at time one, with respect to the normalized volume measure of Brownian motion on TL. It is may be helpful to mention that the notations of [5] are somewhat different and TL here corresponds to Tb in [5], and AL to .fib. We now see that (3.9) is smaller than f (L,q, M, p, s) dew-[ exp{(2M71
)ns+1 pL [exp{-(AL A M)s}]
+ p)s + M}(l + CL + CL
P
It now follows that the left member of (3.7) is smaller than: limL-oo limM-.oo, p--.o limn-o lim,_,oo s log(f (L, r/, M, p, s) )
= lifL.oo FIMM.oo lim,.oo 1 log PL[exp{-(AL A M)s}] = -c(d, v)
,
S
The last equality comes from the upperbound we derive for PL[exp{-(AL A M)s}] in [5], see what follows there (1.39), Theorem 2.2 and Lemma 3.3.
Sznitman: Brownian motion among Poissonian obstacles
373
We should also mention that in the notations of [5], AL plays the role of \KT his proves our claim (3.7) and finishes the proof of Theorem 3.1. As a counterpart to the lower bounds of Theorem 1.1, we have
Corollary 3.3. Suppose d > 2, and x > 0, (3.10) limt_oot-d/(d+2) log(µt(td/(d+2)x, oo) ) vwd_Iad_1x)
< -(c(d, v) +
,
log(/1t(t°'x, oo)) < -vwd-iad-1 x
(3.11)
,
d
d
2
2
(3.12) limt_oot-1 log(pt(tx, oo)) < -(2 + vwd-lad-1 x)
Proof: (3.10) is an immediate consequence of Theorem 3.1. The proof of (3.11) and (3.12) is straightforward, it relies on the already observed fact that: (3.13)
IWt (Z)I > vwd-lad-'IZt - ZoI , from which we deduce pt(t' x,oo) < exp{-vwd-lad_1xt'} P0[Zf > xta]. (3.11) and (3.12) now follow easily.
Remark 3.4. 1) It should be observed that (3.10) does not hold in the one dimensional case (see Theorem 1.3). 2) From Theorem 1.1 ii) and Theorem 3.1, we see that, when d > 2:
-k(d, v, a)x < lim t-d/(d+2) log p ® Po[sup IZSI > xtd/(d+2)/T > t] s
< -vwd_1ad-lx . This strongly suggests the possibility of a large deviation property for the law of t-d/(d+2) sup,
IV. An application We are now going to apply our results to the study of the large t behavior of Eo[exp{h Zt - vlWf I }]
,
when
IhI <
vwd-lad-1
This quantity was investigated by Eisele and Lang, in their study of survival probability for Brownian motion with a constant drift evolving among Poissonian obstacles. They showed that (4.1)
lim 1 logEo[exp{h Zt - vlWW I }] = 0 , t-.oo t
when
IhI < a(d, v, a)
>0,
when
I hl > a(d, v, a)
,
Sznitman: Brownian motion among Poissonian obstacles
374
for a critical value a(d, v, a) in the interval [vwd_lad-1, k(d, v, a)], where
k(d, v, a) is defined in (0.6). We are now going to apply the results of section III, in the d > 2 case, and section I in the d = 1 case, to refine (4.1) when IhI < vwd_lad-1.
Theorem 4.1. When d = 1, and IhI < v: (4.2)
lim t-113 logEo[exp{hZt - vlWW I }] = -c(1, v - IhI) _ -3(ir(v
t-+00
When d > 2, and IhI <
-
Ih1))213
.
vwd_lad-1:
lim t-d/(d+2) logEo[exp{h Zt - viWf I fl _ -c(d, v) t-00
(4.3)
Proof: It is enough to study Eo[exp{hZ' -vlWW l}] for h > 0. This quantity is bigger than (4.4) f 00 exp{hx} ditt(x) = Eo[exp{-viWW I }] + h 0
J0
ehxpt(x, oo) dx
,
and anyway smaller than: /'oo
2Eo[exp{-vjWW I}]+h J
ehxµt(x,oo)dx
.
0
It follows that we simply have to study the second term to the right of (4.4).
It is the sum of A a1 deJ- htd/(d+2)
exp{td/(d+2)hx} µt(td/(d+2)x, 00) dx
J0
and
oo
exp{hx} µt(x, oo) dx
a2d e-L h J
,
with A a positive constant.
Atd/(d+2)
Using (3.13), we see that a2 is smaller than 00
J
dx hexp{x(h - vwd_lad-1)}, td/(d+2)
so that (4.5)
limt__o, _d/(d+2) log a2 < -A(vwd_1 ad-1 - h)
.
We see our claims (4.2), (4.3) will be proved if we show that for large enough A:
(4.6)
log a1 < -c(d, v)
,
when d > 2
,
Sznitman: Brownian motion among Poissonian obstacles
lim t1/3 log al = -c(1, v - ]h`) ,
(4.7)
t-00
375
when d = I
Observe that Ftt(td/(d+2)x, oo) is a decreasing function of x. Using Riemann sums, and Laplace method, we see that when d = 2, thanks to (3.10),
l1mt_",)t-d/(d+2) log al < sup {hx - c(d, v) - vwd_lad-lx} = -c(d, v) xE[O,A]
since h <
vwd_jad-1
This proves (4.6). On the other hand when d = 1,
using Theorem 13, we find 2
lim t-1/31oga1 = sup {hx - v(x V 4o) t---oo xE[O,A]
2(4 V x)2
= c(1, v -
provided A is bigger than [r2/(v - h)]1/3. This yields (4.7) and finishes the proof of our claim. O
Bibliography BOLTHAUSEN, E.: "Localization of a two dimensional random walk with an attractive path interaction", preprint. DONSKER, M. D., VARADHAN, S.R.S.: "Asymptotics for the Wiener sausage", Comm. Pure Appl. Math., 28, 525-565, (1975). EISELE, T., LANG, R.: "Asymptotics for the Wiener sausage with drift", Prob. Th. Rel. Fields, 74, 1, 125-140, (1987). GRASSBERGER, P., PROCACCIA, I.: "Diffusion and drift in a
medium with randomly distributed traps", Phys. Rev., A 26, 36863688, (1982).
SZNITMAN, A.S.: "Lifschitz tail and Wiener sausage I", in J. Funct. Anal., 94, 223-246 (1990). "Long time asymptotics for the shrinking Wiener sausage", Comm. Pure Appl. Math., 43, 809-820, (1990). "On the confinement property of two dimensional Brownian
motion among Poissonian obstacles", to appear in Comm. Pure Appl. Math.
