Stochastic processes: selected papers of Hiroshi Tanaka

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8T0CHA8HC elected Papers of Hiroshi Tanaka

Edited by

Makoto Maejima & Tokuzo Shiga

World Scientific

STOCHASTIC PROCESSES f Selected Papers of Hiroshi Tanaka

This page is intentionally left blank

STOCHASTIC PROCESSES Selected Papers of Hiroshi Tanaka

Edited by

Makoto Maejima Keio University, Japan

Tokuzo Shiga Tokyo Institute of Technology, Japan

iS% World Scientific U

New Jersey *London L ondon•Sine • Singapore * Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

STOCHASTIC PROCESSES Selected Papers of Hiroshi Tanaka Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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1 \

Hiroshi Tanaka

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vii

Preface

Hiroshi Tanaka is known for his outstanding contributions to the theory of stochastic processes. Since his first paper on diffusion processes in 1957, he has obtained brilliant mathematical results including construction of diffusion processes with continuous coefficients, Tanaka formula for Brownian motion, Skorohod equation on a convex domain, Boltzmann equations, diffusion processes in random media and so on. On the occasion of his retirement from academic positions, we planned to collect his important contributions to the theory of stochastic processes in one volume, making his works available in a unified form to the mathematical community. Besides his selected papers in this volume we asked Professors Henry McKean, Marc Yor, Shinzo Watanabe and Tanaka himself to write some essays on Tanaka's mathematics and personality. Among Tanaka's works the most popular one would be the celebrated Tanaka formula for Brownian local time, which was never published. The story around the birth of Tanaka formula can be found in the essays of Marc Yor and Shinzo Watanabe. As editors of this volume, we are grateful to World Scientific Publishing Co. Pte. Ltd. for the production of the volume. Our special thanks go to Professors Henry McKean, Marc Yor and Shinzo Watanabe for their writing interesting essays, to Professors Nobuyuki Ikeda and Yuji Ito for their valuable suggestions in the process of planning this volume, to Professors Yozo Tamura, Hideki Tanemura and Yuki Suzuki for their assistance throughout its preparation.

Makoto Maejima Tokuzo Shiga

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ix

Contents

Preface

vii

H. Tanaka: An Appreciation (by Henry McKean)

1

From Local Times to Random Environments . . . (by Marc Yor) . . . .

2

Contributions and Influences of Professor Tanaka in Stochastic Analysis (by Shinzo Watanabe)

5

Some Comments on My Mathematical Works in Retrospect (by Hiroshi Tanaka)

10

[5]t Additive Functionals of the Brownian Path (with H. P. McKean) . . . 19 [7] Note on Continuous Additive Functionals of the 1-Dimensional Brownian Path [8] Existence of Diffusions with Continuous Coefficients

47 54

[14] Propagation of Chaos for Certain Purely Discontinuous Markov Processes with Interactions 69 [15] An Inequality for a Functional of Probability Distributions and Its Application to Kac's One-Dimensional Model of a Maxwellian Gas . . 83 [16] On Markov Process Corresponding to Boltzmann's Equation of Maxwellian Gas 89 [19] On the Uniqueness of Markov Process Associated with the Boltzmann Equation of Maxwellian Molecules

101

[22] Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

118

t Numbers in brackets refer to the Bibliography of Hiroshi Tanaka (see pages 425-428)

X

[24] Stochastic Differential Equations with Reflecting Boundary Condition in Convex Regions

157

[27] Some Probabilistic Problems in the Spatially Homogeneous Boltzmann Equation [29] Limit Theorems for Certain Diffusion Processes with Interaction [32] Central Limit Theorem for a System of Markovian Particles with Mean Field Interactions (with T. Shiga) [33] Propagation of Chaos for Diffusing Particles of Two Types with Singular Mean Field Interaction (with M. Nagasawa)

172 . . 182

202 223

[34] Stochastic Differential Equations for Mutually Reflecting Brownian Balls (with Y. Saisho)

238

[35] Limit Distribution for 1-Dimensional Diffusion in a Reflected Brownian Medium

254

[36] Limit Distributions for One-Dimensional Diffusion Processes in Self-Similar Random Environments

270

[38] Stochastic Differential Equation Corresponding to the Spatially Homogeneous Boltzmann Equation of Maxwellian and Non-Cutoff Type

292

[41] Limit Theorem for One-Dimensional Diffusion Process in Brownian Environment

311

[49] On the Maximum of a Diffusion Process in a Drifted Brownian Environment (with K. Kawazu)

328

[51] Recurrence of a Diffusion Process in a Multidimensional Brownian Environment

336

[52] Localization of a Diffusion Process in a One-Dimensional Brownian Environment

341

[54] Diffusion Processes in Random Environments

353

XI

[55] Environment-Wise Central Limit Theorem for a Diffusion in a Brownian Environment with Large Drift

361

[56] A Diffusion Process in a Brownian Environment with Drift (with K. Kawazu)

373

[57] Limit Theorems for a Brownian Motion with Drift in a White Noise Environment

396

[58] Invariance Principle for a Brownian Motion with Large Drift in a White Noise Environment (with K. Kawazu)

406

[63] Some Theorems Concerning Extrema of Brownian Motion with d-Dimensional Time

415

Bibliography of Hiroshi Tanaka

425

Permissions

429

1

H. Tanaka: An Appreciation

by Henry McKean

It is a pleasant thing for me to write this little note to accompany the selected papers of H. Tanaka collected in this volume. When I was young and fresh out of school, I visited Kyoto for a year (1957/58) at the invitation of K. Ito. My great-grandfather (whom I never knew) was an amateur of Japanese art, generally, and of Okyo in particular. Part of his collection came down to my family, so even when I was quite little, I was fascinated by Japan and things Japanese. The double chance to see Japan and to work with Ito was an irresistible piece of luck and I was pretty well prepared for that. What I was not prepared for, but happily surprised by, was the enthusiastic crowd of young Japanese probabilists under Ito's wing who came faithfully, once a week, to hear the new developments in Brownian motion and diffusion stemming from the ideas of W. Feller and E.B. Dynkin — "sowing the seeds of diffusion in the mathematical fields of Japan" as Ito put it. H. Tanaka stood out in this company for the quickness of his understanding, for his mathematical taste, and for his quiet but strong personality. That was our first acquaintance. Later he visited me at MIT in Cambridge; in between, we collaborated on additive functions of the Brownian path ([5] *, 1961). Tanaka's mathematical production is marked by the elegant use of common ideas employed in novel ways and with uncommon skill. No better example can be found than his 1963 paper ([7]) in which Ito's lemma is used with a bit of extra audacity to prove a wonderful formula, simple but highly effective, for the Brownian local time. I would like particularly to cite the deep series of papers ([15], [16], [19], [27]) on Boltzmann's equation for the Maxwellian gas and the Markov process that underlies it, a subject dear to my heart. This had been initiated by Kag and studied somewhat by me, but it was Tanaka who really understood it correctly, and everything of Tanaka's shows like elegance, simplicity, depth. This note is to express to Tanaka my admiration and friendship and to wish him health and happy productive years still to come.

+

Numbers in brackets refer to the Bibliography of Hiroshi Tanaka (see pages 425-428)

2

From Local Times to Random Environments...

by Marc Yor

It is both a pleasure and an honor to have been asked to write a few words about the works of Professor H. Tanaka. At least two aspects of Professor Tanaka's works have been of much interest to me and, more generally to many probabilists in France, for quite some time : a) The celebrated Tanaka formula, expressing the local time at 0 of linear Brownian motion as the last term of an extended ltd formula. Not only does this formula extend to Brownian local time taken at every level a(€ R), which makes it a very powerful tool to deduce the Markovian properties of these local times with respect to a, i.e. the Ray-Knight theorems ([1], [2]), but as shown by P.A. Meyer in his "Cours sur les integrates stochastiques" (Lecture Notes in Math., 511, Seminaire de Probability X, 1976), it extends easily to the framework of continuous semi-martingale local times. The works of P. Mc Gill ([3]) and T. Jeulin ([4]) for instance, and later those of L.C.G. Rogers and J. Walsh ([5], [6]), concerning the spatially indexed filtration of Brownian excursions, give ample evidence of the power of Tanaka formula. It is an interesting fact that Tanaka formula appears as a "personal communication" in H.P. McKean's "Stochastic Integrals" (1969) whilst more general discussions describing every additive functional of (possibly multidimensional) Brownian motion appeared earlier, e.g., in Professor Tanaka's paper (Zeitschrift fur Wahr.,1963), and a number of papers by probabilists in the then Soviet Union. There, I see some similarity with the role Problem 1 on p. 72 of Ito-McKean's book (1965), which establishes the existence of principal values of Brownian local times, has taken in the later development of these principal values (see, e.g. the discussion by T. Yamada [7], p. 414). In the study of intersections of 2 - or 3 - dimensional Brownian motion, variants of Tanaka formula - the Tanaka-Rosen formulae - have also played some part in understanding Varadhan type renormalizations (see, e.g., [8]). b) The study of many properties - most of them asymptotic - of diffusions in random, especially white noise, environments. There are at least sixteen papers by Professor Tanaka, which have been devoted to this topic ; through his perseverance, many facts about these diffusions

3

have been exposed in great clarity, in particular via the study of the so called "valleys". For quite some time, I wished to establish some relationship between the (logf) normalization in the asymptotics of these diffusions, and the following radial version of Spitzer's theorem for the windings of two dimensional Brownian motion (Zt)t>0 : 4 (log*)

f* ds 2

Jo

(iaw^T 2

|Za| '->oo

where T denotes a stable ( | ) variable. More generally, I thought some link might exist between the one-dimensional diffusions in random environment (: RED) and some two-dimensional diffusions; a careful analysis of the scale and speed changes involved in the construction of these RED processes, together with a skew-product representation for a pair of independent Bessel processes led the authors of [9] to the recovery of some of Professor Tanaka's results via the asymptotics of local time functionals. A synthetic discussion of Sinai's walk via stochastic calculus is presented by Z. Shi

([10]). To conclude, let me convey the admiration of a number of collegues in France for the remarkable homogeneity of the works of Professor Tanaka which led him from the asymptotics of one-dimensional diffusions to those of the RED processes.

References [1] Ray, D. : Sojourn times of a diffusion process. Illinois J. Math. 7 (1963), 615-630. [2] Knight, F. : Random walks and a sojourn density process of Brownian motion. Trans. Amer. Math. Soc. 107 (1963), 56-86. [3] McGill, P. : Markov properties of diffusion local time : a martingale approach. Adv. Appl. Prob. 14 (1982), 789-810. [4] Jeulin, T. : Ray-Knight's theorems on Brownian local times and Tanaka formula. Seminar on Stoch. Proc. 1983, Birkhauser (1984), 131-142. [5] Rogers, L.C.G. and J. Walsh : The intrinsic local time sheet of Brownian motion. Prob. Th. Rel. Fileds. 88 (1991), 363-379. [6] Rogers, L.C.G. and J. Walsh : Local time and stochastic area integrals. Ann. Prob. 19 (1991), 457-482. [7] Yamada, T. : Principal values of Brownian local times and their related topics. In : N. Ikeda, S. Watanabe, M. Fukushima, H. Kunita (eds). Ito's stochastic calculus and probability theory. Springer (1996), 413-422. [8] Rosen, J. and M. Yor : Tanaka formulae and renormalization for triple intersections of Brownian motion in the plane. Ann. Prob. 19 (1991), 142159.

4

[9] Hu, Y., Z. Shi and M. Yor : Rates of convergence of diffusions in drifted Brownian potentials. Trans. Amer. Math. Soc. 351 (1999), 3915-3934. [10] Shi, Z. : Sinai's random walk via stochastic calculus. To appear in "Milieux Aleatoires", Collection : Panorama et Synthese (Soc. Math. France).

5

Contributions and Influences of Professor Tanaka in Stochastic Analysis by Shinzo Watanabe

The aim of this note is to express my hearty admiration and respect to Professor Hiroshi Tanaka for his works in probability theory which have influenced much and are playing important roles in its modern developments. As we can see by his great contributions collected in this volume, his works have been widely ranged in the field of stochastic analysis with two high peaks around main themes; stochastic models associated with nonlinear equations, particularly Boltzmann equations, and diffusion models in random environment. For me it is difficult and also, I think I am not well qualified, to make comments on his works in these main themes. I would rather give some historical account, combined with my own experience, of his contributions and influences in the area of stochastic analysis and the theory of stochastic differential equations (SDE's) in general. I should say, however, that motivations of Professor Tanaka must have been always to challenge new and unsolved problems; developing or organizing general theories may not have been his major concern. Around 1960, when I started studying the theory of stochastic processes as a graduate student at Kyoto University, Professor Tanaka was a Research Associate at Kyushu University and was just making his debut as a young researcher in this area. At that time, people were interested in and studied extensively the theory of Markov processes, as was initiated and developed by Kolmogorov and Feller, among others. For example, my teacher, Professor K. Ito had been working jointly with H. P. McKean on sample paths description and construction of onedimensional diffusion processes which attracted much attention of a group of young probabilists in Japan. Certainly, Ito-McKean's theory is a really important and beautiful achievement in probability theory. On the other hand, we know now well that the essential part of Ito's theory of SDE's had been established by that time and his AMS Memoir on the SDE theory was already published in 1951. However, students of Ito at that time could have had few occasions of learning the SDE theory directly from him at Kyoto University. Rather, I came to know Ito's theory through the guidance of several members in the group of Japanese probabilists like N. Ikeda, M, Nisio and Tanaka. Tanaka's formula for the local time of one-dimensional Brownian motion was my first exciting experience of usefulness of Ito's stochastic calculus. As a graduate student, I read, in a seminar of Professor Ito, Trotter's paper [T] which established for the first time the existence of Brownian local time. Also, he kindly

6

showed me a part of the manuscript which he was jointly preparing with McKean (this now appeared as famous Ito-McKean's book [IM]) in which was given a proof of Trotter's theorem. Both Trotter's paper and Ito-McKean's manuscript were very difficult for me. However, Brownian local time plays a key role in the Ito-McKean construction of sample functions for one-dimensional diffusion processes so that it was necessary to have some deep understanding of Trotter's theorem. A few years later, when I was still a graduate student, Professor ltd was abroad and Professor Ikeda was our adviser and sometimes our joint collaborator, Ikeda got a letter from Tanaka who was then visiting McKean at MIT, and Ikeda informed us that Tanaka found a nice proof of Trotter's theorem by applying the notion of Ito's stochastic integrals. Of course, we could not completely understand what exactly it was judging from the little information. On the other hand, Ikeda and I knew and read a paper by Skorohod [S] for reflecting Brownian motion on a one-dimensional interval in which local times at boundary points play an important role so that we could immediately have a vague idea what it might be. Later, McKean's book [M] appeared and I could then finally and completely understand the splendid idea of Tanaka. Skorohod's paper regarded Tanaka's formula as a stochastic equation and established its uniqueness of solutions. This is the origin of Skorohod equations to which, we know, Tanaka has also made an essential contribution ([24] in this volume). Now Tanaka's formula is an indispensable content in every standard text book of stochastic analysis. It is now defined most generally for any continuous semimartingale. I have to confess that, at first, I could not well see the importance of such a generalization. Gradually, however, through the work of Le Gall [L] who applied generalized Tanaka's formula to the pathwise uniqueness problem for solutions of SDE's and also through more recent works in [BEKSY] and [EY] which discuss Tsirelson's discovery that the natural filtration of Walsh's Brownian motion on n > 3 rays cannot be a Brownian filtration, for example, I came to recognize that such a generalization (as treated nicely in the text books by RevuzYor [RY] or Rogers-Williams [RW], among others) is really important and useful. Tanaka's formula may be considered an extension of Ito's formula so that it may be reasonably called ltd-Tanaka's formula, as well. An important and interesting aspect of this formula is that it produces a new process, namely, the local time: If we consider a similar extension of Ito's formula, we may obtain another interesting new process. An important idea to extend Ito's formula along the line of Tanaka's formula was given by M. Fukushima in his theory of Dirichlet forms and symmetric Markov processes associated with them. He introduced a class of stochastic processes with zero energy and extended Ito's formula by using this notion. Important examples of new processes thereby obtained contain the Cauchy principal value of the Brownian local times introduced and studied by T. Yamada and M. Yor, among others (cf. [Ya], [Yo]). As I remarked above, Ito's theory on SDE's has already been established around 1960 but the only books available to study this theory at that time were

7 those by Doob [D] and Ito [I 1], [I 2]. These are most valuable books which still keep high values at present. Fortunately in Japan, there was still another valuable research monograph written by Tanaka jointly with M. Hasegawa ([TH]) in 1964. It was one volume (Vol.19) in the series of research monographs written in Japanese, printed mimeographically and published by Seminar on Probability, an organization formed voluntarily by younger members of probabilists in Japan. It was a time when the communications and circulations were much less convenient than the present and these research monographs, some of which contained top class results at the world research level, have not been known to the world outside Japan. These promoted so much the progress of the probability theory in Japan, particularly, they helped those who just began to study in this field. For me, it was really lucky that I came across this monograph by Tanaka right after I had learned, from books by Doob and Ito, the basic elements of Ito's theory such as stochastic integrals, Ito's formula, SDE's with Lipschitz continuous coefficients and diffusion processes associated with them, and so on. It taught me, among others, the importance of studying separately the uniqueness problem and the existence problem for solutions of SDE's. In this monograph, Tanaka discussed SDE's with continuous coefficients; the existence of solutions was established by following Skorohod's method; for the uniqueness problem, he only wrote that it is an interesting open problem whether the solutions are unique or whether there exist solutions which define a Markov process. However, he could immediately solve ([8] in this volume) the question concerning the existence of Markovian solutions in the nondegenerate case. This Tanaka's result was further improved by D. W. Stroock and S. R. S. Varadhan who established the law uniqueness of solutions in the nondegenerate case, and by N. V. Krylov who obtained a Markov selection theorem for solutions in the general case (cf. [SV] for these results). As far as the theory of Markov processes is concerned, what is necessary is only the existence of weak solutions and their uniqueness in law which is also equivalent to the existence of solutions and the well-posedness of martingale problems. However, in the study of stochastic flows generated by SDE's and in the innovation problems (the problems concerning filtrations and noises) associated with solutions, for example, the notion of strong solutions and relevant notion of pathwise uniqueness are also important. The terms like strong solutions, weak solutions and pathwise uniqueness did not appear explicitly in this monograph but I think that Tanaka might have been the first in Japan who recognized the importance of these notions. Actually, Tanaka is the first to give a simple and most understandable example (known as Tanaka's example, cf. e.g., [IW], [RW]) of SDE which enjoys uniqueness in law but no pathwise unique solutions. In this monograph, the pathwise uniqueness of one-dimensional SDE's with Holder continuous coefficients of order a > 1/2 was obtained. Later, when I was working on continuous state branching processes, I needed some uniqueness result for SDE's with Holder continuous coefficients of order a = 1/2 and I suddenly remembered the proof of Tanaka in this monograph. By modifying his proof, I could manage

8

to obtain the necessary result. The proof of pathwise uniqueness I gave was still not general enough but Yamada found a nice improvement of Tanaka's proof so that an almost best possible condition for the pathwise uniqueness was obtained. However, it was not obvious for us that the pathwise uniqueness implies the law uniqueness so that it can define a strong Markov process, and we could not find any existing literature concerning this problem. Such were our main motivations of joint work [YW] and we owe much to Tanaka in this joint work. Among many important contributions and influences of Tanaka in connection with SDE's, I would choose one more topic. This is concerned with his SDE based on a Poisson point process, which he introduced to describe a stochastic model associated with Boltzmann's equation of Maxwellian gas without cutoff ([16] in this volume). The notion of Poisson random measures (Poisson point processes) has been established by Ito, among others, in the study of the Levy-Ito theorem for the paths structure of additive processes and a general theory of SDE's based on them has been discussed by Ito, Gihman and Skorohod, among others. It seems to me, however, that there had not been any significant application of the theory before Tanaka: I believe that Tanaka's equation is the first truly significant example of SDE's based on Poisson point processes. I remember well when Tanaka visited us and gave a series of lectures on this topic at Kyoto University around 1970. I was really fascinated by his splendid way of defining and applying a SDE based on a cleverly chosen Poisson point process. Later, I discussed with S. Takanobu on some application of SDE's based on Poisson point processes of Brownian excursions to solve SDE's for diffusion processes in a domain satisfying the most general Wentzell's boundary conditions. The idea was motivated by the work of Tanaka and I remember I was always wishing to imitate Tanaka's method, even in a small way, in this problem.

References [BEKSY]

[D] [EY]

[IW]

M. T. Barlow, M. Emery, F. B. Knight, S. Song et M. Yor, Autour d'un theoreme de Tsirelson sur des nitrations browniennes et non browniennes, Sem. Prob. XXXII, Lecture Notes in Math., 1686, Springer, Berlin (1998), 264-305 J. L. Doob, Stochastic Processes, John Wiley & Sons, New York, 1953 M. Emery et M. Yor, Sur un theoreme de Tsirelson relatif a des mouvements browniens correles et a la nullite de certains temps locaux, Sem. Prob. XXXII, Lecture Notes in Math., 1686, Springer, Berlin (1998), 306-312 N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second Edition, North-Holland/Kodansha, Amsterdam/Tokyo, 1988

9 [I 1] [I 2] [IM] [L]

[M] [RY] [RW]

[S] [SV] [TH] [T] [Ya]

[YW]

[Yo]

K. Ito, Kakuritsuron (Probability Theory), in Japanese, IwanamiShoten, Tokyo, 1955 K. Ito, Lectures on Stochastic Processes, Tata Institute of Fundamental Research, Bombay, 1961 K. Ito and H. P. McKean, Diffusion Processes and their Sample Paths, Springer, Berlin, 1965, 1974-Printing in Classics in Mathematics, 1996 J. F. Le Gall, Applications du temps local aux equations differentielles stochastiques unidimensionnelles, Sem. Prob. XVII, Lecture Notes in Math., 986, Springer, Berlin (1983), 15-31 H. P. McKean, Stochastic Integrals, Academic Press, New York,1969 D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Third Edition, Springer, Berlin, 1999 L.C.G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, Vol. 2: Ito's Calculus, John Wiley & Sons, New York, 1987 A. V. Skorohod, Stochastic equations for diffusion processes in a bounded region, Theor. Prob. Appl., 6 (1961), 264-274 D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, Berlin, 1979 H. Tanaka and M. Hasegawa, Kakuritsu Bibun Hoteishiki, (Stochastic Differential Equations), in Japanese, Seminar on Prob. 17, 1964 H. F. Trotter, A properties of Brownian motion paths, Illinois J. Math. 2 (1958), 425-433 T. Yamada, Principal values of Brownian local times and their related topics, in Ito's Stochastic Calculus and Probability Theory, ed. N. Ikeda et al, Springer, Tokyo (1996), 413-422 T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, Jour. Math. Kyoto Univ. 11 (1971), 155-167 M. Yor, Some Aspects of Brownian Motion, Part II, Some Recent Martingale Problems, Lectures in Math. ETH Zurich, Birkhauser, Basel, 1997

10

Some Comments on My Mathematical Works in Retrospect

by Hiroshi Tanaka

In this short note I would like to give some comments on my works on probability theory, first by mentioning briefly the initial stage of my mathematical development and then by concentrating on those works which are related mainly to Boltzmann's equation and diffusion processes in random environments. When I was a graduate student at Kyushu University (1955-1958), the theory of Markov processes with related potential theory went through much development. As a student of Professor Gisiro Maruyama I began to study probability theory toward the above field by reading papers on Markov processes by A.N. Kolmogorov, W. Feller, J.L. Doob, K. Ito and E.B. Dynkin. I also learned many things from N. Ikeda, T. Watanabe and T. Ueno, who also began to work in this field and were taking the initiative in research activities of young probabilists in Japan, in particular, in the Probability and Statistics Summer Seminar, the April Probability Symposium and the publication of Seminar on Probability (in Japanese). I was encouraged by Ikeda through frequent conversation; I owed much to Watanabe for the problem of the paper [1] t ; the motivation of the paper [4] was to realize Ueno's idea on invariant measures of recurrent Markov processes ([Ul]). In 1957-1958 H.P. McKean visited Kyoto University, where K. Ito was at the time. He gave interesting lectures there, sometimes in other places as well, and inspired young probabilists in Japan. Ito and McKean were developing their theory on one-dimensional diffusion processes which is now found in the book [IM]. I learned some of the essential points directly by listening to their lectures and also through Ito's seminar which I sometimes attended. By analogy with their results in one-dimension Ito and McKean introduced a class of multi-dimensional diffusion processes X{t) admitting the same system of harmonic measures as Brownian motion B(t) and anticipated the construction of such X(t) by means of time-change from B(t). I carried out the actual construction but under some condition on mean exit times of X(t) from bounded domains (I arrived at additive functionals of Brownian motion as a multi-dimensional substitute, though less explicit, of one-dimensional local time integrals due to Ito-McKean). McKean also obtained almost the same result. The final result was published as the ^Numbers in brackets refer to the Bibliography of Hiroshi Tanaka (see pages 425-428)

11

joint paper [5] which was written up while I was visiting MIT (1960-1962) at the invitation of McKean. I was much influenced by him and afterward became interested in Boltzmann's equation through his paper [M]. After writing the paper [8] on the existence of diffusion processes with continuous coefficients, I was interested in the approach from stochastic differential equations (SDEs). The uniqueness result which was lacking in my paper [8] was given later by D.W. Stroock and S.R.S. Varadhan [SV] in a clear way by introducing martingale problems. I had some misunderstanding on (or expected too much from) the pathwise uniqueness in SDEs and thus obtained no results, but from the works of A.V. Skorohod (Chapter 3 of the book [Sk]) and I.V. Girsanov ([G]) I was able to acquire some knowledge on strong and weak solutions of SDEs and often used SDEs in my later works. In the research of probability theory in Japan around the mid 1960s there was a trend of taking up some nonlinear problems from the view-point of probability theory; in particular, N. Ikeda, M. Nagasawa and S. Watanabe were discussing nonlinear problems in connection with branching processes in a series of interesting papers (e.g. [INW]). I myself attempted to treat quasilinear parabolic equations making use of SDEs ([10]) but it had no connection with concrete problems. In 1966 McKean ([M]) introduced a class of Markov processes (with nonconstant transition mechanism) that can be associated with certain nonlinear parabolic equations such as Burgers' equation and Boltzmann's equation. I soon became interested in [M]. Probably the willingness to study Boltzmann's equation had also come from my experience of reading R. Kubo's book "Statistical Mechanics" (published in 1952, Japanese) when I was a sophomore at Kyushu University. By reading [M] carefully and comparing it with Kac's earlier paper [K], I could understand its main part: The spatially homogeneous Boltzmann's equation (nonlinear) can be derived from the master equation (linear) through the propagation of chaos and this idea is applicable to certain nonlinear parabolic equations. But the rigorous verification had been carried out only in few cases such as Boltzmann's equation of Maxwellian and cutoff type by using an analytic method. I wanted to discuss the problem by a more probabilistic method. It was easy to write down the SDE (based on a Poisson random measure) describing the Markov process associated with spatially homogeneous Boltzmann's equation but the actual proof of the existence and uniqueness of solutions was usually hard and interesting particularly when the total scattering cross section (t.s.c.s. for short) is infinite. The papers [16] and [22] treated the case of Maxwellian and non-cutoff type. The idea behind was in an extension of the previous work[15] on Kac's one-dimensional model of Maxwellian molecules in which a sort of mean square deviation from the Gaussian distribution was effectively used. In solving the SDE there was a difficulty due to the infiniteness of t.s.c.s. but the Maxwellian type provided a special method, so to say, an iteration method in the sense of law in solving the SDE (this was not applicable to Boltzmann's equation in general). Since the argument of [22] seemed technically complicated, I tried to simplify

12

it with some improvement on the SDE in [38]. Let (QuPi) be a copy of the basic probability space (£l,P) on which a solution X(t,co) is to be constructed. Then the SDE (written in the form of an integral equation) was improved t o X{t,u) = X(0,u) + /

6(X(s-)w),X(s-,a;1),

X)N(dsdXdio1),

J{0,t]xAxQi

where TV(-) is a Poisson random measure on (0, co) x A x ^ with intensity measure ds{i(d\)Pi(du)i)y (A, fi) being a suitable measure space. The presence of the copy X(s—,LJI) of an unknown X(s—,ui) matches the nonlinearity of the problem. I thought that there was a gap between the uniqueness of solutions to the SDE of [16] and the uniqueness of solutions to the Boltzmann equation itself, so I wrote the paper [19] to fill this gap but later was able to obtain a better result, namely the uniqueness of probab'ility measure solutions of the Boltzmann equation itself (unpublished) by making use of Varadhan's result ([V]) concerning the uniqueness of probability measure solutions of certain parabolic equations. When writing the paper [6] with K. Sato, in 1962, which was motivated by Ueno's idea on Markov processes on boundaries ([U2]), I became aware that a reflecting Brownian motion X(t) for a multi-dimensional domain can be expressed as B(t) + $(£) where B(t) is a standard Brownian motion and $(£) is an integral with respect to the local time on the boundary for X(t). However, I could not regard this expression as an equation in which X(t) and <£(£) are considered as unknown, because I did not anticipate the uniqueness of solutions at that time. Later I noticed that the uniqueness can be obtained easily from the convexity assumption on the domain, and so I could formulate Skorohod's problem for convex domains ([24]). The convexity assumption was soon removed by P.L. Lions and A.S. Sznitman ([LS]). Thanks to their result 1 was able to construct the motion of a finite number of mutually reflecting Brownian balls as a solution to Skorohod's problem in [34] (the infinite case was discussed later by H. Tanemura In 1984 I stayed in Zurich for three months at the invitation of H. Follmer and Nagasawa. I attended the probability seminar organized jointly by Follmer (ETH Zurich) and Nagasawa (the University of Zurich), which was fresh and active, with many young probabilists. This visit gave much influence to my later works. A series of joint works with Nagasawa ([31] [33] [39] [42]) began around this time; there we discussed some types of many particle systems with mean field interaction which arose from Nagasawa's idea on the Schrodinger equation. At that time T. Brox, a postdoctral fellow at the University of Zurich, was working on a diffusion analogue (appeared in [B]) of Ya.G. Sinai's random walk in a random environment ([S]). I came to know the results of Sinai and Brox for the first time (as I knew later there is also S. Schumacher's work [Sch] on a similar subject) and was much impressed by their works since it seemed to me that the results gave a new type of limit theorems involving a scaling quite different from

13 those I had ever encountered. I thus became interested in stochastic processes in random environments. A diffusion process in a Brownian environment or Brox's diffusion for simplicity is a continuous time version of Sinai's random walk in a random environment and described by the (symbolic) SDE

dX{t) = dB(t) - \w'(X{t))dt,

X(0) = 0,

where B(t), t > 0, is a one-dimensional Brownian motion independent of a Brownian environment w(x), x € R. For a precise definition, e.g. see [B] or [41]. Brox's diffusion is sometimes called a Brownian motion in a white noise environment. The arguments of Brox were based on many of Sinai's ideas ([S]), among which the notion of a valley — a depression in English translation ([S]) — played an essential role together with the self-similarity of the random environment. Reading Brox's preprints I was able to obtain a better understanding of Sinai's result. Since I preferred to work with Brownian motion rather than with random walks, I considered most of the problems in the framework of diffusion processes. Most of the results obtained for diffusions in random environments were preceded by the corresponding ones for random walks; however, working with diffusion processes was more fun for me; it often seemed to clarify methods and results since one could use a lot of fine properties of Brownian motion. I give some comments on my works concerning the following problems. (A) Some refinements of the results of [S][B][Sch] (localization). (B) A generalization of the problem (A) to a wider class of random environments. (C) A Brownian motion with drift in a white noise environment. (D) A multi-dimensional problem. (A): When I heard from Brox that (\ogt)~2X(t) — b(t) tends to 0 in probability as t —> oo (where &(£), for each fixed t, is defined as the bottom of a suitable valley and admits a probability distribution independent of t), I got interested in the limit theorem of the next stage: Can one obtain a non-degenerate limit distribution by a suitable rescaling of (\ogt)~2X(t) — b(t) ? I could not immediately anticipate that the multiplication by (logt)2 simply is enough to have a limit distribution. The paper [41] discusses this problem. The result is that X(t) — (log t)2b{t) admits a non-degenerate limit distribution as t —v oo which can be expressed in terms of independent Bessel processes /3+ and /?_ appearing in (ii) below. This result seemed interesting particularly when compared with the fact that an individual diffusion process with frozen environment w is null-recurrent (a.s.). I must remark that-there is also an earlier work [Go] by A.O. Golosov on the same subject for random walks. Following [Go] and also for terminological convenience I call a result of this type localization (in some occasions the term localization is used in a wider sense ). The key points of the argument in [41] were (i) to refine Brox's method and (ii) to describe the local behavior (law) of

14

the environment around the bottom 6(e) in terms of two independent Bessel processes (3+ and p- of index 3. This part, (ii), was most interesting for me, in which I used the construction of a reflecting Brownian motion on [0,oo) by means of the associated Poisson point process of excursions ([I][IW:pp.123-125]) and then a certain reversal of time. Theorems of D. Williams [W] and J.W. Pitman [P] were also used. The same problem as [41] was again discussed in [52], in which the part (i) was replaced by a use of a theorem of Y. Ogura ([0], see also p.756 of [52]) and consequently the whole story of the proof became transparent. (B): In [47] we proved that the localization of the type discussed in [41] holds for a certain class of random environments that are asymptotically strictly stable. But after having written [52] I began to anticipate that the localization will hold if we take for the random environment {w(x),x £ R} any Levy process satisfying the following condition: There exists a (0 < a < 2) such that every positive sequence {A^} tending to oo contains a subsequence {AJ.} along which {\~1w(\ax),x £ E} converges in law to some nice non-degenerate process that may depend on {X'k}. So I wanted to find a good class of Levy random environments for which the localization of the type discussed in [41] holds. For this purpose I was often led to the study of some properties of Levy processes themselves and this was also fun for me. For instance, I studied superharmonic transforms of absorbing Levy processes in the half line (0,oo) in order to grasp the asymptotic law of a certain part of the environment around some bottom ( a summary in Japanese is in Cooperative Research Report 51, The Institute of Statist. Math.; what I had for the proof of (ii) of Theorem 7 therein is incomplete). (C): The limiting behavior of the process X(t) described by

dX(t) = dB(t) + ^dt -

~w'{X(t))dt

can not be treated by the method of Sinai-Brox-Schumacher. In 1986 when I was at Minnesota, B. Toth pointed out to me that this would correspond to the random walk model discussed in an earlier work [KKS] by H. Kesten M.V. Kozlov and F. Spitzer. However, an adaptation of their method to our case seemed very hard to me. On the other hand, it was already recognized that Krein's spectral theory of strings (e.g. see [KK]) provides a powerful method in the study of various problems on one-dimensional diffusion processes (in non-random environments); for example, see S. Kotani-S. Watanabe [KW] and the references therein. In 1989 K. Kawazu and I had a chance, at a small symposium, to discuss with Kotani on the problem of limiting behavior of the above X(t) from the view point of the applicability of Krein's theory and learned the method of Laplace transform of a hitting time, i.e., Kotani's formula (unpublished, see [56]), which he had obtained previously in finding the limit distribution of the maximum of the Brox-Schumacher diffusion. We thus became interested in using Kotani's formula and Krein's theory for our problem and were able to obtain the first result [56].

15

But it gave only a partial answer; namely, the case 1 < K < 2 was not discussed, in which the limit distribution was expected to have the whole real line for the support. So I was misled, for a good while, to a false conclusion that the method of Laplace transform (i.e., Kotani's formula) would not work. However, luckily it worked as was discussed later in the second paper [57] where, in addition to [KK], I also used Krein's more specific result [Kr] (see also [DM:p.266] [KW:p.239]). The same result was obtained also by Y. Hu, Z. Shi and M. Yor ([HSY]) by a method quite different from ours. Finally I remark that an environment-wise central limit theorem was proved in the case K > 2 (see [55]); the proof needed a series of detailed computations in spite of the easy-looking problem. I do not know whether a similar result holds for K = 2. (The right hand side of the equality in the 13th line of p.1809 of [57] should be (27?rr1/2j!ooexp{-2-V}^. I remark also that the formula at the top of p.201 of [36] is not correct; it was corrected in [35].) (D): P. Mathieu ([Ma]) considered a diffusion process in a multi-dimensional Brownian environment. The paper [51] proved its recurrent property. The process X(t) corresponds to one of random walk models in R. Durrett [D], but is not a diffusion analogue of the random walk of J. Bricmont and A. Kupiainen [BK], a multi-dimensional random walk corresponding to Sinai's one. From [Ma] and [D] we see that the process X(i) has the localization property of the type discussed in [S] but the existence problem of a limit distribution of (logi)~ 2 X(t) seemed unsettled. In this connection I studied some properties of sample functions of a Levy's Brownian motion with a multi-dimensional time (e.g. [61]).

References [BK] [B] [D]

[DM] [G] [Go]

J. Bricmont and A. Kupiainen, Random walks in asymmetric random environments, Comm. Math. Phys. 142 (1991), 345-420. T. Brox, A one-dimensional diffusion process in Wiener medium, Ann. Probab. 14 (1986), 1206-1218. R. Durrett, Multidimensional random walks in random environments with subclassical limiting behavior, Comm. Math. Phys. 104 (1986), 87102. H. Dym and H.P. McKean, Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Academic Press, 1976. I.V. Girsanov, An example of nonuniqueness for solutions of K. Ito's stochastic integral equations, Theor. Probab. Appl. 7 (1962), 336-342. A.O. Golosov, Localization of random walks in one-dimensional random environments, Comm. Math. Phys. 92 (1984), 491-506.

16

[HSY]

Y. Hu, Z. Shi and M. Yor, Rate of Convergence of diffusions with drifted Brownian potentials, Trans. Amer. Math. Soc. 351 (1999), 3915-3934. [I] K. Ito, Poisson point processes attached to Markov processes, Proc. 6th Berkeley Symp. Math. Statist. Probab. I l l , 1970, 225-239. [IM] K. Ito and H.P. McKean, Diffusion Processes and Their Sample Paths, Springer, 1965. [INW] N. Ikeda, M. Nagasawa and S. Watanabe, Branching Markov processes I, II, III, J. Math. Kyoto Univ. 8 (1968), 233-278, 365-410; 9 (1969), 95-160. [IW] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, 1981. [K] M. Kac, Foundations of kinetic theory, Proc. 3rd Berkeley Symp. Math. Statist. Probab., 3, 1955, 171-197. [Kr] M.G. Krein, On some cases of the effective determination of the density of a non-homogeneous string from its spectral function, Dokl. Akad. Nauk SSSR 93 (1953), 617-620. [KK] I.S. Kac and M.G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl. (2), 103 (1974), 19-102. [KKS] H. Kesten, V. Kozlov and F. Spitzer, A limit law for random walk in a random environment, Compositio Math. 30 (1975), 145-168. [KW] S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, 235-259 in Lecture Notes in Math. 923, Springer, 1982. [LS] P.L. Lions and A.S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. 37 (1984), 511537. [Ma] P. Mathieu, Zero white noise limit through Dirichlet forms, Application to diffusions in a random medium, Probab. Theory Related Fields 99 (1994), pp. 549-580. [M] H.P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1907-1911. [O] Y. Ogura, One-dimensional bigeneralized diffusion processes, J. Math. Soc. Japan 41 (1989), 213-242. [P] J.W. Pitman, One-dimensional Brownian motion and three-dimensional Bessel process, Adv. Appl. Probab. 7 (1975), 511-526. [Sch] S. Schumacher, Diffusions with random coefficients, pp.351-356 in: Particle Systems, Random Media and Large Deviations, R. Durrett, ed., Contemporary Math. 4 1 , 1985. [S] Y.G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl. 27 (1982), 256-268. [Sk] A.V. Skorokhod, Studies in the Theory of Random Processes, AddisonWesley, 1965.

17

[SV] [T] [Ul] [U2]

[V]

[Wi]

D.W. Stroock and S.R.S. Varadhan, Diffusion processes with continuous coefficients, I, If, Comm. Pure Appl. Math. 22 (1969), 345-400, 479-530. H. Tanemura, A system of infinitely many mutually reflecting Brownian balls in Rd, Probab. Th. Rel. Fields 104 (1996), 399-426. T. Ueno, On recurrent Markov processes, Kodai Math. Sem. Rep. 12 (1960), 109-142. T. Ueno, The diffusion satisfying Wentzell's boundary condition and the Markov process on the boundary I, II, Proc. Japan Acad. 36 (1960), 533-538, 625-629. S.R.S. Varadhan, Invariant measures of Markov processes, Lecture at Kyoto Univ., 1980; A brief summary of it taken and written up by S. Kotani in Japanese is in Sugaku Vol.33, No.3, 1981, p.276. D. Williams, Path decomposition and continuity of local time for onedimensional diffusions, I, Proc. London Math. Soc. (3) 28 (1974), 738768.

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19 M E M O I R S O F T H E C O L L E G E O F SCIENCE, U N I V E R S I T Y O F K Y O T O , S E R I E S A

Vol. XXXIII, Mathematics No. 3, 1961.

Additive functionals of the Brownian path By H. P.

MCKEAN, JR.

and Hiroshi

TANAKA

(Communicated by Prof. K. Ito, March 14, 1961)

1. INTRODUCTION. A functional f(0 = fttw) of / ( > 0 ) and a several-dimensional Brownian path w: t^x(t) is said to be additive if 1.1 1. 2

\{ft w) depends upon t and x(s): s<^t alone. 0 - f(0) < f < + co

1.3 1.4

f(*±) = f(0 f(0 = f(*) + f(/-s,K\+)

where wt x s

t>s,

is the shifted path wt : t-> x(t -f s) ; for example,

f ( 0 = \ f( ( ))ds

is an additive functional if / > 0 is bounded and

Borel. K. Ito and H. P. McKean, Jr. [13] proved that in the 1-dimensional case such an additive functional is an integral 1.5

f(0 = \i{t,b)e{db)

of the standard Brownian local times 1. 6

t(f, a) = i j m m e a s u r e (s.\a

a<x(s)<jbys
with respect to a non-negative measure e, finite on bounded intervals. Brownian local times are not available in rf>2 dimensions, but f can still be expressed as a (formal) Hellinger integral 17 1 "

f/n _ [ measure (s: x(s) €db, s
K ;

with a non-negative measure e which is smooth in the sense that

20

480

H. P. McKean, Jr. and H. Tanaka

each bounded open D is the union of an increasing series of sets Bn{n~>l)t closed in D, such that 1.8a

the charge distributions e\Bn have bounded potentials \ Gde

< # ( w > l ) , where G is the Green function of D; 1. 8b for large n, depending upon the path, the Brownian particle lies in Bn until it leaves D, i.e., P.ix(t)eB„, * < m i n ( s : x(s)£D)y n\ + « , ] = 1, where P*(B) is the probability of the event B as a function of the starting point of the Brownian motion. The correspondence embodied in 1. 7 between the class of smooth measures e and the class of additive functionals f is one to one and onto. 1.8a implies that e(B) — 0 unless B has positive logarithmic capacity in the 2-dimensional case or has positive Newtonian capacity in the 3 dimensional case; thus, a smooth measure cannot attach positive mass to a line in 3 dimensions, nor, in 4 dimensions, to a surface. But it can be singular relative to Lebesgue measure; the simplest example in 3 dimensions is the uniform distribution on the spherical surface | a | = l . Choose d— 3, e(db)=\b\adb, and let G be the Green function of D: | A | < 1 ; then 1. 9a

p=\

Gde

< c o n s t a n t (2+a)- 1 < +oo except at 0 = + oo 1. 9b

@(
tf>-2 - 3 < a < a < —3 ,

-2

a > -5/2 ,

and it follows that e is smooth for ct^>— 2. e is not smooth for a < — 2 ; in fact, choosing Bn\ D as needed for 1.8, 1.10

E0[\SJx(s)\«ds, < T Jo

<[

ds

\

x(t) e Bn,

t<s\

Plx{s)edb,m?K\x{f)\
G(0,b)de<:+vot

'£s

21

Additive functionate of the Brownian path

481

thanks to 1. 8a, and 1.11

lim lim Plx(t) € £ „ , * < £ ] - 1

thanks to 1. 8b, while, as is not hard to prove, 1.12

P0[jj*(s)|* o j = l ,

contradicting 1.10. V. A. Volkonskii [15, 16] also studied additive functionals, establishing a special case of the above for a wider class of motions; the method used below is similar to his. Given a 1-dimensional diffusion with the same hitting probabilities as the standard Brownian motion: 1.13

P £ [min (t: x(t) = « ) < min (t: x(t) = &)]

= £=£

a<^
o—a W. Feller [8] explained how to express the associated generator @ as a differential opearator based upon a speed measure e, positive on open intervals: 1.14

®u

=

gW e(da)

u-ia) = lim »+«

+

=

limu b a

*

(b)-u+(a) e(a, b)

and K. ltd and H. P. McKean, Jr. [13] found that its sample paths could be expressed as standard Brownian sample paths run with the stochastic clock f"1 which is the inverse function of the additive functional (local time integral) f = 11 de associated with the speed measure. V. A. Volkonskii [15] also studied such time substitutions; his method is less explicit because it does not use local times but has the advantage that it can be applied in higher dimensions. As will be explained below, a i > 2 dimensional diffusion with Brownian hitting probabilities has as its generator the closure of a differential operator

22

482

H. P. McKeant Jr. and H. Tanaka

1.15

© a = - « e(db)

based upon a (smooth) speed measure e, positive on the neighborhoods of H. Cartan's fine topology [2] ; moreover, the associated motion is the standard Brownian motion run with the inverse function f~l of the additive functional f associated with e, and this correspondence between the class of diffusions with Brownian hitting probabilities and the class of smooth measures e positive on fine neighborhoods is one to one and onto. 2.

BROWNIAN MOTION.

Choose c?>2, let Ed = Rd it d=2, let it be the one-point compactification Rd+ov if c?>3, introduce the space of continuous sample paths w: [0, +co)->E r f with 2.1

w(t)£Rd

t^m^

where mM = mM(tt;)
Brownian motion with generator ($ =

2.2

(w: m
t>0;

for example, the passage time mQ=inf(t:

x(t)eQ)

to a closed or

1 QJ/2 is often used as the generator of the Brownian motion, but for our purpose it is simpler to omit the factor 1/2.

23

Additive functionals of the Brownian path

483

open ^-dimensional figure Q is a Markov time and so is m ^ m\n(t:Vf(x(s))ds=l\

if 0 < / < l is a Borel function.

Given such a Markov time m, if wit is the shifted path 2.3

w+:

t-*x(t+m)

and if Bm+ is the field of events BeB 2.4

Bf\(w:

m
such that

/>0,

then the Brownian particle starts afresh at time t=m, i.e., 2.5

aeEd,

P J > A € B | « n + ] = P*(B)

B£B,

b = x(m)2.

Blumenthal's 0 1 law [ 1 ] : 2. 6

P.(B) = ?

BeB0+

-L

=

f\Bs 8>U

is a special case of 2. 5. A. R. Galmarino3 has pointed out that a non-negative Borel function m of the sample path is a Markov time if and only if 2. 7a 2. 7b

m(«) < f #(«, w) — x(st v)

s
imply m(«) = m(f) and that an event B€B is a member of Bm+ if and only if 2.7 coupled with u&B implies v€B. As a simple application of this test, note that Bm+ measures both in and the past x(0/\m) (0>O)4 because 2.7 implies 0/\m(u) = 0/\m(v) is the exit time min (t: x(t) G 3D), then the hitting probability 2. 8

hm{a> db) = Palx(mdD) € db~]

a € D,

dbeBD

is the classical harmonic measure of db as viewed from the point a, and, if GD{a, b) is the classical Green function of D, then E a [measure (t: x(t) e db, t < m & D ] = G^fl, &)<*& 5

2. 9 2 3 4

5

x(m)sH°° in case m = + <=<>; it is understood that FooC^CO^00* £ > 0 ] = 1 . private communication. « A 6 is the smaller of a and b.

£.(/) = j/#/>..

a,beD;

24

484

H. P. McKean, Jr. and H. Tanaka

for the proofs, see J. Doob [5] and G. Hunt [9], G. Hunt [10] has called a non-negative Borel function p excessive on D if 2.10 Ea[p(x(t))> t < m 9D ] t p{a) /JO, aeD. An excessive function can be split into its greatest harmonic minorant h plus the potential \GDde of a non-negative (Riesz) measure e> indeed, Hunt's excessive functions are the same as the superharmonic functions of F. Riesz [14]. J. Doob [5] proved that an excessive function is continuous on the Brownian path (t<^m%D) and that a potential tends to 0 along the Brownian path (t ] m9D). 3.

THE ASSOCIATED MEASURE OF AN ADDITIVE FUNC TIONAL.

Consider an additive functional f of the Brownian sample path as described in seition 1, interpreting 1.1 to mean 3.1

f(£, w) is measurable Bt for each t > 0 .

The purpose of this section is to associate with f a unique non-negative measure e such that} for each bounded open D(ZRd, 3.2

l-pA

= a\Gp«de

pa = E . [ < r ^ m 9 ^ ] , a > 0 ,

where G is the Green funetion of D and the integral is extended over D; it will follow from 3. 2 that e is smooth. Ccnsider, for this purpose, the additive functional 3.3

L(t)=[tAmDp.(x(s))Kds)

t>0,

Jo

and let us begin with the following simple lemmas: a) b)

1— pa is the potential of a non-negative measure ocea. £.[L(m9D)]<+oo.

c)

f „ t f as

d)

l~pa

e)

E.\mDf(x(s))\a(ds)] = \ Gfdea

a[0.

- a£.[f(m 3D )] = cc\^Gdea . if f > 0 is a Borel function.

25

Additive functionals of the Brownian path f)

Pa^aidb) = e{db) is independent of cc and of D.

g)

e is unique.

485

Because 3.3

E.(l-pa(x{t)l

t<mdD) ^ E.(l~e-a^m^-^\

/<m9D)

1— pa is excessive; it is, in fact, a potential thanks to 3. 4

f . h^(a, db)\l-pa-] = E^l-e'^^^™^)

JO

D f D,

and this completes the proof of a). As to b), pm is continuous on the Brownian path because of 1, and, since f is continuous, b) follows on letting n \ + oo in 3. 5

\-pa

> <xE.\ Ik =

E.[^De-^^-^Mdt)]

=

e-^m^wi-"^

2

Kz-^e-^1^

f(Ik) 1

l(k-l)2-»,k2-») fe2-"<m9D

Because of c), which is obvious, 3. 6

^ ( 1 - A . ) = lim£.rr%-*C«.D)-KO]f ( d 0 l , 5+0

LJo

J

and, using the method of 3.5 and Et[_^t(miDy\
« - ( l - A i ) = UmJS.[j e maD ^WO)f.(rfO] = ^ [ 1 ^ ^ ]

proving d). K. It6 (private communication) pointed out the following neat method for proving e). Choose closed BC^D such that ea(dB) = 0 and let e1 and e2 be the charge distributions of the potentials A = £.rf m 9 V#f»] and / » 2 - E J f m 8 D ( l - / y f ( i l in which / is the indicator function of B. Because p1 is harmonic outside B and differs from ct~\l —pt,) = E.\3a,(m9Dy\ by a harmonic function inside B, ex is

26

486

H. P. McKean, Jr. and H. Tanaka

not smaller than the restriction of ea to B, and, for the same reasons, e2 is not smaller than the restriction of ea to D—B. But A+A= a_1 (l—A»)> whence 3.8

D

Pl^E.[\™*

fdu]

=

\BGdea,

and since such figures B generate the class of Borel subsets of D, e) follows. As to f), pa^>0 because K™I>D) and> choosing 0<^/3<^<2, e) implies

3. 9

j Gdea = E. [l^pjprfJ

= j GpJPjie, ,

i.e.r de=p~^dea is indeqendent of cc; it is also independent of D because if D^>D and if pa=Q outside D, then, with an obvious notation, 3.10

\GpJ4e = E.\^Dpj\t'\

^{j>4]-^{c:>4] = J Gpa^di-\dh^D

\ Gpapnde

= \ Gfiappde . g) is immediate from 3.2. To establish the smoothness of e, take bounded open D and put B„ = Df\(p1'^n~1). Because 1—A is a potential, B„ is closed in D and increases to D as w f -h °° ; moreover, according to 3.2, 3.11

f Gde < n [ Gp.de = »(1 -pt) < n ,

which is 1.8a, and, since, along the Brownian path, 0
P.[inf(*: *(0 * * . ) < m * > ] = P.[ inf A(*(0) < w _ 1 ] 10

which verifies 1. 8b.

w f + oo ,

27

Additive functionals of the Brownian path 4.

487

UNIQUENESS.

The following simple lemma is useful in later sections: two additive functionals f2 and f2 with the same bounded mean 4.1

p = £ [ f ( n O ] = l i m 6 ~ \ l - p , ) = f G
f = f„ f,

are the same for *<m 9 D . An argument similar to 3. 5 implies

4.2

£.(j"aD[f,.(m8D)-f,(0]f*(
= JGAfc<||/>||L<+°°; thus, putting f = f 2 -fi, 4.3

£.[f(m 8D ) 2 ]

= 0, and hence 4.4

0 = E.£Km*p)\BtAmD+l = f (0 t < m9D

= £.[f(m 9 0 )-f(/)]B / A m a z , + ] + !(/)

as desired. It is a simple matter to deduce from this that two additive functionals with the same associated measure are the same; indeed, the difference p of two solutions of 3.2 satisfies —p~a\

Gpde,

which implies 4. 5

0 > - f p2de = <$(pde) = \ Gpdepde ,

and it follows that two additive functionals with the same associated measure have the same pat hence the same oc-y(l—pA) = E.\Ja(m9D)'] ( ^ c t - V + oo), and hence the same fa. But this means that the two functionals are the same up to time mdD, and, to finish the proof, it is enough to make D swell out to Rd.

28

488

H. P. McKean, Jr. and H. Tanaka

5. CONSTRUCTION OF AN ADDITIVE FUNCTIONAL FROM ITS ASSOCIATED MEASURE. Given a smooth non-negative measure ey our task is to find an additive functional f with e as its associated measure. Consider, for this purpose, a non-negative measure e on a bounded open figure D with bounded potential p = \ Gde and finite energy Gr(e) = 1 Gdede, let p„ be the potential \ Gf„db of 5.1 let

/ „ = n{p-E.\_p(x{n-')),

n^ < % ] ) ,

fn(t)=\tAmDfn(x(s))ds,

5.2

JO

and let us construct a functional f associated with e as a limit of f„ with the aid of the following simple lemmas: a) p is a Brownian excessive function; esp. 0 K^Wdoldt

increases to p inside D as n\

0 + CO,

d)

limGf(0— fndb) = 0, w/z^re @ is the energy <$(e)=\Gdede.

= A.(*(0) + f.(0 =

f »(™9Z>)

m az> >^>0 ™*D
i.e., l„ is a martingale with respect to the fields BtAmD+ \„ is continuous in t. f) P. [max | ln(t, wt) - IJf, wt) I >«?] < constant x £~2s-d/2V^(f„db -fjb) • g) P . [ lim f.ft wt) = \{t, wt), t > s > 0] = 1 .

; moreover,

ft f +DQ

where the limit is taken as n | 4- °° via suitable nx<^n2 £ > s ) . h) E.[f(m, D )]=A Because p is a potential, it is excessive ; a) is obvious from this, b) is obvious from the bound

29 Additive functionals of the Brownian path 5. 3

E.[p(x(t))7 t < m 9D ] < \\P\LR(t < tn9D) J 0

489

* t + <*>,

and c) follows from b ) ;

5.4

pn^E.[\^Dfn(x(s))ds] = n(

n~l < ;w9D], 5 < tn9z>]

dsE.lp(x(s))-ExCs£p(x(n^))}

Jo

= M( ds[_E.\_p{x{s)), 5<m a o ]-£.[^(A;(5+w" 1 )) J s + « _ 1 Jo

- «(

E.[>0r(s)), 5 < m9D]<2s f £

w t + °° •

Jo

An application of c) establishes 5. 5

®{e) = \pde > (/>„<& = ( pfndb > ®(fndb) \ @(e)

» ! + °°,

and this implies d ) : 5. 6

lim ®(e-fndb) Because t/\mdD

5- 7

= lim [@(e)-2 ( ^ e + © ( / w ^ ) ] - 0 . is a Markov time,

E. [f „(m9D) | Bt A mao+ LJ*Ani3D

J

i.e., I„ is martingale, and since p„ is continuous and tends to 0 along the Brownian path (t f m9D), I„ is continuous, e) is now established, and f) follows from Doob's submartingale extension of Kolmogorov's inequality [4], the Schwarz inequality 5.8

{[Gde.de^

<^{e^{e2)

(see H. Cartan [2]), and the resulting 5.9

£.[|l M ( + oo,M,;)-I m (+oo,

wt)\2,s<mdD]

<E.X_EX,SLIfn(mdD)-fm(tn9z>)|2], < E . [ j G(x(s), b)(fm-fJ(Pn-PJdb,

s <m9D] s <m9D]

30

490

H. P. McKean, Jr. and H. Tanaka = j j G(ay b)eMm(da)E.[_G{x(s), b), 5 < m B D ] d * ) e*Jdb) = < ®(enmr ®(E.lG(x(s)t b), s < m^dej)* 1/2 d/2 1/2 <©(0 constant xs- ®(enm) .

(fH-/Jdb

2

Choose nx0] = l, where ls is a continuous function of t. Because p= lim pn is continuous and tends to 0 along the Brownian path (t f m9z>), it follows that fs(t) = \im\„(t, Ws) is continuous (f[>0) and additive (*<™az>W)); moreover E-Cf M (+-) 2 ] = 2 J G A / „ ^ < 2 | | / > | | i < + -

5.10 implies 5.11

E.[f.(f-5)] = lim£.[f K (+oo, wt)-f„(+<*>, «,+)] » + +»

- lim [ £ . ( A ( « ) ) ) s < m 9 D ) ^ . ( ^ ( / ) ) ) * < m 9 D ) ] = E.(p(x(s)), K m 8 D ) - £ . ( ^ ( 0 ) , / < % ) , and, to finish the proof of g) and h), it is enough to define 5.12

f(f) = lim f,(f)

and to make 5 J, 0 and f | + °° m 5.11. Now take a smooth measure e, choose Bn f Z) as needed for 1.8 with the additional property that e j Bn has ./wwte energy &(e\BH) = \

Gdede6, let fM be the additive functional associated

JBnxB„

with e\Bn as in 5.12 above, and note that E.]AmzDfdU ] = \B Gfde

5.13

/ > 0

as in e) of section 3. It follows that if / is the indicator function 6

®(e|.5„)<;+co is achieved as follows: take DZ)D with dD at a positive distance

from 3D, choose B„ \ D as needed for 1.8, and let B„^B„C\D; ( }B„

de [ lBn

Gde^neCB,)^«(

Grfe^n z ( inf

inf G ^ s u p t DXD

D

JB«

BX.O

G)"

1

^^.

then

&(e\BH)
31

Additive functionals of the Brownian path

491

of Bm, then the functionals \ fd\n (n^>m) and \ fd\m have the Jo

Jo

same (bounded) mean: 5.14

£ ( f >f„] = E.[\^fdq

= \BGde

and are therefore identical up to time m3D according to the first uniqueness lemma for additive functionals of section 4. But this means that f„ = \m up to the exit time from Bm, and since this exit time =mdD for large m, it is legitimate to define a functional f for t
f(0 = f.(0

t< inf (t: x(t) £ Bn\ n>l.

Introducing pa=E.[_e~a]{mD)~\ 5.13 implies 5.16

aE\m%-«^ri-\W\{dm

1-A, = lim "u™

Lo D

- lim aE.\ \ pad\ «++«>

and using the method of 3.5,

LJo

J = lim a [ Gpade = a \ Gpade ; J

»I+M

JB„

J

in brief, e is the measure associated with f as in 3.2. Because two additive functionals with same associated measure are the same, it follows that if e is smooth and if A C A C a swell out to Rd, then the functionals f„ associated with e„ = e\D„ as in 5.15 above determine an additive functional 5.17

Kt) = U*)

t<m*Dn,n>l

depending upon e alone, for which 3. 2 holds for each D. f is additive for almost all Brownian paths, but of course it can be modified on a negligeable class of paths so as to be Borel in the pair (t, w) and to satisfy 1.1, 1.2, 1. 3, 1. 4, as identities; with this modification, f is the additive functional associated with e. 6. DIFFUSIONS WITH BROWNIAN HITTING PROBABILITIES. To avoid confusion, let w, x, m, Bt B, Bm+, etc. be used to describe the Brownian motion, introduce the same paths, times, , • • events, and fields with the new names w, x, xh, B, B, B^+) etc., and take a new motion D with probabilities Pa(6).

32

492

H. P. McKean, Jr. and H.

D is said to be a diffusion time m : 6.1

Tanaka

if it starts afresh at each Markov

aeEdy

Plw^eB\B^^Pb(B)

BeB,

b^x{m);

it is said to have Brownian hitting probabilities if, for each bounded open D, 6.2

PaLx(mdD)

e db] = Palx(m^D)

e dV\ = hZD(a, db) a£D,

db
Given such a diffusion with Brownian hitting probabilities 6.3a

Ele~a^7

pa =

solves 6.3b

1-fa^cc^Gfade,

where e^>0 is independent of cc and of D. measure

of the diffusion;

the speed measure of the Brownian

motion is the Lebesgue measure db. + co] = l and pa^>0,

e is the so-called speed

Because of 6.2, A [ n i 3 D < ^

which implies that e is smooth, i.e., that it is

the associated measure of some additive functional p a t h ; moreover, e is positive

f of the Brownian

on the neighborhoods of H.

fine topology, and this implies that its associated additive

Cartan's functional

satisfies 6.4

P.[f(s)
6.5

P.(6)

=P.I>(f-1) € £ ]

in brief, D is identical stochastic

1

clock f" ;

B

f"1 of f, it turns out that eB;

in law to the Brownian motion run with the

moreover, each smooth measure

e, positive

fine neighborhoods, is the speed measure of a diffusion with hits. T h e proofs are carried out in the next few sections. 7

E.(f) = \fdP..

on

Brownian

33

Additive functionals of the Brownian path 7.

493

SPEED MEASURES

Beginning with the speed measure e of 6. 3b, if a aeRd if tfiE = min(2: \x(t)~a\^£)y then m0+ = limtfte satisfies

7.1

Ea(e~^+) = S ^ "

0

^ ]

= £a(e-*°+Y,

and

* = *V

and since E a (e _m ° + ) = 0 implies P fl [th 0+ = +oo] = l, violating the fact that the dot motion has Brownian hitting probabilities, it follows that i*a[ifi0+ = 0] = l. Because

7. 2

1- A, = «A[j* to «-(*»»-0*] = «£.[[* 8 % W 0)^] ,

1 — A satisfies 7.3

l-A(«)>o:£fl[j^A^] = j^(«,^)(l-A)

aeDCD,

and, using 7.3 and A£Tho+ = 0] = l to establish 7.4a

\h6(a,db){l-p„)\0

7. 4b

J A16_ai _e(«, dft)(l - A ) t 1 - P«

it appears that 1 —A is

D ]D

tne

£I0,

potential « \ G
charge distribution aig* It remains to verify that p~xdea^de is independent of a and of D, which is done with the aid of the additive functional \a(t) = \ pa{x(s))ds {t
an(

i the method of section 3 ; in outline,

Jo

7. 5

1 - A, = « 4 [f„(* 80 )] = « j Gcfe.

implies

7. 6

£[[*'7Ws))f.(«fe)] = J Gfdea

as in e) of section 3, and 3. 9 follows as before, etc.. 8, TWO DIFFUSIONS WITH BROWNIAN HITS AND THE SAME SPEED MEASURE ARE THE SAME. Consider a pair of diffusions with Brownian hitting probabilities and the same speed measure.

494

H. P. McKean, Jr. and H. Tanaka Because 6. 3b = 3. 2 has unique solutions as noted in section 4,

both diffusions have the same pa = E.[e~am7iD~\ and hence the same 8.1

G0+ : / - £.[\^ D f(x)Udt)] Ut)=[pMs))ds

= j Gfpede

t
Jo

Choose £ > 0 and introduce the Green operators 8. 2

6. : f - & [ j * V ^ » / ( * ) f . ( t f ) ]

« > 0;

then 8. 3

aG«+GJ = at.[\^"Udt)(G.f)(x)

]

= at. \£"\m ^%-<*™-^Mm%iM ] = a&[j*V^*Y(*)f.(&)J>M.] = £.[jto(l-e-^)/4]-G0+/-G./( 8.4

d. = ( J 0 + - a C d - ,

and, using the bound 8.5

G0+l = J G£(fe = £-\l - A) < £_1 < + oo ,

the obvious iteration of 8. 4 implies 8.6

CJ- = 2 ( - ) " « " ^ S l 1

<*<£-

Because G^ is a Laplace transform in its dependence on a, 8.6 implies that both diffusions have the same Green operators (<*>0) and hence the same 8.7

E.\[™™e-atf(x)dt~\= LJo

J

lim l i m G , / . D\Rd

8

*°

But this implies that both diffusions have the same 8. 8 Palm edb,t
35

Additive functionals of the Brownian path

495

9. SPEED MEASURES ARE POSITIVE ON FINE NEIGHBORHOODS. Given a point a of bounded open D and a (Brownian) excessive function p on D, the set of points b €D at which \p(b)—p(a)\<^n~1 is said to be a fine neighborhood of a; the corresponding topologyis called the fine topology of H. Cartan [2]. E. B. Dynkin [7] has pointed out that a point a is in the fine interior of BC^Rd if and only if almost all Brownian paths starting at a remain in B for some positive time. Because an excessive function in 1-dimension is concave and hence continuous, the 1-dimensional fine topology is the same as the usual one; in higher dimensions it is different. Given a diffusion with Brownian hitting probabilities, its speed measure e has to be positive on fine neighborhoods; for the proof, it is enough to verify that if ZC^Rd is bounded and Borel and if e(Z) = Of then Z has no fine interior. Choose an open ball D and a decreasing series of open figures Dn so as to have 9. la

D ^ A ^ A ^ etc. ^ Z

9. lb

t J

GoP.de 10

w f + °°, A = ^.D?-* 8 *] ,

DM

and let 9.2

Ut) = \'h(*(s))ds

t<m,D.

J 0

Because of 9. 3a

l - £ f l [ e - ^ a ™ ^ ] = ( GDn&de < [ GDp4e J 0 n\ +oo, a € Z

9. 3b

A > 0,

it is apparent that 9.4a

1 = Pa\m,Dn \ 0, n t +oo] - F f l [limi(m 9 D J = «] = lim ( AaDB(fl, d6)e-'6-"' »t+wJ

« 6 Z.

But this final expression depends upon the (Brownian) hitting

36

496

H. P. McKean, Jr. and H. Tanaka

probabilities hdD alone; thus, 9.4b

Pa [ lim x(mdD) = d\ = 1

ae Z

for the Brownian motion also, and since almost no Brownian path meets its starting point at a positive time, it follows that 9. 5

PalmdDn 1 0, « t + °°] = 1

a € Z;

an application of Dynkin's description of fine neighborhoods completes the proof. 10. SPEED MEASURES GIVE RISE TO INCREASING ADDITIVE FUNCTIONALS. Because the speed measure of a diffusion with Brownian hits is smooth, it has associated with it an additive functional f of the Brownian path; moreover, e is positive on fine neighborhoods, and this is reflected in the fact that f is increasing as in 6.4:

Ks)
Pa(m > 0) - PB[f(/) - 0 for some t > 0] = 1 m = inf (*: fit) > 0)

is vaccuous; note that A is Borel and that P.(m^>0) = 0 or 1 according to Blumenthal's 01 law. Because A is either void or fine open and e is positive on fine neighborhoods, it is enough to show that e(A) - 0, and, for this, it is enough to show that for each bounded open D, the points of D at which P . [ f ( m 9 I J ) > 0 ] < l have e-mass 0. But this is immediate on letting cc \ + oo in

10. 2

a~\\ -pm) = J Gpje

p. = E.\_e~^m^

.

11. WHEN IS f(+oo)= + cx>? In making up the stochastic clock f_1 for use in 6.5, two cases arise according as f(+oo)^ + oo or not, and it is desirable to have a test for this. Because ^ = P . [ f ( + o o ) < ^ + oo] satisfies

37

Additive functionals of the Brownian path 11.1

497

p{a) = Pfl[f(m9D) < + oo, f(+ oo, „ + j < + oo] = Ealp(x(miD)y]

= j hdD(a} db)p(b)

a£D

for bounded open D, it is harmonic and since p>0, it must be constant; moreover, letting first t and then w t + °° in 11.2

P. [f(+ c o ) < «] < E. Xl(t) < n, p(x(tm < P- W) < *t]p,

it appears that p-^Cp2 and hence that p^O or 1. Our test states that p = \ if and only if one of the following conditions is met: a)

d^>3 and l~p1=\Gp1de

admits a solution 0i
d

whole of R 7 where G is the function of Rd. b) ©&=A admits a solution 0 < ^ ! < 1 on the whole of Rd, where ®p1 is the negative of the Radon-Nikodym derivative of the Riesz measure of px with respect to e (see section 13 for the meaning of ©). c) 3 and Rd^A\JB, where A is thin at oo in the sense that P.\_x(f) 6 A for some * > « ] | 0 {n\ oo), and \ G ^ < + oo.8 JB

Beginning with p = ly A = £-[^ and, since GD t G as D\ Rdy 11.3

1-A = lim

+

] *s positive and fCl,

E.le-^^-^n^-p,

D\Rd

= limd J( Gpp.de = J [ Gp.de . DtR

Because G = + co in case d=2, it follows that p = \ implies a), that a) implies b) is clear, that c) implies p^l is evident from

ii. 4

E. rr~/tff ] = ( ° d e

<+°°>

where / is the indicator function of B> and now it remains to verify that b) implies c). But, if 0 < A 3)

is thin a t ^ if and only if (Wiener's test)

S 2-B(tf-2)C(^»)<

+ co, where A„ is the meet of A with the spherical 2 " _ 1 < ; | 6 | < 2 " and C is the ddimensional Newtonian capacity; for the proof in the case of the (/-dimensional random walk, see K. It6 and H. P. McKean, Jr. [12].

38

498

H. P. MeKean, Jr. and H. Tanaka

harmonic part E.[p1(x{m7iD))~\ inside Dy then h1—p1=\GDp1de

inside

d

D, and, as D f R , hx decreases to a non-negative (and hence constant) harmonic function A(°°) such that A ( ° ° ) ~ A = \ Gpxde. Rd is now split into A=(A<^-A(oo)) and B = (p1'>-—p1(^)) and the fact that A(°°)—A excessive is used to ensure that px has a limit along the Brownian path as t\ + 00, permitting us to conclude from H.4a

A
11.4b A ( - ) = lim E . [ A ( * W ) ] = E.[limA(x(0)] that .A is thin at oo. Because 0 11. 5

J Gc/e < 2/»1(«J)_1 J G/r^e < + °° ,

and this completes the verification of c). 12. PERFORMING THE TIME SUBSTITUTION. Coming to the actual time substitution t^f'1 which is supposed to send the Brownian motion into the diffusion D, let f be modified on a negligeable class of Brownian paths so as to have 12.1 12.2 12.3

0 = f(0)s f(*)>f(s) t>s

as identities, let x'1 denote the sample path 12. 4

wr 1 : * -> x~\t) = ^"X/, w_1) = *[f "'ft w), w] ,

note that this path is continuous even if f(+ c°)3), and let us check that the motion D with sample paths 12.5a

w:

t^x(t)

and probabilities 12.5b

Pm0) =

Pm(w~le6)

is the diffusion with Brownian hitting probabilities and speed measure e. Beginning with the proof of the diffusive character 6.1 of this

39 Additive functionate of the Brownian path

499

motion, the problem is to check that if th is a Markov time and ii6 GB, then 12.6

Palwl

6 ^ | B A + ] = Pb{B)

b^x(m).

Given such a Markov time m, 12.7

m(w) ^ f-^thCw-1), w)

is a Markov time for the Brownian path ; indeed, using Galmarino's test, if 12.8a m(«) < t 12.8b x(s> u) = x(s, v) s)> *>] = x'\s9v-x) s
mfr^iiilrXf,

and an application of 12. 9b implies 12. l i b tfi(w) = f-^thCw-1), u) - f-'OnCtr1), ») = m(«;); *'« ftm/, 12. 8 implies, 12. l i b , as needed to conclude by Galmarino's test that m is a Markov time. Given A 6 B^t if A=(w: w'1 e 4) and if 12.12a m(«) < / 12.12b x(s, u) = x(s, v) s
40

500

H. P. McKean, Jr. and H. Tanaka

and it follows that v16 A, or, what is the same, that v e A; thus, by Galmarino's test, A € Bm+, and now it appears that 12.14

Pfl[i^te^] = P.[urle 4 (ii^ejJn = Ea[A, P.iw-1 G 6)1 b~x{m) = Em[w-* e A, P>(6)1 = EalAy Pb(B)-] b = x(m),

= x-'iMto-1),

w-1)

completing the proof of 12. 6. D is now identified as a diffusion; that it has Brownian hitting probabilities is clear, and to complete the discussion, it suffices to verify that it has e as its speed measure. But this is clear because 12.15

mrf = min (t: x~\f) 6 BD) = f(m8D),

and f has e as its associated measure.

13. GENERATORS. Given a diffusion with Brownian hitting probabilities, the Green operators 13.1

Ga: / - » & [ r V * ' / ( i ) < # ]

<*>0

map into itself the space C(Ed) of real, bounded, fine-continuous functions having ordinary limits at ©o in case < / > 3 ; in fact, if d>3, then P.[min\x(t)\^>n] = P. [min | #(£) | > « ] tends to 1 at oo, '2:°

<2:°

1

so that G t f / tends to ar ^oo), and, if a 6 i ? d (tf3>2) and if the ball D^a is so small that l - ^ ( « ) = l-E.[e~* , a 9 / , ;]<»- 1 , then, inside the j6we neighborhood B=Df\{pa^>l — n~1), the difference between 13.2a

u = GJ

= A[j* to «--y(i)^+i r - fl, * to j o + %--y(ift ^ M ] = ^.[J*toe--yW^+e~a^8l,«(i(ifi8D))] and the harmonic (and hence continuous) function

41

Additive functionate of the Brownian path 13.2b

501

h = A[«(i(tft 8l> ))]

is not greater than 13.3

constant x (1 — /*) < constant x r 1 . Because

13.4

Ga-G&+(a-^)GaG^

=0

a,j8>0,

it is evident that Ga maps our space of fine continuous functions onto some subspace D(&) independent of oc and that its null-space CS^CQ) is likewise independent of a. But, for fine-continuous /eG-XO), 13.5

0 - lim /3CJP/ = £ J l i m £ ( + V " / ( £ ) d f l et+-«

Lpt+»

Jo

/

J

according to E. B. Dynkin's description of fine neighborhoods ; thus the null-space is trivial, Ga is invertable, and another application of 13.4 verifies that fy=a — G"1 acting on D(f&) is independent of oc. (S is the so-called generator; it is closed in the sense that if uneD(d$) and <&uH=fu converge pointwise under fixed bounds to u and f£C(Ed), then «€D(<&) and <&«=/. Consider the differential operator 13.6

&u = ^ ^ l

| 6 | < + oo

- 0

rf>3

6 = oo ,

acting on the class Z>(0) of functions u€C{Ed) such that a) inside each D> u is the sum of the harmonic function u

h— u

\ hzD(a, db)u{b) and the potential \ GDde of its Riesz measure e . b)

\GD\deu|

is bounded.

Dw, as described in 13.6, exists and belongs to C(Ed). <& is the closure O of O *w //?e topology of bounded pointwise convergence as will now be explained. c)

Choose a fine-continuous function 0<^/>„
(f>0).

42

502

H. P. McKean, Jr. and H. Tanaka

iuT 1 ) is a diffusion with Brownian hitting probabilities and speed measure den=pHxde, 13. 7

J5.[min (t: ^(f"1) 6 3D)] = £.[t(m 8 D )] - j G^A/fc < + oo ,

and it follows from 13.2a that if u is in the domain of its generator dbn) then 13.8

u=

-i{j^(^)(i(0)f«(^)]+^[«(^tn9D))]

= — I GD{$>nu)pnde + a harmonic function, which is a special case of a formula of E. B. Dynkin [6] ; in brief, 13. 9

-e\db)

= (<&„u)p„de u G /)(<£„).

Choose u = GJ €/>(<&), then 13.10

un ^ & [ £ V ' / W f c 1 ) ) * ]

tends pointwise under the bound | | / | L to u. 13.11

&u„ = ^

^

- />Aw„ =

Because u„€D((&„),

pn(un-f)

satisfies all the conditions for un to belong to D(0), and, what is more, Oun converges pointwise under the bound 2||/|| oo to u—f=<&u; thus, w6D(0) and £iu = Gu, i.e., 13.12

5 ^ <S.

As to the proof of Q C 4 it is enough to show that if u£D(£l), then Gx(l — Q)u = u, and, for this, it is enough to deduce from w6D(Q) and Ou = u that M ^ O . Given such a u€D(£i) with £lu = u and choosing u„€D(£i) so as to make w„ and £1MM converge pointwise and boundedly to u and £lu~u, 13.13

i ^ - A , = - ( GJdude = K = j WO,

^)MM(6)

-E.I\mD(£iu„)(x)ds\

43

Additive functionals of the Brownian path

503

implies 13.14

E. [(«„ - AJ(i(0), / < m 9D ] - (u„ - hn) = E.[^D(£iun){x)ds\

/>0,

which, in turn, implies 13.15

E.[_(u-h)(x(t)),

t<mdD]-(u-h)

= A[j[A,fcto(S»)Wds]

*>0

h = J hD{-, db)u{b), and, letting D f Rd so as to make /z tend to a bounded (and hence constant) harmonic function h(oo)} 13.16a 13.16b

A[rfi8O<+oo] = P . [ f ( m J < + o o ] = 0 w(oo) - A(oo) - 0 d> 3

d =2

implies 13.17

£ . [«(£(*)] - « = £ - [ ( ' «(*)<& 1

* > 0,

and the desired w = 0 follows. TVze Green operators leave invariant the space C{Ed) of bounded functions continuous in the ordinary topology of Ed if and only if, for each D, the mean exit time 13.18

p - E.[_mdD] = \ GDde

is continuous inside D and tends to 0 on dD; in this case, the generator © coincides with the differential operator £l acting on the class of functions u€C(Ed) such that £iu GC(Ed); the reader will easily supply the details of the proof. Here is an example in which the Green operators do not map d C(E ) into itself. Choose d=Z and e^fxdb, where / = l + 2 / « and the f„ are the indicators of little non-overlapping open balls Dn converging to as n | + oo but not covering 0 itself and so small that

44

504

H. P. McKean, Jr. and H. Tanaka

13.19

P0 [m 3D < - oo ] < 2 -

« > 1.

Because of the first Borel-Cantelli lemma, />=P.[jTi9z>„ *-°-] = 0 at the origin, and it is also clear that ^ = 0 on the rest of Rz.

But then f = l f(x(s))ds

is a continuous additive functional,

Jo

f(s) an<* *(f_1) is a diffusion with Brownian hitting probabilities and speed measure e. Given a neighborhood D of 0, 13. 20

£ 0 [min(* : *(f_1) e 3D)] = £o[f(m9D)] »2:i

LJo

J

= 2 ( GD(0, b)db/wo\ume (D„) — -|-co ,

But, as E. B. Dynkin [6] has pointed out, this cannot happen for all small neighborhoods if the Green operators map C(Ed) into itself. 14. DISCONTINUOUS ADDITIVE FUNCTIONALS. V. A. Volkonskii [16] has studied discontinuous additive functional ; in the present Brownian case their structure is very simple. A functional t of the Brownian path which satisfies 14.1

t()f, w) is measurable Bt for each / > 0 .

14.2 14.3 14.4

0
t>s

is the sum of a continuous additive functional f and a discontinuous additive functional \ with

14.5 14.6a

P.[i(/) = K0 + ) ^ > 0 ] = 1 P-[j(0 + ) > 0 ] = 0 or 1

14.6b

C(E) = 0, where E is the set of points at which

45

Additive functionals of the Brownian path

505

^•[i(0-h)^>0] = l and C is the Newtonian {logarithmic) capacity in 3 (=2) dimensions.* Consider the (discontinuous) additive functional }„ (t) = the sum of the jumps of t of magnitude ]>« * taking place before time t, note that i M (0+)>0)efi 0 + so that -P.[i M (0+)>0] = 0 or I according to Blumenthal's 01 law, let En be the (Borel) set on which ^•[t»(0 + ) > 0 ] = l, and introduce the least positive jumping time m of in. If P.(m<^ + cx>)>0 at some point, then 14.7

( > y m ) 0 1 0 , and it follows that E„ contains a subcompact A of positive capacity, having a (regular) point at which P.[x(t)eA, i.o., * J 0 ] = 1." But then P - [ i K ( 0 = + °°, C>0] = 1 at that point, contradicting in^Ct and it follows that 14. 8a 14.8b

P.UJf) = i„(0 + ), / > 0] = 1 C(En) = 0 .

The rest is clear: \= lim \„ satisfies 14.5, the remainder f = t — j is a continuous additive functional, 14. 6a is immediate from Blumenthal's 01 law, and C(E)= limC(£„) = 0. Massachusetts Institute of Technology Kyushu University

9

G. Choquet [3] found that if E(ZRd is Borel, then inf C(B) : B open, B^E = supC(A) : A compact, A CLE; this common value is the capacity of E. 10 ? . [ j f ( O e £ at some positive time] is positive or ^ 0 according as C ( £ ) > 0 or n o t ; see, for example, G. Hunt []9]. 11 See O. D. Kellogg [11]| for the classical significance of regular points and J. Doob [5] for the probabilistic interpretation.

46

506

H. P. McKean, Jr. and H. Tanaka REFERENCES

C1 J [2]] [3 3 [4] [ 5] [ 6] [ 7] [ 8] [ 92 [10] [11] [12] [13] [14] [15] [16]

R- Blumenthal: An extended Markov property. Trans. Am. Math. Soc. 85 (1957), 52-72. H, Cartan: Theorie generale du balayage en potentiel newtonien. Ann. Univ. Grenoble 22 (1946), 221-280. G. Choquet: Theory of capacities. Ann. Inst. Fourier Grenoble, 5 (1953-1956), 131-295. J. Doob: Stochastic Processes. J. Wiley and Sons. New York (1953). : Semimartingales and subharmonic functions. Trans. Am. Math. Soc. 77 (1954), 86-121. E. B. Dynkin: Infinitesimal operators of Markov processes. Teor. Veroyat. i Primenen. 1 (1956), 38-60. —-— — : Natural topology and excessive functions connected with a Markov process. Doklady Akademii Nayk CCCP, 127 (1959), 17-19. W. Feller: The general diffusion operator and positivity preserving semi-groups in one dimension. Ann. Math. 60 (1954), 417-436. G. Hunt: Some theorems concerning Brownian motion. Trans. Am. Math. Soc. 81 (1956), 294-319. -: Markov processes and potentials (1). Illinois J. Math. 1 (1957), 44-93. O. D. Kellogg: Foundations of Potential Theory. J. Springer. Berlin (1929). K. Ito and H. P. McKean, Jr.: Potentials and the random walk. Illinois J. Math. 4 (1960), 119-132. — ___—_ ; Diffusion, to appear. F. Riesz: Sur les fonctions subharmoniques et leur rapport a la theorie du potentiel. Acta Math. 48 (1926), 329-343. A. Volkonskii: Random substitution of time in strong Markov process. Teor. Veroyat. i Primenen. 3 (1958), 332-350. — — : Additive functionals of Markov processes. Doklady Akademii Nayk CCCP. 127 (1959), 735-738.

Note added in proof. The authors came to know that A. D. Ventsel' [19] obtained almost the same result as in sect. 3-5 of the present paper and that A. Meyer [18] studied the same problem for a more general class of Markov processes. As for signed additive functionals, E. B. Dynkin [17] constructed them using stochastic integrals in case of a Brownian motion. [17] [18] [19]

E. B. Dynkin: Additive functionals of a Wiener process determined by stochastic integrals. Teor. Veroyat. i Primenen. 5 (1960), 441-452. A. Meyer: Representations integrates des potentiels. C. R. Paris 251 (1960), 22 79-2280. A. D. Ventsel': Non-negative additive functionals of Markov processes. Doklady Akademii Nayk CCCP 137 (1961), 17-20

47 Z. Wahrscheinlichkeitstheorie 1, 251—257 (1963)

Note on Continuous Additive Functionals of the 1-Dimensional Brownian Path By HIROSHI T A N A K A

1. Introduction A functional a{t) = a{t, w) of t ^ 0 a n d a Brownian p a t h w: t -> x{t) is called additive if 1.1a

a(t, w) depends on t and x(s):0

^s

1-lb

a{t, w) = a{s,w) -+- a(t — s, wf),

^ t only, 0 ^ s ^ t,

where wf is t h e shifted p a t h wf : t ->x(t -{- s). Additive functionals play a n important role in transforming a Markov process, such as t h e substitution of time by t h e inverse functional of a non-negative additive functional (K. I T O a n d H . P . MCK.EA:N [7], V. A. V O L K O N S X I I [10]) and t h e

transformation of a Brownian motion to a diffusion with a drift (E. B. D Y N K I N [2], M. MOTOO [8]). As for t h e structure of additive functionals of t h e several dimensional Brownian motion, A. D . V E N T S E L [9] proved t h a t such a functional a can be written in t h e form: 1.2

a(t,w) = f(x(t))

-f(x(0))

+

$g(x(s))-dx(s), o

under t h e assumption t h a t a(t, w) has a finite expectation for each ( < oo. Here, t f and g are Borel functions on t h e state space, g satisfies f g (x (s) ) 2 ds < oo (t < oo) o with probability one and t h e integral in 1.2 is a stochastic integral of K . ITO [5]. I n this note, it will be proved t h a t , for every continuous additive functional a of t h e 1-dimensional Brownian path, a(r, w): r = min (t, m) has finite moments of all orders ^ 0, where m is t h e first time x (•) hits t h e complement of a bounded interval. This result combined with a slight modification of A. D . VENTSEL'S proof imphes t h a t any continuous additive functional of the 1-dimensional Brownian path has the representation 1.2 with continuous /. I wish t o t h a n k Professor H . P . MCKEAN" for helpful suggestions. 2. Existence of moments Given t h e space W of continuous sample paths w : t e [0, oo) -> wt e R1, write Wf, = x(t, w) (or = x(t) — xt for short), and introducing t h e corresponding coordinate fields Bt — B[x(s): s < t] a n d B — B[x(s): s < oo], let Pa{') be t h e Wiener measure on B with initial position a e R1. [x, B, P.] is t h e so-called (standard) 1-dimensional Brownian motion. W e use t h e notation Ea{f, B) for t h e

48 252

HIROSHI TANAKA:

integral over B e B of a B-measurable function / with respect to Pa and write Ea(f, B) = Ea{f) when B = W. The first passage time inf {t: x(t) = r} through r and the first exit time inf{t: x(t) $ 1} for an interval / are denoted respectively b y ntr and m/. The following strong Markov property* will be used often: if tn is a Markov time**, then for a e R1 and B e B 2.1

Pa[(Pa(w+e

B\Bm+)

= Px(m)

(B), m < oo] = Pa(m

< oo),

where w£ is t h e shifted p a t h M>+ : £ —>- x(t -f- m) and J3m+ is t h e field generated b y t h e events B e B such t h a t B C\ (w : m
a (t, w) is Bt-measurable

for each t ^ 0 ,

2.2b

|a(f, to)| < oo and a(t,w) is continuous in t {a.e.),

2.2c

a(t,w) = a{s,w) -\- a(t — s,io£),

0 ^s

St

(a.e.),

where (a.e.) means {for every w outside a certain set which has _FVmeasure zero for all a e R1). Now, consider a continuous additive functional a of t h e 1-dimensional Brownian path, and p u t a (t, w) = m a x | a {s, w) | . Then, we have t h e following Theorem 1. For any bounded interval I = [r±, ri\ we have 2.3

Pa [a (m) >t]
e~^,

a e I,

where m = m j , and c i , c% are some positive constants depending on r\ and r 2 but not on ae I. In particular, a(t A m)*** has finite moments of all orders ^ 0. Proof. First we note t h a t 2.4

—a{t,w)-\-a{s,w?)Sa{t

+ s,w)^La{t,w)-\-a{s,w}),

s,t>0

(a.e.),

which follows from t h e additivity 2.2 c of a. Let 1 be a bounded open interval containing i", and, for each ae I, choose a constant ta< oo such t h a t P a [ a ( m ) > ta] < 1/2, where m = m j . Then, using 2.4 and the strong Markov property 2.1, we have l/2^Pffl[a(m)>« > Pa[a{m)

> ta, mb < m ] ,

be!

^ Po[o(m(tt>i),w;£B) - a(m6,wj) > t a , m b < rit] ^ Pa[a(m(w+6), w+b) >2ta,mt,< m, a(m&) < t«] = Pa[mb < m, a(m&) < M -P&[a(m) > 2* a ]. * See G. A. HUNT [4] or R. BLUMEKTHAL [1].

** m is called a Markov time if m ^ 0 and ( w ; n t < i ) e B [ for each t ^ 0. *** a A & (« V b) is the smaller (larger) of a and &.

49 Note on Continuous Additive Functional of the 1-Dimensional Brownian Path

253

Now, choosing a neighborhood Ua of a such that for all b e Ua Pa[mb< it follows that 2.5

m,a(m 6 ) 3 / 4 ,

P6[c(th)>2y<2/3,

beUa.

Because a(m) ^ a(m), 2.5 is true for a e / and beUaC\ I when th is replaced by m. Therefore, using Heine-Borel's covering theorem, it is clear that 2.6

-Pa[a(m)>T]<2/3,

ael,

for some constant T < oo independent of a. Next, put o"w = inf {t: ~d(t) ^inT} if a(£) ^nT for some 2 < oo, and = oo if there is no such t (n ^ 1). Because an (n IS 1) is a Markov time and On ^ ffn-i + ^i(w+(7ft-i)> (a-e-)> w ^ 2, by 2.4, it follows from 2.6 and by induction that Pa[a(m)>nT]

=

Pa[(xn<m]

^ PalOn-l

+ 0*l{w+On-l) < ™]

= -E^C-Pa: (*„_!) (cri < m), (Tn-i < m] ^(2/3)Pa(
3.1

-Pa [ J V (»»)
Lemma. Suppose the stochastic integral J*gn(xs)dxs has a version of continuous o mr additive functional (n — 1, 2, ...). Under the notation e[a,h) = Ea[jh2{xs)ds], o suppose, for each compact interval I, that e(a, g) and e(a, gn) are finite for ael and that e(a,g — gn) converges to zero uniformly in ael as n^oo. Then, the stochastic t

integral J g(xs)dxs has a version of continuous additive functional. o Given g satisfying 3.1, putting gn(b) = g(b) for \g{b)\ w c show that the assumptions of the lemma are satisfied. Theorem 1 t

applied to the additive functional f(() = §g2(xs)ds implies that e(a, g) is finite, o

50 254

HIROSHI TANAKA:

and using the additivity of j ^ 0 and the strong Markov property, we see t h a t e is concave and hence continuous on I . B y the same reason e (a, gn) is continuous on / , and since e(a, gn) increases to e(a, g) as n t oo on I, e(a,g — gn) = e(a, g) — — e(a, gn) tends to zero uniformly on / as n -»oo Thus the lemma is applicable. F r o m now on, we take a version of continuous additive functional for a stochastic integral. Theorem 2. A continuous additive functional path can be written in the following form:

a of the 1 -dimensional

Brownian

t

3.2

a{t, w) - f(xt) — f(x0) + jg{x8)dx8, 0^t
Sftw) = a(t,w) - [f{xt) ~ f{x0)],

3.3b

/ = //=

t ^ m,

-E.[a(m)].

Denote b y G(a, b) the Green function: G(a, b) = 2(a A b - n) (r a - a V 6)/(r 2 -

n),

and b y L% (a) {r\ < a < r%) the space of those functions on I which are square integrable with respect to t h e measure G{a, b)db. L^ia) is a Hilbert space with inner product {0±, &2)a — j G(a} b) &i (b) 0 2 {tydb for each a; i t is independent of a e ( r i , r 2 ) as a set. For each 0 G L%(a) let 3

j* 0(xs)dxs o 3.4a

and note t h a t (see K. ITO [5])

3.4b

^ ^ ( m j ^ f m ) ] = ^ a [Ji,<£2)«. o

Ea[§(t

A m)] = Ea[$* (t A m)] - 0

Now, consider t h e functions p±{a, 0) = Ea[\ 3(m) ± 3
p ± ( a , 0) = Ea[\Hn)

± 3<>(n)|2] + Ea[P+(x(n),

0)]

and hence £>± is concave in (ri, r 2 ) and p± (a, 0) I 0 as a j. r1 or a t r 2 • Therefore, there exists a non-negative measure ft % finite on compact subsets in {r\, r 2 ) such that 3.6

p±(a,0)

= fG(a,b)f,i(db),

ae(rur2);

51 Note on Continuous Additive Functional of the 1-Dimensional Brownian Path this fi% is uniquely determined b y p±; in fact dfx% = d(—D+p±)*. 3.7

p(a, 0) = J* G(a, b)fi9{db),

255

Hence we have**

ae{n,r2)

dfx.p = (1/4) {dfji% — dyi^) =

d(—D+p);

H& is the unique signed measure finite on compact subsets in {r\, r%) and satisfying 3.7. Next, we prove t h a t there exists a unique function g(b) independent of a and belonging to L2 (a) such t h a t 3.8

X>{a,0) = {0,g)a,

0eL2(a),

ae{r1,r2).

This results from t h e following three steps ((1) —(3)). (1) If 0 vanishes identically in J = (Q1} Q2) c / , then p0 has no mass inside J . I n fact, in this case %0 = 0 u p to t h e exist time xtij and hence from 3.5

i.e., p is linear inside J , proving d/i9 = d( — D+p) = 0 there. (2) If %E is t h e indicator function of a Borel set Eel, 3.9

p(a>XE) =

then

$EG(a,b){il(db),

where 1 is t h e function identically equal to one on / . First, from 3.4b we note t h a t p+(a, 0) + p-(a,

0) = 2p+(a, 0) + 2 J" G(a,

b)0*(b)db

and hence from 3.6 t h a t 3.10

dfi% + dpi = 2d[4

+

20*db.

Now, if E is an open interval with p% (dE) = 0 and if 0 = %E , then from (1) and 3.10 dp® = 0 on I — E and dpt-# = 0onE and hence dp0 = 0dpt follows from t h e identity dp0 + dp^-^ = dfit. Thus we have 3.9 for such an E. But, because \p{a, %E)\ ^ |/.# a [3(m) 2 ] • ||#ff||a*** by Schwarz's inequaUty and 3.4b, p(a, %E) is a signed measure in E. Hence 3.9 must hold for any Borel set E in I because it holds for open intervals E with ^(dE) — 0 and these intervals generate all Borel sets in / . (3) F r o m (2) we have 3.11

p(a,0)

= JG(a,b)0(b)fi1(db),

ae{rltr2)t

if 0 is a linear combination of finite numbers of indicator functions. On t h e other hand, for each a, p (a, 0) is a linear functional on L2 {a) because of t h e bound \p(a, 0)\ ^ j/-^a[3(tn) 2 ] • ||||a, and hence b y Riesz's theorem there is a unique * D+ means the right derivative, and d( — D+p±) is the measure induced by the function - D+p± of bounded variation. ** I owe this reasoning to H. P. MCKEAN. *** ||0>|j f l - VW70)a.

52 256

H I R O S H I TANAKA:

ga e L2(a) such t h a t p{a, 0) = (0, ga)a for 0 e L2(a). Comparing this with 3.11, it follows t h a t dfii — gadb and t h a t ga must be independent of a, which we denote b y g. Thus we have 3.8 with a unique g e L2{a). Next, we sketch the proof t h a t 3 {t) = $g (t) for / <^ m (a.e.), following A. D. VENTSEL [9]. P u t t i n g "1 = 3 — %g, it is sufficient to show t h a t Ea[Fi(x{h

A m)) — Fn{x{tn

0 ^ h

A m))3(t A m)] = 0,

< oo,

< • "
0 ^ * < oo,

for any bounded Borel functions i^i, . . . , Fn, and b y t h e additive property of 1 and the Markovian property of Brownian motion, it is also enough to prove t h a t 3.12

K[F(x(t

A m))I(* A m)] = 0

for any continuous function F on / . F r o m 3.4b and 3.8 we note t h a t Ea [3,p {m)l> (m)] — 0 and hence b y t h e additive property and 3.4a t h a t 3.13

Ea[$+(t A m)I<* A m)] = 0 ,

0eL2{a).

Now, if J 1 is continuous on / , writing i'7 = i^o + & where FQ (ri) = i^0 (^2) = 0 and & is a linear function, we have Ea[h(x(t A Ttl))l(* A m ) ] — 0 b y 3.13 and 3.4a. On t h e other hand, if q is an eigenfunction for t h e problem (1/2)g" = Xq with q{r{) = g(?*2) = 0, then applying the formula of stochastic integrals (K. ITO [6]) t o q{xt) and using 3.13, we have Ea[q{x{t

A m))g(< A m)] = 4/je? ffl [g"(a;(S A m))S(3 A m)]ds 0 f

.

= Aj\E B b(ar(* A m))3(3

A m P ,

0

and hence #«[g(a;(( A m))3(* A m)] = 0. Now, Ea[F0{x{t A m))I{t A m ) ] = 0 follows b y approximating FQ uniformly b y a linear combination of q's. Thus 3.12 holds. Finally, t o obtain 3.2, take an increasing sequence of bounded intervals In = [rni, rn%], n ^ 1, with union R1. We have obtained already t

3.14a

a(t,w) =fn{xt)

— fnfao) -f J9n{xs)dxs,

t^nn

(a.e.),

0

3.14b

/n = — #.[a(nM)],

n» = m / t t .

P u t ^ i f « ; = 0 and for n ^ 2 n-l

/ n ( a ) = y {rk2 — rjfci)-i [(r t 2 - a)/*+i fati) + (« - r*i)/*+i frfca)]. Then, because Zn {xt) — 2n (#o) can be written as the stochastic integral J* V (xs) dxs, 0

3.14a remains valid when fn and gn are replaced respectively by fn = fn — ln and gn ~ gn -f- ln. On t h e other hand, from 3.14b, it follows t h a t \n = fn+i inside In and hence gn = <7w+i inside In. Thus, defining / — fn and g — gn inside In, we obtain 3.2. 3.1 is clear from t h e construction of g. Because a and t h e stochastic integral term are continuous in t (a.e.), / m u s t be continuous on R1. Thus t h e theorem is completely proved.

53 Note on Continuous Additive Functionals of the 1-Dimensional Brownian P a t h

257

Note added in proof. As for the multidimensional case, t h e author found, in the course of proof reading, A. B. SKOROHOD'S paper {Teor. Verojatn. Primen 6, 430-439 (1961)) and A. D. VENTSEL'S paper (Doklady Akad. Nauk SSSK n. Ser. 142,1223-1226 (1962)). These papers treats t h e same representation as 3.2 for continuous additive functionals, and especially the latter paper treats the most general continuous addive functionals of a Brownian motion, but the proof given here is different from theirs. The multidimensional version of Theorem 1 was obtained also by H . P. MCKEAJT (private communication) where / (in the theorem) must be replaced b y a suitable fine open set.

References [1] BLUMENTHAL, R.: An extended Markoff property. Trans. Amer. math. Soc. 85, 52—72 (1957). [2] DYNKLN", E . B . : Some transformations of Markov processes. Doklady Akad. Nauk SSSR n. Ser. 133, 2 6 9 - 2 7 2 (1960). [3] — Additive functionals of a Wiener process determined b y stochastic integrals. Teor. Verojatn. Primen. 5, 441—452 (1960). [4] H U N T , G. A.: Some theorems concerning Brownian motion. Trans. Amer. m a t h . Soc. 81, 294—319 (1956). [5] I T 6 , K . : On stochastic differential equations. Mem. Amer. math. Soc. No. 4. [6] — On a formula concerning stochastic differentials. Nagoya math. J . 3, 55—65 (1951). [7] —, and H. P . M C K E A N : Diffusion. (Forthcoming.) [8] MOTOO, M.: Diffusion process corresponding to \ 2 d2Jdxi' + 2 bi{x)djdxi. Ann. Inst. statist. Math. 12, 37—61 (1960). [9] VENTSEL, A. D . : Additive functionals of several dimensional Wiener process. Doklady Akad. Nauk SSSR n. Ser. 1 3 9 , 1 3 - 1 6 (1961). [70] VOLKONSKII, V. A.: Random substitution of time in strong Markov process. Teor. Verojatn. Primen. 3, 3 3 2 - 3 5 0 (1958). Mathematicae Institute Kyusyu University Fukuoka, J a p a n (Received February

15,1962)

54 Memoirs of the Faculty of Science, Kyushu University Ser. A, Vol. 18, No. 1, 1964

EXISTENCE OF DIFFUSIONS WITH CONTINUOUS COEFFICIENTS By Hiroshi TANAKA (Received Nov. 30, 1963) Introduction. The object of this paper is to prove the following theorem. Let aiJ, bl for \<>i, j2), such that for each x~(xl, ***,xd) in Rd aij(x) is a positive definite symmetric matrix. Let B(Ra) be the Banach space of all real-valued bounded Borel functions on Rd, with the supremum norm. Then, (1) there exists a family of linear transformations Tl acting on B(Rd), 0 ^ < + °o, such that

o-i

r/>o if f>o lim Ttf(x)=f(x), no

x^Rd,

f^C(Rd)}

and, such that whenever f is of class C2 with compact support in Ra,

lim^lir/^)-/^)!!

uniformly for x in Rd. semigroup {T1}.

(2) There exists a diffusion

process on Rd with

We also consider the case when bl (l
55 90

Hiroshi TANAKA

Lipschitz continuous), and the construction of fundamental bolic differential equations (the coefficients are assumed tinuous). Recently, Ito's stochastic integral equation with continuous or having certain discontinuity was treated by Girsanov [3].

solutions for parato be Holder concoefficients merelySkorohod [10] and

Our construction is based on Hille-Yosida theorem in semigroup theory, and relies heavily on the followings: (1) A result of Morrey [6] in elliptic differential equations. (2) McKean's estimate of an a priori kind concerning the continuity of sample paths (4.2 in section 4). (1) plays an important role in the present application of Hille-Yosida theorem. As for the construction of diffusions, we first construct diffusions inside balls, and then obtain a (global) diffusion in Rd connecting these local ones. (2) will be used for proving some properties of local diffusions which will be needed for the later constructions. The present result extends Nelson's result [7] to the case of continuous coefficients. I wish to thank Professor McKean for helpful talks and for communicating the result (2). § 1. Notations and preliminaries. All functions we consider are real-valued. We shall use the following notations. If D is a bounded domain in Euclidean d-space Rd(d>2), p(x, dD) denotes the distance between x and the boundary dD of D, and C{D) —{f:

f is bounded continuous in D} ,

C(D) = { / : / i s continuous on the closure D of D} , C0{D)=-{f: / e C ( D )

and /(»)-><> as

C0{D) = {f:

and / = 0 on

f^C(D)

p(x,dD)->0}t dD}.

The supremum norm of a bounded function / o n D (or D) is denoted by II/IU (or U/lls). If ^ is a positive integer, Cn(D)={f: f and all its partial derivatives of order <^n are defined and belong to C(D)}. The first and the second partial derivatives of u is denoted by u,a and u,ap respectively (l
56

Existence of Diffusions with Continuous Coefficients

is given in Rd,x=(xl, l-2a l-2b l*2c

•••,xd)^Rd;

aij is continuous in Rd

91

here we assume that (l
1

b is measurable and bounded on each compact set in Rd

(l
ij

(a ) is symmetric positive definite: d i,5=l

Let D be a bounded domain, and denote by fc(s, Z>) the modulus of continuity of {aij) on D\ 1-3

k(s,D)=

2

max

|o y (i»i)-a«(af 2 )| !

s>0

We say, for convenience, that

are approximating

operators of A on D, if the coefficients are defined in

some domain D containing D and satisfy the following conditions l-4a-d. l-4a l-4b

c#eC 3 (i>) and 3

6*neC (Z)),

I[c#-aw||j>-»0 as w->+oo

[16ib is bounded in n>l

ll&*-&*lkj>->0 as n^+oo l*4c

(l
and

(l
If kn(s, D) is the modulus of continuity of (aV) on D^defined in the same way as 1-3, then kn(s,D)
s>Q, n=l,2,

with some function k(s) such that l'4d

•••,

k(s) J, 0 (s j 0).

(«*„') is symmetric and for some constant M

0<[flaiWJ-1^

M1 = 1, X G 5 .

i,i=l

Such operators An can be obtained as follows. Let
0 and /

57

Hiroshi TANAKA

92

l*5a

l-5b

oi J (a;)=(a'^„)(«)=Ja^(x~y)
, &UaO==(&'*?J(aO -

Then, if the support of

aiJ^C(D)

and bl is bounded measurable

l*6b

there exist constants M, ML and M2 such that \\aiJh<Mlt 0< [ 2 cf'frMA

||615
(l
(l
j
\JL\ = 1 , x^D .

•i,3

Then, 1) if u has continuous partial derivatives up to the second order, and if u and Au belong to L2(D), then 1-7

[ S

\\u,ap\\lDolh
D0cD,

where the constant Lx depends only on

(

the dimension d, M, Mlt M2, D0t \\u\\2, D, \\AU\\2,D, the modulus of continuity k{s, D) of (aij) on D, and the distance of D0 from dD.

2) If, in addition to the hopotheses in 1), Au satisfies 1-9

\\Au\\2,B(x,r^L(r/pfc

with X={d/2)-l+p,0
\\uh0
0
dD),

for all B{xtr) = {y:\y-x\
\u,a(x)-u,a(y)\
then

(l
where the constants L2, L 3 and Li depend only on the quantities

1-8 and on

58 Existence of Diffusions with Continuous Coefficients

93

p. and L. In particular, if An is bounded, 1-9 is satisfied with L=co1/2pQd/2\\Au\\x> and / / = l / 2 where to is the volume of the unit ball in Rd and p0=(diameter of D)/2. § 2. Extension of A. Given A with the coefficients satisfying 1*2, we shall define an extension A of A. First, for each x0^Rd, we denote by $U(.£) the collection of all functions u defined in some neighborhoods (which may depend on each u) of Xo with the following property: for each we£U(^)j there exist a bounded domain U containing xo, approximating operators An of A on U and a sequence of functions une.C2{U) such that un and AnUn=fn converges uniformly in U to u and a certain limit f^C(U) respectively. We want to define A by Au(xa)==f(xo) for weSDtfo(A). But, for this, we must prove that the above limit / does not depend on the choice of (U, An, un). To prove this, suppose there exist two (U,An,un) and (U't Afn, wO with the property in the definition of ^)x0(A). In U we consider the equation ^•1

Anun~jn.

For this equation, as is easily verified, all the conditions assumed in Morrey's theorem are satisfied; in fact, the corresponding quantities 1*8, L and p. can be chosen to be independent of n, because the coefficients of An satisfy 1*4 and \un\B and |[/n||z, are bounded in n>l. Thus, if D0 is a domain such that Do^xo and D0l. Hence, we have 2-2

SUp[|lM B ||| f D a +2||« 1ll JI,i> 0 + 2 71^1

a=l

ll*n,./i||S.2).] 4 <+00.

arfi — l

By the same reason, 2-2 remains true when all the un are replaced by uL Now, introduce in C2(D0) the inner product: (uf v} = (u, v ) 2 t D o + S ( w , „ , v>a)%,Da+ S ( w m j 9 ,

v,ap)2,Do,

and denote by H the real Hilbert space obtained by the completion of C2(D0) with respect to the metric \\\u-v\\\~(u-v,

u-v)*.

Then, 2-2 means that

59 94

Hiroahi TANAKA

{un> n>l} is a bounded set in H, and therefore contains a subsequence converging weakly. By Banach-Saks theorem, one can choose a sequence of certain convex combinations vk of elements in that subsequence so that vk converges strongly in H. vk can be written as follows: h

h

l=k

1=1

By the same argument, there exist some convex combinations v&: l=k

t=k

converging strongly. Put vk~vk—v£. Then, each of vk,vk>a and vKa$ (l*,«£ as is verified by taking Fourier transform. Hence one has \\Avk\\2tD1l—»0 as k—»+oo. On the other hand, using the assumptions on un and u'n and on the approximating operators it follows that /*

U:

h:

Ik

Avb= S CdAtUt- 2 c'klAlui + S c«(A-ilOwi— 2 ^(A-A£)w? converges to / — / ' ( / ' = \im Anu'n) in the norm of L2(D0). Thus / = / ' almost everywhere on D0, and hence f(x)=f(x) for X&DQ by the continuity. We now define A by Au{x)=f(x)t x^U for w e S ^ A ) where U" and / are the same ones in the definition of ©^(A). %Xi{A) is the local domain at Xo of A, Note that if u is defined on a domain D and «e3) x (^f) for every x&D, then Au(cc) is continuous in Z>. LEMMA 2*1. If u^^Xo(A) and has a local maximum at xQ then Au(xo)<0. Proof. The proof is similar to that of Lemma 1 of Ventsel [13]. Suppose, on the contrary, that Au(xo)>0. By the definition of ©^(A), there exist a bounded domain U containing x0, uneC2(U) (n>l) and approximating operators An such that un and Anun converge to u and Au uniformly on U respectively. Then, there exist e, d>0 such that Anun(x)>s for \x—xa\
60

Existence of Diffusions with Continuous Coefficients

95

Then for all sufficiently large n, Anun(x)>0 for \x—x0\<3t and hence, un can not have a local maximum inside |sc—Xol<
= max un(x)—ed2l max An\x—Xo\2J-%t \x—xo\ = 3

}x—xo}^3

and hence u(x0)< max

tt(«)-2Edi[S{K,llBc»..«)+3||6*||BC«..o}]~1.

This is true for all sufficiently small 3>0, and therefore contradicts the hypothesis, proving. Au{xQ)<0. §3.

The equation {X—A)u=f and the semigroup on Co(l)).

In this and later sections, D denotes an open ball B(a, r) with center a and radius r. Take approximating operators {An} of A on D, and let us prepare a simple lemma. LEMMA 3-1. Let 0 < r 0 < r and DQ—{x:\x-a|
Let un, n>l,

be the

solution of ( Anun^=~f t un=0

in Do, /eC(Do) on dDo.

Then, there exists a function co(s) (s>0), independent of n>l and r 0 , such that co(s) [ 0 as s j O and \Un(%)\^\\fn\\D<>mco(p), p—p{x,QD). Proof. It is sufficient to prove the lemma in the case / = 1 , because |w n (a;)|<||/||5 0 -K(a;)[ where urn is the solution of 3*1 corresponding to / » = 1 . Take any xo^dD0, and let l=(lpq) be an orthogonal transformation which sends the vector r^Ca—#o) to the unit vector (1,0, •••,0). Let iW be the constant in l-4d and M2— sup \\blW-5, and consider the function

h(x-Xo)=YihP(xp-xl),

cl=<JT-M-M2,

c 2 =2M.

Then, using the orthogonality of I and l-4d, it follows that 3*2

Vxo(x)^:0 and Anvx<>(x)<>—2,

61 96

Hiroshi TANAKA

for x in {y:0<,li{y—x0)<2ro} and hence in D0. 3*3

Thus

An{vxa—un)<,-l

in D 0 . We now prove that un0 on dDQ by 3-1 and 3-2. Therefore, An{Vxo~un)(xO>0 and this contradicts 3*3. Since un>0, one has 00 and f^C(D) 3-4

(l—A)u~f

in A

u=0

the equation on QD,

is uniquely solvable in Co0), and the solution u satisfies 3-5

Mx^-l/llir,

and is the uniform

limit of the sequence of the solutions un of

3-6

(X-A„)u=f

and

in Df

u>0

if

/^0,

u=Q on &D.D

Proof. We first notice that any solution of 3-4 or 3*6 satisfies 3-5, which, in fact, follows from Lemma 2-1. If / is of class C 1 on D, then it is wellknown that the solution &C2(D) of (A—An)u=f in D and w=0 on 3D exists uniquely (for example, see [5: p. 84]). For general f^C0), it follows from the bound 3*5 that the solution un of 3-6 is the uniform limit (on D and in n^l), as k | +oo, of the solutions un>k of (X~An)u=fk in D and w=0 on dD, where fk is a sequence of functions of class C1 on D converging to / uniformly on D. We now prove that un,n converges to the solution 3-4 uniformly as n | + ° ° . Because of the bound 3-5 for uv,n, 1-9 with An replaced by Xun,n—fn holds with constants L and ju independent of n, while An satisfies 1-4. Therefore, the Morrey's theorem applied to the equation Anunin—Xun,n ~fn implies that Un,n and its all first derivatives are bounded, on each subdomain D 0 with D 0 c D , by certain constants independent of n; in particular, { \ n } constitutes a family of equicontinuous functions on D0. Using Lemma 3-1 and noting that Do is arbitrary under the condition 30
62

Existence of Diffusions with Continuous Coefficients

97

{un,n} which converges to a certain limit u uniformly on D. This limit u solves 3*5 because M6C 0 (5)fl{ C\ &X(A)}, and the uniqueness of the solution follows from Lemma 2-1. Finally, because un,k converges to un uniformly'on D and in n>l as k] +oo, and because every (uniform) limit function in the (compact) set {Un,n} is the unique solution of 3*4, un converges to u uniformly on5asn|+oo. Q.E.D. Denote by ^(AD) the space of functions u^.C0(D) such that %E®^(i) for each xe.D and Au^C0(D). We define AD by ADu(x)=Au(x)f x&D, for 3-2. If ^>(AD) is dense in C0(D), then (1) there exists a unique semigroup of operators {T£, 0 0 } acting on CQ(D) with generator AD such that 3-7 lim \\T£f-f\\D=0, f<ECQ(D) THEOREM

3-8

T£f^Q

if

/^O,

and

\\T^I\\D<1.

(2) Let {Tn,D} denote the semigroup with generator AniD. 3-9

Km \mf-Tl,Df\\n=Ot

f^C0(D),

Then

t>0.

Proof. Under the assumption, Theorem 3-1 means that, for each ^>0, (A—AD) maps the dense subset ^>(AD) of C0(D) onto Co(D) in one to one way, and the inverse (X—A^Y1 is a positive operator with norm
{Tin}

and the corresponding: diffusion.

To the semigroup {Ti,D} there corresponds a unique system of transition measures {Pn,D{t, %, •), 0 0 , x&D} on D with total mass ^ 1 such that n,z>f(x)=ff(y)Pn,D{tf

x, dy),

/eC0(D).

These transition measures give rise to a diffusion on D.

To be precise, let D

63 98

Hiroshi TANAKA

be the one-point compactification DU{oo} of D (oo is an extra point), and introduce the sample space WD of continuous paths w: i e [ 0 , + oo)-^>x{t,w) —x{t)—xt^D such that x(t,w)^.D for t<a™ where 0(w)<; + oo, and the corresponding coordinate field BD. Then, probabilitymeasures VZ,D{'), X^D, on B D can be introduced by P^Ax(t)^A3=Pn,i>(trx,A)r P£>l>(t)=oo, * > 0 ] = 1 .

x<=D,

AcD,

[Wz>, BD, J*n,Dt xE:D~\ is a strong Markov process25, which is called AntD~ diffusion. i5„,irdiffusion can also be constructed using the stochastic integral equation of K. Ito [4] as follows. Let a%(x) be the positive definite square root of the matrix {2a%(x)}t extend a\i(x) and bl(x) (l
4-1

3# 0i =»'+S C aiJ{y?>)d?s+ CbUy^ds, 3=1J

0

J 0

where &=(&» •••»??) is a Brownian motion in Rd.

Then

)

P n % ( a ; [ e A ) = P ( # £ 4 ) T0:y^$D} . Using the above stochastic integral representation of the sample path, H. P. McKean3) showed that for any open set U and compact set F c U 4-2a P . M ^ G Uc)=o(t), t[0, o(t) being uniform in x&F and in n>l, and that for any £>0 there exists a constant <5>0 independent of n such that 2) Strong Markov property results from the fact that the corresponding semigroup maps C0(D) into itself. 3) 4.2 are the consequences of the following McKean's sharper estimate concerning stochastic integrals: if (aiJ'(0) is a positive definite dxd-matrix with eigenvalues bounded above and helow by positive constants, and each component is a measurable function of the sample path {f*(s);s<£} of the (Z-dimensional Brownian motion (£0* and if a{t) is the d-dimensional vector with components 2 aij(s)^(ds) (stochastic j=i>J 0

integrals), then Pt\a{t)\>e1
where K, c>0 depend only on the bounds of eigenvalues, e>0 and d.

64

Existence of Diffusions with Continuous Coefficients 4-2b

P„%[

sup

99

|x(t2)-K(t1)|<e]>l-e.

0<(l,t2
\ta-tt[£8

Let Gi,D be the Green operator: 4-3

A>0 4 \

Gl!>f{x)=K*IC™e-xf(x )dt +00

/0 Then, u~Gi.Df

e~xTlDf{x)dt,

if

/eCo(D).

(/eCoCD)) is the solution of 3-6. 4>1 implies that

4-4

UGinf-f

Wn^O,

/eCo(O),

uniformly i n w > l a s - 2 T + ° ° . But, because by Theorem 3-1 for each ^>0 Giof converges uniformly on D to the solution u* of 3-4, 4-4 holds when Gi,Df is replaced by u*, and in particular, ^>(AD) is dense in C0(Z>). Thus, Theorem 3*2 is valid without the continuity assumption of &\ § 5. Construction of the diffusion with generator A. Let {T£} be the semigroup constructed in Theorem 3*2. By the RieszMarkoff theorem, for each #eZ) and t>0 there is a unique regular positive measure PD{t,x} •) with total mass < 1 such that

nf(x)=Jf(y)PD{t,

x, dy),

PD{t+s, x, ')-=jpD{s,

f<ECQ(D)

x, dy)PI)(t} y, • ) .

Defining Pj.it, x, ™)=l-PB(t, PI)(t,ootA) = l =0 {P${t, x, •), x^D}

x, D), ,

X
,

otherwise,

turns out to be a transition probability on D.

5-1. Pn{t, x, •) has a uniform local character: if U is open in D and F is a closed subset of U then t~lPD(t, x, £/"c)—>0 uniformly in x^F as LEMMA

no. 4) E*,i>[/;£]=J f(w)PZAdw) and E ^ C / j ^ E ^ E / j W i , ] . notations without suffix n.

We also use the similar

65 100

Hiroshi TANAKA

Proof. If U$>oot take another open set Ui such that FczUi and U^aU, and choose / > 0 in CQ{D) such that f-\ in C/i and = 0 outside U. Then, from 3*9 and 4-2a, Pn{t,x, Uc)
lim

TlDf

n—• H-oo

< 1 - lim Pn,n(ttx,

Ui)

< hm Pn% [>(«)€= Ctf ] =o(t), o(t) being uniform in x^F.

The case when

C/BOO

is treated similarly.

By this lemma, using Ray's theorem", one can introduce a family of probability measures {?£(•), x&D} on BD so that [Wu, B flt P^, a ; e ^ ] is a diffusion process55 with transition probability Po{t, x, •)• We call this process ^-diffusion. Of course, G»f ( / G C 0 ( D ) ) defined similarly to 4-3 is the solution of 3-4. Next, take an open ball Dy such that Dxz^D, Then, if {An} are approximating operators on Du they are also approximating operators on D and Ant ^-diffusion up to a^ is identical in law to A^.z^-diffusion up to the exit time aD= inf {t:x(t)QD}, because for f^C0(D) 3

5-1

««(») = En.D!

D

xt)dt

/

*>Q,

n>l

solves the (resolvent) equation 3*6 for AW|I>-diffusion. But, this identity in law is not obvius for AD- and ADi-diffusions. Let WDI be the space of continuous paths w:[0, -\~oo)-+%(t, such that

lim x(t, w)=oo.

w)-x(t)=Xt^Di=D1U{cn}

WDI endowed with the metric

p(w, w')~

sup 'pixfy, w), x(t, w')) 0<,t< + co

is a complete metric space, where p is the metric of the compact set Di. Each of Pn,Di(') and Pjbi(*) is a probability measure on the topological Borel field of WDI, and because of 3-9 and 4-2b an application of Prohorov's theorem [8:p. 181] yields, for any bounded continuous function F on WDI, 5) See [9] or [2:Chap. 6].

66

Existence of Diifusions with Continuous Coefficients 5-2

101

limE£D,(F)=ED,(F).

LEMMA 5*2. For any e>0 and # e D there exist continuous functions ht and h2 on WDI such that §
and is the solution of 3-1 with / = 1 , using Lemma 3*1 one sees that there exist r 0 and r 2 such that ES.D(O'O—0*2) <e, n
Qi{ max |#s—a|)ds,

/

LEMMA 5*3. AD-diffusion

i = l , 2.

up to (rro is identicalin

law to

ADrdiffusion

UP tO 0D>

Proof. It is enough to prove that for / > 0 in Co(A) an

u{%)=El1[J

e-z*f(x[)dt

X>0, x<=Dt

is the limit of un{x) defined by 5.1 as n] +co, because un is the solution of 3-6 and hence by Theorem 3*1 converges to the solution Ghf of 3*4. Fix x^D, and for given e>0 choose ht and h% as in Lemma 5*2. Then hi(w)

/

e-xtf(xt)dt,

i=l,2f

o

is bounded continuous on WDX and hence by 5-2 tfk,Di(Fi)->W)l(Ft)t i=l, 2, as n—>+oo. But, because e~^f{xt)dt
l&MFi-FiXBWfWD!,

o one sees that l\mun{x)~u{x). 6) a is the center, and r is the radius of D.

Q.E.D.

67 102

Hiroshi TANAKA

Let Ed be the one-point compactification Rd\J{oo} of Rd, W the space of continuous paths M>:£e[0, +00)—>x(t, w)=x(t)=Xt^Ed such that x{t, w)GRd for tr M , where 0
x<=Dt

AcdD.

5-4. i) G , ^>0, maps B{R ) into C(Rd), and if /eC(tf d ), u~Gxf is a solution of (X~A)u~f in Rd. ii) USD maps B{dD) into C(D). Proof, i) For each D, let An,n be the approximating operators on D obtained by the method of 1.5 with

x

d

un,D(x)=ElJ

r°°e~*tf(xt)dt 0

converges, as n] +00, to the function uD(x) = ExD r~e~*tf(xt)dt

=E*|~

rDe-*tf{xt)dt

and, if / > 0 , uD(x) increases to u(x) as D ] Rd. On the other hand, if in addition, f^Cl(Rd), un,D is a bounded ( I I W ^ I I O ^ - ' I I / I I D ) solution of (X—An)u=f in Z>, and hence applying the estimate in § 1 to the quation (/I—An)un,D—f in a fixed domain D0 {un,D;n>l, Dz)DQ, fcCl(Rd) and 1I/II^<1} turns out to be equicontinuous in D0. Therefore {Gxf; f<=C\Rd) and H/lUd0, Gx is one to one from C(Rd) onto the common range and that A-(G*)_1 acting on the on the common range is also independent of A>0. /t— (G*)"1 is called the generator of the diffusion [W, B, P*]. Summarizing the results in this section, we have

68 Existence of Diffusions with Continuous Coefficients THEOREM 5*1. There exists Green opertor

[W, B, P*] on Rd such that

a diffusion

d

maps C(R ) into itself

103

and the generator

is a contraction

the of

A. If bl is continuous (l
is a C 2 -function with compact

and if /

support, then / restricted on D belongs to ^>(AD) provided D contains t h e support of / .

Noting t h e uniform local character of t h e transition

function

of ^[-diffusion, we have t h e following THEOREM 5*2. d

R

If W is continuous

such that the corresponding

(1-^i^d), l

semigroup

T

there exists a diffusion (acting

on the space

on d

B(R ))

satisfies lim | for

any C2-function

t

i.jii

dx'dx^

f with compact

iti

dxl

support.

References. [ 1 ] S. BOCHNER, Harmonic analysis and the theory of probability, 1955. [ 2 ] E. B. DYNKIN, Theory of Markov processes (English translation). Pergamon Press. 1961. [ 3 ] I. B. GiRSANOV, On stochastic integral equations of Ito. D. A. H. CCCP. 138 (1961), 18-21. [ 4 ] K. ITO, On stochastic differential equations in a differentiable manifold. Nagoya Math. J. 1 (1950), 35-47. [ 5 ] S. ITO, Fundamental solutions of parabolic differential equations and boundary value problems. Jap. J. Math. 27 (1957), 55-102. [ 6 ] C. B. MORREY, J R , Second order elliptic system of differential equations. Contribution to the theory of partial differential equations. Ann. Math. Studies. 33, 101-159. [ 7 ] E. NELSON, An existence theorem for second order parabolic equations. 88 (1958), 414-429. [ 8 ] Yu. V. PROHOROV, Convergence of random processes and limit theorems in probability theory (English translation). Theory of prob. and its appl. 1 (1956), 157-214. [ 9 ] D. RAY, Stationary Markov processes with continuous paths. 82 (1956), 452-493. [10] A. B. SKOROHOD, Hcwie^oBaHHH no TeopHH cjiynaPiHhix npoueccoB, (1961). [11] H. TOTOKI, A Method of construction of measures on function spaces and its applications to stochastic processes. Mem. Fac. Sci., Kyushu Univ. A. 15, (1962), 178-190. [12] H. TROTTER, Approximation of semigroups of operators. Pacific J. Math. 8 (1958), 887-919. [13] A. D. VENTSEL, On lateral conditions for multidimensional diffusion processes.

69 Propagation of chaos for certain purely discontinuous Markov processes with interactions Dedicated to Professor Kosaku Yosida on his 60th birthday By Hiroshi TANAKA

§ 1. Introduction. Given a measurable space (Q, $) such that each single point set is in $ , we consider a nonlinear equation for probability measures on Q: <1-1)

^

- = 2 \ qn{x){IIn{x, u(t)t -)-d(x, •)}«(*, dx) , at «=iJo with the initial data / in 9ft, the set of all probability measures on (Q, $). We assume that each q%{x) is a nonnegative ^-measurable function on Q, q(x)= CO

2 q.n{x)
<1.2)

I7n(%,f, H= \ " \ n"(X> XU ' ' ' > Xn> r)fidxd

• - - f(dXn) ,

(feW,re%) where (i) I7n(x, •)e9ft for each fixed n, xi, • • •, xn, and (ii) II*(x,xif '••,xn>n is measurable in (x, xit • • •, xn) for each - T e $ and symmetric in (xu • • •, Xn) for each x and P. The precise meaning that u(t) is a solution of {1.1) with u(0)—f in 9ft is understood as follows. u(t) is in 9ft for each t>0 and satisfies {1.3)

u(t,r)=[

e-w/idx)

H-t'dst «(*, dtf) nS= 1ff,(tf) ( e- u - 8 , 9 l z , /7nC?/,wCs),^),re^,t>0. Jo JQ Jr The usual method of successive approximation provides us with the smallest solution of (1.3) among substochastic solutions, but of course, this smallest solution is not necessarily stochastic. We denote by XI the set of those / 6 9ft for which the smallest solutions of (1.3) belong to 9ft for each t>0; it can happen that the class II becomes empty. Probabilistic significance is that the equation (1.1) corresponds to a Markov process describing the motion of one particle under the interactions with infinite

70 260

Hiroshi TANAKA

number of similar particles. This class of Markov processes was first introduced and the explanation was given in the works of McKean [2] [3] through the socalled propagation of chaos, the notion which Kac [1] discovered for his 1-dimensional model of Maxwellian gas. In this direction, there are also works of D. P. Johnson [4] and T. Ueno [8], in which the propagation of chaos for the equation (1.1) were proved under a certain growth condition on {qn(x)}n^i when each qn(x) is bounded. The purpose of this paper is to prove that the propagation of chaos for (1.1) holds in the sense described below if the initial distribution / is taken from the class U. The method is similar to that of H. Tanaka [7] in its outline. For each n>2, we consider a Markov process X(t) of n particles under motion. If each qu{x) is bounded, it is the Markov process on the state space Qn=Qx "• xQ with generator An given by n-l

(1.4)

{An
where
71

Propagation of chaos for Markov processes

261

for functions; the latter omission is the same for sets. The supremum norm of a bounded function (p is denoted by ||^H, and the integral over a set r of a function
r or simply by if, 9> if r is the whole space. 1°. {q, n}-minimal transition function. Given a measurable space (R, (&), we consider a kernel A(x,T) over (R, (S) expressed in the form: A(x, r)=q(x){II(x, r)~d(x, T)} ,

xeR,

fe®,

where 0
u(t,x,r)=e-«{x)td(x,r)+

\tds[ Jo

q{x)e-*{x)*u(ts, y, r)ll(xt

is obtained by the usual successive approximation as follows. Mt, x, r)=e-ql9)td{xt

(2.2a) <2.2b)

pk+i{ttx,r):=MttiBtn+\id8\

.

JR

pQ(t, x, r)=e-"wtd(x, pk+1{t,x,r)=pC)(t,x,r)+\tds\

and hence p{t,x,r)

T)

pk{s,x,dy)q{y)\ Jo

<2.4)

f)

increases to p{t, x, -T) as / c t ° ° ; the p^'s are also obtained by

<2.3a) (2.3b)

Set

q(x)e-^*pk(t-s,y,r)n(x,dy) JO

Then, pk{t,x,r)

dy)

Jfl

e-^^n{ytdz)

,

Jr

JR

is the minimal solution of the forward equation:

u{t,x,r)=e-*(xnd(x,r)±\ds\

e~^){t~8)lJ(ytdz)

u{s,x,dy)q(y)\ Jo

Jfi

J/

.

1

p{t,x,r) is called the { l such that

72 262

Hiroshi TANAKA

Pk,%(fi,x, ')=Pm.n(t,x, then p,n(t,x, 2°.

•) tends to p(t,x,

A formula

solving

•) for

•) as n t

00

the equation

all Jc>m and

n>l,

(1.3).

S. Tanaka [7] and T. Ueno [9] e x -

pressed t h e (minimal) solution of t h e equation (1.3) in an explicit f o r m ; a p a r t of their results is sketched here as a preparation for t h e n e x t section. F i r s t we introduce t h e set T= U Tn, each set Tn being defined inductively as follows, (a) Ti consists of single element 0.

(b) If Tm is defined for m
t h e n Tn is

defined as t h e set consisting of all objects r's of t h e form T=[TO, TI, • • -, r&] where TJB Tnj, no+ni ••• ~\-nk=nt tion for measure-valued t-iunctions [fo,fit

k>l. f0(t),

N e x t , we introduce t h e following nota•••,/*(£).

• " , / * ] ( * , ^ ) = I ds\ fo(s,dx)qlc(x)\ Jo Je JQk • • • /*(«, dxk) [ e-iw-'Wkix,

fx(s,dxx) » ! , - . , xt, dy) .

Given t h e equation (1.3), we now define t h e measure-valued ^-function Ur—Ur(ty indexed by r e T inductively as follows. ue=Mt,r)=r.

(i) (ii)

U^—[UTQ, Mrj, • • • , Urk]

for

r = [r 0 , TI, ••-,

rt].

Then t h e (minimal) solution u(t) of (1.3) is expressed as tt(i)=Ettr(t)=2w«(0 here un(t)~

;

2 «r(i) is defined inductively by rern

n-l

(2.5)

Un~ S

2 «o- •"'

§ 3.

[UnQ, Unv "•, Unk] ,

Ul — Ug .

n

k^1

A formula concerning the minimal solution of (1.3). In this section t h e formula

given in the second half of t h e

preceding

section is discussed from another point of v i e w ; t h e resulting formula (3.2) will be t h e basis for the proof of t h e propagation of chaos. Let
Xi , •••,xi

W r i t i n g IIx{xi, xix, • • •, Xi ,
,dx)
and regarding it as a functions of max n

xi+u

•• •) (ii, • • • , i w > w )

variables, we collect t h e

functions-

73 Propagation of chaos for Markov processes

263

of k variables in the formal series: os os 2 2 m=l N=l

m 2 QN{Xi){nN{Xif i=l

Xm+1} •'•,

Xm+N,
The result is k—1 m 2 2 qk-m{Xi)I7k-m(Xi,

Xm+1,

k • ' • , Xk,
for h>2 and —q{xi)

tf(*)=gGci)-h

(3.1b)

H(x, r)=q(x)~i

••• +q{xk) , m

2 qk-m{xi)llk-m{Xi, xm+i, ••-,#&, Xr)

for T c Q m with l<mfc =0 for all r if &=1 . An easy calculation shows that U is substochastic, and so the {q, i7"}-minimal transition function P{t,x, •) is substochastic. Denote by 0k the space of bounded measurable functions on Qk, and by 0 the space of those functions on Q whose restrictions on each Qk belong to 0k. The restriction of
(3.2)

For any

= 2 •m=k

where u{t) is the minimal solution of (1.3), and ( for some with symmetric f o r / r u n n i n g on 9Jt as the following formula indicates: (3.3)

m=i

Je%m

m* ,

74 264

Hiroshi TANAKA

where 3™ is the family of all subsets of {1, *••,&} with m elements, and /jf •denotes the fe-fold outer product of / / defined by fj{T)=— ^S(xj,r) for Je!$m mseJ (see [6]). Let 3 be the space of all sequences £—(ft, ft, - * •) with each & in 5^f and define a multiplication £&? by (£&?)*= 2 ft?i. Then, 8 is a Frechet space with the seminorms |[£IU— S A. The mapping ®:0-*8

llftlli, and il£<8>?IU
defined by

(%»)*(/)= for y>=($p*)**i satisfies H®$pll*^il$p|U for all fc. Now the proof of Theorem 1 is carried out in 3 steps. Step 1. Suppose q(x) is bounded. In this case, the linear operator A : 0 - » 0 defined by k—1

(A^p)*(a;if ** *, « * ) = S

TO

2 gfc-»(iw)i7*-»(iCi, £Cm+i, • • •, Xk,
— 2 flf(i»«)p*(a!if • • *, Xk)

(the first term vanishes for &— 1) is bounded, and the semigroup with generator A is nothing but {?"}. can easily prove that
Xm+1, " • , Xk,
Since one

,

one can define a linear operator %:3->3 by %!BAtp. Then obviously the semigroup {%*} with generator 21 satisfies %t($
&W8ty)=&t$)®&ty)

(3.4)

.

Step 2. If g(as) is unbounded, we set qin)(x)=qk(x)An

(for

ZP<W),

=0 (for

n

&>w> and qlnl(x)~2,Qkn){x). Let P«(ft a;, P) be the {q», #n}-minimal transition function and {£»} the associated semigroup on 3, where qn and JT» are defined by (3.1) with qk{x) and q{x) replaced by qin)(x) and qln)(x) respectively. Since qln)(x) is bounded for each n, the semigroup {£»} is multiplicative by Step 1. On the other hand, by the convergence lemma mentioned in 1° of §2, the transition function Pn(t,x,r) tends to P(t,x,r) as wT°° for each r
75

Propagation of chaos for Markov processes

265

(@TV)*Q* = lim Q* = limC£L%>)*(/). Therefore, one can define 2? on 3 by £'©fp=@T'p and so-defined {%.'} becomes a multiplicative semigroup as the limit of such ones. Step 3. For each t>0 and integers fc, w with l>=(S'ft® • • • ® W » ( / ) where &(/)=(.;•>, 0,0, •••), and hence by the multiplicative property (3.4) of {£*} we have (3.5)

fnAt)=

S

/.i.itf)® • • • 0 / ^ . i W •

On the other hand, by the forward equation for P(t, •, •) we have for r<^Ql PG.^n^e-^'afei,/1)

xeQ1

P(s, a?, dy)qm-i{yi) \ e~9(2,u"8,77m_i(?/i, y2, •••, ym, dz)

= I ds S I JO

if

m=2j Q m

J

r

if

a e Q " , w>2 .

Integrating both sides of the above by / * and then using (3.5), /i.itf, n - < / , e-*(-"3(-, P)>=«*(t, P) fn,i{t,n=

\ ds'jt \ Jo

«=2jQm

= \ ds S J0

J

2

I fnl

•••rym,dz)

r

(s, dyi) • • • / - ,i(s, dym)qm-i{yi)

m=2 » ! + • • - + « m = n JQm X \

= S

e-q{!S)(t~a)IIm-i{yi, y2,

fn,m{s,dy)qm-i(yi)\

S

[/»i.i, •••,/.„.!]

e-qWt-a)IIm-i(yi,

2/2, • • • , 2/m, dz)

for w > 2 .

Comparing this with (2.5), we conclude that u{t)~ 2 / » i(t), and this means that M=l

(3.2) holds for k=l. 2/..*(«)= 2

For k>2, using (3.5) again, we have 2

/^.iCO® ••• ®/» 4 .i(0= (i/-.i(t))*=«(t)* ,

completing the proof of the theorem.

76 266

Hiroshi TANAKA

§ 4. The motion of n particles. Let Q be the adjoined space Q^{A), A being an extra point €Q. In this section let n>2 be fixed. The motion of n particles we consider in this section is a Markov process X(t)=(Xi(t)t - - -, Xn(t)) with state space Qn=Qx -•• xQ whose probabilistic development is determined in the following way. Suppose the process starts at x=(xi, • • • ,xn)eQn and set I={i:xieQ}, Ir~{i;xi=J}. Then, (1)

Xi(t)=d

for

ieT

Xi(t)=xi

for iel,

and all

t>0 ,

t<Si,

where Si is a random variable with P*{Si>i}=exp {-t 2 r(xi)}

#=i

(t>0)

(n—l—N)\ (n-l-Ny.

(2) At time Si, one of the particles with index in I jumps according to the following probability law. For each iel PAXi(Si) e r, Xj(Si)=xjt

j^i)

= S n~N 2 ' qN(xi)HN{xi, xh, ••, XiN, T)/ 2 r{x3), T c Q . For r={J},

the left hand side of the above is replaced by {r(xi)— 2 n~N2' N=I

qN(xi)nN(xi, Xi,, • • •, Xi„t Q}i S r(xj) . "

jei

Here, we set IT^ixi, Xilt • • •, XiN, r)—0 for PczQ if at least one of Xi, Xilt • • •, XiN is J, and 2 ' is taken over all ordered N-tuples (ii, ••',IN) such that i
(4) If S(fi>)
77

Propagation of chaos for Markov processes

267

at time S as in (1) (2) and (3). Repeated arguments of this now leads to the process X(t) denned for all t>Q. The following explicit construction of the process X(t) will be used later. For each x=(xi, • • •, #n) e Q" we set a{t, x)=exp {—t 2 r(xi)} ,

r(J)=0

*=i

P(itx)=n-NqN{xi)l,Zr(x3)t

r{i,x,r)=

i=(i,ii,

• • • , ^ ) , qN{A)~0

/ nN{xi, Xilr •••, XiN, JT), for raQ 0 , for r={d] if all Xi,Xiu •••,XiK J are in Q 1 , for r={J} if at least one of \

Xi,Xilt

••-,XiN

is A ,

and then construct a discrete parameter Markov process Uk—{Rk, Yk, Zk), over a suitable probability space {Qt Px) with the following properties. (i) 0<jRfc
k>lt

Zk={Zk{i), i e 3}={CZ*.i(i), • • •, £*..(i))f * e 3 } ; here 3 denotes the set of all i=(i, ii, ••-, IN) such that i,ii, * • •, IN are different integers taken from 1, - • •, n. (ii) The initial distribution is given by C4.1)

P*{Ri>t, Yi=i, Zi.JJ)e rm,t, \<m
le 3}

=a(t,x)P(i,x) (m,I) n r(i,x,rm,t) (m,I) n a(a?m, r»pI) m=|l|

m^|/|

where \l\=l for l={l,h, •••,IN). (iii) For ^={.Rfc+i>£, F i + i = i , ^ + 1 , 4 ) 6 ^ , 1 , l < m < n , / e S } , the conditional probability Px{A\Ui, ••-, Uk}=P*{A\Uk} is given by the right hand side of (4.1) with x replaced by Zk(Yk) almost surely on the set { 2 r(Zk,m(Yk))>0}t while n

we set Uj=Af for j>k+l

m=l

on the set { 2 r(^,m(rO)=0}, A' being an extra point

(death point). In order to construct the process X(t) by means of {Uk}, we need to extend the scope of the time parameter of {Uk} further to o o + l f o o + 2 , - " , 2 0 0 + 1,200 + 2, • • • , ( » - l ) o o + l , ( » - l ) o o + 2 f ••• .

We set

78 268

Hiroshi TANAKA

f (4.2)

A

if #{fc:|r k |-i}=°o

aw(oo)=

( ZkH),i(Yk[i)) if

ft(i)=sup{l
| y*|=i}
and then construct {U^+kh^.z, ••• so that its distribution conditional to {Ui, U2, • • •} coincides with that of the process obtained exactly by the same way as in (i) (ii) and (iii) but with the replacement of x by A:(OO)—(a;i(oo)f . ••, #»(co)). Repeating this kind of construction again and again, we obtain a Markov process {Uk} with time parameters 1,2, • • - , 0 0 + 1 , 0 0 + 2 , • • » , 2 o o + l , 2 o o + 2 , • • • , ( w - l ) o o + l , < ^ - l ) c o + 2 ,

Now the process X(t) is constructed over (O, Px) by jr(*)=z*_i(r*_i)

if

~{A, . . . f J)

if

1

E J ^ ^ E ^

1

E i ? 3 < i < + co . j < 71 CO

For a subset J of {1, *--,w}, we define the first jumping time T{I) of the particles with index in I by <4.3)

T(I)=

E Rj

= + co

if if

fc(^=min{fc:|Ffc|€i"}<woo \Yk\$I

for all fc
it is distributed according to the exponential distribution with mean 1/ E r(xi). Next, we introduce a (q, 77)-function on the space Qn as follows: for x=(xi, ••-,Xk)eQk k

qn{x)=^r{xi) -ff»(#, ^^ffwCA:)" 1 ^"^"™ 1 ^

rff- 2 Qk-m(Xi)nk-m{Xi,

for

TcQm

with

=0

for

m

with

=0

for all r

r
if

Xm+1, • • ' , Xfc, *r)

l<m
fc=l,

and denote by P*{t, x, f) the {qn, J7n}-minimal transition function. m PROPOSITION 2. If (p is a nonnegative bounded function on Q (m
then

where xO'oo) is defined

79

Propagation of chaos for Markov processes

< \

E{Xl,...,Xn){
PROOF. Taking a family (4.4)

<J={IO,

•••.h)

269

.

of subsets of {1, - — ,%} such that

7o={l, • - , m}, L-i^Ij

d<j
and recalling (4.3), we set inductively T(IP, 7p_i, • • •, IP')=

2 j£k[lp,Ip—i,

Rj •••, Ipi)

if fc(/P, • • •, Ip')=min {k>k(Ip, • • •, JWi): I Yk\ 6 JP'}<w°o , if and then

T(Ip,Ip-u •••,/,') = + «> I U P , • • • , V + i ) = + oo or if | r * | 0 / P ' for fc(Ip, • •-, JP'+i)
A0={T(Iv)
Yw^elp,

T(IP, 7P_i)<£, r ^ v r ^ e f p - t

• • •, T(IPt • • -, Ji)*} , where /,• denotes the set of all ordered (p'-j-l)-tuples (i, ii, • • •, v ) such that ielj-i and {ii, ••• ,iP'}=Ij—Ij-i (p'=#(./}—i>-i)). If we set P0, ff)= ( t
p(4,
where

X(IP)=(XJ,

,n,(da>) ,

j e IP) alone for each £ and ff, and

2 <7*WMff*, x(IP-IP^i),

»eip

N=${Ip~Ip-i)f0i={Io,

•••,ip~i} and r(IP)= 2 r(a?<) .

Two ff~{/o,-•-,/?} and <x'={Jo, I'u •• •, lrP'\ both satisfying (4.4) are said to be similar, if p=p' and # J j = # / J for 1 < J < J > . Using (4.5) we can easily prove by induction on p that — for similar o" and a'f and so, if
{{1,2,-..,m}, {1,2, •..,?},.-•,{1,2, . . . , j } , {1,2, ...,fc}}

(m
> We set (p(xx, • • •, xm)=0 if at least one of %i, • • •, xm is A.

80 270

Hiroshi TANAKA

(4.7)

S

?(t,e>)>

a': similar to a

(4.8a)

0(t, I)= P«-«•»-<*-*|n~ft; S Qk-Kxi)IlN(xi, xuu •••,»*, $(ts, Jo (n—«)! <=i

tri))ds

(ffi={{l, • • • , m } , { l , - , I h • • • , { ! , •••,J}},P>1) (4.8b)

Wtla)-e-',,(/o)io(!Bif • - , »m), ff={{l, •••,)»}}.

Therefore, the function
.. a Is of thee form • • • , iCft))— < with fixedd fck

&(tt
m
(1.6)

ra—k (<7={{1, • • -, m}}

satisfies (4.9)

( ( ctC*. fan. • •. ZfcT)= \I e-«« #r*** *i."•• •••• •*> **> s^V.„1 * " ^ ^ - ^ ¥?*(*, (»i, • ••,**))= *i.

~i

(n-ky

J

x S2 Qk-j{Xi)nN(xi, Xj+i, -•• ,Xh, m •i=i

which is nothing but the backward equation associated with {om • On the other hand, by the definition of Aa two events Aa and Aa- are mutually exclusive unless a and a' are the same, and so we have

tf

a is of the form (4.8)

= £=±, completing the proof of the proposition. § 5. Propagation of chaos. In the preceding section we fixed n, the number of particles under motion, but here we let n t °o and prove that the propagation of chaos holds if the initial distribution / of (1.3) belongs to the class U. Let un(t) be the probability distribution at time t of n particles under motion starting with the initial distribution / " , that is un(t,r)=\

JQ W

P W l l ...,*n){X(t)er}f(dxi)

•-•f(dxn)

,

81

Propagation of chaos for Markov processes

271

where {X{t)tPimlr ••-.*„)} is the Markov process on Qn introduced in §4. m THEOREM 3. For each m > l , / e U and a bounded function

\

_

J Qm X Q » ~ m

(p(Xl, ' • •, Xm)Un(t, dXl, • • •, dxn) ~~> <«(<)*, 9> a& M T ° ° ,

where u(t) is the solution of (1.3). PROOF. By the convergence lemma in 1° of §2, Pn(t, x, P) tends to P(t, x, P) as w t ° ° for each t, x and P
;F)>Q* > m,P)

=
-,P)>Q* ,

then F«.k(t,r) tends to Fk(t,r) as u t ° ° . Therefore, by Theorem 1 and Proposition 2 and using Fatou's lemma, we have 1> lim £ Fn,k{t, P)> 2 Fk(t, r)=u{t)m{P) . Since / e U implies w(£)m(Qm) —1, we have the equality if r=Qm and therefore

in the above,

s u p E ^ , f c C « , r ) < s u p E K , ^ , Q m ) ^ 0 , Z->OD, r<=Q m . Thus we have by Proposition 2 ! i m w . ( t , r ) > l i m ZFnAt,P)=u(t)m(r),

P
•where vn{t, P) denotes the left hand side of (5.1) with ^ = M # i , • • •, »»). On the other hand, Una v»(i, r ) = Hm {vn{t, Qm)-vn(t,

= i ~ am vM(t,

Qm-D}

Q™-r)
n—too

and so we must have lira vn{t, r)=u{t)m{r),

which was to be proved.

References [1J Kac, M., Foundation of kinetic theory, Proc. Third Berkely Symp. on Math. Stat. and Prob. 3, 171-197. [2] McKean, H.P. Jr., A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. 56 (1966), 1907-1911. [3] McKean, H. P. Jr., An exponential formula for solving Boltzmann's equation for a Maxwellian gas, J. Combinatorial Theory 2 (1967), 358-382.

82 272

Hiroshi TANAKA

[ 4 ] Johnson, D.P,, On a class of stochastic processes and its relationship to infinite particle gases, Trans. Amer. Math. Soc. 132 (1968), 275-295. [ 5 ] Tanaka, H., Propagation of chaos for certain Markov processes of jump type with nonlinear generators, I, II. Proc. Japan Acad. 4 5 (1969), 449-452, 598-600. [ 6 ] Tanaka, H-, Purely discontinuous Markov processes with nonlinear generators and their propagation of chaos, to appear in TeopHH BepoHT. H ee npHMeH. [ 7 ] Tanaka, S., An extension of Wild's sum for solving certain non-linear equation of measures, Proc. Japan Acad. 44 (1968), 884-889. [ 8 ] Ueno, T., A class of Markov processes with bounded, non-linear generators, Japan. J. Math. 3 8 (1969), 19-38. [ 9 ] Ueno, T., A class of purely discontinuous Markov processes with interactions,"I, II. Proc. Japan Acad. 45 (1969), 348-353, 437-440. (Received December 11, 1969) Department of Mathematics Faculty of Science University of Tokyo Hongo, Tokyo 113 Japan

83 2. Wahrscheinlichkeitstheorie verw. Geb. 27,47-52 (1973) © by Springer- Verlag 1973

An Inequality for a Functional of Probability Distributions and Its Application to Kac's One-Dimensional Model of a MaxweUian Gas Hiroshi Tanaka 1. Introduction Let & be the class of 1-dimensional probability distributions / with 0 by e [ / ] = inf£{|X-Y|2}, where the infimum is taken over all pairs of random variables X and Y defined on (fi, P) and distributed according to / and g respectively; here g is the Gaussian distribution with mean 0 and variance )-X(G)')) (Y(CO)- r ( a / ) ) ^ 0 for almost all (a>, co') (P®P). Suppose the contrary holds. Then for some e > 0 at least one of the following events has

84 48

H. Tanaka

positive P(x)P-probability: {(cD,a)')eQxQ: X(a))-X(w')<

- 3 s and Y{w)-Y{a)')>3e]

f

{(co,CQ )(=QxQ: X(o))-X(3e

(1)

and Y(oj)-Y(co')<-3e}.

(2)

We assume that the event (1) has positive probability for simplicity. Then, for some integers j x , j 2 ,kuk2 with jx + 1 <j2 and fct + 1 < k2, the event f A e < ^ ( « > ) s a i + l)e,J2fi<-X'(Q>')^0"2 + l ) e | ^=<(a>, to): > I

fc1£
+ l)e, fc2£
has positive P®P-probability. If we set A = {w:jle<X(ai)£{jl

+ l)e9k2e
+ l)E}

A' = {a>:j2e<X(a))£ti2 + l)etk1e)gl(kl + l)E}9 then A = AxA' and hence P(4) > 0, P(A'} > 0. Next, we take an irrational number X and denote by T the Weyl automorphism: a>eQ->(o + X(mod 1). Then there exists an integer n^O such that P ( A n T " M ' ) > 0 . If we set B = y 4 n r _ M ' , B' = TnB, q>=T\ then P(B) = P(B')>0 and B n B ' = 0 . We now define

I

X(
for weB

Xf^M)

for coeB'

X(co)

for G>$BvB'. 1

Since 9 : B-+B' is measure-preserving , X * has still distribution /, and we have B

B'

2

+ I |x(co)-y(«)| P(^aj) (BuB') c

= I {|*(«»(a>))- y M | 2 + \x(w)- r() B

+

j" |X(to)-y(co)| 2 P(dco)

(BuB') c

< J { | * M - y(co)|2 + |X(M)- y() B

+

J | X M - y ( a ) ) | 2 P 0 c o ) = £{|X(co)-y(co)| 2 }; (BuB')c

the inequality part in the above employs the following elementary fact: if ax , and denote by Sy the smallest closed interval such that P(y,Sy)=l. We claim that Sy and Syf are non-overlapping for almost all (y, / ) with respect to g®g. (2,1) 1

I owe the use of the Weyl automorphism for constructing

85

An Inequality for a Functional of Probability Distributions

49

Since P(y,-)®P(y',*) is a regular conditional probability distribution of (X(m), X{co')) given (Y(co)9 Y"«)) = (y, y'), we have from Step 1 ttg(dy)s(dy')ttx(x,x'fy,y')P(y,dx)P(y',dx') = E{X(X(a>),X(co'l Y(a>\ Y(to'))} = l where x is the indicator function of the set r={(x,x',y,y')ei? 4 : ( x - * ' ) ( y - / ) £ 0 } . Therefore, for almost all (y, y') with respect to g®g, we have

\\x(x,x\y,y')P(y,dx)P{y\dx')=i. So, if we set ri = {{x,x')eR2\ x ^ x ' } , then P ( y , - ) ® W » ' ) is supported by /J for almost all (y, y')e/^; but this is a complicated way of saying that (2.1) holds. Step 3. From Step 2 one can prove easily that Sy is a single point for almost all y with respect to g. Now, this fact combined with the inequality of Step 1 implies that X is an increasing function of Y (a.s.); this is possible only when X = F~1(G{Y)) almost surely. The "if" part is obvious, since the infimum in the definition of e [ / ] is actually attained by some pair. Theorem 2. Let X and Y be independent random variables with distributions f± and f2£&> respectively, and assume that E{X} = E{Y} = 0. Then, for any real constants a, b such that o=t=0, fo#=0, e[aX + & r | < a 2 e [ X ] +

fe2e[y],

(2.2)

unless both X and Y are Gaussian. The proof of this theorem is based upon Theorem 1. It is obvious that e[aX + f > y ] ^ a 2 e [ j r | + fe2e[y] holds, and so assuming the equality holds in the above, we will prove fi = gi, where gt is the Gaussian distribution with mean 0 and variance of — a 2 (/*), i = 1,2. If X} and X2 are independent random variables with distributions g2 and g2, respectively, then with the obvious notation it follows from Theorem 1 that a2^[X^b^lY^=a2E{\F1-'(GdX1))-X1\2}

+

-£{|aFr 1 (G 1 (X 1 )) +

b2E{\F2-1(G2(X^^

feF2-1(G2(X2))-(aAT1+^2)|2}.

Since aX1 + bX2 is also G-distributed, we have again from Theorem 1 aF1-1{G1(Xl)) + bFf1(G2(X2))

= F-l(G(aX1 +bX2)) a.s.,

(2.3)

where F is the distribution function of aX-\-bY. By the right continuity of the functions involved, (2.3) yields aFr1{G1(x))^bF2-i{G2{y))

=

F-1(G(ax^by))

for all x,yeR}. This functional equation for unknown Fu F2i F can easily be solved; the result is F1 = GiiF2 = G2, completing the proof. 4 Z.Wahrscheinlichkeitstheorie verw. Geb., Bd. 27

86 50

H. Tanaka

We next list some simple properties of e for later use. 1. If/„ converges to some fe@> as nf oo in such a way that lim sup

j" x2fn{dx) = 0,

(2.4)

then lim e [ / J = e [ / ] . The condition (2.4) is satisfied if for some p > 2 the absolute p-th moments of fn are bounded in n. 2. Let f&G0* and 0L2(f$) = a2 for O ^ 0 < 1 , and assume that f
e[f/,Aitffl)]^Je[/JM
tm=2^{x2-xG-1{F(x))}f(dx) for a continuous probability distribution / in &. The inequality (2.2) will now be applied to give a simple proof of the central limit theorem. Let {Xn}n=l2i be a sequence of independent identically distributed random variables with mean 0 and variance 1. Then the so-called central limit theorem states that the distribution of ^„ = n~^(Xl-\ \-Xn) tends to a Gaussian distribution as n]co. Here we prove that e[^„]->0 as n-»oo assuming E {X?} < oo 2 . This condition implies that £ { ^ } = — £ { X , 4 } + 3 ( l — \ < c o n s t . (independent of n). (2.5) n \ nj Putting f]k = £2*> w e first prove that e[^]J,0 as /C|GO. The decreasing property of e [j/J is obvious by the inequality (2.2), and so we denote by / the limit of e [*/J as /c-> GO. If/ is a limit distribution of nk as k-*• oo via some subsequence kx < k2 < • • •, and if n and f are independent random variables with distribution f9 then it follows from (2.5) and 1 of § 2 that e M = lime[ifc_] = / and p-»»

G ^ - — = lime[ij2fc»] = /, L I

J

P-*»

therefore by Theorem2 the limit I must be 0. Next, we write an integer rcj^l as m

H— £ nfc where nk = sk2k with e ft =0 or 1. Then, using the inequality (2.2) we have k=0

1 m e [£J = — Z "k e [^J' n

anc

* hence e [£„]-•() as was to be proved.

k=o

3. c Decreases along Solutions of Boltzmann's Problem for Kac's Model of a Maxwellian Gas Given fltf2e& and 6(=[0,2n\ we denote by Be(fuf2) the probability distribution of Xx cos 0-\-X2 sin 0, where Xx and X2 are random variables with distri2

This condition is assumed just to simplify the proof. Without this c[<^n] still tends to 0.

87 An Inequality for a Functional of Probability Distributions

51

butions / and f2 respectively. We also put B{f1,f2)=~2{Bd(fuf2)d6. 2n o In Kac's one-dimensional model of a Maxwellian gas, the distribution u{t, dx) of molecular speeds at time t>0 is determined by the solution of Boltzmann's problem ^f^=B{u(t),u(t))-u(t,-).

(3.1)

The solutions of this equation can be obtained by Wild's sum [2]. If the initial distribution has a density, then so does the solution, and it is known that the entropy increases along the solution with time, while the solution itself tends to a Gaussian distribution as fjoo. McKean [1] gave detailed discussions on this subject; he gave also other functionals which are (or, at least are expected to be) monotone along the solutions of (3.1) together with an interesting conjecture about them. But, among these functionals, the entropy and Linnik's functional are the only ones which were used effectively in the investigation of the asymptotic properties of the solutions of (3.1). In this section, we prove that the functional e decreases monotonically to zero along the solutions of (3.1); this statement itself implies automatically that the solutions of (3.1) tend to Gaussian distributions as t]co. Theorem 3. Let u{t) be the solution of (3.1) with initial distribution fe^. Then, (i) e \u(tj\ is decreasing in t, and (ii) iff has finite fourth moment, e [«(£)] decreases to 0 as tfoo. The following corollary is an immediate consequence of the above theorem and 3 of §2. Corollary. Let &0 be the subclass of 3? consisting of continuous probability distributions, and put e 0 [ / ] = J x G - 1 [F(x)]/(dx). Then the functional e0 is increasing along the solutions of (3.1) with initial distributions e^0. The proof of Theorem 3 will be given in several steps. Proof 1. Let ^„{f), w ^ l , be the (finite) set of probability measures from 0> defined inductively as follows: (i) ^ ( / ) consists of a single element / and (ii) ^ ( / ) is the set of all probability measures of the form B(fuf2) with fe^if), f2e3Pn2{f), ni+n2 = n. Then, the solution u(t) of (3.1) with initial distribution / can be expressed as Wild's sum «(0 = e - £ ( l - e - ' r 1 p 1 1 ( / ) ,

(3.2)

n= l

where pn(f) stands for a convex combination of elements in ^„{f), n ^ 1 ([2], see also [1]). 2. I f / d e n o t e s the even part of/, say f(dx) = ^(f(dx)+f( — dx)), then it is easy to see that B(f1,f2) = B(fl,f2). Therefore, if/ and f2 have the same second 4*

88 52

H. Tanaka

moment, it follows from 2 of § 2 and Theorem 2 that In

0

+ S^~ }"{«[/»] cos^ e + c [ / 2 ] sin* 6} d6S-eUll 'U^ , In o I because e [ / ] ^ e [ / J Therefore we have e [ p „ ( / ) ] ^ e [ / ] , and hence by Wild's sum (3.2) and 2 of § 2 we see that

e[w«]^e[/]

(r>0);

(3.3)

the equality holds if and only if / is a Gaussian distribution. (3.3) implies the part (i) of the theorem. 3. If §x4f(dx)
3

1

4

-dT=T

2

T

°

*2(/)'

which implies that a(t)-*3o-4 as r->oo, and hence a(r) is bounded. Next, let ux be a limit distribution of u(t) as t|oo via some subsequence £x <£ 2 < • • •. Since a(f) is bounded, we have e[uQ0] = lime[w(t n )]^lime[w(r)] by 1 of §2. If u^{t) denotes n-too

t-nx>

the solution of (3.1) with initial distribution u^, then an application of Wild's sum shows that um(t) = lim «(*„ + *) and hence e[Mco(0] = lim e[M(tn + i)] = c[u 0O ]. n-*oo

J-tco

Therefore u^ must be a Gaussian distribution from the preceding step, as was to be proved. Note. After sending the manuscript to the editor, I was informed from T.Yanagimoto of a simple proof of Theorem 1 based upon the following Hoeffding's formula: iff denotes the joint and Fx and Fy the marginal distribution functions of X and Y, then E(XY)-E(X)E(Y)

= J

]

\_F(x,y)-Fx(x)FY(yj]dxdy

— 00 — 00

provided the expectations on the left hand side exist.

References 1. McKean, H.P.: Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas. Arch. Rat. Mech. Analysis 21, 343-367 (1966) 2. Wild, E.: On Boltzmann's equation in the kinetic theory of gases. Proc. Cambridge Philos. Soc. 47, 602-609(1951) Hiroshi Tanaka Department of Mathematics Hiroshima University Hiroshima, Japan (Received January 10,1973)

89 ON MARKOV PROCESS CORRESPONDING TO BOLTZMANN'S EQUATION OF MAXWELLIAN GAS

Hiroshi Tanaka

il.

Introduction.

The basic equation in the kinetic theory of dilute

monoatomic gases is the famous Boltzmann's equation.

In the spatially homogeneous

case, the initial value problem of this equation was solved for a gas of hard balls by Carleman [1], for Maxwellian gas with cutoff by Wild [14] t and for bounded total collision cross-section by Povzner [8] (in modified spatially inhomogeneous case), but it seems that no results (for existence and uniqueness) have been obtained for Maxwellian gas without cutoff.

On the other hand, H. P. McKean [5] Introduced a

class of Markov processes associated with certain nonlinear (parabolic) equations such as Boltzmann's equation, and brought a new light in the field of investigation of such equations by probabilistic methods (see also [6]). Then, there appeared works by D. P. Johnson [3], T. Ueno [11] [12], Y. Takahashi [9] and H. Tanaka [10], mostly concerned with Boltzmann's equation of cutoff type and certain nonlinear equations with similar structure.

Especially, Ueno [12] constructed Markov

processes which describe motions of infinitely many interacting particles, while Takahashi [9] introduced interaction semigroups and discussed their relationship to branching semigroups. Maxwellian gas ;

In this paper we are exclusively concerned with non-cutoff

our purpose is to construct a Markov process in the sense of

McKean [5] corresponding to the 3-dimensional Maxwellian gas without cutoff by solving appropriate stochastic differential equation (the equation (2.10) in §2). The theory of stochastic differential equations was initiated by K. Ito [2] and, in the case of diffusions, equations similar to (2.10) were considered by McKean [7 ] in connection with certain nonlinear parabolic equations. summarized ;

full proofs will be published elsewhere.

The results are only

90 479

We consider a monoatomic dilute gas composed of a large number of molecules moving in the space and assume that there are no outside forces. be the number of molecules with velocities at time

t, where

N

where

au(

3^°=

dx

Then under the assumption

satisfies the following Boltzmann's equation :

j {u(t,x*)u(t,y*)-u(t,x)u
S =(0>ir)>t[0,2TT)

Denote by

u(t,x)

Nu(t,x)dx

within the differential element

is the total number of molecules.

of spatial homogeneity,

(1.1)

x

Let

S x,y

and

8,0

are points in (0,TT) and

the sphere with center

—r^- and diameter i.

[0,2TI)

respectively.

|x-y|, and on this

sphere we consider a spherical coordinate system with polar axis defined by the relative velocity and

0

x-y.

x*

and

be the colatitude of

y*

are the post-collisional velocities.

x*(the angle between two vectors

and the logitude of

x*, respectively.

energy

are always situated on

x*

and

y*

x-y

and

Let

6

x*-y*)

By the conservation laws of momentum and S

and constitute a diameter of »y and 0. For each x and y the origin x

S x

, and so

y*

of the longitude y* Q

is also determined by

6

>y IJJ may be arbitrary chosen within the requirement that

as functions of

(x,y,6,0)

should be Borel measurable.

x*

and

A nonnegative function

is determined by the intermolecular force and is called the differential

collision cross-section.

In the model of gas of hard balls

Q

is a positive

constant, while in the Maxwellian model in which molecules repel each other with a force inversely proportional to the fifth power of their distance, |x-y[QC|x-y\ ,6) turns out to be a function

Q M (6)

of

6

alone ; in the latter case

Q M (8)

is a

5_ decreasing function of collision

6 with

QMCe)

%

const. 6

, 9+0 , and so the total

cross-section is infinite (non cutoff) (see [13]).

consider in this paper.

This is the case we

91 480

§2.

Markov processes and stochastic differential equation

In order to indicate our problem clearly, we first explain how a Markov process in the sense of McKean [5] hard balls by example.

is associated with Boltzmann's equation, taking gas of

The equation (1.1) for gas of hard balls is usually treated

in the following form ; •

3u(c,x) at

(2.1)

{u(t,x*)u(t,y*)-u(t,x)u(t,y)}|(y-x),H))d£dy,

s2*ii3 where on

x*=x+(y-x, I) (., y*=y-(y-x,K.) I,

,2 S .

I£ S ,

and

dS,

is the uniform distribution

We set

u(t,r) =

u-( t, x) dx

u(t,y) =

jp<x)u(t,dx)

re 45 a n

(2.2)

where

C, (R )

,

yecb(R°)

denotes the space of real valued bounded continuous functions on

(the notation (2.2) will be used throughout in this paper).

R~

Then, from (2.1) we

have

(2.3)

{y(x*)-y(x)}|(y-x,£)|dJlu(t,dx)u(t,dy), y C C , ( R 3 ) .

3t

2 3 3 S XR J xR J Povzner's result [8] may be stated as follows : (probability measure on

Now, keeping

equation for

v(t,*) :

f

3 fi i4 R ) such that Mxj f(dx)<0°, there exists a unique solution

u(t,*) (probability measure) of (2.3) such that bounded.

given an initial data

f

and

u(t,*)

u(t) = |x| u(t,dx)

is locally

as above, we consider the following

92 481

3v(t,y) _ 3t

{yCx*)-y(x)}!(y-XlJ!.)|dAv(t,dx)u(t,dy) 2 3 3 S xR J xR J

(2.4)

j»6C (R3)

v(0,-) = 6(z,-),

where

6(z,*)

denotes the unit distribution with mass at

z.

Then, using the

result of [8] it is not hard to prove that (2.4) has a unique solution, which we denote by

P f (t,z,*).

We can also prove that

(2.5a)

P,(t,z,*)

is a probability measure on

(2.5b)

u(t,r)

p f (t,z>r)f(dz),

(2.5c)

p f (t+s,z,r)

The above properties of follows.

Let

P f (t,z,dy)P u(t) (s,y,r)

P-(t,z,*)

enable us to construct a Markov process as

f! be the space of step functions

taking values in R , and
r€d3(R J )

z(t)

[0,™)

and

the o-field generated by (measurable) cylinder sets in 3

ft .

defined on

For each probability measure

construct a probability measure

f P f (*)

on

R on

fi i4 such that (£1,6)

so that

|x| f(dx)<°°, we can {ft,z(t),Pf}

is a Markov

process in the following sense. (2.6a)

P f (z(0)£ dxJ = f(dx) ,

(2.6b)

P f {2(t+s)€rjz(T):0< ! T< E t}=P u(t) (s f z(t),r)

t

a.s. (Pf) .

This is the Markov process in the sense of [5] associated with the gas of hard balls (2.1).

93 482

Now, our problem can be stated as follows :

construct a Markov process

x(t)

in the sense of [5] which is related to the Maxwellian gas without cutoff as

z(t)

is to (2.1).

So we take up the equation (1.1) with

rewrite it as in the form (2.3).

(2.7)

iHlSJQ.

QM

specified as in §1 and

A formal but careful calculation yields

{S>(x*)-JP(x)}QM(9)sined0diJJu(t,dx)u(t)dy) 3..„3 S >
Considering the singularity of function) from the space

QM(e)

1 3 C (R )

we do not treat (2.7) directly ;

at

6=0, it may be natural to take

1 of C -functions

with compact supports.

However,

instead we consider a suitable stochastic

differential equation as will be described below. Let

S=(0,l)x(0,Tr)x[0,2ir)

da®Q(d6)®diJj, where

and

X

the measure on

Q(d6)=Q (6)sin9d9.

remark that the only property of

defined by

dX =

Although we have thus specified

Q(d8)

(2.8)

S

Q(dO), we

we need later is

6Q(d8) (0,71)

and so our results remain valid for arbitrary measure condition (2.8). denote by

J*

Let

X

be the product measure

the class of Borel sets in

A family of random variables

{p(A,u)}

is called a Poisson random measure on

dtgtiA

(0,°°)xS _

Q(d8)

subject to the

on the space

which have finite

(0,™)xS

Each

p(A,w)

associated with the measure

is distributed according to the Poisson

distribution with mean

X(A).

and

A-measure.

defined on a probability space

the following two conditions are satisfied.

(i)

(0,°°)xS

(fl,6,P) X, if

94 483

(ii)

If

AltA2,-~€$,

A

then the family

rtA^Ustk)

{p(A, .tu)}, , „

p(A,w) = I p(A, ,w)

(2.9)

and

k=l

A=(jAk6^r,

is independent and

(a.s.)

k

Since a Poisson random measure always admits a suitable modification for which (2.9) holds for all

w, we may suppose, if necessary, that

stronger version of (2.9).

For a Poisson random measure

$1

where the notation random variables in

{p(A,oi)}

satisfies this

{p(Atu)).,*£

we set

= o{p(A,to): AC(t,«0*S},

o{

}

{

}

stands for the smallest measurable.

o-field that makes all the

Next, we regard the unit interval (0,1)

as a probability space by considering the Lebesgue measure (precisely, its restriction) on the Borel field of (0,1), and on this probability space we sometimes 3 consider an R -valued stochastic process (y(t,a), 04t<°>} with path functions which are right continuous and have left hand limits. ct-process

for simplicity ;

Such a process is called an

similarly a random variable defined on the probability

space (0,1) is called an a-random variable.

Now our stochastic differential

equation associated with (2.7) can be written as follows :

(2.10a)

x(t,w) =x(0,w)+

a(x(s,w),y(s,a),6,^)p(dsda,w).

(0,t]xS

(2.10b)

( y ( t , a ) , 0<=t<°°} in law t o

(2.10c)

For each

i s an ct-process which i s equivalent

{ x ( t , w ) , ()<£<»}. t>p,

the

o-field

a f x t s . t o ) , p(A,tj) i s independent of

/R

: o^s^t,

AC(0,t]xS}

95 484

Here

a(x,y,B,^)=x*(x,y,0,i/>)-x,

random measure associated with {x(t,w), QSX
do=dad6dq|s and of course A

{p(A,w)}

is a Poisson

defined over a (basic) probability space (fi,<^,P).

is unknown process to be determined by the equation (2.10a) under

the additional conditions (2.10b) and (2,10c).

Solving (2.10) gives rise to an

answer to the problem of finding Markov process associated with the Maxwellian gas, as will be seen in the next section.

53.

Solving the stochastic differential equation. Main results

Let

f

distribution

be a probability measure on

By a solution of (2.10)

3 f, we mean an R -valued stochastic process

over a suitable probability space (i)

3 R .

x(t,io)

(£1,(B,P)

with initial

{x(t,w), 0<X.} defined

and satisfying the following conditions:

is right continuous and has left hand limits with

probability 1, (ii)

P{x(0,u)£ dx} = f(dx),

(iii)

the relations (2.10a), (2.10b) and (2.10c) hold for some Poisson random measure

{p(A,w)}

associated with

defined over the probability space

A

and

(fi,(Q,P).

In solving the existence problem, the usual method of successive approximation can not be applied directly, since the function it can be shown that

a(x,y,B,i^)

a(x,y,6,ijj)

is not smooth.

However,

has a nice property similar to the Lipschitz

continuity (Lemma 1 ) . Owing to this property a kind of successive approximation method works, but in the results the usual pathwise uniqueness will be replaced by weaker one, the uniqueness in law. for initial distribution distribution

Here, we say that the uniqueness in law holds

f, if any two solutions of (2.10) with initial

f (which may be defined on different probability spaces) are identical

in law as stochastic processes.

In what follows, a periodic function on

a(x,y,8,ij0 R

viewed as function of

with period

2ir.

IJJ alone is regarded as

96 485

Lemma 1.

There exists a Borel function

ij; (x,y,x,y,)

of

x,y,x,y £ R

such that |a(x,y,e,^)-aCx)y,e,^+^oCx)y,x,y))|<:K{|x-x| + |y-y|}-e,

where

K

is an absolute constant.

In fact, ^ =0.

If

x?*y

can be defined as follows.

and

If

°

lx-yl

Z

the rotation around

2

—~-

determined by the three points

x, x

f

2

such that x+y —^-

straight line passing through the point

x+y-z).

px = x , where

x

JL is the

and perpendicular to the plane (when

x = y , we define

p

by

lies on the sphere

S^ ^ x,y

) + M

and hence we can define

.

IJJ (,0<^> <2TT)

formula

x (x,y,9,0) = x (x.y.e.^Cx.y.x.y)).

One of our main results is I |x| f (dx) J]x|f(d:

Theorem A. Assume that

(i)

there exists a solution distribution

.

Then,

{x(t,k>)j 0
_of_ (2.10) with initial

f,

(ii)

the uniqueness in law holds for initial distribution

(iii)

the probability distribution for any

(iv)

pz =

Now we set

x*(x,y,e,0). - g j - (px*(x > y > 9,0)- J&-

Then

x=y, we set ty (x,y»x,y)

x^y.+ »ty_

,* *\

|x-y|

p

or

x^y, we set

x

and denote by

x=y

u(t,*)

^f_ x(t,w)

f,

satisfies (2.7)

^eC^R3),

(a)

(conservation of momentum) E{x(t,u>)}

(b)

(conservation of energy)

provided

2

jI| x| x| | f (f(dx)<°°. dx)

is independent of

E{ |x(t,to) j }

t,

is independent of

t

by the

97

486

Here is an outline of the proof of the existence part (i). Choose a sequence {E, (a)} of independent a-random variables with the uniform distribution on n n n= o, 1, • * * (0,1), and set

%

- a{£. (a), O^k^n}.

3 we take an R -valued random variable random measure

{p(A,(j)}

{p(A,w)>, and set

x(0,w) \

associated with

x (t,u)=x(0,w)

-measurable a-process

Over a suitable probability space

for all

{y (t,a), O ^ K ™ }

with distribution so that t^O.

x(0,ui)

f

(ft,(ft ,P)

and a Poisson

is independent of

Then, taking arbitrary

^ -

which is equivalent in law to the process

{x (t,io), 04t<»}, we put

x1(t,to)=x(0,Ll))+

J

fr

a(xQl(s,o)),y (s,a) ,8,i(0p(dsda,w)

(0,t]*S In g e n e r a l , for

n>X we put

x n + 1 (t,a))=x(0,ui)+ ,ui)+

where

{y (t,a) , 0<.t<°°}

is an

-i(t>o!)>y (t,a)), 0^t<™}

f(y

x (t,u)), 0<.t<«}, and n —

f

J a(x n (s,oj),y n (s,aj) , 6 . i H ^ p C d s d o ,u) , J aUn(s,iu),yn(i (0,t]xS J T -measurable *) a-process such that the process

is equivalent in law to the process

{(x _.(t,w),

&-$., !+*„(* i(s,w),y — , (s ,a) ,x (s ,u) ,y (s ,a) ) , i|/ =0. n n~J. o n — l n J. n n o

Then by Lemma 1

(3.1)

) a(x n _ 1 (s),y n _ 1 (s) > e,^ n _ 1 )-a(x n (s),y n (s),0,^^)) < ; K{ix n (s)-x n _ 1 (s)| + iy n (s)-y n _ 1 (s)|}e,

and hence E|x n+1 (t)-x n (t)i
/

E|x (s)-x

x (s)\di

which implies

n=l

*)

J*-measurability is

imposed only to make the construction of {y (t,a)}

easier.

98 1+87

Therefore,

x(t)=lim x (t)

exists

( a . s . ) , and hence t h e same f o r

Again using ( 3 . 1 ) , i t can be shown t h a t equal t o

a ( x ( t ) ,y ( t ) , 6 , ifH-ip )

ljj.m a ( x n ( t ) , y n ( t ) .e.ifH-ijj^)

with some

tj> si|> ( t , a , w )

y(t)=lim y

(t).

e x i s t s and i s

except on a n e g l i g i b l e

set.

Now setting

i

p(A,w)=

I

X A (t,a,e,i|^$Jp(dtdo,u),

(0,«)*S we obtain (2.10).

The uniqueness part (ii) follows from the following Lemma 2. Let

f

be the same as in the theorem and

solution of (2.10) with initial distribution [0,T] and let

A

random variable process

f.

{x(t,oj) , 0^t<»}

any

Take arbitrary finite interval

3 be a partition of [0,T] : 0*t
{X (t), O^t^T}

X A (t)=X(0)+

{p(A)l and with distribution

f, we define a

as follows :

J

a(X(0) ,Y o (a) ,6,iJ0p(dsda) ,

Q<X^x

(0,t]*S

x A (t)=x A (t . »

/

a ( X A ( t 1 ) ,Y 1 (a) ,e,i|0p(dsda) ,

t-^t^

(t-^tJxS

* Vt)=Wl)+ J (t

where

Y (a) • • • ,Y o

Y. (a)

, (a)

n-r

a(X

ACVl}

,Y

n - l ( a ) » 8 »*)P< d s d a > >

t

n^i

= >

tJxS

are ct-random variables defined successively so that each

n — 1

••" •

has the same distribution as

distribution of

{X (t), O^t^T}

distribution of

{x(t), 0 <£
• •—

X (t,) .

—

• .

<

Then, any finite dimensional

converges to the corresponding finite dimensional as_ |A|= max Itv-ti-.il l
tends to 0.

99 488

{x(t,u), 0£:t
The following theorem shows that the process

is the Markov

process we were looking for. The proof is almost the same as that of Theorem A.

Theorem B. the process

(i) Let {y(t,a), 0<=c<°3}

{x(t ,03) , 0<=t<™}

there exists a solution

be any a-process equivalent in law to

constructed in Theorem A.

z(t,oi)

Then, for each

xfR

for the stochastic differential equation

f

2(

(0,t>S and the uniqueness in law holds for this equation, (ii) Set

p f ( t , x f r ) - p { z ( t > u ) e rl , and l e t

r£
{ x ( t , w ) , 0<X<°°} be t h e s o l u t i o n of ( 2 . 1 0 ) .

Then

p{x(t+s,u>)e r|^ t >=p u ( t ) (s ) x(t > i l )),r) where

( 8 t = a { x ( s , w ) ,p(A,w) : O^s^t,

a.s

AC(0,t]*S}

References

[1]

T. Carleman, Problemes Mathematiques dans la Theorie Cinetique des Gaz, Uppsala, 1957.

[2]

K. Ito, On stochastic differential equations, Mem. Amer . Math. Soc. No. 4 (1951).

[3]

D. P. Johnson, On a class of stochastic processes and its relationship

[4]

M. Kac, Foundation of Kinetic theory, Proc. Third Berkeley Symp.

[5]

H. P. McKean, Jr., A class of Markov processes associated with nonlinear

to infinite particle gases, Trans. Amer. Math. Soc. 132(1968),275-295.

on Math. Stat, and Prob. 3, 171-197.

parabolic equations, Proc. Nat. Acad. Sci. 56(1966), 1907-1911.

100 489

[6]

H. P. McKean, Jr., An exponential formula for solving Boltzmann's equation for a Maxwellian gas, J. Combinatorial Theory 2(1967), 358-382.

[7]

H. P. McKean, Jr., Propagation of chaos for a class of non-linear parabolic equations, Lecture series in Differential Equations, session 7, Catholic Univ. (1967).

[8]

A. Ya. Povzner, On Boltzmann's equation in the kinetic theory of

[9]

Y. Takahashi, Markov semigroups with simplest interactions I, II

[10]

H. Tanaka, Propagation of chaos for certain purely discontinuous

gases, Mat. Sb. 58(1962), 63-86.

(to appear in Proc. Japan Acad.).

Markov processes with interactions, J. Fac. Sci., Univ. of Tokyo, Sec. I 17(1970), 259-272. [11]

T. Ueno, A class of Markov processes with interactions I, Proc.

[12]

T. Ueno, A path space and the propagation of chaos for a Boltzmann's

Japan Acad. 45(1969), 641-646 ;

II, ibd. , 995-1000.

gas model, Proc. Japan Acad., 47(1971), 529-533. [13]

G. E. Uhlenbeck and G. W. Ford, Lectures in Statistical Mechanics, Amer. Math. Soc. Providence 1963-

[14]

E. Wild, On Boltzmann's equation in the kinetic theory of gases, Proc. Camb. Phil. Soc. 47(1951), 602-609.

Department of Mathematics Hiroshima University.

Reprinted from Proc. 2nd Japan-USSR Symp. on Probab. Th., 478-489, Lecture Notes in Math., 330, Springer-Verlag, 1973.

101

Proc. of Intern. Symp. SDE Kyoto 1976, pp. 409-425

On the Uniqueness of Markov Process Associated with the Boltzmann Equation of Maxwellian Molecules Hiroshi TANAKA

§1.

Introduction

The spatially homogeneous Boltzmann equation of Maxwellian molecules takes the following form: (1.1)

JMh*L dt

= f (u'ui - uu1)Q(dd)dedx1 , J

where Q(d6) — QM{&) sin Odd with a positive decreasing function QM{8) such that QM{6) ~ const, x 6~5/2, 0 J, 0, and the integration is carried out over (0, x) X (0,2x) X R3. In this paper we consider the following weak version of (1.1):

-A
(i.2)

at 3

where Q°(/? ) is the space of real valued C°°-functions on R3 with compact supports and (Kp)(xt JCJ = | {y>(x?) —
Xt = X0 + f

102 410

H. TANAKA

where S ~ (0, n) x (0, 2K) X (0,1), a{x, x,, 0, e) = x' — x and (i) p{dsd6deda) is a Poisson random measure on (0, oo) x S associated with dsQ(dd)deda, (ii) {Yt(<x), t > 0} is a stochastic process defined on the probability space {(0, 1), da}, and is equivalent in law to the process {Xt, t > 0} to be found. An alternative form of (1.3a) is (1.3b)

Xt = X0 + Z<*(s,X„Zs)

,

where a(s, x, a) = a(x, Ys(a), 0, e) for a = (6, e, a) and {Zt, t > 0} is the Poisson point process on S corresponding to the Poisson random measure p, i.e., defined by the relation p(A) = £ iA{s, Zs) for A e &(R+ x S). It has been also proved in [4] that the uniqueness in the law sense for solutions of (1.3) holds, however this uniqueness does not mean the uniqueness of the associated Markov process. The purpose of this paper is to fill this gap by showing that path functions of any Markov process associated with (1.2) can be represented as solutions of (1.3) after a suitable extension of basic probability space. Our task is to find a Poisson point process {Zt, t > 0} on 5 with characteristic measure Q(dO)deda such that (1.3b) holds. For this we have to find the compensator of the point process {Z„ t > 0} on Rz — {0} denned by Zt = Xt— Xt_; this will be done in § 3 and it will follow that Xt~XSi-\J]s
Markov process associated with (1.2)

We begin with the definition of transition function associated with (1.2). Denote by 0* the family of probability distributions / onjR3 satisfying |;t| f(dx) < oo, and for / € 0* we put J R* (Kf
{K
.

Definition. {ef(t, x, •):{€&*, t > 0, x e Rz} is called a transition function associated with (1.2), if the following five conditions are satisfied.

103 Boltzmann Equation

411

(e.l) For fixed / e 0>, t > 0 and x e R\ ef(U x, •) is a probability measure on R3. (e.2) For fixed Ae&(R3)y ef(tyx,A) is jointly measurable in

(f,t,x)e&>xR+

XR\

(e.3) For each t > 0 and / e &, there exists a constant c depending only upon / and / such that f (e.4)

\y\ef(s,x,dy)

< c(l + fjcf> ,

If we set u(t, •) = \

x € R3 .

0<s
f(dx)e,(t,*x, •)> then

J.R8

50 e Cs°(K3) .

<> = 9>0) + J <ef{s, x, •), Kuls)
(Kolmogorov-Chapman equation)

*,(*, x, •) =

e / s , x, dy)eu{s)(t ~ s,y, •) ,

0 < j < r,

where u is the same as in (e.4). Given a transition function {e^(t, x, •)} associated with (1.2), we can construct a Markov process on R*. To be precise, let Q be the space of Revalued functions on R+, and denote the value w(0 of <w( e Q) at t by Xt(m) or Z e . We put 3§ = a{Xt: f < co} and @t = ff{Z3: 5 < f}, where
ef(tXix>dxx) J AX

' '' I

e

u(tn-i)\tn

euitl)(t2 —

tx,xl9dx2)

J A3

*n-D Xn-D dXn)

.

JA„

We put Pf{ • ) = f f(dx)P}( • ), and denote by Ef (or E}) the expectaJ n* tion with respect to Pf (or P p . Then the following Markov property is proved by a routine method. For a nonegative ^-measurable function 0 on Q we have

104 412

H. TANAKA

(2.1a)

E}{0(6ta>)\<%t] = E%o{0} ,

PJ-a.s. ,

(2.lb)

Ef{0(Ota>) 10 t } = E*;t){0} ,

Pra.s. ,

where &t: Q —> Q is the shift operator defined by Xs(9ta>) — Xt+a(a>) for 0 < S < oo.

Lemma 1. If tp: R3 —* R is Lipschitz continuous, then (i) there exists a constant cx depending only upon the Lipschitz constant of
0 and f € £?, there exists a constant c2 depending only upon t, f and the Lipschitz constant of

0 < V5 <

t

,

f(dx)ef(s, x, •) .

Proof. From |*' — x\ < 6 \x — x^/2, < const. |* — xx\$ and hence | ( ^ ) U , JC2)| < const. |* — xx\

we have

\
6Q(d6)de = cx \x — xx\ , J (0,a)X(0,2«)

proving (i). (ii) follows from (i) and (e.3). The following lemma is an immediate consequence of the above lemma and (e.4), Lemma 2.

I]f R is Lipschitz continuous, then

<es{t, x, •),?>> = 9>W + I <*/(•*> *> ' ) , KU(s)9>ds . Lemma 3. Exf{\Xt — Xs\] < const. (1 + |*|) |r — s\ for 0 < s, t < T, where const, may depend upon f and T but not upon x. Proof.

For 0 < s < t we have

E}{\xt - xs\} =

ef{s, x, dy) \

euW(t - s9 y, dz) \z - y\ .

We now put
105

Boltzmann Equation

< cA

dt\

413

euw{T,y,dz)(l

Jo

Jfls

S

ef(s, x, dy) f

+ \z\) .

Therefore we have E}{\Xt - Xs\]
dz[ <2r

Jo

< const. (1

eu{s)(r, y, dz)(l + |z|)

^ ( J + T,x,dz)(l Jfls + |JC|)(*

+ |z|)

~ s) .

Theorem 1. The stochastic process {Xt> P}} has a modification which is right continuous and of bounded variation on each finite t-interval. Proof. Let Yt be any component oiXt. Let <2+ = {fk}k^i D e t n e of nonnegative rational numbers, and for any partition of [0, / ] : J : 0 = t0 < t, < • • • < tn = t ,

tjZQ+

set

(0<j
we put

Vi= Ui = V{-

±\Yh-Ytl_,\, (Yt -YQ)

= 2±

{YtJ - Yti_X

,

and then Vt = sup Vi,

Ut = sup Ui,

tzQ+i

where the supremum is taken over all such partitions A. For each t e Q+ and A: > 1 we write {rlt •••,/**} H [0, *] = {rs}l&j&n with 0 = r0 < • • • < zn = / and then denote by Jfc the partition of [0, /] with partitioning points {Tj}i£j£n. Then Vt is clearly the increasing limit of V{k as k\ oo, and hence E*{F ( } = lim E}{V?) = lim £ £*{| Y„ < const. (1 + |*|)* < oo ,

y ^ J

t e 0+ ,

and similarly £/{£/ t } < °°> ^ e i2+- On the other hand, since Vt and Ut

106 414

H. TANAKA

are non-decreasing in t e Q+, Yt = Y0 + Vt — Ut, t <= g + , has right hand limits. Therefore the limit Xt = lim Xt9 si t SS.Q +

teR+

exists and gives a desired modification of Xt. By virtue of Theorem 1, the probability measures Pxf and Pf can be constructed on the space of IP-valued right continuous functions on R+ having bounded variation on each finite t-interval. From now on, Q denotes this restricted space and so the ff-fields & and 3St are the ones on this new Q. We call {Q, 38, Xt, Pf:f€^} a Markov process associated with (1.2). § 3.

Point process {Zt}

3.1. In general suppose we are given a probability space (J2, 3?', P), a Borel subset S of Rd and an extra point d not belonging to S. An 5 U {devalued process {Zj( 0} defined on (Q, &, P) is called a point process on S, if i) Zt(a)) is jointly measurable in (t, a>), and ii) {t: Zt(ai) € S} is a countable set. Given a point process {Zt, t > 0} on S, we put PiA) = S ;u(*, zt) ,

Ae #((0, «>) x 5) ,

P«(» =

B € #(S) ,

2 X*(Z*) ,

*> 0 .

Given also a 0} is called a Poisson point process on S with characteristic measure X, if for any disjoint family Ax, • • -, An of Borel sets in (0, oo) x S such that l(Ak) = f rffcfcl < oo (1 < k < n) we have

p{P(^fc) = mk,k = i , . . . , « } = ft WlUk)

Q{Ak) mk

} )

for mj, • • •, m n e N . Then the following characterization of Poisson point processes is wellknown. Theorem 2. Suppose we are given a point process {Zt, t > 0} on 5, a a-finite Borel measure X on S and an increasing family { ^ } of suba-fields of &. If, for each B € 38{S) with X{B) < oo, {pt(B) - X(B)t,

107 Boltzmann

Equation

415

t > 0} is an {^^-martingale, then {Zt, t > 0} is a Poisson point process on S with characteristic measure X. In the above case, we also call [Zt9 t > 0} an {^J-adapted Poisson point process. 3.2. Suppose we are given a Markov process {Q, &, Xt, Pf-.fe^} introduced in § 2, and put Zt = Xt — Xt_. Then {Zt, t > 0} is a point process on R% = R3 — {0}. The purpose of this subsection is to find the compensator of {Zt). Fixing / e &>, we introduce the following notations. ^

= the completion of & with respect to Pf ,

SF\ = {A € &: Pf(A 0 B) = 0 for some B e &t] ,

We now think of the unit interval (0,1) as a probability space by considering the Lebesgue measure on ^((0,1)), and we take an .Revalued stochastic process {y ( (a), t > 0} defined on this probability space (0,1) with the following properties: (a) Sample paths of [Yt] are right continuous and have bounded variation on each finite ^-interval, and (b) {Yt, t > 0} is equivalent in law to the process {Xt, t > 0, Pf}. Next, we put S = (0, x) X (0, lit) X (0,1) and denote by a = (0, e, a) a generic element of S. On S we consider the Borel measure X defined by dX = Q(dd) ®de® da. We also put a(x, xu 0y 0, e) = x' — x , a(t, x, a) = a(x, Yt(a), 0, e) n(x, xl9 A) = f

for a = (0, e, a) ,

dx,^(A)Q(d0)ds

,

A e #(KJ) ,

J(0,
nu(x, A) =

n(x, JC15

^4)M(^1)

,

A e ^(i?o) •

J S3

Then, = <-*, iKfl('»y> •))> holds for any non-negative Borel function ^ on i? 3 withi^(0) = 0, where «(/, •) =

f(dx)ef(t, x, •)•

Finally we put Pt(
108 416

H . TANAKA

(3.1)

f ds u(s,dy) \ nuW(y,dz)\
< °o

-))}ds

is an {&t}-martingale with respect to Pj. In other words j\l9(s,Xt,a{s,Xs,-))yds is the compensator of pt( 0 we consider a partition of [0, t]: A: 0 = t0
** = ^{tjih^X^X^

- Xtk_x)] .

Then, with the notation
eAh-.19x,dy)

[

fc = l J - R 3

= S I

euitk^(tk

-

tk_l9y9dz)y*'v(z)

Jii3

*/('*-i>*» dy)\

k = l J R*

* * ds {

Jo

euitM_t)(s,y,dz)Kultt_i+a)p*>v(z)

,

J RS

= /i + i" , where «

r

ctk-tk-i

Jt-1 J f l l

JO

f JiJ3

X x.fe - y)^ U ( (i _ 1+S )P fc 'Hz) , and I'J denotes a similar formula obtained from l'd with the replacement of XA

JJ

lO

for[z-y|<£

109 Boltzmann

Equation

417

by x.fe - y) = 1 - x.(z - y). Lemma 4. // ^ e Q(i* + X -R3 X Rl), then for any e > 0 /j -> 0 as |Zf| = max (tk — ^.j) -* 0 . Proof, By Lemma 1 there exists a constant c independent of y, z, k and s such that \Ku(tk_l+s)
for|z|<e/2

p(z) = VZ \z\je - 1 for e/2 < |z| < e ,1 for|z|>e, £v(z) = cp(y - z)(l + \z\) , then we have xXz) < P(Z) , ft = l J f l s

Jo

JiJs

Since the support of
y, dz) in the above may be performed only on some compact set B of R3. Moreover, %y(z) is Lipschitz continuous as a function of z for each fixed y, and the Lipschitz constant is bounded as far as y is on B, Therefore, by Lemma 2 we have I « . ( M & , y . W b ) = I dr \ J

Jo

fl=

eMtk_l}(T,y,dz)Kuitk_l+l)^(z)

J£>

< const,

dt JO

^ ^ ( z ^ y ^ z X l + |z|) , Jfli

and hence |/i| < const. 2 f ^ f e . , , *, <*y) p 'fc_1
Jo

Jo

X [ eui^J?9y9ddQ.

+ \z\)

J R3

= const. 2 P* tk~1 ds Pdt f c / ^ . ! + r,x,dz)(l fc=>lJo

Jo

Jfl»

+ |z|)

110 418

H. TANAKA

< const. 2 (tk - tk^)2 — 0 ,

as | J | -» 0 .

Lemma 5. //

Define J(J), 0 < 5 < *, by

zt(0) = 0 ,

A(s) = *,_,

for ^

< ^ < *4 (1 < A: < «) ,

and for e > 0 (small enough) put

==

J-fig

0>Cs, >, z — y + w)x,(z — 30«fc Zi, dw) .

Then ^ = E \

e^k-i, *, dy)

ds

e^^is,

y, dz)

x f u{tk_x + s, dzd${tk-» y, z, zd Jfc = l J 0

Jo

= f da£?(f Jo

dsnfe, Yt(a), dz)

U[0,(]Xi?g

X f1 ^«r£:j{ f Jo

<&«(*,_, K,( a ),
Uco.oxsg

J

= E'f<\
Btfpfa)} = £?{£ <*, ?(*, X„ a(s, X3i -))>*} •

111 Boltzmann

Equation

419

Proof. We first consider the case

as |A\ -» 0. Since \
Xt^\
,

where K is some P^-integrable random variable. Therefore by Lebesgue's dominated convergence theorem, E/iPtty)} = Mm E'f{pH9)} Ml-o

= £ ? [ [ <J, <&} > and this proves (3.2) for smooth *r = £<^,9(r,X I ,fl(T,Jr T , .))>dr + f~' ), •),?&, XA8),a(T,Xt(e&)t -)))dt , where & , then we have qt{
Ef{qt_t(9>)o&s\^t} Pra.s. .

112 420

H. TANAKA

Thus {qt{
Xt = X0 + Z . S i Z , ,

Pra.s. .

Proof. For
Derivation of stochastic differential equation

Notations are the same as in the subsection 3.2. We now know the compensator of the point process {ZJ, and so the representation (1.3b) of {Zt} by means of certain Poisson point process {ZJ might be a consequence of general works due to Grigelionis [1] and Karoui-Lepeltier [2]. However we give the construction of {Zt} in detail, because we wish the whole proof to be self-contained. The construction given here seems to be simpler. Theorem 4. For each fixed f € 01 we can find a probability space {Q, $', P}, an increasing family {^t}> a mapping % from Q onto Q and an {#^-adapted Poisson point process {Zt, t > 0} on S with characteristic measure Xy having the following properties. (i) ff"1^) C &t, iz-l{3F) C # and P{*~KA)\ = Pf{A] for Az&. (ii) Xt°K — X0o7t + S ^ f l C ^ ^ . o f f . Z J , P-a.s., or what is the same Xtox

= Xaoiz +

a(XsoX, Y3(a), 6, e)p(dsdv) ,

P-a.s. ,

J «U]xS

where p(A) = 2

%A{t,

Zt) for A e #((0, oo) X S).

Lemma 7. There exists Q(t, y, z, A) such that ( i ) for fixed t > 0, y € JR3 and z e R3Q, Q(t, y, z, •) is a probability measure on 5, (ii) for fixed A e &(S), Q(t, y, z, A) is jointly measurable in (t, y, z), (iii) for any nonnegative Borel function
0, y e i? 3 , Z(do)
num(y, dz)Q(t, y, z, do)
For fixed t > 0, y e R3 and A e $(S) we put

113 Boltzmann Equation

421

<«(y, B) = £ **(«(*, ?, o))X{do) ,

£ e #(*}) .

Then £>(*, y, z, A), as a function of z, should be the Radon-Nikodym derivative of «|}(0(y, •) with respect to n u(() (y, •)- A nice version of Q(t, y, z, A) as stated in the lemma can be obtained by making use of the convergence theorem of martingales. In order to prove Theorem 4 we must prepare some probability spaces together with various quantities defined on them. 1°. {Q, J5", P}: This is, of course, the basic probability space on which our Markov process {Xt} has been given. Let B0 - {z € R>: \z\ > 1} ,

Bn = L

€

R*: — * _ < |z| < 1 } « > 1 >

and define {J^J-stopping times Tnk, n > 0, /: > 1, by rn0(<») = inf {* > 0 : Zt(a>) eBn} , Tnk(w) = inf {t > T^.fa):

n>0

Zt(a>) e B J ,

n ^ 0 , k > 1.

2°. {i7, ^"', P'}: This is the probability space obtained by taking Q' = (0,1), &' — #((0,1)) and P' = the Lebesgue measure (restricted on &'). On this probability space we choose a sequence of independent random variables {£Bjfc(a/): « > 0, /: > 1}, each being uniformly distributed on (0, 1). Moreover we choose a jointly measurable function Y(t, y, z, a>') on R+ x R* X Rl X Q! such that, for each t > 0, y e i?3 and z e JRS* ^ ( ^ y * ^ •) is an 5-valued random variable defined on {£', &', P'} and with probability distribution Q(t, y,z, •)• 3 °. {i5, # , P} : We choose an arbitrary probability space {Q, # , P} on which there is defined a Poisson point process {Zty t > 0} on 5 with characteristic measure X, and put # ( == a{Zs: s < t}. Now we can construct all that we need. (a)

Q = Q x Q' X Q ,

(b)

^

& = & <8> &'
P = P , (g> P' (g) P .

= the a-field on £ generated by all sets of the form

(4.1)

(AH B)xA'

where B e ^* t , A e ^

t

X A ,

and

^={r„ft
114 422

H. TANAKA

for some M
\*\lnJc;,-A-nk,Z,nk,l~nk)

7f

t=

{0

iit*Tnk

II t = I

nk

forVfo*),

where Xnk = XTnk_ and Znk = XTnk - XTtik_.

(e)

Zt

ifa(tyXt_,Zt)

0

otherwise

= 0

z, = z; + zj'.

Then, (i) of Theorem 4 is obvious with the self-evident notation jr (projection), and the rest will be proved in the following two lemmas. Lemma 8. {Zt, t > 0} is an {#^-adapted Poisson point process on S with characteristic measure X. Proof. Since the {# ( }-adaptedness of {Zt, t > 0} is clear from the definition of Zt, it is enough to prove that E{pt(
P-a.s.

for any Ji-integrable Borel function
^ i s ^so enough to prove that \jPM

~ Ps(P{A]

for any set A of the form (4.1). Since

we have (4.2)

f

{p{(0 - P . ( 0 W r ® P0

= f

X(sATnMY(Tnk,XnkiZnk,Znk))d(Pf®P>)

2 (n,fc)€Jtf J {AOB)XO'

= P'{A'}[da\ Jo

s #,¥l , (a,-,z„«))ff,*'

J-ifiBsO^t

= i*^'} P da f JO

dPfdmu(T}(XT_, dz)9{Y(T, * t _ , z, a))

J MnB)X(s,£lXfl||

*> x(z) # 1 for z * 0, and =0 for z = 0.

115 Boltzmann Equation

= f

d(Pf®F)

= f

d(Pf ® P') f

J UT\B)XA'

f

423

dtnuM{Xt_,dz)Qix,XT_,z,da)x{z)(p{a) dzX(do)x(a(z, Xt_, o))
J(*,t]XS

in the above we used Theorem 3 and Lemma 7. We have also (4.3)

^M'isp)

- p's\
= f

d(Pf 0 P') f {pftf(
= f

<*(P, ® *") f <*P f

JunB)X-l'

J^

rfW* Avl(<M

J (8,0x5

X { 1 -Z(fl(r,*,_,cr))te(
= LdP f Jx

^ ( ^ ) { 1 - Z(fl(rJt.,(j))Wff) .

J(»,e]xs

Combining (4.2) with (4.3), we obtain [ A P M - P*(
{ri(0 - ^ ( ^ ( P , (x) P')

= (*-*)<>!, ?>P{i}, as was to be proved. Lemma 9. a(/,y,0) = 0.

Zto% = a(t,Xt_,Zt),

P-a.s.,

in which we have put

Proof. Since the sets {t: Z\ e 5} and {t: Z " e 5} are disjoint with /-probability 1, we have a(f, * , _ , £ , ) =
=

a

\Tnk, Xnk, Y{Tnk, XnJc, Znk, £nk)) ,

P-a.s. .

Putting

&*(t,y,z) = xa£d\z - a(t,y, Y(t,y,z^nkW)))\ and then applying Theorem 3 and Lemma 7, we have

,

116 424

H. TANAKA

E\X(o,ti\Tnk) \Znk — a{Tnk, Xnk, Y(Tnk, Xnk, Znk, f

nk))\}

<,£{ s 0-\s,xs_,zs)) £ £'( f = -

dsu(s, dy)nu(s)(y, dz)&»'(s, y, z)\ dsu{s, dy)nuw(y,

dz)Q{s, y, z, da)xBn{z) \z - a(s, y, a)\

dsu(s, dy)X(d<j)xBn(a(s> y>ff))\a(s> ?> °) ~ a(s> y,
as required. Combining the above two lemmas with the corollary to Theorem 3, we now complete the proof of Theorem 4. § 5.

Concluding remark

The existence and uniqueness of solutions for the stochastic differential equation (1.3) were discussed by Tanaka [4]. The main results of [4] read as follows. (A) Let / € & be fixed. Then, on a suitable probability space {Q, # , P} with an increasing family {&t} °£ sub-<7-fields we can construct an {#J-adapted Poisson point process {Zt, t > 0} on 5 with characteristic measure X and an {#J-adapted right continuous process {Xt> i > 0} on R3 with initial distribution / such that (i )

J* E{\Xs\}ds < oo for each t < « ,

(ii) (1.3b) holds with probability 1, where {Yt(a), t > 0} is some right continuous process defined on the probability space {(0,1), da] and is equivalent in law to the solution process {Xt, t > 0}. Moreover, the probability measure on the path space induced by {Xt, t > 0} is uniquely determined from /. (B) Let / € & and x € R3 be fixed. Then, we can construct a Poisson point process {Zti t > 0} on 5 with characteristic measure X and an {#J-adapted right continuous process {X?, t > 0} on R3 such that (iii)

r^{|JSrj|}ds < oo for each t < oo,

(iv) X* = x + Z^t a(s, Xx„ Z8\ a.s., where a(s, y, o) is the same as used in (1.3b). Moreover, the probability measure on the path space induced by {Xxt, t > 0} is uniquely determined by / Bud x, and ef(t, x, A) = P{Xxt € A) ,

,4 € @(R?) ,

117 Boltzmann Equation

425

gives a transition function associated with (1.2) in the sense of § 2. Combining these results with Theorem 4, we obtain the existence and uniqueness of Markov process associated with (1.2).

References [ 1 ] Grigelionis, B., On representation of integer-valued random measures by means of stochastic integrals with respect to the Poisson measure, Liet. Mat. Sb., XI (1971), 93-108, (Russian). [ 2 ] Karoui, N. E. and Lepeltier, J. P., Repr6sentation des processus ponctuels multivaries a l'aide d'un processus de Poisson, Z. Wahrscheinlichkeitstheorie verw. Gebiete 39 (1977), 111-133. [ 3 ] McKean, H. P., A class of Markov processes associated with non-linear parabolic equations, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1907-1911. [ 4 ] Tanaka, H., On Markov process corresponding to Boltzmann's equation of Maxwellian gas, Proc. 2nd Japan-USSR Symp. Prob. Th., Lecture Notes in Mathematics, 330, Springer, 478-489. [ 5 ] Uhlenbeck, G. E. and Ford, G. W., Lectures in statistical mechanics, Lectures in Appl. Math., vol. 1, Amer. Math. Soc, Providence, 1963. DEPARTMENT OF MATHEMATICS HIROSHIMA UNIVERSITY HIROSHIMA 730, JAPAN

118

Z. Wahrscheinlichkeitstheorie verw. Gebiete 46,67-105(1978)

Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete © by Springer-Verlag 1978

Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules Hiroshi Tanaka Department of Mathematics, Faculty of Science, Hiroshima University, J-730 Hiroshima, Japan

Introduction The Boltzmann equation in the kinetic theory of dilute gases is the equation that governs the time evolution of the number density u(t,x) given by the number of molecules with velocitiesedx at time t the total number of molecules

u(t,x)dx

for a gas composed of a very large number of molecules moving in space according to the law of classical mechanics and colliding in pairs. Here we assume spatial homogeneity. When there is no outside force, the equation is du

It

J

{u'u'l—uu1)\x

—

xx\rdrd(pdx1,

f>0,

xeR1

( 0 , O D ) X ( 0 , 2 7T)XR3

where u = u(t,x), u1=u(t,x1\ u' = u(t,x') and u[=u(t,x'1). If we denote by SXXi the sphere with center (x + x^/2 and diameter | x - x j , then x' and xi (the velocities of molecules after "collision") are always on the sphere Sx x or more precisely SXtXi=Sx,tXi according to the conservation laws of momentum and energy. We consider a spherical coordinate system with polar axis defined by the relative velocity x — xl9 and put 8 = the colatitude of x'

I \-P-

(0.1)

?o

119

68

H. Tanaka

where V(p) = const• p ~ 4 and p0 is the positive root of

'-'"-R4F" £H F r o m the relation (0.1) we have \x —xl\rdr = QM(0)sin9d9 with some positive decreasing function QM(9) of 9 such that QM(9)~const-6~5/2 as # | 0 . Thus we have the following Boltzmann equation of Maxwellian molecules —= ^

J

(u'u[-uu1)Q(d9)d(pdxl,

(0.2)

(0,x)>c(0t 2TT)XR3 t

where Q(d0) = Q M ( 0 ) s i n 0 d 0 . F o r these matters, see Uhlenbeck and Ford [20]. The fact that Q(d9) does n o t involve |x — x j is a consequence of the inverse fifth power force, and in this sense the situation is simplified. But difficulties arise from the non-cutoff type IJ<2(
=
ZeCg(R3);

(0.3)

here CQ (R3) is the space of real valued C^-functions on R3 with compact support, (^{)(JC,JC1)=

J

{t(x')-t(x)}Q(d9)d
(0.4)

(0, 7t>x(0, 2*r)

and denotes the integral of £ with respect to a probability measure solution u = «(£,-) to be sought. The main objectives of this paper are the followings: (I) T h e construction of the M a r k o v process associated with (0.3) by solving certain stochastic differential equation. (II) T h e trend to the equilibrium for (0.3). Chapter I is devoted to the construction of the associated Markov process. A part of the present results was summarized in [ 1 8 ] ; here we will give full proofs. The idea is to use the following stochastic differential equation X(t) = X(0)+

| (0,

a{X{s-),Y(s-,al9,(p)N(dsd9dq>dal t]xS

(0.5a)

120 Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

69

or what is the same thing, X(t) = X{0)+ 5 > ( X ( s - ) , Y(s-lp(s));

(0.5b)

here S = (0,7r) X(0,2TC) X ( 0 , 1) and (i) {p(t)>t>0} is a Poisson point process on 5 with characteristic measure Q(dd)dq>da, and N(dsd6d(pdrx) is the corresponding Poisson random measure defined by N(A) = YitA(s,p(s))1 for Ae<%(R + xS), (ii) { y ( t , a ) , t ^ 0 } is a right continuous R e v a l u e d stochastic process defined on the probability space {(0,1), da} and is equivalent in law to the solution process {X(t), t ^ O } , (hi) a{x,xl,9,(p) = x'-x and a(X(s-\ Y{s — ), o) = a(X{s-),Y(s-, a), 6, q>) for 2 the space of probability distributions / on R 3 satisfying §\x\2f(dx)
1

R3xR3

\x-y\2F(dxdy),

P(flJ2) = V^fJl), where the infimum is taken over all probability measures F in R 6 satisfying F(A xR*)=fl{A) and F ( R 3 xA)=f2(A) for any , 4 e ^ ( R 3 ) . For fe^2 satisfying J|x 2 — m\ f(dx) = 3v>0 where m is the mean vector of/, we put Qf(dx) = (2nv)~312 exp{ — \x —

m\2/2v}dx,

e(/)-e(/,9/). It will be seen that p gives a metric in 0^. In the one-dimensional case the functional e was introduced in connection with the study of Kac's one-dimen1A denotes the indicator function of A throughout

121 70

H. Tanaka

sional model of Maxwellian molecules by Tanaka [17], and a part of the results in [17] (concerning some basic properties of e itself) was then extended to the several dimensional case by Murata and Tanaka [12] and to the case of Hilbert spaces by Kondo and Negoro [8]. The main results of Chapter II are as follows: (A) The nonlinear semigroup {Tt} on @>2 is non-expansive with respect to the metric p: p(Ttfi> Ttfz)£pifiM

t^Q, / l f / 2 € ^ 2 .

(B) Iffe&2 satisfies j | x — m\2f(dx) = 3v>0 where m is the mean vector off, then t(Ttf) decreases to 0 as f\ co, and hence in particular Ttf converges to g / as rf oo. The (rigorous) entropy arguments in dealing with the trend to equilibrium require the existence of initial densities with finite entropy. According to our method, though it works only for Maxwellian type, we need less restrictions on initial distributions for proving the trend to equilibrium. Also, the result (A) will provide a typical example of a semigroup of nonlinear operators which are nonexpansive with respect to certain metric. I wish to thank T. Ueno; I came to be interested in Maxwellian molecules through conversations with him.

Chapter I. Associated Markov Process § 1. Definition of Markov Process Associated with (0.3) Let us denote by ^ the family of probability distributions / on R 3 satisfying J \x\f(dx)
Definition, {ef(t,x,'):fe0>lit^O,xeR3} is called a transition function associated with (0.3), if the following five conditions are satisfied. (e.l)

For fixed fe0[, f£0 and

JCGR3,

ef(t,x,*) is a probability measure on R3.

(e.2) For fixed ^4e^(R 3 ), ef(t,x,A) is jointly measurable in ( / , f , x ) e ^ x R + x R3, the Borel structure on ^ being the one induced by the usual vague topology on ^ . (e.3) For each t^O and fe£?u there exists a constant c depending only upon t and / such that

5\y\ef{s,x,dy)£c(l+\x\),

O^s^t, xeR 3

R

(e.4) If we put

u{t,-)=\f{dx)ef{t,x,-\ R3

(Km®(x)=l(KQ{x,Xl)u(t,dXi), R3

122 Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

71

then for £ e C £ ( R 3 )

(ef(t,x,-),0

= Z(x) +

o

]<ef(s,x,-XKuis)Ods.

(e.5) (Kolmogorov-Chapman equation) ef{t,x,-)=

$ ef{s,x,dy)eu{s){t-s,y,-),

O^s^r,

R

where u is the same as in (e.4). In the cutoff case nQ(d9)
solutions to (0.3) can easily be obtained

from Wild's formula ([21, 10]); a similar formula can also be used to obtain ef(t, x, *) defined for all probability distributions / on R 3 . In the non-cutoff case with which we are concerned in this paper, the restriction fe0[ is imposed in the above definition since our present method works only under this restriction. To proceed, let Q be the space of Revalued function on R + , and denote by Xt(a>) (or Xt, for short) the value co(t) of a>(eQ) at t. We put @ = a{Xt: t
P r a.s.,

O^s^t.

Thus we obtain a (temporally inhomogeneous) Markov process {Xt,Pf,fe0[}. This is a Markov process which is associated with (0.3). In the above we have assumed the existence of {ef(t,x,*)}9 but we do not know its existence in advance; the analytical proof of the existence seems to be difficult. What we are going to do in Chapter I is, as stated in the introduction, to employ the method of stochastic differential equations in order to obtain an associated Markov process.

§ 2. Preliminaries from Poisson Point Process

Suppose we are given a complete probability space (Q, J^ P), a Borel subset S of Rd and an extra point d not belonging to S. An Su{5}-valued process {p(t,co), t>0} defined on (Q,^,P) is called a point process on S, if i) p(t,co) is jointly measurable in (t,co\ and ii) the set {t:p(t,o))GS} is countable. Given a point process {p(t),t>0} on 5, we put JV(^) = I tA{t,p(t)\

Ae&{(0, oo) x S)

t

Nt(B)=

£

lB{p(s% Be^(S), t*Q;

123 72

H. Tanaka

N(*) is the associated random measure. Let X be a given a-finite Borel measure on S. Then, a point process {p(t), t > 0 } is called a Poisson point process on S with characteristic measure X, if for any disjoint family {A1,...,An} of Borel sets in (0, o o ) x S such that X(Ak) = j dtdX< co ( 1 ^ / c ^ n ) we have

P{N(Ak) = mk, \
^ T m

k-

for m 1 , . . . , m w e N . The following characterization of Poisson point processes is well-known. Theorem 2.1. Suppose we are Borel measure X on S and an each Be@(S) with X(B)0} is a Poisson point case, {p{t), t>0} is said to be

given a point process {p{t), t>0} on S, a o-finite increasing family {J^} of sub-o-fields of 3*. If, for {Nt(B)-X(B)t,t^Q} is an {^-martingale, then process on S with characteristic measure X. In this {^-adapted.

We often deal with integrals of the form £ A{s,p(s),co)= s£,t

j"

A{s,o,(o)N{dsda),

(0,r]xS

where {p{t), £>0} is a given { # J - a d a p t e d Poisson point process on S with characteristic measure X and iV(*) is the associated Poisson random measure. When A (t, a, co) is predictable 2 satisfying the integrability condition J E\A(s,a,co)\dsX(d(j)
t)xS

£(I^(s,p(s),co)}-£{ s^I

j" A{s,a,w)dsX(da)},

(2.1)

|0,I]xS

and if in addition A(s, <J,CO), z^s^t, £{expOC X = exp { (z,

are ^ - m e a s u r a b l e in co for some T, then

A{s,p{s),w)WT} (e''^ (s - *• w» - 1) d s X(d a)},

j

CeR.

(2.2)

t]xS

For X(t) = X(Q)+ X A(s,p(s),co) and ^ e C 1 ^ ) we have £{X(t)) = £{X(0))+ Y {aX(s-)

+ A(s,p(s),co))-c:(X(s-))}.

(2.3)

The following lemma is also elementary, but we give the proof for completeness. 2

A real valued function A (t, o, co) on R + x S x Q is said to be predictable, if it is measurable with respect to the predictable a-field; the latter is defined as the smallest cr-field on R+xSxQ with respect to which all real valued functions a{tta,(o) with the following properties (i) and (ii) are measurable. (i) For each fixed t^.0, a(t,o,
124

Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

73

Lemma 2.1. Let A(t,0 and assume that there exists A{a,ciy) such that \A(t,a,co)\SA(a,(o) E{^A(a,w)l(do)}
for 0 ^ ^ 7 ; (2.4)

Then, for any e > 0 there exists a partition A of [0,T~\: A: 0 = t0
=T

such that \A\ = ma.x(tk — tk__1)<£ and E{

{

\A{Uo,o))-A{A{i),o,a))\dtX{do)}<£,

(0, r j x s

where A (t) is defined by A (0) = 0 and

Proof For convenience we redefine A{t,a,co) for t>T by putting A(t,
for k2-"
+ l)2~n (k = 0, ±1,...).

Then by (2.4) we have for each t I

lim j \A{t + s,CT,a>) - A(<5„(t) + s, a, co)| ds = 0 almost everywhere with respect to X®P, and hence lim

j

\A{t + s,o,co)-A{Sn(t)-\-s,a,co)\dsdtA(d
n~KX> (0, D x R x S x f l

Therefore, there exist se(0,1) and nl
j

such that

\A{t + s,
or equivalently HmE{ j |A(t,CT,co)-A(dBk(t-s) + s,CT,co)|dtA(dff)}=0. But this formula clearly implies the existence of a partition A as stated in the lemma. § 3. Two Lemmas We state two lemmas. The first one is of particular importance. 3.1. We set a(x,x1,9,(p) = x' — x,

125 74

H. Tanaka

and as a function of q> we extend it to the periodic function on R with period 2 71. This function depends upon the choice of the origin q> = 0 in a spherical coordinate system on the sphere SXX{. We can easily see that no choices of the origin
) in the variables x and x l 5 but we can prove that a(x,x1>8,(p) has a sort o/Lipschitz continuity which is enough for our later developments. Lemma 3.1. There exist a constant c and a Bor el function
+

on R 12

+ (p0{x,x1,y,y1))\

\x1-y1\}e.

Proof, (i) When x = xu we put (po(x,x,y,y1) = 0. Since a(x,x,0,
\a(y,ylAq>Ul-^^e

=

^iilx-yl + lxi-y^O. (ii) When y = yy, we obtain a similar result with (pQ(x,xiiy,y)=0, (iii) We assume that x + xx and y^yx. Let / be the straight line which passes through the point (x + x t )/2 and is perpendicular to the plane determined by the three points (x + xJ/2, x and x*, where >=J*-*il

y-yx

x~t-xl

\y-y,\

1

2

°

We denote by p the rotation around / sending x to x*. Also we define the transformations x and x from R 3 to itself by \y-yx\

/

x+xA

2

x + Xj

2

Then we have px = x*,

TX*=x>,=-y- + -^—,

and xxp sends the sphere S

txt=y,

to the sphere Sy

. So, if we put

A{x,x1,0,
=

A{y,yi,d,(p0{x,x1,y,yl)).

126 Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

75

We then have the following formula. xxpA{x,x1,9,(p) = A(y,yl,6i
y1)). We now claim that \a(xtxu6,| + K - ^ | } f t

(3.1)

To show this, we first notice that \a{xtxlte,q>)-(pA(xtxltet
Z~\x-x*\0, \(pA{xtxlt8ttp)-x*)-(zpA(xtxl9et
^H\x-xi\-\y-yx\}0^H\x-y\+\xi-yi}8, TpA(x,xlf9,(p)~Tx*

—

(3.3)

xxpA(xyxl,6)(p)~!i'cx*

=A(y,yl>e,(p +

= a(}',)'i,0>

(3.4)

From (3.2), (3.3) and (3.4) we then have \a{x)x1,6,(p)-a{y>y1,0,
+ <po{x,x1,y,y1))\

^||3c-**|0+i{|x-yl+l*i-yil}ft which combined with the following inequalities proves (3.1): \x-x*\£\x~-y\

+ \y-xj

^|x-y| + x + xx ^Ix-yl

+ \x

-x*\

y + y} 2 2 + lx-yl + frt-y^

x-x,

\y-yi\

\y-yi\

3.2. In this paper we often consider stochastic processes having sample paths in the following space W: W=the space of Revalued right continuous functions on R + having left limits. In W we consider the Skorohod topology. Then it is well known that W is a completely metrizable and separable space (see [7, 14]) and that the topological Borel field fflw on W coincides with the usual coordinate c-field. We think of the unit interval (0,1) as a probability space by considering the Lebesgue measure (strictly speaking, its restriction to $(0,1), the cr-field of Borel sets in

127

76

H. Tanaka

(0,1)). A stochastic process defined on this probability space and having sample paths in W is called an a-process for simplicity; similarly a random variable on this probability space is called an a-random variable. We sometimes want to have a-processes constructed as in the following way. Lemma 3.2. Suppose we are given two processes Xl-{X1{t),t^:0} and X2 = {X2(t), t^.0] defined on a common probability space (Q,^P) and having sample paths in W. Let Y 1 = {Yl(t,a),t'^.0} be an cc-process which is equivalent in law to X j . We assume that there exists an a-random variable n which is independent of\l and uniformly distributed on the interval (0,1). Then we can construct an a-process Y 2 = {Y 2 (£,a),t^0} in such a way that (i) the joint process (Y^Yj) is equivalent in law to (X 1 ,X 2 ) and (ii) there still exists an a-random variable which is independent of\2 and uniformly distributed on (0,1). Proof. Denote by U the probability measure on Wx W induced by the joint process (Xj,X 2 ), and by U1 the one on W induced by X j . Since W is a complete metric separable space, there exists a transition function P(w,A) of X 2 given Xj with the following three properties: For each fixed weW, P(w,*) is a probability measure on W.

(3.5)

For each fixed Ae$w> P(*,A) is a Borel function on W.

(3.6)

For any AltA2s^8w UiA.xAJ^SP^AJU.idw).

(3.7)

Since any complete metric separable space having the same cardinality as R is Borel isomorphic to R (see [14]), there exists a Borel isomorphism 0 from W into R. We fix such a

ae(0,1).

Then Y(w,a) is jointly measurable, and for each fixed weW the distribution of F(w,-) on W is P(w, •). Taking two (arbitrary) independent a-random variables rji and n2 with the uniform distribution on (0,1), and regarding Y 1 (a) = {Y1(t,a), £^0} as an element of W, we can define a W-valued a-random variable Y2 by Y2(a) = Y(Y1 (a), n^//(a))). Then the joint process (Y l5 Y 2 ) is clearly {/-distributed, and n2{n(a)) is a uniformly distributed a-random variable independent of Y 2 .

§ 4. Stochastic Differential Equation We use the notations introduced in §3, such as the function a t x , ^ , #,
128

Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

77

measure on S defined by dX = Q(d6)d0} on S with characteristic measure l and (iii) an Jvmeasurable R3valued random variable X. Let N(dsd6d0}. Then X and {p{t),t>0}(N{dsd8d(pda)) are independent. We now consider the stochastic differential equation dX(t) = a(X(t-),Y(t-\e,q>)dN,

X(0) = X,

(4.1)

whose precise meaning is X{t) = X +

j" a{X(s-\Y(s-,a\e,q>)N{dsd9d
a.s.,

(4.2a)

(0, t] x S

or equivalently X(t) = X + X a{X(s-),

Y(s~),p(s)l

a.s.,

(4.2b)

where {X(t\ t^O} is to be sought as an {^}-adapted process with sample paths in W under the condition that {Y(t,a), t^O} is an a-process equivalent in law to {X(t), t^.0}; the notation a(xy Y,o) for an Revalued a-random variable Y is defined by a(x, Y,a) = a(x, Y{<x)A
(4.3)

In the right hand sides of (4.2a) and (4.2b) we may (and sometimes do) replace the left limits X(s —) and Y(s —) by X(s) and Y(s), respectively. However, the use of the left limits seems to be suited for the intuitive meaning of the motion: a particle changes its velocity by the interaction with another similar independent particle. We use the following notation. For fuf2e0[ we put Pi(/i,/ 2 )=inf

J \x-y\F{dxdy\

(4.4)

FeFR3xR3

where F = F(/ 1 ,/ 2 ) is the class of probability measures F on R6 satisfying F{A xR3)=f1{A) and F(R3 xA)=f2(A) for any v4e^(R 3 ). Then it is clear that the infimum in (4.4) is attained at some FeF{f1,f2). Also, it can be proved that p± gives a metric in ^J; in fact the triangle inequality can be proved as follows. Given fl7f2tf3e^lt we take F lG FC/i,/ 2 ) and F2eF{f2,f3) such that Pi{fiJ2)^=i\x~y\Fi(dxdy\

Pil/2J3) = Ilx-ylF 2 (dx£2y).

We can easily construct a probability measure F on R 9 satisfying F(A x R3) = Fl(A) and F(R 3 xA) = F2(A) for any Ae@(R% and we have

PiUkiJiHPiU'2J3)^i^-y\P^dydz) + ^Ix-zlFidxdydz^p^J,).

i\y~z\F(dxdydz)

129 78

H. Tanaka

The existence and uniqueness of the solution to (4.1) are now our objectives. We begin with the uniqueness part. Denote by / the probability distribution of the initial value X. Lemma 4.1. Assume that E{\X\}<<x>9 that is, fe&[. constant, A a partition of the interval [0, T ] :

Let

T be any

zJ:0 = t 0 < £ 1 < - - - < t „ = 7: and define a process {XA(t),0^t^T}

XA(i)=XA(tk)

+

£

positive

(4.5) by

a(X,(tk),Yk,p(s))fortk
(4.6)

where Y0,..., Yn__l are a-random variables defined in each step so that Yk has the same probability law as XA(tk). Then we have the following assertions. (i) The probability law of the process {XA(t)t Q^t^T} is uniquely determined by f (and so does not depend upon the choice ofY0,..., Yn_2). (ii) Let X* be another ^-measurable random variable with probability distribution / * in &u and define {X*(t), O^t^T} by a rule similar to (4.6) replacing X by X#. Then, enlarging the probability space if necessary, we can construct two processes {X(t), O^t^T] and {X(t), O g t ^ T } which are equivalent in law to {XA(t\ O^t^T} and {X*(t\ Q^t^T], respectively, and satisfying E\X(t)-X(t)\^e^Pl(ff#) with c0 = 4nc$6Q(d6), o

(4.7) where c is the constant appearing in Lemma 3.1.

Proof (i) By (2.2) we have

cxp{iC-XA(tk)^(t~tk)\(e^a^^y-B-^-i)uk(dy)Q(dO)d
=

tk£t£tk+1, where uk{dy) is the probability distribution of XA(tk). formula.

CeR3,

Then (i) is clear from this

(ii) Enlarging the probability space if necessary, we can assume that there exists an /*-distributed random variable X such that E\X — X\= p ^ff*). For each k ( 0 ^ / c < n ) , denote by uk and uk the probability distributions of XA(tk) and XA(tk), respectively, and then take a-random variables Yk and % with distributions uk and uk, respectively, in such a way that EJYk — Yk\=p1(uk,uk) holds. We now put X(Q) = X and X(0) = X, and assume that {X(t)} and {X(t)} are defined for 0^t^tk. We first define a(x, Yk,a) for <7 = (0,
<po(X(tk),Yk(<x),X(tklYk(0L)),

130 Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

79

using the function
£

a(X(tk),Yk,p{s)),

£

a(X(tk),Yk,p(s)),

tk<s^t

for tk
+ 1.

By virtue of (2.1) and Lemma 3.1 we have for

tk
E\X(t)-X(mE\X(tk)-X(tk)\ + c1(t-tk){E\X(tk)-X(tk)\

+

pl(uk,u*)}

S{l+2Cl(t-tk)}E\X(tk)-X(tk)\,

(4.8)

where cx = 2iic\dQ(dQ). Now (4.7) follows from (4.8). The proof is finished. o Lemma 4.2. Let T be any positive constant and A a partition of the interval [0, T ] given by (4.5). Let Yk, 0^k
+

I

a(XA(tk),Yk,p(s)),

tk
(4.9)

tk<S^t

Then we have the following

assertions.

(i) The probability law of {XA(t), O^t^T] is uniquely determined by f and uk, 0^k
tk
+1

(0^k
(4.10)

n

where cx =2nc\QQ{dO). o

In particular, if pl{uk,uk)<efor

E\X(t)-X*(t)l^e^lPi(f,f#)^8}.

0^k
then (4.11)

The proof of this lemma is similar to that of Lemma 4.1 and so is omitted. Lemma 4.3. Let {XA(t), O^t^T} be the same as in Lemma 4.1. Then, any finite dimensional probability law of {XA(t), O S i f ^ T } is convergent as |zl|-»-0. More

131

80

H. Tanaka

precisely, if • : 0 = s o < s 1 < ••• <sm = T is another partition of[0, T], then we can construct two processes {X(t), O^t^T} and {X(t), O ^ r ^ T } which are equivalent in law to {XA(t\ OfSt^T} and {Xn{t\ O^t^T}, respectively, and satisfying E\X(t)-X(t)\Sc2(\A\

+ \n\),

O^tgT,

(4.12)

where c2 = 2n\QQ{dQ)^p{2n{\+2c)T\eQ{de)\E\X\. (4.13) o o J Proof We may assume that • is a sub-partition of A without loss of generality. First we construct {XA{t), O^t^T} as in (4.6) using auxiliary a-random variables Y0>...,Yn_1, and then put X(t) = XA(t\ O^t^T. Each Yk can be arbitrarily chosen under the restriction that it is equivalent in law to XA(tk). Now we require, in addition, that each Yk satisfies the following condition: There exists an a-random variable which is independent of Yk and uniformly distibuted on (0,1). The process {X(t\ O^t^T}

(4.14)

must be constructed more carefully. We put

X(t)=X + Z a(X, Y0,p(s)),

0Zt£Sl.

Assuming that X{t) is defined for 0St^sk, we define X(t) for sk),X(sk)); this is possible by virtue of (4.14). Putting a{X(sk\ %,a) = a{X{sk), Yk{a),6,cp + q>0),

a = {9,cp,a),

X

by

a(X(sk),Yk,p(s)).

In this way we can construct X(t) for 0 g t^ T, and it is not hard to see that thus constructed {X(t), O ^ f ^ T } is equivalent in law to {Xn(t), Q^t^T}. We assume that sk^t<=sk+1 for a moment. Since X(t) = X(sk)+

£

a(X(tk.),Yk.,p{s)),

sk<s^t

using Lemma 3.1 we have E\X(t)-l(t)\ ^E\X(sk)-X(Sk)\+E{

J

\a(X(tk,lYk,,
(Sfc, I ] x S

SE\X(sk)-X(sk)\Ht-sk)2nc]9Q(d9){E\X(tk,)-X(sk)\+Ea\Yk,-Yk\} o = E\X(st)-X(sk)\ + c0(t-sk)E\X(tk,)-X(sk)\,

(4.15)

132 Probabilistic Treatment of the Boltzmann Equation of MaxwelHan Molecules

81

n

where c0~4nc\6Q(d6). o

On the other and

E\X(t)\^E\X(tk)\+(t-tk)E$\a(X(tk), Yk,o)\X(do) s ^ElX(tk)\Ht-tk)E\{lX(tk)-Yk\/2)0X{d
+ c'(t-tk)E\X(tk)\,

where c' = 2n$9Q(d6), o

tt
+ l,

and hence by Gronwall's inequality

E\X(t)\^E\X\ec,t,

O^t^T.

Therefore E\X(sk)-X(tk.)\^(sk-tk.)E\X(tk,)\ Sc"(sk-tk,),

c" =

E\X\c'ec'T,

and hence E\X(t)-X(t)\

+ d'\A\ + c"\A\} e<°«-s-\

S{E\X(sk)-X(sk)\

sk
+ u

which implies that E\X(t)~X{t)\Sc"\A\{eCot~l\

O^t^T,

as was to be proved. In what follows, a process {X(t)} E{ sup |X(s)|}
(4.16)

is said to be integrable for simplicity, if

0£s£l

Lemma 4.4. Given an ^-measurable random variable X with E{\X\} < oo, we assume that there exists an integrable solution {X(t),t^0} of (4.2). Let T be any positive constant, A a partition of [0, T ] and {XA(t), O^t^T} a process of Lemma 4.1. Then, any finite dimensional probability law of {XA(t), O ^ t ^ T } converges to the corresponding one of {X(t), O^t^T} as \A\->0. More precisely, on a suitable probability space (Q^^P) we can construct two processes {X(t), O ^ t ^ T j and {XA{t), Q^t^T}, which are equivalent in law to {X(t), 0 < i r ^ T } and {XA{t\ O ^ r ^ T } , respectively, and satisfying E\X(t)-XA(t)\Sc2\A\,

O^t^T,

(4.17)

with the constant c2 given by (4.13). Proof. Define A{t\ A(0) = 0, A(t) = tk

Q^f£T,by for tk
(0Sk
(4.18)

82

H. Tanaka

and put XA{t) = X+ X a(X(A(s)), Y(A(s)),p(s)). Also, we define {X*(t), O^t^T]

by

#

X (0) = X, X*(t)=X*(tk)+

£

a(X*(tk),Y(tk),p(s))

for tk
(Og/c
(4.19)

where, in each step, a(X # {tk\ Y{tk\ a) is defined to be equal to a(X * (tk), Y(tk, a), a,

+

Ciit-tJEWtJ-X*^

^{l + cl(r-g}£|^(rJ-JT*(fJJ| + c1(f-tJ£|2r(rJk)-^(a

(4.20)

where Cj is the same as in (4.10). Now, if we put e(A) = E

J (0,

\a{X{t\Y{t),rj)-a{X{A{t)\Y{A{t)),o)\dtX(dc\

T]xS

then E\X{t)-XA{t)\Ss{A)

for O^t^T, and hence (4.20) yields

£|^(0-X#(t)|^e(zl)(^T-l), #

(4.21)

ClT

E\X{t)-X {t)\Se(A)e .

(4.22)

#

Since (X (£)} is equivalent in law to {X%(t)} which is defined by a rule similar to (4.19) with respectively, and satisfying (4.10). Since uk and u* are the probability distributions of X(tk) and X(tk), the uses of the triangle inequality for pl and the estimate (4.22) result in pduk,uk*)SE\X(tk)-X*(tk)\ igE\X(tk)-X*{tk)\

+

E\X*(tk)-X(tk)\

+ e(A)e^.

Therefore, (4.10) yields E\X(t)-X*(t)\<{l+c0(t-tk)}E\X(tk)-X*(tk)\ + Cl{t-tk)s(A)eClT,

tk
+ 1,

which implies that E\X{t)-X*{t)\^e{A)eCiT{eCoT-l),

O^t^T.

(4.23)

By virtue of (4.22) and (4.23), on a suitable probability space (Q, &, P) we can construct two porcesses {X(t\ O^t^T) and {Xd{t\ O^t^T} equivalent in law to

134 Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules {X(t\

O^t^T]

83

and {XA(t), O ^ t ^ T } , respectively, so that they satisfy

E\X(t)-XA(t)\SE\X(t)-X*(t)\+E\X*(t)-X(t)\ ^e(A)e3ciT,

O^tST,

(4.24)

Now by an application of Lemma 2.1 we can make both s{A) and \A\ arbitrary small, and hence the right hand side of (4.24) tends to 0 as \A\ -*0 via some subsequence {Am}. Combining this fact with Lemma 4.3, especially with the estimate (4.12), we can easily prove the assertion of the lemma. The proof is finished. Making use of methods similar to those employed in Lemma 4.3 and 4.4, we obtain the following lemma in which {Y(t)} is an a-process given in advance (we do not require that it is equivalent in law to the solution process). Lemma 4.5. Given an ^-measurable random variable X with E\X\ < oo and also an integrable a-process {Y(t)} which is continuous in the mean, we assume that there exists an integrable solution {X(t)} of X(r) = X + 2 > ( * ( s - ) ,

Y(s-),p(s)).

Let T be any positive constant, A a partition of [0, T] and {X A(t), O^tf^T} the process obtained by (4.9) with Yk = Y(tk). Then, any finite dimensional probability law of{XA{t), 0 ^ t S T} converges to the corresponding one of{X(t), 0 S t S T} as \A | ->0. More precisely, on a suitable probability space (Q,^,P) we can construct two processes {X(t), O^t^T} and {XA(t), Q^t^T} in such a way that they are equivalent in law to {X(t), O ^ t ^ T} and {XA(t), O ^ t ^ T], respectively, and satisfy + zY(*)eClT,

E\X(t)-XA(t)\Sc3\A\

O^tSX

(4.25)

where c2^n] M=

eQ(d6)exp
+2C)T]

9Q(d6)l{E\X\ + M),

Ea\Y(t)\,

0|!£T £r(zJ)=max

EJY(tk

+

1)-Y(tk)\.

The following uniqueness theorem follows immediately from Lemma 4.1 and 4.4. Theorem 4.1. The uniqueness in the law sense holds for integrable solutions of (4.1), that is, the probability law on W of any integrable solution of (4.1) is uniquely determined by its initial distribution f iffe^. More precisely, if{X(t)} and {X*(t)} are any integrable solutions of (4.1) with initial distributions f and f* ( e ^ ) , respectively, then on a suitable probability space (Q,&,P) we can construct two processes {X (t)} and {X * (t)} in such a way that they are equivalent in law to {X(t)} and {X*(t)}, respectively, and satisfy E\X(t)-X*(t)\Sem,Pl(fJ*),

t£0.

(4.26)

135 84

H. Tanaka

Next we deal with the existence theorem concerning (4.1); the precise statement of this is given as follows. Theorem 4.2. Let fe^ be given. Then, on a suitable probability space {Q,^,P} with an increasing family {^t} of sub-o-fields we can construct an {^-adapted Poisson point process {p{t)} on S with characteristic measure X so that (4.1) has an integrable solution with initial distribution f In order to prove this theorem it is convenient to introduce another stochastic differential equation which will turn out to be essentially the same as (4.1). It is expressed as dX{t) = a{X{t-\Y{t-\e,(p

+
X{0) = X,

(4.1*)

and a solution {X(t)} of this equation should be found as an {^-adapted process with sample paths in Plunder the conditions that {Y(t)} is an a-process equivalent in law to {X(t)} and that (p* = (p*(t,<x,ca) is an {^-predictable process. Always, q> + (p* should be interpreted mod27r. Now, for
j

tA(t,dy

Ae@((Q,<Xj)xS).

(0, c c ) x S

Then by Theorem 2.1 {N*(dtd(r)} is again an {J*}-adapted Poisson random measure corresponding to X, and (4.1*) is nothing but (4.1) with N replaced by N*. Therefore, for the proof of Theorem 4.2 it is enough to prove the following theorem. Theorem 4.3. Let {p(t)} be an {^-adapted Poisson point process on S with characteristic measure X, and {N(dtdo)} the associated Poisson random measure. Then, for any ^-measurable R2-valued random variable X with £|X|
(4.27)

We then define {X^t)} by X1(t) = X+ X

fll(X0(s-),

Y0{s-\p(s)),

where X0(s) = X, Y0(s) ~ Y,
(p0{Xn_1(t-lYn_1(t-),Xn(t-),Yn(t^)),

= a(Xn(t-),Yn(t-,a),e,
+ cp*(t,a,co)),

Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

85

for c = {6,(p,ct), and define {Xn+1(t)} by Xn + ,{t) = X+ X an + l(Xn(s-\

Y„(s-),p(s)).

(4.28)

Thus we obtain a sequence of processes {Xn(t)}, n^.1. By Lemma 3.1 we have K+l(Xn(s-),Yn(s-),o)-all(Xft_l(S-lYn_1(s-),a)\ ^C{l^,(s-)-A'1I_1(s-)| + |yB(s-)-i;_1(S-)|}ft

(4.29)

and hence E\Xn+l(t)-Xn(t)\
J

{|X )1 ( S -)-A' 1I _ 1 ( S -)| + |y B ( s -)-y B _ 1 ( s -)|}0dsA(rf«T)

(0,t]xS

= c 0 j£|X„( S )-X„_ 1 ( S )|d S . o Since £|.Xr1(f)-.Xr|^c0E|.X'|* we have E\Xn+1(t)~Xn(t)\SE\X$c0ty^/(n

+ $l

and hence £ { s u p |X n+1 (s)~X n ( 5 )|}

ZE{ll\a,

l(Xnis-\Y,(s-),p(s))-a.(X._1(s-\Y._1(s-),pm

+

sir

££|X|(c 0 t)" + 1/(n + l)!. Therefore, Xn(t) converges almost surely to some limit X(t) as n -»• oo uniformly on each finite t-interval, and hence 7n(t) also converges almost surely to some a-process {Y(t)} which is obviously equivalent in law to {X(t)}. These convergences together with the inequality (4.29) imply the almost sure convergence of an(XH_ x(t —), Yn_1(t — ),) = a{X(t-),Y(t-\Q,(p + q>*). Now letting n\oo in (4.28), we see that (4.1*) is satisfied by the triple (X{t\Y(t\
t^O,

(4.30)

137

86

H. Tanaka

where c'v = Vvn\dQ{dQ). o

When v = l,2, we have

E{X(t)}^E{X}, 2

2

E{\X(t)\ }=E{\X\ },

t^O,

(4.31)

^0.

(4.32)

Proof. Let {XA{t), 0 < U S T] be the same as in Lemma 4.1. Then, by Lemma 4.4, in order to prove (4.30) it is enough to show E{\X4(tY}^ec'vtE{\X\vl

OSt^T.

(4.33)

We first notice that jE{|Xd(t)|v} is bounded in te[0, T ] ; in fact this follows from the fact that the v-th moment of I

\a(XA(h),Yk,p(s))\,

tk
+1

conditioned on the cr-field ^tk is given by

t 4

I vi,

- f ^ - f ft ((t-h)\\a(X,(tk\ Yk,oTX(do)).

•••, V j i ;

1

We next write ak = a(XA (tk), Yk, p(s)), ak = a(XA (tk), Yk, o) and then apply (2.3) to (4.6); the result is \XA{t)Y = \XA{tk)Y+

£

{\XA(s-) +

si\XA(tkW + v I

\ak\{\X,(s-)\

Therefore, we have for tk
atf-\XA(s-)n + \ak\V-1,

tt
+1

+ vE

J" (tk,

la.UlXM

+

la.iy-'dsXidv).

t]xS

On the other hand, since |a k |{|X J (s)| + |a f t |} v _ 1 is dominated by (e/2){\XA(tk)\^\Yk\}{\XA(s)\^\XA(tk)\

+

\Yk\y^

zy-Ho/iwxM^lXAitkW + mi, we have for

tk
E{\xA(t)n ^E{\XA(tk)n

+ cv

J

E{\XA(s)\v + \XAtJ\v +

( ( k , ( ] x ( 0 , 1)

= {l + 2cv(t-tk)}

E{\XA(tk)\v} + cv]

E{\XA(s)\v}ds,

cv=y-xv7i]eQ(de). 0

Now an application of Gronwall's inequality yields (4.33).

\Yk\ldsd*

138 Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

87

When £{|X| 2 } < GO, we take the expectation in \X(t)\2 =

\X\2+^{\X(s-)^a(X(S-),Y(s-),p(s))\2-\X(s-)\2} sgt

to obtain E{\X(t)\2} = E{\X\2}+

{ E{\X(s) + (0,

= E{\X\2} +

a(X(slY{s),cr)\2~\X(s)\2}dsX(da)

t]xS

l{\x'\2-\x\2)Q{de)d
^EIIXI^+J 1 ^^ 1 ^ 1 2 " 1 ^ 2 "^ 1 1 ^^^^^^)^,^,)^ = E{\x\2}, proving (4.32), where u(s, •) denotes the probability distribution of X(s) and the last two inegrals are performed on (0,7t) x(0,2n) xR 3 x R 3 x(0, £]. The equality (4.31) can also be proved by a method similar to the above. The proof is finished.

§ 5. The Transition Function and the Markov Process Associated with (0.3) In this section we show that the solutions of (4.1) give rise to a Markov process which is associated with (0.3) in the sense of § 1. As in the preceeding section, (p(t)} and {N(dtda)} stand for an {J^J-adapted Poisson point process on S with characteristic measure X and the associated Poisson random measure, respectively. By virtue of the uniqueness in the law sense for solutions of (4.1) we may write i^ for the probability distribution on (W,$w) induced by any integrable solution of (4.1) with initial distribution fe0[. Given fe t?u we take a Pydistributed a-process {Y(t)} and consider the stochastic differential equation dX{t) = a{X{t-\

Y{t-),e,(p)dN,

X{0)=x.

(5.1)

Although this equation has the same expression as (4.1) (except for the initial value), it should be noticed that {Y(t)} of (5.1) is a given a-process and so, of course, is not required to be equivalent in law to the solution of (5.1). As in the case of (4.1), the stochastic differential equation (5.1) is essentially equivalent to dX(t} = a{X{t-\Y{t-\e,q>

+ (p*)dN,

X(0)=x,

(5.1*)

in which q>* = (p* (t, a, co) is an {^-predictable process. The existence of a solution of (5.1*) can be proved by a method of iteration similar to that used in solving (4.1*), and from Lemma 4.5 and (i) of Lemma 4.2 it follows that the probability distribution on (W,<%w) of any integrable solution of (5.1) (or (5.1*)) is uniquely determined by / and x; we denote this probability distribution on (W,$w) by Pxf. Also we denote by Xt{w), or Xt for short, the value w(t) of weW at time t, and put @t = (r{Xs: s^t}, &= v &t{ = @w). Here the notation Xt should not be confused with X (t); the former is defined on W while the

88

H. Tanaka

latter is on Q. Combining (4.11) with (4.26) and then using Lemma 4.5, we have the following assertion; F o r any x, yGR 3 a n d / ge3\ we can construct two processes {X{t)} and {X * (t)} on a suitable probability space (Q, #", P) in such a way that their probability distributions on the path space W are Pxf and Jj* respectively, and that E\X(t)-X#(t)\SeCiC{\x-y\^e^Pl(f,g)y

t^O.

This assertion immediately implies the following lemma, in which we put = Pxf{XleA},

ef(t,x,A)

Ae@(R3).

(5.2)

Lemma 5.1. (i) For any Be@tw, P^{B) is jointly measurable in (f, x ) e ^ x R 3 . (ii) For each A e&(R3),ef(t,

x, A) is jointly measurable in (f,t,x)e0[

xR

+

xR3.

It can be easily verified that the function ef(t,x,A) of (5.2) satisfies (e.3) of § 1. (e.4) can also be verified by first applying (2.3) to a solution X(t) of (5.1) and then taking the expectation. Theorem 5.1. For any xeR3,fe0[,

Ae&(R3)

and 0^to

Pxf{XtieA\<%lo}=eui!Jt1-t0,Xto,A\

PJ-a.s.,

(5.3)

Pf{XtleA\ato}

Pra.s,

(5.4)

= eu{to)(t1-t0tXto>A),

where u(t0) = u(t0,')

=

Pf{Xtoe-}.

Proof. For a fixed £ 0 ^ 0 , if we put p#(t)=p(t0

+ t),

&* = {4>,G},

t>0, ^*

=

<j{p*(s),0<s
then {p* (t)} is also an {^* }-adapted Poisson point process on S with characteristic measure h Let N # {d td a) be the associated Poisson random measure. Taking a Pfdistributed a-process {y(t)}, w e define a i^ (to) -distributed a-process {Y*(t)} by Y*(t)=Y(t0 + t), f^O, and then consider the stochastic differential equation dX*(t) = a(X*(t~),

Y*(t~),

X*(0) = yi

(5.5)

where (p * — (p# (t, a, co) is a suitable {J^ # }-predictable process. By a method of iteration similar to that used in solving (4.1*), we can construct a family {X*(t),yeR3} of integrable solutions of (5.5) together with {^ # }-predictable processes (p * =

\

f *(*> <*><»)

„

,

for

°- f< ^° for t>t0. for 0 ^ t < £ 0

Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

89

Since J^0 and {J2;*} are independent and the process {X*(t)} is i^toj-distributed, we have for tl >t0 PiX^tJeAl^J -P{X%o){tx-t0)eA\^tQ} =P{X*(t1 -t0)eA},

y=X(t0),

a.s.

t

=Pultf {Xt.toeA}=eu(tJt-t0>X(t0lA),

a.s.

(5.6)

On the other hand, from (i) and (ii) it follows that {X°(t)} is {^-adapted, 2 The purpose of this section is to prepare some lemmas concerning e and p, defined below, for the use in later sections. We denote by ^ 2 the space of probability distributions on R3 with finite second moments. F o r / and g in ^ we put <E(F)=

J"

\x-y\2F(dxdy),

R3xR3

z{f,g) = inf <E(F)f

p(f,g) = V^Oi),

FeF(f,g)

where F(/, g) denotes the family of probability distributions F on R6 satisfying F(A x R 3 )=/(^) and F(R 3 x A) = g(A) for any Ae@{R3). Since F(/,g) is compact with respect to the topology induced by the usual convergence as probability distributions on R 6 and since (£(F) is continuous on ¥(f, g), the infimum value e(/, g) is attained at some Fe¥(f, g). As in the case of pt of §4, it can be proved that p gives a metric on ^ 2 . However, when we speak of a convergence in ^ 2 , we always mean that it is the usual one as probability distributions unless ^-convergence is explicitly stated. The proof of the following lemma is elementary and so is omitted. Lemma 6.1. (i) The p-convergence implies the usual convergence in 3P2. (ii) Iffn^f inland if lim sup N->co n^l

J |x| 2 / n (^) = 0,

(6.1)

\X\>N

then {fn} is also p-convergent to f. In general, f„->f and g„^>g in 0>2 imply e(/,g)^lime(/„,gj. n— co

141 90

H. Tanaka

Let £ > 0 be fixed. We denote by g£ the Gaussian distribution exp(~\x\22s)dx on R 3 and put

(2ney312

^ . ) = {/*fl i |/G^ 2 and / ( { | x | > l / f i } ) = 0}. Then we have the following lemma. Lemma 6.2. For each pair (f g ) e ^ E ) x ^ j £ ) , there exists a unique Ff< ge¥(f g) such that 1&(Ff g) = t(fg). Moreover, the mapping 0 from ^ ( E ) x ^ j e ) into the space &(R6) of probability distributions on R 6 , defined by
and G(F) = e(/,g), then

F ( 4 xB) = J^ ( x ) (i4)g(dx) for any . 4 , B e ^ ( R 3 ) with a suitable Borel mapping ij/ from R 3 into itself,

(6.2)

where 5 W J C ) (*) denotes the ^-distribution at i/^(x). In fact, (6.2) was proved in [12: Theorem 1] in the special case when g is the Gaussian distribution with the same mean vector and variance matrix as those of/, and the proof in [12] is also adapted, without any change, to the more general case when g has a strictly positive density with respect to the Lebesgue measure. Next, assume that Ft and F2 are in F ( / , g) and satisfy <&(F1) = &{F2) = t{f,g). Then, F = (F 1 +F 2 )/2 also belongs to ¥(fg) and satisfies (£(F) = e(/,g), and so by (6.2)

with some Borel mappings ij/, \j/^ and \JJ2. But this formula clearly implies that \J/ = t//1 = t//2, g-a.s., and hence F1 =F2. This proves the first half of the lemma. Finally, to prove the second half, we assume that/ n - • / a n d gn -*• g in ^ £ ) , and write J^ = Ff . Obviously {Fn} is relatively compact in ^*(R 6 ). Let F be any limit point of {Fn}. Then by (ii) of Lemma 6.1 we have e ( / g ) = lim e ( / „ , g j = lim <£(FB) = <E(F), which implies that F = Ff>g by the uniqueness part of the lemma, proving the continuity of 0. Thus the proof of the lemma is finished. Lemma 6.3. Let (Q, &, P) be an arbitrary probability space and suppose that we are given sub-families {fm,ojeQ} and {gw, coeQ] of^satisfying the following conditions. (i) For each Ae&(R3)tfa(A) and g<°{A) are ^-measurable in o>. (ii) The probability distributions f= $f<°P(do)) and g = $ g01 P(da>) belong to 0>2. Then we have e(/g)££{e(/»g»)}.

(6.3)

142 Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

91

Proof, F o r each e > 0 and fe^2 let fE stand for the probability distribution / * g e , where * denotes convolution and f(A) =f(A n {|x| £ 1/e}) + / ( { | x | > 1/fi}) S0{A), Then we h a v e / £ - » / a s ejO, lim N-.cc

sup 0 < E < 1

j

Ae@{J!L%

\x\2fE(dx) = 0, and hence e(/£5gE)-»-

\X\>N

e(/,g) as e | 0 for any f,ge^2 by (ii) of Lemma 6.1. Next, denote by F£w the unique probability distribution on R 6 such that FE°>eF(f?,g£) and ®(FEta) = e(j?,gf). Since each mapping in

is measurable (the last mapping is continuous and hence Borel measurable according to the preceeding lemma), F™ is also measurable in to. Therefore <£(ij") = e t / f , g " ) is J^-measurable in co, and it follows that lim£{e(ir,gr)}=£{e(r,gto)},

(6.4)

because the integrand e(f^,g^) is dominated by 2$\x\2f Wfl>) = £ { e t / r , g f ) } . Now letting e | 0 in the above and then noting (6.4), we obtain (6.3). To state the last lemma of this section, let us denote by Cx r , the circle with center x ( e R 3 ) , of radius r and lying on a plane which is perpendicular to a unit vector /. Also we denote by UXtTtl the uniform distribution on Cxr[; this can be regarded as a probability distribution on R 3 and so UXtrtle&2. Lemma 6.4. For any x,yeR3, *(Ux,rJ,Uy!S<m)<\x-y\2

r , s > 0 and unit vectors I and m, we have + r2±s2-rs{l

+ \(l,m)\}-

Proof. In proving the lemma, without loss of generality we may assume that x = 0, / = (0,0,1) and m = (0, -siny,cosy) with O^y^Tt/2. Let Q=[0,2iz), P be the Lebesgue measure in [0,2n) multiplied by 1/2n and put X(co) = (rcosco, rsinto, 0), Y(co) = y + (scosco, ssincocosy, ssincosiny),

COEQ.

Then X and Y are random variables that are uniformly distributed on CXtfil and CyjS m, respectively. Therefore, *(Ux,rJ,Uy,sJSE{\X-Y\2}=^

$

\X(co)-Y(a))\2dco.

143 92

H. Tanaka

By elementary calculations we see that the last term is equal to \y\2 + r2 + s2 — rs(l +cosy), completing the proof.

§ 7. Non-expansive Property of the Associated Nonlinear Semigroup with Respect to the Metric p Let X = {XtiPf,fe@[} be the Markov process associated with (0.3). We associate with each t ^ O a n d / e ^ the probability distribution Ttf on R 3 defined by Ae09(R3).

(Ttf)(A) = Pf{XteA},

Then, the Markovian property of X implies the semigroup property of {Tt}, that is, Tt+sf = TtTsf,

M^0,/e^.

Since fe&2 implies Ttfe3?2 by Theorem 4.4, {7^} is also a nonlinear semigroup on ^ 2 . The purpose of this section is to prove that Tt is non-expansive with respect to the metric p on &2. First we prepare a lemma of an approximation type. Namely, we prove that the Markov process associated with (0.3) can be approximated in an appropriate sense by the one associated with ^<M,£> = <«®«,KE£>,

£eCS>(R 3 )

(7.1)

for small £ > 0 , where KE£ is defined by (0.4) with the replacement of Q{d9) by Qe{dO) = t{E>n)(9)Q{dd). As in §4, we take an {^{-adapted Poisson random measure {N(dtd<j)} corresponding to the measure X. Then the Markov process associated with (7.1) can be obtained by the family of solutions of dX(t) = a£(X(t-),Y(t-),6,(p)dN,

(7.2)

where aE(x,xi,6,(p) = l(E7t)(9)a(x,xl,6,
+

X tk<s£t

a£(XA(tk),Yk\p(s))

for tk
+l

(0^k
144 Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

93

where Y£,..., Y„e_ t are a-random variables defined in each step so that Yk6 has the same probability law as XEA(tk). Then, if E{\X\2} < <x>, we can construct two processes {XA(t), O^t^T} and {XEA(i), O^t^T] in such a way that they are equivalent in law to {XA(t), OSt^T} and {XA(t), O ^ t ^ T } respectively and satisfy E{\XA(t)-XA(t)\2}^const]eQ(d6). o Here, const depends on T but neither on e nor on A. The following approximation lemma is an immediate consequence of the above and Lemma 4.4 Lemma 7.2. (i) Let T be a positive number, se(0,n) and fe&2. Then, on a suitable probability space (Q,&,P) we can construct two processes {X(t), O^t^T] and {X£(t), 0 ^ t ^ T} in such a way that they are equivalent in law to solutions of (4.1) and (7.2), respectively, with initial distribution f and satisfy E{\X(t)~Xe(t)\2}Sconst]9Q{d$) o with const depending on T but not on s. (ii) p{Ttf Tr(E)f)^0

as elOfor

each t £ 0 and fe@2.

Before stating the theorem of this section, we introduce some notations. For each 6e(0, n) and x ^ e R 3 , we put IIX

Xi 0

= UZ rJ = ihQ uniform distribution on C z

r/;

where z = {x + x 1 + ( x — x^jcosd}/!, r = \x — x1\(sin9)/2a.ndl = (x — x1)/\x — x j , and regard II x XltB as a probability distribution on R 3 . For any probability distributions /, g on R 3 and 6e(0, n), we define another probability distribution {f°.g)9 on R 3 by

(f°g)e(A) = J nXiXlJA)f(dX)g(dXl),

AB®(V).

R3xR3

Obviously <(/°g)*f> =

\ (0, 2 7 t ) x R 3 x R 3

^{x')d
£eC?(R3),

and hence ( / ° g ) 9 e ^ 2 provided / , ge0>2. We write [ / ] e = ( / ° / ) f l for short, and put W g ) = e(/,g)-e([/]..[gU Theorem 7.1. For each t^0Ttis is, p(Ttf,Ttg)Zp(f,g),

non-expansive on 0>2 with respect to the metric p, that

f,g<=&>2.

145

94

H. Tanaka

More precisely, for anyf,ge&2 we have e e (/,g)^0,

(7.3)

t(TJ>TtgUtU:g)-2n\ds]t9(TJ9Tag)Q{d0i. o o Our discussions are devided into two cases according whether

(7.4)

n

\Q{dQ)
or =oo.

Case I. First we discuss the special case in which q = 2n\Q(dd)< co. In this case, o for each pair of probability distributions f and g on R 3 we can define a probability distribution / o g on R 3 by

fog = (2n/q)](fog)eQ(d6). o With this notation the equation (0.3) is equivalent to d (u,0 =
^eC^(R 3 ).

(7.5)

A unique (probability) solution u(t) of (7.5) for any given initial distribution / can be obtained by a method of iteration, and the solution is explicitly expressed by the so-called Wild sum ([21]): 00

u(t) = e~qt £

{l-e~qt)n-1fitt\

Here/ ( n ) , n ^ l , are probability measures on R 3 defined inductively by

f{1)=f, fln)=^nifik)°f(n-k\

n>l.

On the other hand, from what we have proved in Chapter I we know that 7J/is also a solution of (7.5) with initial distribution/, at least i f / e ^ . Therefore, we have 00

Ttf=e~« £ ( l - e - « r - 7 ( " \

f^i-

The proof of the theorem in Case I will be based on the above Wild sum and the following three lemmas. Lemma 7.3.

146 95

Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules where l + cos0 l-cos0 —-^ (x-y) + — = — ( * i - J ' i ) sin'

{|x-x1|2 + | > ' - y 1 | 2 - | x - x 1 | | y - 3 ; 1 | - | ( x - x 1 , y - y 1 ) | } .

Proof. It is enough to apply Lemma 6.4 with the replacements: x -»• {x + x j -h (x — x j) cos 0}/2,

y -^ {y + y i + (y — y i) cos 0}/2,

r - • | x - x j (sin0)/2,

s -* | y - y j ( s i n 0 ) / 2 ,

/ -+(x-x1)/|x-x1|,

m-^^-^iVly-yil-

Lemma 7.4. Lef/ x ,/ 2 , gi and %i belong to @v Then we have the following (i) e [ ( / 1 o / 2 ) f l , ( g 1 o g 2 ) f l ] g l ± ^ e ( / l ! g 1 ) + 1 ~ ^ ° S (ii) c ( / 1 o / 2 , g 1 o g 2 ) ^ y e ( / 1 , g 1 ) + ( l - y ) e ( / 2 , g 2 ) ,

e(/ 2 ,g 2 ),

inequalities. 6e{09n).

where

y = {2iz/q)] 2~ : ( 1 + cos 9) Q(d9). o Proof. We choose two pairs {X1} Y±] and {X 2 , Y2] of random variables so that they satisfy the following three conditions. (a) E{\Xl-Y1\2}=c(fl,g1\ E{\X2-Y2\2} = i{f2,g2). (b) F o r i = l , 2 , X{ i s / r d i s t r i b u t e d while Yi is g r distributed. (c) {Xu Y J and {X2, Y2} are independent. Then we have

and hence by Lemma 6.3 and 7.3 e[(/i°/2)8»fei 0 S2)J ^£{e(i7jri(X2i9,i7ri,r2ifl)}=£{*e(X1,X2)y1,y2)}

= £• l l ^ ^ - y j + l Z ^ ^ - y j +•

sin'

•E{\Xl-X2\2-+\Y1-Y2\2-\Xi-X2\\Yl~Y2\

-l(x 1 -x 2 ,y 1 -y 2 )i}. We now use the inequality

(x 1 -x 2 ,y 1 -y2)^^ 1 -x 2 i|y 1 -y 2 i

(7.6)

147 96

H. Tanaka

to obtain C[(/l°/j)fl.bl°«2)J 1-COS0

1 + COS i

<E

-(Xl-*i)+-

(X2-Y2

sin'

+^ T -£{!x 1 -x 2 | 2 + |71-y2|2-2(x1-x2,y1-72)} (l + cosS) ;

^.-m+Mr^Wi-m2}

^ 4^ - ^ H ^ p r . - n i * } 1-cosfl

1 + cos 0

This proves (i), and (ii) follows from (i) and Lemma 6.3. The proof of the lemma is finished. Lemma 7.5. For any f and g in &2, we have

e(f{n\gin))S*(f,gl

m\.

(7.7)

P r o o / S i n c e (7.7) is evident for n~l, it is enough to prove that (7.7)holds for n=m assuming that it holds for n<m. Making use of Lemma 6.3 first and then (ii) of Lemma 7.4, we have e(/(m\gn = e

/

<m

m 1

1

~

£ 1

(k)nfim-k) fm°f

_1_-| ^ . ^

e(/*»o/*-»>,g«)og<--'))

k=l m - 1

<

T

I

{K(/< fc) ,g (k) ) + ( l - y ) e ( /

(m — k)

r,(m-k}\

^e(/,g), as was to be proved. Now the proof of the theorem in Case I is completed as follows. (7.3) is immediate from (i) of Lemma 7.4. To prove (7.4), we notice that CO

Tt+sf=e-'iS

^ n= 1 co

(l-e^r-HTjn

Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

97

and then apply Lemma 6.3 and 7.5. The result is t{Tl+sf,Tt+sg) ^e^t(TJ,Ta) s + (l-2e-«

+ +

e-^(l~e-^tl(TJf2\(Ttgn e-2«s)e{Ttf,Ttg),

and hence lim {t{Tl+sf,Tt+sg)~t{Ttf,

Tt g)}/s

S - q {HU Tt g) - e [(T;/)<2>, (TJgp]} ^-q(2n/q)]{t(T[f,Ttg)-<>llTtne,lTfgle]}Q(de) o =

-2iz]ze(Ttf,Ttg)Q(dO). o

The inequality (7.4) now follows from the above, since e (Ttf, Ttg) is continuous in t. The proof in Case I is finished. n

Case II. We deal with the case when \ Q(d6) — GO. For each ee(0, n\ the result in o Case I is applicable to the semigroup {7/E)} which is associated with QE(d6), and hence t{Tt^f, Ttvg)^(f,g)-2n]ds]

€e(TVf, Ts^g)Q(d9). 0

(7.8)

E

On the other hand, making use of (i) of Lemma 7.4 and (ii) of Lemma 7.2, we have pllT^ne,[TJUSp(VE>f,Tsf)->0 as 4 0 and hence pKT^ne,lT^gU^pllTsn0,lTsg\l

40.

Therefore we have te(Ts{E)f,Ts(e)g)^-£e(Tsf,Tsg) as 4 0 , the convergence being bounded. Now, letting e|0 in (7.8) we obtain (7.4). Thus the proof of the theorem is completed.

§ 8. Theorem of Ikenberry and Truesdell on Time Evolution of Moments The result on the time evolution of the moments for solutions to Boltzmann's equation of Maxwellian molecules goes back to Ikenberry and Truesdell [4]. In [4], however, the existence of solutions of the Boltzmann equation is not discussed rigorously. Here we state and prove the theorem of Ikenberry and Truesdell in our setting, for completeness. We state also a corollary; this will be useful in the next section where a more precise result on the trend to equilibrium will be obtained in connection with our metric p. The method of [4] is to use harmonic polynomials. For each fc^Owe choose 2k + 1 linearly independent (homogeneous) harmonic polynomials {^(x)}|,|
149 98

H. Tanaka

degree k in R 3 and put U*)H*l2r£«

for n = (r,k,l),

•=(0, o, o ) W ~ !•>

where r — 0, i, ...,k = 0,1,..., and \l\^k. The degree of £n is |n| = 2r + /c. Then it is well-known that any homogeneous polynomial of x with degree n can be expressed by a linear combination of £n{x) with |n| = n, and therefore when dealing with moments of a probability distribution / on R 3 it is sufficient to consider only the (harmonic) moments ^ ( / ) = < / , 0 Lemma 8.1. (i) Let h(x), \x\ = 1, be a spherical harmonic of degree k and y be a unit vector in R 3 . On the unit sphere S2 = {\x\ = 1} we take a spherical coordinate system with polar axis y, and denote by y and \jt the colatitude and the longitude, respectively, of a point xeS2. Then 2ir

J h(x)d\j/ = 2nPk(cosy)h{y), o

(8.1)

where Pk denotes the Legendre polynomial of degree k. (ii) If £{x) is a (homogeneous) harmonic polynomial of degree k, then
=

Pk(oos0)i(x).

Proof (i) is known as the mean value theorem for spherical harmonics; for the proof it is enough to check (8.1) for each h in the list Pfc(cosy) Pk{m) (cos y) sinm y cos m \J/ P{km)(cosy)sinmy

sinm\j/,

m-1,...,

k,

(8.2)

because (8.2) forms a basis for the vector space of spherical harmonics of degree k ([3]). (ii) follows from (i), since £ can be expressed as £{x) = \x\k h(x/\x\), x e R 3 , with some spherical harmonic h of degree k. Theorem 8.1 (Ikenberry and Truesdell [4]). Given a probability distribution f on R 3 with j\x\vf(dx)< co for some integer v ^ l , we put l*n{t) = iin{Ttf) for n such that |n| S v. Then, for any n with |n| ^ v w e have ^/Ut)=lX,»2Mt)/UO-A.rt,«,

(8-3)

where £ ' means the summation taken over all pairs (n^Dj) satisfying |n 1 | + |n 2 | = |n| and jnj, |n 2 j^l. ft, is given by /SL = 2 n j | l - ( o o s | ) W n ( c o s ^ ) - (sin^)""p 4 ( s m ^ J e ( d 0 ) > O t and Pni
150 Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

99

Proof. The proof will be given in four steps. Step 1. We prove that / d\\"\ I Q\ 4>= [cos-j Pk (cos-J £n(x), In fact, from the relation iI,, 0 , e = iT xcosW2) ,

n-(r,/c,/). -*CO»<«/2M/2

{t

follows that

x cos (0/2), — x cos (0/2), 0/2 » — |x|

IC0S2/

= (cos-J

^"xcoB{fl/2), -xcos(9/2), fl/2»Sfc/

P ft (cos-j4,(x).

Step 2 is to prove that

o = (K £.)(*,)>) is a homogeneous polynomial in x and y of degree |n|. Since we can write £n(y + x) = ZfyC*)C;y), where ^ and £( are homogeneous polynomials of degree mi and n{, i

respectively, with m{ + ni = |n|, we have

=L(y+')> ss

EMj;)i

On the other hand, y\i can be expressed as

and hence from Step 1 we have <^.,.fl.{.>=IWy) ^ 4 (COS-) Pj ( C O S - ) U X Therefore

=E|mEm {(co4)""^' ( C0 4)~ 1 } cU '°' )U *~ 3 ' ) ' This is a homogeneous polynomial in x and y of degree |n| with coefficients depending upon 6 in such a way that they are 0(6) as 0|O. Thus Kn(x,y) is a homogeneous polynomial in x and y of degree |n|.

151

100

H. Tanaka Step 3 is to prove that Ko(x,0)=

~ilntR(xl

Kn(Q,y) = - f t ' i ( y )

(SA)

where ^=27rj|l-(cOS^'n|pk(cOS^Je^0),

r; = -lit J ( s i n - )

Pk ( s i n - ) Q(rf0),

n = (r, fc,/)-

In fact, the first expression of (8.4) follows immediately from Step 1. As for the second, noting n0 y e = IIy OiK_0 and then using the result of Step 1 we have

= ( c o s ^ — J Pk ( c o s ^ — J ^n(v) = (sm-)

^(sin-j^ty).

This implies the second expression of (8.4). Step 4. If we set JR{x,y) = Kn{x,y)-Kn(x,0)-Kn{0,y), then by Step 2 the polynomial Jn(x, y) consists only of those terms which have at least degree 1 in x as well as in y , and therefore it can be expressed as

•a*,*)=r«1.««.,MWjo. Now, keeping in mind the moment estimate (4.30), we obtain ^ n ( t ) = <(7;/)(x)(7;/),Kn(x,y)> = W)®(7;/U(x,y)> + <{Ttf) <8> (7;/), Kn(x, 0) + KB(0, y)>

where /?„ = /?„ + /?„• This completes the proof of the theorem. It should be noticed that, in the right hand side of (8.3), there appear only the moments of degree less than |n| except for {in(t) and that the coefficient /?n of fxn(t) is positive (we exclude the trivial case 2 = 0). As a consequence we have the following corollary which is also found in [4]. Corollary. Let/be a probability distribution onRz with finite absolute moments ofall degrees, and assume that $\x-m\2f(dx)

= 3v>0,

m=\xf{dx).

(8.5)

Let g be the Gaussian distribution (2nv)-*l2exp(-\x~m\2/2v)dx

(8.6)

152

Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

101

and put /^n = Mn(g)- Then, for each n fiB(t) converges to ftB exponentially fast as t-> oo.

(8.7)

In particular, TJ converges to g as t->cc. Proof Clearly (8.7) holds for |n| =0 and 1 (the case of |n| =1 is nothing but (4.31)). So we assume that (8.7) holds for 0 ^ |n| < k and prove it for |n| = k. First we notice that g is invariant under Tt (see 2 of Appendix). This implies I ' « 1 . - ^ ! ^ a - A ^ = 0.

(8.8)

For any n with \n\ = k we put P„(0 = X'^ni,n2Mn1(0Mn2(0- Then the induction hypothesis implies that /2n(£) converges, exponentially fast as t->co, to Z'^ni.nz^m^nj which is equal to /?niun by (8.8). This fact combined with (8.3) implies that M„(0 = e~^nV„(o) + Je~^n<'~s)Mn(s)^s->/*„> o

exponentially fast as t-»oo.

So the proof is finished.

§ 9. Proof of the Trend to Equilibrium In this section we make use of the results of § 7 to prove the trend to equilibrium without assuming the existence of higher absolute moments. Fundamentally, our theorem is an extension of the result [17] in Kac's one-dimensional model to the case of Boltzmann's equation of Maxwellian molecules. Theorem 9.1. Let fe^2 and assume that (8.5) is satisfied. Let g be the Gaussian distribution (8.6) and put e(/) — e(/, g). Then, t(Ttf) decreases to 0 as ff oo. In particular, TJ converges to g as t]co. The proof is based on the following lemma. Lemma 9.1. Let f and g be the same as in the theorem and put €e (/) = e„(/, g). Then, ee(f)>0for0<6
(9.1)

So, assuming the equality holds in the above, we prove t h a t / = g. We now recall the proof of (i) of Lemma 7.4. Then, in order to have the equality in (9.1), the inequality (7.6) must be the equality

(x1-z2,y1-r2)=|x1-x2||y1-y2|)

a.s.,

which is equivalent to Xx—X2

^l~^2

l*i -*i\

lyi-r2l

a.s,

(9.2)

153 102

H. Tanaka

On the other hand, both Yx and Y2 are g-distributed in the present case, by Theorem 1 of [12] (or by (6.2)) there exists a unique Borel mapping if/ from R 3 into itself such that Xi = ij/(Y^,i = li 2, almost surely (the uniqueness of \j/ was remarked in the proof of Lemma 6.2). Therefore, (9.2) yields iMyi)-*(y 2 )

yx~y2

MyJ-^WI

\yi-y2\

for almost all yl,y2GH3 y0eR3

with respect to the Lebesgue measure. Thus for some

\y~yo\ must hold for almost all y. Therefore,

Myi)-Myo)\, , W ^ h M , l*-y.l {n-yo)\y*-y0\ 1/1

>

(y2 yo)

~

J2I

and hence

myi)-Hy0)\

My2)-Myo)\

l^i-yd

1^2-yd

a.e.

This combined with (9.3) implies that if/(y) —}j/(y0)-{-const {y — y0\ a.e., and hence il/(y)=ybecauseE{Xl}=E{Y1} and E{\XX\2} =£{i7 1 | 2 }.Thusweobtain/=g,as was to be proved. Proof of the Theorem. Since t{Ttf, TtQ) — t{Ttf g) = e(T(/), the decreasing property of t(Ttf) in t follows from Theorem 7.1. To prove that z{Ttf) tends to 0 as t] 00, we first assume that J \x\4f(dx) < 00. Then by the corollary to Theorem 8.1 J |x| 4 (Ttf) (dx) is bounded in t, say, by M. We denote by # the family of probability distributions/on R3 satisfying $\x-m\2?(dx)

$xf(dx) = m,

= 3v,

j \x\*J(dx)£Mt

and put ^ = {/e#: e(/) ^ s} for e >0. Then ^ is compact with respect to the metric p. Moreover, using the triangle inequality for p and (i) of Lemma 7.4 we can see that I*«(/) — M£)l^2e(/,g) for f,g£$E and hence ee is ^-continuous on ^ for each 0e(O,7t). Since efl is strictly positive on ^ by Lemma 9.1, we have inf e,(/)>0,

MO, 7i),

and hence * ( B ) = inf 2n]te(J)Q{d0j>O. /6#e

0

(9.4)

154

Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

103

On the other hand, from (7.4) we have

t(Ttf)^(f)-2n]ds]i>e(Tsf)Q(de). o

o

Because Ttfe$E with e = e{Ttf) if e ( 7 J / ) > 0 , the above inequality combined with (9.4) implies *(Ttf)St(f)-]$MTsf)lds (9.5) o for t such that t(Ttj)>0. But, this inequality clearly implies that e(7^/)->0 as t-> oo. Finally we remove the restriction j \x\4f(dx) 0 and fe8P2 satisfying (8.5) we can choose f£ in such a way that \xf{dx)

= m,

and p(f,f£)<£

$\x-m\2fE(dx)

j \x\4fe(dx)
= 3v,

hold. Then, using Theorem 8.1 we have

*lTtf)4{p(UT&+p{Ttf„a}2 S{e + y*(TtfE)}2 and hence lim e{Ttf)Ss2-

The proof of the theorem is completed.

r-*cc

Remark. Making use of the corollary to Theorem 8.1 in full, we obtain a much simpler proof of Theorem 9.1. I f / h a s finite absolute moments of all degrees, then M n ^ / ) - " ^ as r~> oo for every n and hence e(T;/)-fO as t-+ oo. The general case when / belongs to 0>2 and satisfies (8.5) can be treated by choosing fe with finite absolute moments of all degrees in such a way that $xfe(dx) = m, \\x — m\2fE{dx) = 3v and p(ffE)<e hold. However, our first proof based upon the inequality (9.5) seems to be interesting. Appendix 1. In the introduction we regarded the equation (0.3) as a weak version of (0.2). This is justified by the formula J"

£(x)(u'u\—uul)Q(d6)dq>dxdx1 £eC£(R 3 ),

- | {KQfaxJuMuixJdxdx!, R6

which holds, at least if w(x) is smooth enough, according to the following lemma. Lemma. Let ^{x,xuy,yi) Then, for each 0e(0, n) j" (0, 2 7 t ) x R 6

be a continuous function on R12 with compact

£(x,x1,x',x'i)d
j (0, 2 J I ) X R 6

£,(x',x'1,x,x1)d(pdxdx1.

support. (1)

155 104

H. Tanaka

Proof. We denote by and *F are continuous in 0, for the proof of (1) it is enough to show ]
(2)

0

for any continuous function Q0(9) with compact support in (0, n). For x # x x we put l = (x'-x)/\x'-x\ and define ye(0,7r/2) by cos y = (xl-x,l)/\xl~x\. Since 9 = n-2y, Q0(9)cosy becomes a function of \(xl~x,l)\ and \xx— x\. Thus we can write Q0(9)co$y=F(x,xl,l) with some function F on R 3 x R 3 x S 2 satisfying F(x,x1,l) = F(x',x'1,l) = F(x,x1,-l).

(3)

We then have j(0)()o(0)sin0^ o =4 |

^(x, x 1, x', x't) <2O(0) cosy siny dy d

(0, J T / 2 ) X ( 0 , 2 n ) x R 6

= 4 { dxdxj R6

=2

J »eS2:(xi-Jc,I)>0|

j

£(x,xl,x',x'1)F(x,xl,l)dl

^(x,x1>x',x[)F(x,x1,l)dxdx1dl;

(4)

R6xS2

in the last line of the above x' and x[ are defined by x' = x -f (x : — x, /) / xj =Xj —(xx — x,/)/. Since dxdx1=dx'dx[ j"

(5)

for each fixed / e S 2 , the last integral in (4) is equal to

^(x,Xj, x', xi) F(x, Xi, /) dx' dx[ d /,

(6)

R6xS2

where x and x1 are defined by the same rule as (5): x = x'+(x' 1 — x',/)/ xl=x'l—{x'l

—x',l)l.

Now, from (3) it is clear that (6) is equal to I

£(x',x[, x,xjF(x,X!,!)dxdXi

dl.

R6xS2

But this is equal to the right hand side of (2) by the same reason as (4) holds. The proof is finished. 2. Let g be the Gaussian distribution (8.6). Then g is invariant under Tr F o r the proof, we first notice that the density function g satisfies g'gi =gg1 and hence by the above lemma <9®9,K£>=0,

£eC*(R3).

156 Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules

105

jt

This implies 7Jg = g at least if j" Q(d 9) < oo, because the uniqueness of the solution o for (0.3) clearly holds in that case. Therefore, in general, we have Tt g = lim 7^(£,g = g, proving the invariance of g under Tt. Moreover, (g°g)<, = g also follows from g'g'i = SSi a n d the above lemma. References 1. Arkeryd, L.: On the Bolztmann equation, Part I: Existence, Part II: The full initial value problem. Arch Rational Mech. Anal. 45, 1-16, 17-34 (1972) 2. Carleman, T.; Problemes Mathematiques dans la Theorie Cinetique des Gaz. Uppsala 1957. 3. Erdelyi, A. (Editor): Higher Transcendental Functions, Vol. II, New York-Toronto-London: McGraw-Hill 1953 4. Ikenberry, E., Truesdell, C : On the pressure and the flux of energy in a gas according to Maxwell's kinetic theory I. J. Rational Mech. Anal. 5, 1-54 (1956) 5. ltd, K.: On stochastic differential eqations. Mem. Amer. Math. Soc. No. 4, 1951. 6. Kac, M.: Probability and Related Topics in the Physical Sciences, New York-London: Interscience 1959. 7. Kolmogorov, A.N.: On Skorohod convergence.Theor. Probability Appl. 1, 215-222 (1956) 8. Kondo, R., Negoro, A.: Certain functional of probability mesures on Hilbert spaces. Hiroshima Math. J. 6, 421^28 (1976) 9. McKean, H.P.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. 56, 1907-1911 (1966) 10. McKean, H.P.: An exponential formula for solving Boltzmann's equation for a Maxwellian gas. J. Combinatorial Theory 2, 358-382 (1967) 11. McKean, H.P.: Propagation of chaos for a class of nonlinear parabolic equation. Lecture series in Differential Equtions, session 7: Catholic Univ. 1967 12. Murata, H., Tanaka, H.: An inequality for certain functional of multidimensional probability distributions. Hiroshima Math. J. 4, 75-81 (1974) 13. Murata, H.: Propagation of chaos for Boltzmann-like equation of non-cutoff type in the plane. Hiroshima Math. J. 7, 479-515 (1977) 14. Parthasarathy, K.R.: Probability Measures on Metric Spaces, New York-London: Academic Press 1967 15. Povzner, A.Ya.: The Boltzmann equation the kinetic theory of gases. Mat. Sb. 58, 65-86 (1962). (English Transl.) Amer. Math. Soc. Transl. (2)47, 193-216 (1965) 16. Skorohod, A.V.: Limit theorems for stochastic processes, Theor. Probability Appl. 1,261-290 (1956) 17. Tanaka, H.: An inequality for a functional of probability distributions and its application to Kac's one-dimensional model of a Maxwellian gas. Z. Wahrscheinlichkeitstheorie verw. Gebiete 27,47-52 (1973) 18. Tanaka, H.: On Markov process corresponding to Boltzmann's equation of Maxwellian gas. In: Proc. 2nd Japan-USSR Sympos. on Probab. Theory, Lecture Notes in Math., 330,478^189. BerlinHeidelberg-New York: Springer-Verlag 1973 19. Tanaka, H.: On the uniqueness of Markov process associated with the Boltzmann equation of Maxwellian molecules. In: Proc. Intern. Sympos. on Stochastic Differential Equations, Kyoto, 1976 20. Uhlenbeck, G.E., Ford, G.W.: Lectures in Statistical Mechanics. Amer. Math. Soc, Providence, Rhode Island, 1963 21. Wild, E.: On Boltzmann's equation in the kinetic theory of gases. Proc. Cambridge Philos. Soc. 47, 602-609 (1951)

Received December 19, 1977

157 HIROSHIMA MATH. J.

9(1979), 163-177

Stochastic Differential Equations with Reflecting Boundary Condition in Convex Regions Hiroshi TANAKA

{Received September 16, 1978)

§ 1.

Introduction

A. V. Skorohod [4] considered a stochastic differential equation for a reflecting diffusion process on 5 = [0, oo) (see also McKean [2] [3]). This is the simplest case among stochastic differential equations subject to boundary conditions and can be solved easily. The purpose of this paper is to show that the multi-dimensional version of Skorohod's equation is still easy to solve if we assume that the domain D is convex. Skorohod's equation describing a reflecting Brownian path £ on Z) = [0, co) is (1.1)

£ = w +
where w is a standard Brownian path and £ is to be found as a Z)-valued continuous function under the condition that (p(i) increases only when £(r)=0. The equation (1.1) has a unique solution not only for almost all Brownian paths but also for all continuous functions w with w(0) e D, and the solution is given by (1-2)

f w(t)

for

0 < / < T,

[ w(t) - inf (w(s): T ^ s ^ t}

for

t > T,

«r) =

where T=inf{r>0: w(t)<0}. Our first problem is to consider a multi-dimensional version of the equation (1.1) assuming that D is a convex domain. Although we can not obtain an explicit formula for the solution like (1.2), we are able to construct the unique solution f for any Revalued continuous function w with w(0) e D and to prove that £ depends continuously on w, if D is a convex domain in Rd satisfying certain condition (Theorem 2.1). This additional condition is automatically satisfied if D is bounded or d — 2. This result will lead to a simple solution to our second problem which is concerned with a stochastic differential equation with (normal) reflection having variable coefficients similar to Skorohod's. The following may be stressed, (i) The boundary does not need to be smooth as far as the domain is assumed (ii)

to be convex, The diffusion coefficients may degenerate (however, in this case the path of

158 164

Hiroshi TANAKA

the solution might not behave like an ordinary reflection). Our results are roughly stated as follows. The convexity assumption for the domain makes the situation quite similar to that in the whole space; in fact, the existence of solutions will be obtained assuming only the bounded continuity of the coefficients (Theorem 4.2), and the pathwise uniqueness of solutions will be proved under the same regularity assumption on the coefficients as found in the work of Watanabe and Yamada [9] for the case of whole space (Theorem 4.3). However, it is noted that our methods and results are restricted to the case of reflecting boundary condition; the convexity assumption will not simplify the situation in the case of general boundary conditions such as discussed by Ikeda [1], Watanabe [8], Stroock and Varadhan [6] and Tsuchiya [7].

§ 2.

A deterministic problem

An Revalued function {p(f) = (\ (t) — the total variation of q> on [0, i] = sup5Xf4)-«o(f»_,)|, k

where the supremum is taken over all partitions: can be expressed as (2.1)

0 = t0
n(s) \
J[o,o

with a unit vector valued function n(t); n(t) is uniquely determined almost everywhere with respect to the measure d\
(resp. D-valued)

159

Stochastic Differential Equations with Reflecting Boundary Condition

165

On C(R + , R d ) and C(R + , D) we consider the compact uniform topology. Given a function £ in D(R + , D), a function q> is said to be associated with £ if the following three conditions are satisfied. (2.2) (p is a function in D(R + , Rd) with bounded variation and
For any

n

eC(R + , 5), (n(t) - «*), 0.

EXAMPLE. Let dD be smooth, n(x) the inward normal vector at xedD and ^ G D ( R + , D ) . Then, for any right continuous non-decreasing function p(t) on R + withp(0) = 0

\'hDm)nm)dp{s)

is clearly an associated function of £. Our first problem stated in the introduction can now be formulated as follows. PROBLEM.

(2.5)

Given weD(R + , R d ) with w(0)eD, find a solution £ of £ = w +
When we speak of the equation (2.5), it is always understood that £ e D ( R + , D) and (p is associated with £. As stated in the introduction, in the simplest case D — [0, oo) the solution of (2.5) is given by (1.2). However, in the general multi-dimensional case the existence of a solution of (2.5) is not trivial. An example in which a solution of (2.5) can easily be found is the case when w is a step function as will be seen in the lemma below. For a given point x e R d - i ) we denote by [x] d the (unique) point on dD which gives the minimum distance between x and Z). L-EMMA

2.1. / / w is a step function with w>(0)el>, then a solution of (2.5)

exists. PROOF. Put Tl = inf{t>0: w(t)&D} and define f(f), 0^t£Tlt by £(i) = w(t) for t160 166 Hiroshi TANAKA Tn = inf{r > Tn.x: w(t) + { w(t) + «0« [wCrj + ^ V t H a for Tn_, for r=Tn. Then, £(() solves (2.5) for 0 < r < T „ . Repeating this argument, we can obtain a solution of (2.5) for 0 (i) Ler w, we D(R + , Rd) w/r/i w(0), tf(0)€ D, and & £ be any solutions of £ = w + respectively. I = w + Then we have i«o - Koi 2 < iw(o - w(oi2 + 2\(w(t) Jo - w(t) - w(s) + w(s), (ds)). (ii) If £ is a solution of (2.5), then l « 0 - fWP < |W(0 - W(5)|2 + 2( (W(0 - W(T), 0 ^ 5 < /. (i) We have JoJo JJo£ri£n£r - Z | < 2['( (w(0 - w(0, - w(f) - w(s) + w(s), 4- \ (w(s) - w(s), Therefore 161 Stochastic Differential Equations with Reflecting Boundary Condition \m - l(t)\2 = K O - m\2 167 + 2(w(t) - w(t), q>(t) - «5(0) +1^(0 - m? < MO - m\2 + l U m - fw, v(ds) - - w(t) - w(s) + #(s), But the second term is non-positive by (2.4') of Remark 2.1. (ii) By a method similar to the case (i), we have \at) - m 2 = MO - w(S)\2 + i « ( T ) - f(s), J(s,0 + 2( (w(()-w(T),(p(dT)). J(MJ But the second term is non-positive by (2.4'); the proof is finished. REMARK 2.2. By a method similar to the above, we can prove the following: If w and w of (i) of Lemma 2.2 are replaced by w + a and w + a, respectively, where a and a are Revalued right continuous functions of bounded variation with a(0) = a(0) = 0, then | « 0 " f ( 0 l 2 < K 0 - w(0l 2 + 2 ( ' ( « s ) - fts), a(ds) - a(ds)) Jo + 2\ (w(f) - w(t) - w(s) + w(s), a(ds) + «*)|2 ^ |W(0 - w(s)\* + l\ («T) - «s), fl(rfT)) J (5,0 + 2\ (w(0 - W(T), fl(dt) + q>(dx)). J(s,(] LEMMA 2.3. (2.5) has at most one solution. PROOF. Let £ and | be solutions of (2.5). Then, putting w = w in (i) of Lemma 2.2 we obtain \£(t)-%(t)\2<0. LEMMA 2.4. Ifw is continuous, then the solution of (2.5) is also continuous. 162 168 Hiroshi TANAKA From (ii) of Lemma 2.2 we have PROOF. | { « - « * ) | 2 <• |W(() - W(5)|2 + l\ \w(t) - W(T)| \\(dT), from which the continuity of { follows. 2.5. Let {wn}j&i oe a sequence in D(R + , R 4 ) such that for each n the equation %n = w„ + (pn has a solution for 0 PROOF. Let K be the bound of {\ Then applying Lemma 2.2, we have (2.6a) \(H(t) - (Jt)\2 £ K ( 0 - wm(0l2 + 8X sup \wn(s) - wm(s)\, 0£s£f (2.6b) \Ut)-tM2^\*>M-wM\2 + 2K sup Mt2)-wM\. From the first inequality it follows that {£„}„;>! is uniformly convergent on [0, 7"] and hence the same for { sup H ^ - M ^ i . This implies the continuity of £. We now prove that £ is a solution of (2.5) for 0<,t£T. For this it is enough to prove that q> is associated with £. First, [<pa\(T)<*K implies \ that for 0<.tl I \'\n0) - wo. * tJtt I The first is dominated by K sup \£(t) — £H(t)\ and hence tends to 0 as n-+co; ti£t the second term also does as can be seen by approximating the integral by the Riemann sum. Therefore [%(t) - «*), 9(dt)) = lim ("OKO - « o . **")) ;> o, and the proof is finished. 163 Stochastic Differential Equations with Reflecting Boundary Condition Rd). 169 We proceed to the existence problem for (2.5) assuming that weC(R+, We begin with the special case when D satisfies the following condition. (A). There exist a unit vector e and a constant c > 0 such that (c, n)>c for any n e w ^VJD). CONDITION yedD LEMMA 2.6. Assume that D satisfies the condition (A). a solution £, of (2.5) for any weC(R + , R d ), and for 0<s (2.7a) Then, there exists |{(0 - i(s)\ < KAS„ (2.7b) \ where K and K' are constants depending only upon the constant c in the condition (A);As>t= sup |w(r2)--H<'i)|. For each integer n>l we define u>„eD(R + , Rd) by wn(t) = w(—) k — \ k for -\. Then w„ converges to w uniformly on compacts as PROOF. n->oo, and by Lemma 2.1 there exists a solution ln of £„ = % + sup }w„(t2) - w„((,)|, KH,s,t = W W - Mis), and notice that (e, Ut) - t&)) = (c, wrt(0 - wn(s)) + (e, (e, wn(t) - wn(s)) 4- cK„tStt, that is, (2.8) KM,tl £ {\Ut) - Us)\ + KJic- On the other hand, (2.6b) yields \Ut) - Us)\2 < Alttt1 + 2KniSttA„iSil < A\tm%t + ^ , l i ( + AlJs\ that is, 16.(0 - ZM\ < (l + Y)A^ This combined with (2.8) implies + E K »>^ e > °" 164 170 Hiroshi TANAKA and therefore (2.9) \Ui) ~ « a ) | < KA^t, Kn,Stt < where K is the minimum of N + - H— V1 ) K'A^, as e ranges over the interval (0, c) and K' = (l + K)lc. In particular, \q>n\(T) ( = Kn0T) is bounded and so by Lemma 2.5 £„ converges uniformly on compacts to the solution it, of (2.5) as n->oo. This proves the existence part. The estimates (2.7a) and (2.7b) are also immediate from (2.9). The proof is finished. Next, we introduce the condition (B) for a convex domain D. CONDITION (B), There exist e > 0 and S > 0 such that for any x e dD we can find an open ball BE(x0) = {yeRd: \y — x0\<e} satisfying BE(x0)<=Z) and |x —x0| <<5. We can easily see that the condition (B) is always satisfied if D is bounded or if d = 2. We now assume that D satisfies the condition (B) and for a point xedD put B(x) = DX= {ye^:\y-x\<el2}> r\__ yedDHB(x) r\ H(D), He<ary{D) where H(D) denotes the open half space bounded by a supporting hyperplane H and containing D. Then Dx is a convex domain satisfying the condition (A) with c = (xo — x)l\xo — x\, c = e/25. THEOREM 2.1. (i) Assume that D satisfies the condition (B). Then there exists a unique solution of (2.5) if weC(R + , R d ) , and the solution depends continuously upon w with respect to the compact uniform topology, (ii) Let D be a general convex domain and {w n } n 3 l l he a sequence in C ( R + > R d ) such that £n = wn + (Pn nas a solution for each n. Assume that wn and £M converge to w and £ uniformly on compacts as n^co, respectively. Then t, is a solution of (2.5). PROOF, (i) If we put r o = inf{f>0: w(t)<£D}, then f0(i) == w(t) ( 0 < t ^ T 0 ) is the solution of (2.5) for 0 < t < ; T 0 . Assuming that the solution £„_j of (2.5) is constructed for 0\\ we now extend it beyond T„_{ as follows. Put w<">(f) = w(r B -i + 0» t>0, let {<"> be the solution of {(») = w(") + ^"> on ^„_ l ( r n -i) and again put tn = inf{r £ T„_ i : |{(-)(t - T . . 0 - {W(0)| - e/2}, TH = inf {t > tn: {<">(*„ - 7 ^ ) + w{t) - w(Q £ 0 } , 165 Stochastic Differential Equations with Reflecting Boundary Condition '{»-i(0 WO- W-T.-J , 0 , Tn_v , (iH)(tn - r B _,) + w(0 - w(g, 171 *„ < t < T„. Then £„ is the solution of (2.5) on D for 0 < ( < T n . Repeating this argument, we obtain an increasing sequence {T„} and a continuous function £ defined for f If h>0 is so small that AT{h)<£J2K where AT(h) = max{|w(f) - w(s)|: 0 < s, f < T and \t - s\ < h}, T being an arbitrarily fixed positive constant, then Tnh. In other words, Tn>T for all n>T/h. This fact implies the followings: (2.10) Tn = oo, that is, there exists a solution of (2.5) for 0 < t < oo. (2.11) For 0 < s < t < Tthe solution satisfies (a) |£(0-^)l<(£+l)K4t(, (b) M«-M00<(J+ I)K'4,(. Next, let {ww}n^i be a sequence in C(R + , R d ) converging to w uniformly on compacts and let £n = wn + q>n. Then (b) of (2.11) applied to \ 0 <; s < t < T. Here hn depends upon w„, but it can be chosen to be independent of w„ for all sufficiently large n because wn-^w. Therefore the above inequality on \(pn\ implies that {\(pn\(T}} is bounded and hence by Lemma 2.5 0 be any fixed constant. Then there exists N such that sup max |£„(r)| < N. For such an N both £,„ and £ are the solutions of £„ = *>„ + 166 172 Hiroshi TANAKA we can apply the result of (i) to conclude that £ is the solution of (2.5) for DN and hence for D. The proof is finished. (0 REMARK 2.3. Even if D does not satisfy the condition (B) (so d>3 and D is unbounded), for each xedD we can find an open ball Be(x0) = {y e R J : \y — x0\<e] inside D (but now e or | x - x 0 | _ 1 is not bounded away from zero as x moves on dD). Therefore, in a manner similar to the proof of Theorem 2.1, (i), we can construct the solution of (2.5) for t § 3. A stochastic version of (2.5) The purpose of this section is to remove the condition (B) in the existence of global solutions of (2.5) by taking w from sample paths of a continuous semimartingale. Let (Q, &, P) be a complete probability space with an increasing family {•^Mc>o of sub-er-fields of & \ it is assumed that each &t contains all /'-negligible sets and &t= P\ &t + v Let D be a convex domain as before. 3.1. Let {M(t)} be an Rd-valued process with M(0)ED such that each component is a continuous local &,-mar-tingale and {A(t)} be an Rdvalued, continuous and ^t-adapted process of bounded variation with A(0) = 0. Then there exists a unique &x-adapted solution {X(t}} of THEOREM (3.1) X(t) = M(t) + A(t) + Moreover, for / e C 2 ( R ) with / ' > 0 o n R + and 0<s we have f(\X(t) - X(sW) < /(0) + 2 f / ' I ( * ' « - XKsMMKdr) + AHAT)) Js i + 2 ( 7 " I (A-'(T) - X ' ( S ) ) ( X ' ( T ) - Xi{s))d[_Ml, Mq Js Js i,j i where f\f" are evaluated at \X(x)-X(s)\2 and [M*, MJ"] denotes the quadratic variation process. REMARK 3.1. By a solution of (3.1), we mean a D-valued process {X(t}} which satisfies (3.1) almost surely, under the condition that almost all sample paths of { 167 Stochastic Differential Equations with Reflecting Boundary Condition PROOF. 173 Let r(t) be the inverse function of 0(0 = ' + £ [ M < , M'] + \A\(t), 1=1 and put &* = * m 9 M*(t) = M(t(0), X*(0 - > « T ( 0 ) . Then {M*(/)} is a continuous J^-martingale and {A*(t)} is a continuous ^"*adapted process of bounded variation, satisfying (3.3a) 0£ £ Jc'x'dEAtf*1, M*>] < \x\2dt, xeR', (3.3b) i4*(f) = ( V ( s ) d s , |a*(0l < 1Jo Moreover, once we obtain the ^*-adapted solution of X*(t) = M*(t) + A*(t) + But the last term is non-positive by (2.4'). In order to prove the existence of the solution, we first consider the equation for Dn~D n {|x| ^ ) = WW(0)M(0. Since Dn is bounded and hence satisfies the condition (B), by Theorem 2.1 there exists a unique ^ - a d a p t e d solution {Xn(t)} of Xn(t) = Mn(t) + An(t) + &„(t) for Dn. If we put T„ = i n f { r ^ 0 : | Z M ( 0 i = n}, then {Xn(tATn)} is again the solution of Xn(tATn) = MB(tATn) + An(tAT„) + $„(MT„) for Dn, and so (3.2) can be applied to \Xn(t A T„)-Xn(0)\2. Thus, by taking the expectation we have 168 174 Hiroshi TANAKA E{\Xn(t A Tn) - Xn(0)\2} < 2^JE{\XH(s A Tn) - Xn(0)\}ds + t <^E{\Xn(sATn)-XM\2}ds^2t, and hence E{\XH(t A Tn) - Xn{Q)\%} < 2te<, that is, E{\Xn(t A T„)\2} is bounded in n for each fixed t. Therefore, for each t P{Tn (3.4) as n - oo. On the other hand, the uniqueness lemma in § 2 implies that Tn < Tm and Xn(t) = XJt) hold on the set {M(0)eDn} if n<m. to define {X(t)} almost surely by for t < Tn This fact combined with (3.4) enables us X(t) = Xn(t) on {M(0)e A J n {t < Tn). Thus defined {X(i)} is clearly the J^-adapted solution of (3.1). § 4. Stochastic differential equation with reflection Let D be a convex domain in Rd and {Q, &, P; &t) satisfy the same condition as in §3. We suppose that an i^-adapted r-dimensional Brownian motion B(t) = (B1(t),...tBr(t)) with B(0) = 0 is given; that is, {B(t)} is an J^r-adapted continuous process and for 0 < s < f, £ e R** £ | e i«,fl(,)-B( i ))| i r j = e -(r-.H«l 2 /2 ( a s _ Given an R d ®Revalued function o(t, x) = {aik(t, x)} and an Revalued function b(t, x) = {bl(t, x)}, both being defined on R + x £ ) , we consider the stochastic differential equation with reflection (4.1) dX = or equivalently (4.1') Xl(t) = x* + £ ('ffifc * W ) ^ * ( s ) + ( V ( s , X(s))ds + * ' ( l ) , k=lJo 1 d Jo where x = (x ,..., x )eD. Our problem is to find an ^ - a d a p t e d ^-process {X(t)} under the condition that {#(*)} is an associated process of {X(t)}. It is always assumed that a(t, x) and b(t, x) are Borel measurable in (?, x). 169 Stochastic Differential Equations with Reflecting Boundary Condition THEOREM 175 4.1. If there exists a constant K>0 such that (4.2) || (4.3) \\ \x\*yi\ then there exists a (pathwise) unique &t-adapted solution of (4.1) for any xeD. PROOF. First we prove the pathwise uniqueness of the solution. Let {X(t)} and {Y(t)} be .^,-adapted solutions of (4.1). Then by the first inequality in Remark 2.2 \X(t) - Y(t)\2 and hence (4.4) E{\X(t) - Y(m £ E[' \\a(s, X) - G(S, YWds Jo + £ ( ' \X(s) - Y(s)\2ds + E['\b(s, X) - b(s, Y)\2ds Jo Jo < (2K2 4- l)[E{\X(s) Jo - Y(s)\2}ds. Therefore, E{\X(t)-Y(t)\2}=0. We give the existence proof, first assuming that D is bounded. 2.1, we can define a sequence {X("\t)} of £>-processes by By Theorem X<°>(0 = x X<"\t) = x + ('<7(s, X<"-l>)dB 4- [*b{s, X, n > 1. Jo Jo Then, as in (4.4) we have 1) Because we do not know beforehand that j X(i)— Y(t) | a and the remainder term are integrable, we must employ the following truncation argument. Let Tn be the infimum of r^O at which V\x(s)Jo I Y(s)\*ds+ \0\(t)+ | \W\(t) = n, and derive (4.4) for X{ • A Tn) and Y( • A Tn); it then follows that E{ \ X(t A Tn) - Y(t A 7*w) [lJ —0; now let n \ <x>. 170 176 Hiroshi TANAKA £{|X(0 - X<">(0I2} < (2K2 + l)(*£{|X(->(s) - Xi"-l>(s)\2}ds. Therefore, by a routine argument we see that [' + Pft(s, JO X^)ds JO is convergent uniformly on compacts as n-»oo (a.s.). Consequently, from Theorem 2.1, (i), it follows that {X(n)(t)} is also convergent uniformly on compacts as n->oo (a.s.) and the limit process {X(t)} is an .^-adapted solution of (4.1). When D is unbounded, we put Dn~D n {|xj 0 at which \Xn{i)\ = n. Then, applying (3.2) to |X„(r A T„) — x\2 and taking the expectation, we have CtAT„ E{\Xn(t A T „ - xP) < 2E\ (Xn(s) - x, b(s, Xn))ds Jo + E\ JO Y, XMYdsa i,k and hence by making use of (4.3) E{\Xn(t A Tn) - x\2} S [E{\XJ(s A T„) - x\*}ds Jo + 2K*['E{1 + IX& A TJ\*}ds. Jo Therefore, by Gronwall's inequality we see that E{\Xn(tA Tn)\2} is bounded in n for each fixed t and hence P{TH^t}^E{\Xn(t A T„)|2}/n2-+0 as n->oo. On the other hand, by the uniqueness already proved we have T„ We choose sequences {<;„(*, x)} and {bn(t, x)} such that (i) an-+o and bn->b boundedly and uniformly on compacts as n->oo, and (ii) an and bn satisfy the Lipschitz condition. Then there exists a solution {X„(t)} of (4.1) with coefficients an and b„, for each n. Making use of (3.2) with f(u) = uP, p^.1, PROOF. E{\Xn(t) - Xn(s)\2P] < CP{E^\X„(T) + - Xn(s)\2p-ldT E^Xn(x)-XH(s)\2p-2di}, 171 Stochastic Differential Equations with Reflecting Boundary Condition 177 where cp is some constant depending on p but not on n. Therefore E{\Xn{i) - X „ ( s ) | 4 } ^ c | * - s | 2 for 0<s, t Let p and p satisfy {p 2 («)u- J + p(u)}-idu (4.5) [ (4.6) p2(u)u~l + p(u) Then, for any = oo, is concave. satisfying x) - the pathwise uniqueness of solutions y\), holds for (4.1). References [ 1 ] N. Ikeda, On the construction of two-dimensional diffusion processes satisfying Wentzell*s boundary conditions and its application to boundary value problems, Mem. Coll. Sci. Univ. Ky6to, 33 (1961), 367^27. [ 2 ] H. P. McKean, A. Skorohod's integral equation for a reflecting barrier diffusion, J. Math. Kydto Univ., 3 (1963), 86-88. [ 3 ] H. P. McKean, Stochastic Integrals, Academic Press, 1969. [4] A. V. Skorohod, Stochastic equations for diffusion processes in a bounded region 1, 2, Theor. Veroyatnost. i Primenen. 6 (1961), 264-274; 7 (1962), 3-23. [ 5 ] A. V. Skorohod, Studies in the Theory of Random Processes, Addison Wesley, 1965. [ 6 ] D. W. Stroock and S. R. S. Varadhan, Diffusion processes with boundary conditions, Comm. Pure Appl. Math., 24 (1971), 147-225. [ 7 ] M. Tsuchiya, On the stochastic differential equation for a two-dimensional Brownian motion with boundary condition, to appear. [ 8 ] S. Watanabe, On stochastic differential equations for multi-dimensionai diffusion processes with boundary conditions, J. Math Kyoto Univ., 11 (1971), 169-180. [ 9 ] S. Watanabe and T. Yamada, On the uniqueness of solutions of stochastic differential equations II, J. Math. Kyoto Univ., 11 (1971), 553-563. Department of Mathematics, Faculty of Science, Hiroshima University 172 SOME PROBABILISTIC PROBLEMS IN THE SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION Hiroshi Tanaka Department of Mathematics Faculty of Science and Technology Keio University, Yokohama, Japan 1. 1.1. Introduction The master equation approach to the spatially homogeneous Boltzmann equation initiated by Kac Clj and continued by M c K e a n ( 3 ) , Griinbaum (5) and others is briefly reviewed together with some new results on (I) (II) propagation of chaos (law of large numbers), fluctuation (central limit theorem). We are interested in the time evolution of the velocities x\ (t) n + \ nf> „ « „ „ + * „ • ! « moving »rt,.4v,„ in *„ the + U „ space +. T>3 ... ,X • h (t) of n particles R under certain binary interaction (collision). We assume that o„ a Markov process on R with generator (X^ n '(t),' *»,x'n'(t)) is x n (Kny)<xlf...,xn) = ~ 2_ J { xj,...)-<jp(x1,...,xn)]Q(xi,xj.,e)dede , where x! and x! are the post-collision velocities of the i-th and j-th particles with velocities x. and x. . If S(x,y) denotes the 2-dimensional sphere with center (x+y)/2 and diameter | x~y[ , the x! and x! are always on S(x.,x.) or more precisely S(x!,x!)=S(x.,x.). We now take a spherical coordinate system on S(x.,x.) with north pole x i and denote by 6 (resp. £ ) the colatitude (resp. longitude) of x! . The function Q(x t y,G), which is determined by the interparticle (binary) potential, is assumed to be nonnegative and depend only on |x-yj , x+y and 6 „ The forward equation (1) •— {u(t),j=^= ^ ( t ) » K n ? ^ • ?= (test) function on R 3 n describing the Markov process x' n '(t) is called the n-particle master equation corresponding to the following spatially homogeneous Bolt2mann equation 173 259 (2) ( u ' u { " uu1)Q(x,xlfe)dffdedx1 ,/ §T (O.vrMO^aOxir where u = u(t,x), u 1 =u(t,x,) # u'=u(t,x')> u-j =u( t ,xi ) ; x' the post-collision velocities, i.e., (x,x-,) —» (x' ,xJ ) the notation (u*f) and x,' are by "collision" ; i-n general stands for the integral of a f u n c t i o n ^ with respect to a measure u . As in (7J we consider a weak version of (2); Jt(u^'f) C3) = (u(t)®u(t),Kf) , or equivalently where y> is a (test) function (Ky)(x,x1) = R and f(x)JQ(x,x1,&)d9de J /jP(x') (0,7r)*(o,27t) (K # y)(x,x 1 ) = {CKy)Cx, Xl ) + (Kf)(x 1>X )}/2 . By a weak solution of (2.) we mean a probability measure solution of (3) (=(3 # )) . 1.2. Propagation of chaos. Let ju , n>l\ be a sequence of probability measures, each ..u being a symmetric probability measure on the n-fold product space R n = R x"...x"R . Let u be a probability measure on R . Then a sequence -[u \ is said to be u-chaotic if {Vft.®' ' *®jpm®1®- " -®1) ~* IT (U,*k) as n ~*°° for any ^1' * * * ' y £C Q (R ) , m>l . The relation between the master equation (1) and the corresponding Boltzmann equation (2) was made clear by Kac C13 through the following propagation of chaos : Let u (t) be the ^solution iof (1) with initial value u and_ assume that j u , n>lV _is_ u-chaotic. Then ju n (t), n>l| is also u(t)-chaotic where u(t) isi the (weak) solution of (2) with^ initial value u . We next introduce the normalized occupa-r tion number n k=l where 8 denotes the * ^-distribution at x . We can easily show that 174 260 {u j is u-chaotic if and only if the law of large numbers holds for fu \ in the following sense: n — / S^Y(n) — * u , k k=l where n —> #> (in probability) x' n ' = (xj n ',...,X^ n ') is a u -distributed random vector. fore, since X '(t) is u (t)-distributed provided that X^ (0) Thereis u -distributed, the propagation of chaos is equivalent to the following law of large numbers: (5) X n (°) —> u» n —*°° (i n prob.)c^XnCt) —> u(t), n -> *> (in prob.) where u(t) is the same as before. The proof of the propagation of chaos was first given by Kac C13 for the case of 1-dimensional Kac's model of Maxwellian molecules; McKean f3} gave a beautiful proof for some special case including cutoff model of the Boltzmann equation of Maxwellian molecules. Murata £6} treated the 2-dimensional non-cutoff Maxwellian model. Grunbaum C.5") discussed the case of a considerably wide (cutoff) class; his discussions covered the gas of hard spheres but under some assumption which was unverified though very believable. In §2 of this article the propagation of chaos will be proved in the following two cases. Q(x,y,e)d0 (i) J (ii) 0 (Maxwellian type) Q(x,y,e) i s a function Q(0) of 6 I alone s a t i s f y i n g 1 (TQWdSO 0 It is to be noted that the case (i) includes the gas of hard spheres (Q(x,y,6) = const.I x-y|sinG) while the case (ii) includes the 3-dimen- sional Maxwellian molecules with the inverse 5-th power interparticle repulsive force 1,3. (in such a case Fluctuation. Q(0 )<^ const. &~ J/ ,©4-0) . Since the normalized occupation number is fluctuating about a solution u(t) X (t) of the Boltzmann equation (2), the next problem is to study the asymptotic behavior of 175 261 (6) Y n (t) = Vn(X n (t) - u(t)) as n —9ta . The case of McKean's 2-state model of Maxwellian moleclues was first discussed by Kac C2) and then by McKean £4} in detail. As found in heuristic discussion by C4} for the case of gas of hard spheres, the limit process of Y (t) must be, in general, an infinite dimensional Ornstein-Uhlenbeck process. Rigorous discussions in the case of Kac's 1-dimensional model of Maxwellian molecules were given by Tanaka C7j (equilibrium case) from the point of view of a limit theorem on Markov processes taking values of tempered distributions. Non-equilibrium case was then treated by Uchiyama £8~). Recently, Dchiyama C°0 discussed the equilibrium case of cutoff type including gas of hard spheres. In $3 of this article the fluctuation theory (=central limit theorem) will be discussed in the case (ii) (Maxwellian type) along the same lines as in L7}. The emphasis here is on the non-cutoff property. Fundamentally our method is to derive appropriate convergences of Markov processes ^ (t) and Y n (t) knowing the convergence of their generators (a martingale problem approach will then be useful), except for the treatment of chaos propagation in the case (ii) where the cutoff approximation will be used. Proofs are only outlined; details will appear elsewhere. 2. 1 2* * Case (i). Let Propagation of chaos f0-l» ^ x and ? k =lx| k for k>2 (xfeR3) . Theorem 1. The function Q(x,y,fi), depending only on |x-y|, x+y and 9 , is assumed to satisfy the condition (i) of §1, In addition, we assume that (7) Xf is continuous provided that j 3 i s bounded and continuous. Let u be the initial distribution of X (-) and assume that 1 u \ is a u-chaotic sequence satisfying (u,f6)<~, Then for any (8) £>0 and sup n ^ 1 <^un,f6®l©...(S>l^ T (0 P| s u p / O T n ( t ) , u ( t ) J > e | — > 0 0 as* n —^oo , where P is any metric on IffC (the space of probability 3 measures on R ) which gives the usual vague topology on T(C . 176 262 Remark. u(t) in (8) is the solution of (3) (with u(0)=u) whose existence and uniquencess are guaranteed by the following Arkeryd's result Qicfj : Under the same assumption on Q(x,y,6) as in Theorem 1 except for (7) (which is unnecessary here), for any initial value u(o) = u satisfying such that tion (u, ^ z ^ ^ 0 0 there exists a unique solution Ai(t),?A u(t) is bounded on each finite t-interval. also satisfies u(t) of (3) The solu- ^u(t),f, ) = v u »?k^' ^ = °'-'-»2 • The proof of the theorem is sketched here. Choose a sequence -jf^.* k > l ^ in C Q (R 3 ) such that ||fk|| < 1 and the set of all (finite) linear combinations of f^'s is dense in CQ(R ) , and set f(u, if) = y 2" /u-^fA for u, "u"e7/C» We first consider the special case in which u d^T-) = 1* n > l , where 3C is the ball in R of radius en , the constant c being independent of n . Let TfCr, ^ e the set of u€.Jf£ such that / u , £ p ^ < c and W be the space (endowed with the Skorohod topology) of lTCn~ v a l u e ^ right continuous paths with left limits. Then X (t) is regarded as a Markov process with sample path in W . Denote by P the probability measure on W induced by the process X (t) . We can prove the following lemmas (l -3 ) in which T is an arbitrary positive constant. 1°. E n |/(w(t 1 ),w(t 2 )) 2 /(w(t 2 ),w(t 3 )) 2 Uconst. (t^t-^ 2 for Oit-^ t 2 where const, may depend on T . Therefore, {P~^ 2°. sup P | sup / w ( t ) , f A > N J — > 0 as N n n If l0 4 sup /w(t), F.Voo 0 4 , JP J, then P^-a.s., t J <w(s)®w(s),K^)ds = 0, P ^ - a s ^ e C ^ R 3 ) . (vM,f)-(w(0),f)~ b) DO J Poo is any limit point of a) i-s tight. 0 Since X n (0) —*• u in probability, w(0)=u (P w -a.s.). Arkeryd's uniqueness result applied to 3 implies that Therefore, w(t)-u(t), t>0i Ppo-a.s., i.e., P«p-^ u f.) • This implies (8). General case can be reduced to the spacial case by noticing that u ^(5£ ) —^ 1 as n —> co where 2.2. is defined as before choosing Case (ii). We deal with this case by approximating by cutoff one. define ^£ a c > {u, f^) • constant (f ) Let Qc(e)=Q(fr) (for £ < 6 <7C) and K v ' with Q(6) (-4) of K . Let u_(t) Q(x,y,0] =0 (for O < 0 < £ ) , and replaced by Q&(fl) (cutoff) in the definition be the solution of 177 263 (3£) ^{u(t),?> * ^u(t)<*u(t),K(E) with initial value u . Theorem 2. (i) If (u*%2) iu (t)? is also u(t)-chaotic and lira /u (t),^)l®. ,.®l\ = / u ( t ) , 0 , n-»e° \ where * ' x u(t) is the solution of (3) constructed in (i). 3. Fluctuation We consider only the case of non-cutoff Maxwellian type. assume that a function Q(6) satisfying (ii) of 51 is given. Thus we We set g(x) = (27r)- 3 / V l x , 2 / 2 , g=g(x)dx , e0(x)=ygTT) . We also assume that the initial distribution of X (• ) is the n-fold product gn . In this case we have un(t)=g and u(t)=g , and (for notational convention) it is better to modify the definition (6) of Y (t) as follows. Yn(t) = /H(ln(t) - g}/eQ(.) , The fluctuation theory is to study the asymptotic behavior of Y (t) n —> DO , Note that Y (t) is a Markov process on the state space h e / ( R 3 ) : 1= Yn(x-g)/e0(-) , x = \ J_$^ fx± We introduce some notation. p(x,y) =?(x») +^(y') - SP(x) - y(y) , x n *R 3 } . 178 264 Q (*'^3 = ^ / v^N, jjy( x »y)?(x f y)Q(x,y,0)dfld£ (0,1C)x(0,27c) y # (x) =y(x)/e 0 (x), Let £"(x,y) = jP*(x,y) . F(^) be a function of the form F(7) = f(.....<'l-J»]B>). fCC^R3), y^...,^^, where ^> = -»o(R ) , the space of rapidly decreasing C -functions. the generator L of the Markov process Y (t) is given by Then m v w = ,L(eo"z®(3r g ) • KV<*^j>v 2 — V /n" ' 'LJW ''&/"«*? 7 ' z1 m = + Rn ( 'n ribJ>Z. > j fQdsde , J bn yn ^A,r=l where t h e argument i n ,/ , l 3, D i-f is K^(x-.x-) Te&fx vn ,x ) /n and hence the limiting generator ly) by m L=lim ^L is given (at least formal- in Now the problem may be stated as follows: (a) Find a diffusion process Y(t) with generator L on a suitable space of distributions. (b) Prove the convergence in the law sense of Y (t) to Y(t) . Linearized _collision operator: If we introduce l . c . o . ^ b y g(y}$0(x,y)Q(x,y,£)(Wd€dy , (0,7t)x(0,27tr)xR3 179 265 L can be expressed as where ( f ) denotes the usual inner product in L (R ) . Eigenfunctions of JL : For each integer £*>0 monic polynomials fy , m=0, +l,...,+£ of degree ^ j s | = = H ? | 2(spherical harmonics): we choose 2/ + 1 harso that the family m=0, +!,....+#> forms an ONS in L 2 ( S 2 ) , and set e nCx) = c D e 0 (x)L^ + 2({x/ 2 /2)H™(x) , n = (n,f,m), where n,j? = 0,1,..., m = 0,+l ., L Ctj = the associated Laguerre polynomial of degree n and order £+ -^ , 2 |l/2 < 1 '* Then l2 - nn+l+4)/ 2'* ' is a CONS in L 2 ( R 3 ) . J e n : n = (n,i,m)| are eigenfunctions of £ It is known that eT (Cll] 0.2)), i.e., e n = ~\% • where \ = 2 7 C J I1 0 >0 A P« (t) (10) (0,0,0) ~ (cos|) 2 n ^P £ (cos|) - (sin|) 2 n + -^( S inf)}Q(e)dS if 2n -tj>0 , = A (0,l,m) = *(1,0,0) = ° f o r * = °, +1 , being the Legendre polynomial of degree £ . A < const. (2n + i ) 2 . We can prove that 180 266 lx/2 •' , -A , i.e., 4 Moreover, e 's are also eigenfunctions of = (ID -A)en ( ^ = (2n + ^ + |)eQ . Spaces of distributions: ( = {Z a n ( 2 n n +J?+ i ^ } 1 ^ f°r T - Z V n ^ (12)), ^0^= the Hilbert space obtained by completing >0 respect to H' jl,^ , j&^= the dual space of J^ on j$ (£*£ Stochastic differential equation: determined by with ) . Let B(t) be a Brownian motion in fact B(t) exists as an jy^+£-Brownian motion ( v £ > 0 ) . From the expression (9) of <£ it follows that the Markov process Y(t) with generator L should satisfy the stochastic differential equation (12) dY(t) = s / ^ 5 d B ( t ) +o£l(t)dt . If we set Y n (t) = ( x ( t ) , e n ^ , then (12) yields dY Q {t) =J2\ where dB a (t) - A B T B ( t ) d t , B (t)'s are independent copies of 1-dimensional Brownian motion. Making use of the bound (10) for 2n+| A it can be proved that = uniformly on each finite t-interval in the space for any £>0 xir,±.i almost surely , and we can finally prove the following theorem. 181 267 Theorem 3. The Markov process Y(t) with generator L and sat- isfying ;|ei(Y(o),y>| exists as a diffusion process on dimensional distribution of fe, 'O'J'or *• ^, 1 (t) t for any £>0 , and any finite as a process on Jo-3 + £ the corresponding finite dimensional distribution of Y(t) It is to be noted that the SDE (12) has a solution continuous in J^OLC » while the process V-2o£ B(t) tinuous in converges to as l(t) n —}°° , which is is not always con- Jos+c • References 1. 2. 3. A* 5. 6. 7. 8. 910. 11. 12. M. Kac, Foundations of kinetic theory, Proc. Third Berkeley Symp. on Math. Stat, and Prob., 3(1956), 171-197. M. Kac, Some probabilistic aspects of the Boltzmann-equation, Acta Physica Austriaca, Suppl. X(1973), 379-400. H. P. McKean, An exponential formula for solving Boltzmann's equation for a Maxwellian gas* J. Combinatorial Theory, 2(1967), 358382. H . P , McKean, Fluctuations in the kinetic theory of gases. Coram. Pure Appl. Math., 28(1975), 435-455* F. A. Grunbaum, Propagation of chaos for the Boltzmann equation, Arch. Rational Mech. Anal., 42(1971), 323-345. H. Murata, Propagation of chaos for Boltamann-like equation of noncutoff type in the plane, Hiroshima Math. J., 7(1977), 479-515. H. Tanaka, Fluctuation theory for Kac's one-dimensional model of Maxwellian molecules, to appear in Sankhya, 44, Series A, Part I (1982). K. Uchiyama., Fluctuations of M&rkovian systems in Kac's caricature of a Maxwellian gas, to appear in J, Math. Soc. Japan. K. Uchiyama, A Fluctuation problem associated with the Boltzmann equation for a gas of molecules with a cut-off potential, to appear. L. Arkeryd, On the Boltzmann equation, Part I, II, Archive for Rational Mech. Anal., 45(1972), 1-16, 17-34. C. S. Wang Chang and G. E. Uhlenbeck (1952), In "The Kinetic Theory of Ga'ses", Studies in Statistical Mechanics, Vol. V, North-Holland, Amsterdam (1970). C. Cercignani, Mathematical Methods in Kinetic Theory, Plenum Press, New York (1969). Reprinted from Theory and Application of R a n d o m Fields, 258-267, Lecture Notes in Control and Information Sciences, 49, Springer-Verlag, 1983. 182 Taniguchi Symp. SA Katata 1982, pp. 469-488 Limit Theorems for Certain Diffusion Processes with Interaction Hiroshi TANAKA Introduction Given smooth functions a(x, y) and b(x, y), we write a[x, U] = o(x, y)u{y)dy ( = \ a{x, y)u{dy)\ b[x, u] = I b(x, y)u(y)dy ( = \ b(x, y)u{dy)\ for a function u(y) (or a measure u(dy)). On the analogy of Kac's master equation approach [5] to the Boltzmann equation, McKean [9] considered the ^-particle diffusion process £<">(/) ^ (fi B) (0, • • •, &B)(0) described by the stochastic differential equation (abbreviated: SDE) n — 1 0 ) 1 j=n*i) - + — V E *(fis,(0,£?°(0¥', i <*'<*, tt — 1 J = K*0 (wjCO'-s are independent copies of a 1-dimensional Brownian motion) and proved the following: If the initial values £{n>(0), - • -, ££°(0) are independent random variables with the same distribution u, then for each m the process (fiB)(/)f • •., £ ? ( 0 ) converges in law to (£(*), • • -, l m ( 0 ) as n —> oo where &(*)> &(*)> * * • a r e independent copies of the diffusion process £(f) obtained by solving the stochastic differential equation (2.a) d f ( 0 - *K(0, w(*)]<M0 + fc[«0, «(0]A (2.b) £(0) is independent of w(t) and w-distributed subject to the condition "w(0 = the probability distribution of £(*)"> w(0 being a 1-dimensional Brownian motion. Moreover, u(t) is the weak solution of the nonlinear parabolic equation 183 470 (3) H. TANAKA i l = i - - ^ W x , «]2M) - J-(b[x, at 2 dx% u]u) dx with w(0) = u. Let W be the space of real valued continuous paths. Then each process £SB)(0 is regarded as a PF-valued random variable $[n) and McKean's result implies the following law of large numbers: (4) un = — 2 ^fj ni — * • J" > w ~~* °° ( m P r °t»-); n i-i where ^z is the probability measure on W induced by the process £(/) and the notation dx stands for the ^-distribution at x. The next stage is the central limit theorem in which the asymptotic behavior of (5) Yn = as n —> oo, is studied. The special case when a = 1 and 6(x, >>) = — X(x — y), X > 0, was discussed by Tanaka and Hitsuda [14]. We shall discuss, in this paper, the case when a = 1 with general b(x, y) but the SDE (1) will be slightly modified as follows: (6) d^\t) = dwit) + 1 2 #fi->(o, w > y • Braun and Hepp [1] studied similar limit theorems in the case of modified Vlasov equation starting from a (deterministic) n-body problem. Their result on the law of large numbers is covered by McKean's result, since a may be assumed to vanish in (1). However, the work of Braun and Hepp is very interesting because of their method used in the discussion of the central limit theorem. The purpose of this paper is to study the central limit theorem and the large deviation problem for Un in the case when b(x, y) is general but a = 1, by amplifying the method of Braun and Hepp. The central limit theorem in the time evolution, which studies the asymptotic behavior of Markov processes (?) Yn(t) = V ^ ( - S d^Ht) - M(0) as n —• oo, may also be discussed by employing the method of martingale problem as in the works [12] [13] [15] for Boltzmann's equation; however, the result in the time evolution does not automatically imply the result in the path space. The advantage of the present method is that we can easily arrive at the result in the path space and also can find the /-functional governing the large deviation for Un. This result for the large deviation 184 Certain Diffusion Processes with Interaction 471 problem of Un will lead to a conjecture for a similar problem in the case of Boltzmann's equation. § 1. A method of Braun and Hepp We state a general theorem which is an abstraction of the method used by Braun and Hepp in the proof of Theorem 3.5 of [1]. Let S be a Polish space and Wl be the Banach space of bounded signed measures on S with total variation norm ||-||. Let SD^ be the subset of 9ft consisting of substochastic measures on S. Suppose we are given a function / : S x 3ft >R satisfying the following assumption. Assumption. There exists a (Borel) function f: SXSX Hft,—>R satisfying the following conditions (A.1)-(A.4). (A.l) / ' is bounded. (A.2) ||/'(x, y, p) - / ' ( * , y9 p')\l < const. \\p - pf || for any p, pf e 9ft,. (A3) If pni p e 9ftj and if pn —• p weakly as n —> oo, then lim f'(x, y, pn) = f'(x, y, p). (A.4) f(x, p+v) = / ( x , p) + / ' ( * , v, p) + £ {/'(*, v, p + tv) -f'(x, v, p)]dt for any p e SO^ and veSJi such that p -\- v s Sft^ where f'(x, Notation. v, p) = ( / ' ( * . y> p)v(dy) • For p, p' e 9ft we use the notation 7(^^)-|/(-v^V(^)Theorem 1.1. X e / / 6e a bounded function satisfying the above assumption and Xly X2, • • • be a sequence of independent S-valued random variables with the same distribution X. We set 1 *n ~ n Z-i "X t » rt (-1 185 472 H. TANAKA 7%en /Ae probability distribution of Yn converges to the Gaussian distribution with mean 0 and variance az as n —> oo, where az - £ {/(*, fl + / ' ( * , x, X) - m } 2 ^ x ) , m = £ {/(*, fl + fU x, X)Wx) . Proof. First we consider the special case when /(x, p) is given by f(x,p) = laf(x,y)p(dy) with a bounded function/(x, j ) on 5* X S (for simplicity we use the same notation with confusion). In this c a s e / ' f o y, p) = f(x, y) and hence the variance a2 is equal to j s {f(x, X) -f / f t x) - 2/tf, X)fX(dx). If we set Yn = Vn~{f(kn, X) + / t t 4 ) - 2/tf, Ji)} then the probability distribution of Yn converges to N(0, a2) as n —> oo, and hence it is enough to prove that E{\Yn~YJ}- >0, rc^oo. Setting Zni = / ( * , , Xn) - /(AT,, fl - f(X9 Xn) + / ( ; , fl, we have Yn - ? B = V~n~{f(ln, An) ~f(l K) -fiK K) + / t t ^)} V n <-i We now claim (i) £{Zi,} = o(l), (ii) E{ZniZnj] *->oo, ^ 0(n~2), n~+oo (i ^ j). (i) is immediate from the law of large numbers. As for (ii), setting 186 Certain Diffusion Processes with Interaction 1 473 n z n „ = /(jr„ # ) - /(*«,a) - / f t # ) + / f t J), *»„ = f(xt, xt) + fix,, xs) - fix, xt) - / a *-,), we have Zni = Z n i J + n-lRnli (i =f= j), and hence £{Z n( Z„,} = £{Z n i J Z n J i } + n~lE{ZnijRnii) + /r\E{Z n „A B ,j} + 0(n~2) . The first term on the right hand side of the above vanishes because E{ZntJZnji | Xk> k j= U j} = 0 , while the second and the third terms are 0(n~2) because E{ZniJRnji 1 Xt, Xj} = E[RnjiE{Znij - E[Rnji{n-\n \ Xt, Xj}] - 2)f(Xu X) - f(xt, X) -n-Xn-2)f(Z,X)+f(X,Z)}] = E[Rnji.2n^{fiX, Since E{Z2ni] and E{ZniZnj\ imply X) -f(Xt, X)}] = O(»-0 . are independent of i and j (i =£ j), (i) and (ii> lim £{| y n - Ynf] = lim n- 1 ± E{ZniZnj] = 0. Next we consider the general case. If we set Yn = r» - F„ = V^{/ft, ;„) - f(xn, x) - f(xt xn) + f(x, x)} i = n V Z n i=i Step 1. £"{1 y n — Ynf} —»0, n -> oo. For this it is enough to prove the followings: (hi) E{Zlt} = o(l), (iv) E{ZniZnj} n->oo, = 0(>r 2 ), R - * oo (i ^ y ) . Setting p = X and v = Xn — ^ in (A.4), we have 187 474 H. TANAKA fix, Xn) =f(x,X) + /'(*, K - X9 X) + £ {/'(*' *«-*,* + *(*n - *)) - f'(x, kn - Xt fydt , and hence f(x, Xn) ~ f(x, X) = p(x, Xn~ X)-\- n where ?„(*, y, o>) = £ {/'(*, j>, * + f (Jl, - X)) - /'(*, y, X)}dt - £ {f'(xs X, I + t(Xn - jl)) - / ' ( * , ;, *)}# . Therefore smxukj-fiXuXM <2n~* ± E{„(*«, 2T„ )} 4- 2/r2 2 ' ^W^> *i» % « , *,, «>)} + 0(/r') , where 2 ' is the summation over all pairs (/', k) such that I <j\k))} 4- 0(«"]) -—> 0, n —• 0 0 , because pn(x, y, co) —• 0 as n -> co (a.s.) for each fixed x, y e 5" by (A.3). Similarly £{|/0, ;,) - f(X, X)f] > 0 , n ~> 00 , and hence we obtain (iii). For the proof of (iv) let i i= j . Noticing that 188 Certain Diffusion Processes with Interaction f(Xi9 Xn) = f(Xu m 475 + n-*{f'(Xu Xi9 %) + / ' ( * , , XJ9 # ) } + 0(n->) and also a similar formula f o r / f t Xn)9 we have = Znt Znij -f- « -"nii J where Z nii - /(*«, # ) - /(JTf, *) - / f t tf ) + / f t *) , - f'(k9xi9 %)-/'& x,%)> Therefore E{ZniZnj} - £{Z n i J Z n J i } + n-lE{ZnijRnJi + Z„„U n „} + 0(AT 2 ) which is still 0(n~2) as n —• oo, because the first and the second terms on the right hand side of the above vanish as will be verified below: E{ZntjZnJt} = E[E{ZnijZnJi\Xk> E{ZnijRnH] = E[ZniS{f\X}9 k =£ ij}] = 0 , Xit # ) - / ' f t Xu %)}] r + E[Zni3{f (Xj9 = E[ZniJE{f'(Xj9 XJ9 # ) - / ' f t Xj9 %*)}] Xi9 }%) - / ' f t Xi9 # ) \ X k , k ^ j}] + E[{f\X}9 Xj9 X%) - / ' f t Xj9 %)}E{Znij\XHi - k^i}\ 0. Step 2. The distribution of Yn converges to N(0, + ) , and hence Yn = -^ S {/(*«, *) - /ft « + /ft W - /ft *)} i=1 v « = V n ) , where ^>(x) = / ( x , X) -f- / ' f t x, ^). Since the distribution of V n 2 converges to N(09 a ), it is enough to prove that E lW n *n> o>)\2} = o ( l ) , The left hand side of the above equals n - > oo . 189 476 H. TANAKA llv n *-i > = Jj{ p ^ , 2T1S o>) ' j + !^Z±E{9n(kt Xu )} . The first term of the above tends to 0 as n -~> oo, because cpn(X, Xlt at) -> 0 boundedly (a.s.) by (A.3). As for the second term we first write 9»W, j ; , a)) = * . & j ; , w) + 0(rrl) (by (A.2)) , and then notice that £{*»& X19 o))^„W, X2, a)) ] Xfc, fc =£ 1, 2} - 0 . Thus the second term also tends to 0 as n -*• oo, because £{p n (*, *i, <%„& X2, to)} = £ { i K t t JST„ft))^n(^,X2, ft>)} + ^ 5 5 L £ { | ^ y , jrls ffl)| + |^ B y, x2, o))|} + 0 ( 0 = o(»-1) § 2. (by (A.3)). Central limit theorem In this section we explain how Theorem 1.1 can be applied to the central limit theorem for Yn of (5) when the coefficient a = I. 2.1. Let fFbe the space of continuous paths w: t e [0, 1] ->• R, For simplicity the time parameter t is restricted to 0 < f < 1. W is a Banach space with the maximum norm || • ||M and will stands for the Polish space tfof §1. Let 2ft be the Banach space of finite signed measures on W with total variation norm ||-||, 20^ be the subset of fUl consisting of substochastic measures on W and C\(R*) be the space of C2-functions which are bounded together with their first and second partial derivatives. Given a coefficient b = b(x, y) e C2b{R2) and a signed measure p e SW, we consider the equation (2.1) £(*, W) = W (0 + £ ^(f(j, W ), £(*)>&, 0 < / ^ 1, 190 Certain Diffusion Processes with Interaction 477 where £(s) is the map: w e W —> £(s, w) € if and the notation bp(x, a) = b(x, a(w))p(dw) Jw is used for a map a: W—>R. The equation (2.1) can be solved easily by iteration. We denote by £(t, w, p) the solution of (2.1). When we regard £(t, w, p) as a function of / alone by fixing w and p, we denote it by f(tv, p). Then £(w, p) e W, because (2.2) |f(f, w, p) - tfs, w, p)\ <: \w(t) - w(s)\ + ||/>||.p>IL-|* - s\. Lemma 2.1. £(w, p) has the following properties. ( i ) \\£(w> p) — £(w',p)\\oo < ct\\w — w'll^,, w/zere cs is a constant depending only on \\p\\ and H^H*, (bx = db/dx, etc.). (ii) ||£(w, JO) — £(w>, p')IU ^ ^11/° — i°'ll> wAere c2 is a constant depending only on \\p\\t \\pf\\, \\b\\„, \\bx\\„ and ||£ tf ||„. (iii) Let pn, p e SK, am/ pn-^> p weakly as n —• oo. 77ze/z £(w, / O -»- f (w, />) (w W), « - > oo uniformly on each compact subset K of W. Proof We give only the proof of (iii). By (2.2) and (i) the family of functions £(t, w, pn), n — 1, 2, • - •, on [0, l ] x ^ i s uniformly bounded and equicontinuous provided that .ST is a compact subset of W. Therefore, for any subsequence nx < n2 < • • • we can choose a further subsequence {ni} of {nk} such that (2.3a) £(t, w, pnS) converges to some £(t, w) as k —> oo for any (t, w), (2.3b) the above convergence is uniform on [0, 1 ] X ^ for each compact subset K of W. The assertion (iii) follows once we identify £(/, w) be done as follows. Letting n -*• oo via {ni} in as £('» w> p)- This can £(f, w, pn) = w(t) + | ds \ pn(dwf)b(%(s, w, pn\ £(J, W', p j ) , Jo Jw we have « f , w) = w ( 0 + f * Jo f /KrfivOWfc w), £(J, wO) , Jw and hence the uniqueness of the solution implies £(/, w) = £(f, w, p)- 191 478 H. TANAKA For p, v € Wl we set y(t, w, », p) = Hm — {£ft w, p + hv) - £(*, u>, p)}. ft-0 ft Under the assumption 6 e C?(i?2) it can be proved that the above convergence is uniform in ft w) € [0, 1] X W, and hence TJ{W, v, p) = hm — {£(H>, p -f hv) A-0 £(w, p)} ft also exists as the strong limit in the Banach space W. Moreover, for fixed p and v, y(t) = i}(t9 w) = ?ft w, v, /?) can be regarded as a PK-valued function of t and satisfies ?ft w) = \ pW){bMU w, (°), £ft W, ^ ( f , w) + b$(t9 w, p), « * , < ?)>?(*, W')} + f v(dw')b($(t, w, pi £(/, w', p)) with initial condition )?(0) = 0 (the dot = djdt). 7](t, w, w, p) = j?(f, w, 4 , />) , Setting 3?(w, w, p) = y(w, d<et p) , we have (2.4) #*, iv, w, |o) = f p(dw'){bx(£(t, w, p\ £(*, w', ?))>?(/, w, w, />) + &,(!(*, w, p), f ft wf, p))v(t, wf, iv, ,0)} + w e , w, P), eft #> io)), (2.5) ?(/, w, v, p) = y(rfiv)i?(r, w, w, p). J w Now looking carefully at (2.4) which is a differential equation in the Banach space W (for fixed w and p), we see that ^(/, w, w, p) can be expressed as (2.6) ?(*, w, w, />) = Cft £(w, A f(w, p), ? o Krf" 1 ), where £(/, w, # a ^) is the solution of „ (2.7) Cft w, w, p) = f p{dw'){bx{w(t), w'(0)Cft w, #, p) Jw 192 Certain Diffusion Processes with Interaction 479 with the initial condition £(0, w, w, p) = 0 and pefip)'1 is the image measure of p under the map tip): w e W-* f(w, p) e W. When we regard C(', w> w> p) as a function of t alone by fixing w, w> and p, we denote it by £(w>, #, p). The proof of the following lemma is easy (especially, the proof of (viii) is similar to that of (iii)), so is omitted. Lemma 2.2. £(w, w, p) has the following properties. (iv) K(w, M>, £)![„ < c3 w//A a constant cs depending only on \\p\\9 |[*imi^lUanrf|IMU. (v) ||C(w, w, ^o) — C(w', #, ^)||«, < cA\\w — w'|U w/7A a constant ct depending only on \\p\\ and the supremum norms ofb and its first and second partial derivatives. (vi) ||C(w, w, p) — Q(w, w, JO)||„ < cs||w — H>||OT w/fA a constant c5 depending only on \\p\\t \\bx\\„ and ||2>viL. (vii) |[C(w, #, p) — C(w, M>, ^')IU < 4>Hp — pll w/VA a constant c9 depending only on \\p\\, \\p'[\, [|2>|U U^IU an*/ ||6J M . (viii) Let pn, p € 2)^ a/i^/ pn—^p weakly as n —>• oo. 77je« C(w, iv, />„) > C(w, w, jo) (m W), « - * oo uniformly on each compact subset ofWXW. Lemma 2.3. ^(w, w, p) has the following properties. (ix) ||??(w, w, p) — J?(W, w, /o')|U < c,||/) — p'\\ with a constant c7 depending only on \\p\\, \\pf\\ and the supremum norms ofb and its first and second partial derviatives. (x) Let pn, p €%fl and pn—*p weakly as n —> oo. Then v(w> &, pn) - • y(w, w, pn) (in W), n -> oo uniformly on each compact subset of Wx W. Proof (ix) can be proved directly from (2.4), and (x) follows from (2,6), (iii) and (viii). 2.2. Given a probability measure u in R, we denote by X the Wiener measure on W such that X{w(0) € • } = «(•). Then the solution £(f) of (2) with a = 1 is expressed as (2.8) I0O = £(',HU). To describe the solution of (6) let Q = W X W X • • • and P be the product measure X® X&> • • -. Then, the solution process $w(t) = (fi"}(0» • • •» &"}(0) of (6) with the initial condition 193 480 H. TANAKA £5n)(0), • • -, £J°(0) are independent random variables with the same distribution u (2.9) is realized on the probability space (Q, P) by (2.10) «">(*, o>) = £(*, w(, ^ ) , 1 2, • • •). Therefore, 7 n of (5) is expressed as (2.ii) rn = v ^ l - i ; <w,^> -p), p. = ^ a r 1 , and hence for a (nice) function
(2.12) y n ( 0 = V T T J l £ to which Theorem 1.1 can now be applied with f(w, p) = W* (the dual space) such that \ w' -» 0 . F o r ^ e Q(W)weset (2.13) A (2.14) gfo) = W w e PK, + 4^P} " !*,{(/ + ^MF . where Ep denotes the expectation with respect to p(dw). If f(w, p) = 2))(C(£(H>, ^)» fO^Xj")) anc * hence by Theorem 1.1 we have the following theorem. Theorem 2.1. Z,e/ £<»>(f) = (fi«>(f), • • •, &{t)) be the solution of (6) with the initial condition (2.9) and let Yn be defined by (5) in the introduction. Then (2.15) lim E{ei¥»w} = e~Q^2,
n—taa that is, the distribution of Yn()). § 3. Another expression of A The soluution £(0 of (2) (a = 1) is a temporally inhomogeneous Markov process. This Markov process is called the diffusion process associ- 194 Certain Diffusion Processes with Interaction 481 ated with the nonlinear parabolic equation (3) (a = 1) ([8]). Similarly, for each fixed we Wwe can consider the diffusion process |(f) associated with the nonlinear parabolic equation (3.i) ML = i . * m . - sm ot dxz 2 m + w{t))]m EKX> dx This diffsion process |(f) with initial distribution u is described by (3.2) f(f, wO = w'(0 + f W i f e *"), £ ( ' ) ) * + e [' *(lfc W), W(J))A , Jo Jo where {w'(t), 0 < t < 1, A} is a Brownian motion with initial distribution M as before. Noting that f(f) = £(*) when e = 0, we can prove by a routine argument in differential equations that (3.3) Iim !{#(*, wf) - ft, w')} = i)(ty wf) exists as a uniform limit on [0, 1]XWand (3.4) that jfct, w') satisfies ifct, w>) = f' * f X(dw){bMs, "OSfc *>)#& H 0 Jo J IT + Vffc W 0. ffc W ))T)(S, w)} + £ &(£(*, O , W(j)>fc . Comparing (3.4) with (2.7) we see that (3.5) y(t,w') = ttt,!;(w'),w,it), v = *°&Y1- Theorem 3.1. For each fixed we W let fiWt, be the probability measure on W induced by the diffusion process associated with (3.1) having initial distribution u and denote by EMIWII the expectation with respect to ptWit. Then for A9{w) = lim . ! { £ „ „ » - EJtp)} . .-o g Proof. From (2.13), (3.5) and (3.3) we have = f Kdwf) 195 482 H. TANAKA = l i m l f W ) M l ( > v ' ) ) - 9(«W))} = lim —{£; i B t < (0 - £„(p)}. § 4. An infinite dimensional SDE Let {Y( (4.1) (4.2) QW/ e- \ Y(clipi + c2^2) - CjKfo) + c2F(^2) a.s., and define Y(t) by < 7 ( 0 , / > = ^ ( / M / ) ) ) f o r / i n the space Sf of rapidly decreasing C"-functions on R. We also define Yn(t) by (7) or equivalentlyby = Yn(f(w(t))),fe S?, where r „ is given by (5). Then Theorem 2.1 implies the following central limit theorem in the time evolution scheme: For any 0 < tx < • - - < tm and / , • • - , / m 6 ^ the joint distribution of < r n ( 0 , / > , •••, converges to that of < y ( 0 , / 1 > , - • •, < n O , / m > as n - • oo. SDE for Y(t): Once we know (2.15), we can derive the SDE for 7(/) as in ltd [4], Let u(t) be the (probability measure) solution of (3) with a = 1 and with initial distribution u. We set (*»<»/)(*) = y / " ( * ) + *>[*. " « ] / ' ( * ) , (LuUJ)(x) = j " A(* x)/'(yHKt, dy) , Mf(t) =fWf)) -/MO)) - P (JC.W/X«<*))& , Jo A # ) = MJf) - J' (L„(>)/)(W(^))A . Then Mf(t) is a /^-martingale and it is not hard to prove that (4.3) Ep{\Mf{t)~Ms(s)f\^s} = ^\\r\l^T, a.s. (0 < s < t) , where <2, = a{w(z): 0 < T < s} and |H|„ / e &. 196 Certain Diffusion Processes with Interaction 483 Proof. It is enough to prove that - £ (LuWf)(w(S))ds + £,{/'(Mv)} (4.4) - £ ^{(A«(1)/)'(w(J))Cfc w)}ds -J]^{(A l(f) /)'(H'(j))C(5, ")}<& = 0 for each fixed w z W, where £(?, w) = £(f, w, w, //). Since w(/) (as a process defined on the probability space (W, p)) can be expressed as a Brownian motion + b[w(s), u(s)]ds, an application of Ito's formula yields W O K O K C , w)} - £ „ [ [ F(s>fc} , where i 7 ^) is equal to - 6 [ W ( J ) , M ^ l / ' X w W K f e W) + y/'"(w(5))Cfc W) + / / (wW)[|//(^0{^(w(5), v/(sMs, ") + ^(H>(*), *'(*))«*, w')} + W ) , *(*))] Therefore, writing down (KuWf)' and (Lult)f)' left hand side of (4.4) is equal to ( ^ (2.7)). explicitly, we see that the Efi{G(s)}ds where G(s) = -b(w(s), w(s))f'(w(s)) + b[w(s), u(s)]f"(W(S)K(s, w) + j/'"(H<*))Cfc ") +/'Wj))Ui«(rfMO{^Wj) ( - ^f"'Ms)K(syw) M/«)C(S, M» -f>(w(sK(s,w)§bx(w(s\y)u(Sidy) - b[w(s), u(s)]f"(w(s)X(s, w) - «*, w) J bjz, w(S))f'(z)u(s, dz) 197 484 H. TANAKA = I fKdw^b^wis), w'(s))f'(w(s))&S, w') - bv(wf(s), w(sW(w'm(s, w)} - But from the last expression of G(s) it follows that Ep{G(s)} = 0 and so the lemma is proved. We now write < > m / > - = -/(w(0))> = ds. Jo Jo If we set then by (4.1), (2.14), Lemma 4.1 and (4.3) we have j£{e«i*/ I ^ J = exp { - £ H/'lliwrfr^J , 0 <£ j < *, where &, is the smallest ff-field on Q that makes {B/r)f 0 <£ r < J , / e 5^} measurable. Let {.#(/)} he a (distribution-valued) Brownian motion such that (B(t\f) = Bs(t) a.s. (regularization) . Then d{Y(t\f> = d(B(t),f> + <7(0, *.«>/ + Lult)f)dti or symbolically we have the infinite dimensional SDE dY{t) = J 5 ( 0 + (K*c0 + L * w ) r ( 0 # , where * means the dual operator. This is, of course, consistent with the following formal computation about the generator of the Markov process Y(t). Generators: Yn{t) is a (temporally inhomogeneous) Markov process and its generator L\n) "at time r" is given as follows. Let 0(rj) = , • • - , <9, / m » , flt • • •, fm € S?, Then for y = y' n (x — w(f)) (* = n~l X) $**) w e (Z«(Z>)(,) = E (*, 4rXfMf9 m v> € C 0 "(iT). nave + S <7, KuMQdaV 198 Certain Diffusion Processes with Interaction 485 and hence the generator Lt of Y(t) "at time t" is given (at least formally) by a=l a=\ § 5. Large deviation Let £/„ be defined by (4) and let Qn be its probability distribution, i.e., Qn(A) = P{UneA}, A^0>, where 0 is the set of probability measures on W. For y e ^ w e set v(t) = v(t, •) = v{w(t) e •} and denote by v the probability measure on W induced by the diffusion process (initial distribution = u) associated with the linear equation du 1 d2u d r,r .„„. We then define ., 1X (5.1) _., ff log (dvjdv)dv , /p(v) = J ^ { oo, if v bounded weakly continuous function F on 0° lim — log E{enF^} = sup {f(v) - / » } - /Y00/. If we define a mapping 6: & —> 0* by 6(p) = p° SOO'1, then Un = 6(U%) where £/° = n~l 2 5W • Moreover, it is easy to see that 6 is 199 486 H. TANAKA a homeomorphism from & onto itself and hence by a result of Donsker and Varadhan [3] for independent identically distributed random variables I E — log Qn{C) = HE log P{U°n € 6~\C)} n < - inf I%p) p€8-l(C) where ' f log (dpjdX)dp, if p < X and log (dp/dX) G L\p) { otherwise. oo, We now set ^ = 6(X) and v = 6(p). Since p i« f ' ! W = inf/„(*), proving the first inequality. The second inequality of (i) and the equality of (ii) can also be proved by the same method. § 6. The case of Boltzmann's equation Kac's master equation approach to Boltzmann's equation goes back to 1956. The central limit theorem (fluctuation theory) in the time evolution scheme as in § 4 was discussed in [6], [10], [12], [13] and [15]. Although the methods in the preceding sections can not be applied in a straightforward way to the case of Boltzmann's equation, one may give some conjectures about fluctuations in the path space and large deviation problem for Un. Here we consider McKean's 2-state model of Maxwellian molecules. Let £<">(*) = (£ B> (0, • • •, &B)(0) be a Markov process on the n-fold product space {+1}" with generator (Knf)(xl9 where 2 7 *s t n e •••,*„) sum with respect to the two types of collisions (x't\/xiXj\ \xy or \ Xj / txt\ \XtXj' Let u be a given probability measure on {±1} and assume that the initial 487 Certain Diffusion Processes with Interaction distribution of £ w (t) is w (§) • • • & «• Sample paths of each component process ££°(*) are in the space ^ of right continuous step functions taking values of ± 1 (the time is restricted to 0 < t < 1 as before). Define Un and Yn as in (4) and (5). For each fixed w e W and e > 0 we denote by fiWtt the probability measure on W induced by the Markov process associated with the nonlinear equation (6.D -4-<m,r> = <«(') ® wo + «*.«>). */> having initial distribution w, where (Kf)(x,y) = f(xy) —fix), pt = /*Wi0 is nothing but the probability law of the Markov process associated with the Boltzmann equation 4- / e C 0 ~(JT), 0 < / , < • • • < *m > where A(p(w) = lim — {£ PiIDl . v 0 - £„(?>)}, .jo e and also define /-functional by 'log {dv\dv)dv, if v < v and log (dv/du) e L\v) otherwise, where v is the probability measure induced by the Markov process (initial distribution = u) associated with the linear equation -4-<*<0./> = <«(0 ® «(0. */> , dt v(t) being defined by v(ty •) = v{w(t) e •}. Then from the results in the diffusion case one may conjecture as follows. (6.2) (6.3) MmE{eiY^}^e~ «(?)/* The large deviation for £/n will be governed by the functional IM(y) as in Theorem 5.1. 201 488 H . TANAKA N o t e s a d d e d in p r o o f . R e c e n t l y , m a k i n g u s e of a m e t h o d of S z n i t m a n [11], T . S h i g a a n d t h e a u t h o r p r o v e d t h a t (6.2) is c o r r e c t . References [ 1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Comm. Math. Phys., 56 (1977), 101-113. D . A. Dawson, Critical dynamics and fluctuations for a mean field model of cooperative behavior, J. Statistical Physics, 31 (1978), 29-85. M. D . Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time-Ill, Comm. Pure Appl. Math., 29 (1976), 389-461. K. It6, Motions of infinite particles (in Japanese), Kokyuroku RIMS, Kyoto Univ., 367 (1979), 1-33. M. Kac, Foundation of kinetic theory, Proceedings of 3rd Berkeley Symposium on Math. Stat, and Prob., vol. 3 (1956), 171-197. , Some probabilistic aspects of the Boltzmann equation, Acta Phys. Austriaca Supp. X, Springer Verlag, (1973), 379-400. S. Kusuoka and Y. Tamura, The convergence of Gibbs measures associated with mean field potentials, to appear in J. Fac. Sci., Univ. of Tokyo. H. P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci., 56 (1966), 1907-1911. , Propagation of chaos for a class of non-linear parabolic equations, Lecture Series in Differential Equations 1, Catholic Univ. (1967), 4 1 - 5 7 . , Fluctuations in the kinetic theory of gases, Comm. Pure Appl. Math., 28 (1975), 435-455. A. S. Sznitman, An example of non-linear diffusion process with normal reflecting boundary condition and some related limit theorems, preprints, 1983. H. Tanaka, Fluctuation theory for Kac's one dimensional model of Maxwellian molecules, Sankhya: The Indian Journal of Statistics, 44 (1982), Series A. Pt. I, 23-26. , Some probabilistic problems in the spatially homogeneous Boltzmann equation, Proc. IFIP-WG 7 / 1 Working Conference on Theory and Applications of Random Fields held in Bangalore, January 1982, Lecture Notes in Control and Information Sciences 49, Springer-Verlag. H . Tanaka and M. Hitsuda, Central limit theorem for a simple diffusion model of interacting particles, Hiroshima Math. J., 11 (1981), 415-423. K. Uchiyama, Fluctuations of Markovian systems in Kac's caricature of a Maxwellian gas, J. Math. Soc. Japan, 35 (1983), 477^199. DEPARTMENT OF MATHEMATICS FACULTY O F SCIENCE AND TECHNOLOGY K E I O UNIVERSITY, YOKOHAMA 223, JAPAN 202 Zettschrift fiir Wahrscheinlichkeitstheorie Z. Wahrscheinlichkeitstheorie verw. Gebiete 69, 439-459 (1985) und verwandte Gebiete © Springer-Verlag 1985 Central Limit Theorem for a System of Markovian Particles with Mean Field Interactions Tokuzo Shiga1 and Hiroshi Tanaka 2 1 Department of Applied Physics, Faculty of Science, Tokyo Institute of Technology, Oh-okayama, Tokyo, Japan 2 Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan § 0. Introduction Let S be a measurable space and ^(S) be the totality of probability measures on S. Suppose that to each ve&(S) there corresponds a generator Qv of a Markov process on S. Starting from the family we can consider an interacting rt-particle system Xi")(t)=(Xi")(t), ..., Xj,n)(f)), which is a Markov process on 5®" = s x ... x S with generator (0.1) G ( "V(*I,.•.,*„)= E e £ * ( * i . •••»*-). where 3X stands for the ^-distribution at x, and Q[l) is used instead of Qv when it acts on the i-th variable of <j>{xx,...,xn). Denoting by W the path space on S, let us consider an empirical distribution Un of the n-particle system: 14=-E*IW (•)' (0.2) which is a random measure on W. Our problem is described as follows. Assume that (Xfi(0),...,X%){0)) is independent and identically distributed with common distribution U G ^ ( S ) for any w ^ l . It is known that there exists a probability distribution \i on the path space W such that U„ converges to \i as n-+co in the sense that <£/„,
converges to in probability for any bounded measurable function
dt y = (u{t\Qui[)<j>>, (0: test function). 203 440 T. Shiga and H. Tanaka As the next stage we consider the central limit theorem on the path space, that is to show that (0-4) ^n = ]/n(Un-fi) converges as «->co to a Gaussian field. In the case of diffusion processes there have been many works concerning these problems (cf. Dawson [1], Kusuoka-Tamura [7], McKean [9], Sznitman [13], [14], Tanaka [15], Tanaka-Hitsuda [16]). Above all, Sznitman recently proved the central limit theorem on the path space for a general class of diffusion processes on Rd associated with {Qv; ve^{Rd)} where ( 5) Q 4iia^j^+i^M^, °- {ocij{x,y)}l^iJ^d and {bi{x,y)}l^i^d being uniformly bounded and Lipschitz continuous functions on Rd x Rd and d a ijtx>v']= Z ^ik(x>y)v{dy)iajk{xiy)v(dy)i k = l bi[x,v] = jbi{x,y)v(dy). In the present paper we study the central limit theorem mainly for pure jump type Markov processes. More precisely, let {S,&s) be a measurable space and let Q{x,x';dy) be a bounded measurable kernel on SxSx@s. Then to each ve&(S) there corresponds a generator Qv of a Markov process on S by (0.6) Qv4>(x) = iv(dx')\Q(x,x';dy)(<j>(y)-c}>(x)). Under a certain condition of non-degeneracy we obtain the central limit theorem for (0.4) (see § 2). Furthermore, we remark in § 3 that our proof is valid for McKean's 2-state model of Boltzmann's equation with a slight modification and we have the central limit theorem as conjectured in [15]. We also give some remarks in §4 to the case of diffusion processes as discussed in [13], [14] and [15]; here we assume the diffusion matrix tx(x,y) is the identity matrix but the drift vector b{x,y) is bounded measurable. Our method of the proof is essentially based on Sznitman's paper [13], which treats the case of diffusion processes with a{x,y) = identity matrix. There he calculates a limiting quantities of Cameron-Martin-Maruyama-Girsanov density by making use of symmetric statistics and multiple Wiener integrals. His method is also applicable to our case. However we emphasize that once we obtain formulae on multiple Wiener integrals (the two lemmas in § 1) the proof turns out to be more transparent. § 1. Symmetric Statistics and Multiple Wiener Integrals For the later use we summarize some facts on symmetric statistics and multiple Wiener integrals according to Dynkin and Mandelbaum [3], and present two formulae on multiple Wiener integrals. Central Limit Theorem for a System of Markovian Particles 441 Let {X,&) be a separable Borel space, and let {Xn}™=l be a sequence of independent and identically distributed ^-valued random variables with a distribution v. For eachfc= l,2,... let L2{v®k){L2c(v®k)) be the space of all realvalued (complex-valued) square integrable functions on 3£®k = 3£ x... x3£ {kfold) with respect to v 0 i = v x . . . x v (fc-fold). Denote by L2symm{v®k) the space of all symmetric functions of L2(v®k). Then all of these spaces are separable Hilbert spaces. To hkeL2symm(v®k) there corresponds a symmetric statistic al{hk) defined by (L1) <*)= I hk{Xh,...,Xik) for =0 n^k for n Let { l ^ ^ ^ e L 2 ^ ) } be a centered Gaussian field satisfying (1.2) £ { ' I ( * ) / I M } = (*,WL2(V). 2 For each h* = \, ht(xl,...,xk) = A multiple Wiener integral /k(fc£) is defined by the following relation, d-4) l £ / 4 ( J # = exp ( ^ ( f l - y l W I i ^ ) . For a general /ifceL*yinm(v®*) /k(/ifc) is defined by a standard procedure making use of the fact that the linear hull of {hf: <j>eL2{v)} is dense in L2symm(v®k). Then we have Theorem 1.1 (Dynkin-Mandelbaum [3]). Suppose {/ik}™=: satisfies the following: for each k^l hkeL2symm{v®% and (1.5) $hk{x1,...,xk_1,x)v(dx) =0 for v®k-1-a.e.{x1,...,xk_1). Then ~f1 1 {« 2 ^(/i k );/c^l} converges to < — Ik(hk);k^ 1 > as n-*oo m the sense of convergence of finite dimensional distributions. Let a(x,y)eL2(v®v) and denote by A the integral operator on L2c(v) associated with a(x, v), (1-6) ^<M*Wfl(x,:v)0O')v(d3') (06L2c(v)). Then ^4 is a Hilbert-Schmidt operator, and An(n^.2) and ,4,4* are trace class operators. It is easy to see (1.7) TraceAn = §...§a(x1,x2)a(x2,x3) ...a(xn,x1)v(dxl)... v(dxn) 442 T. Shiga and H. Tanaka for any n^2, and TmctAA* = \\a\\2L2{v@vr (1-8) The following lemmas will be used effectively in proving central limit theorems in the following sections. Lemma 1.2. Suppose that Traced" = 0 for all n ^ 2 . Then E [e*/2(/> ] = e^'^eAA* (1.9) where (1.10) f{x,y) = a{x,y) + a{y,x)-\a{x,z)a{y,z)v{dz). Proof. Theorem9.3 of Simon [12] implies that Trace4" = 0 for all n^2 if and only if (1.11) det 2 (J + ^ ) = l for any complex number fi, where det 2 (J+ fiA) stands for the regularized determinant. In particular we have (1.12) det2(/-i4)^det2(/-i4*)=l. Hence it follows from Theorem 9.2 of [12] that I —A* is invertible. Denoting by F the integral operator on L2c(v) associated with f(x,y), I-F = (I-A)(I — A*) is strictly positive definite because of the invertibility of I —A*. Accordingly any eigen-value of F is real and less than one. Let Xu A2, ... be all eigen-values of F and {ex, e2, •••} be the orthonormal base of the corresponding eigen-vectors of F on L2(v). Then / is represented as follows: oo 00 Since £ A£ independence of {/i(ek)}fL j «PLy(MeJ2-l)J 00 r 1 2 ~\ = ni£[expf(/,W -l)J 00 p 2 On the other hand the following formulae are known (cf. Simon [12], p. 107): oo (1.14) det2(/-F)=n(l-^)^k fc= i (1.15) dGt2{I-A)(I-A*) = det2{I-A)det2{I-A*)e-jTli(*AA\ Central Limit Theorem for a System of Markovian Particles 443 Thus we have by (1.12) (1.16) d e t 2 ( / - F ) - d e t 2 ( / - J 4 ) ( / - y l * ) = e-Trace-4-4*. Therefore (1.9) follows immediately from (1.13)—(1.16). Lemma 1.3. Assume the same condition as in Lemma 1.2. Then I — A is invertible and for any <£eL2(v) (1-17) £[exp{i/^l/l(*)+i/2(/)}] ^exp-KlKZ-^-Vll^-Trace^*}. Proof. It follows from (1.12) that I —A is invertible. Also using the same {ek} of the proof of Lemma 1.2 we have (1.18) £[exp{y'=Il1(0 + i/2(/)}] ni£ k= _ [exp|y,^T(^et)/1K)+y(/ife)2-l)}] e 2 k=iyl-^k f 1 I ^k=i = {det2(J-F»-*exp{-i((/-f)-10,0)L2{v)} ••e±T^AA*exV{-i\\(I-A)-i\\l2iv)}. § 2. The Case of Pure Jump Type Markov Processes Let S be a standard Borel space, i.e., {S,£%s) is Borel isomorphic with a complete separable metric space. Denote by IB(S) the Banach space of all bounded measurable functions on S equipped with the supremum norm || * || G0. Let Q(x,x';dy) be a measure kernel satisfying the following conditions: (2.1) for any fixed (x,x')eSxS and Q{x,x';{x}) = 0; (2.2) for any fixed Ee&s (2.3) q(x,x') = Q(x,x';S\{x}) Q(x,x';dy) Q(x,x';E) is a bounded measure on is measurable in (x,x') with respect to Ms is bounded in (x,x')eS xS; (2.4) there exist constants c1 > 0 and c2>0 such that clQ(x, x'; dy)^Q(x, x"; dy)^c2Q(x, x'; dy) for any x, x' and x" of S. For each finite measure u on S, we write (2.5) Qu(x;dy) = SQ(x,x';dy)u(dx% s and define a bounded operator QM on JB(S) by (2.6) (S,$s) e H * W = JeH(^;dj')(*0')-*W) for 0 e B ( S ) . 207 444 T. Shiga and H. Tanaka Let W be the set of all S-valued step functions, i.e. for each weW there exist 0 = t1 w(*) = w(tj Denote by & restriction of ne&(W), the measure (or a for tt£t and &x the usual cr-field of W. For any T>0 {WT,^T) denotes the (W,^) to [0,T]. Then these are separable Borel spaces. Let set of probability measures on (W,^). \i is called a McKean McKean process) corresponding to {Qv;ve£?(S)} if t (2.8) is a/^-martingale for any e]B(S), o (2.9) &{w(t);ti) = u{t). Here S£(w(t);fi) stands for the probability distribution of w(t) under fi. For any fixed ue^{S), the set of probability measures on {S,$s), there exists a unique McKean measure /j corresponding to {Qv;ve^(S)} with ^(w(0);fi} — u, which will be shown in Lemma 2.3. Now, let us consider an n-particle system, which is a Markov process {X\{t\ ..., Xnn(t)) on S®n = Sx ... xS generated by the following operator: (2.io) <2(nV(*i,...,*,,)=~£ Z J ^ , * ^ ) ^ ^ , . . . , ^ ) «i=l j= 1 S where ^(*i, ...,x„) = (/)(x1, ...,y,...,x n )-^(x 1 ,...,x M ). lhei-lh We assume the initial distribution is u®n = u®...®u, the n-fold product measure of u. Let us introduce an empirical distribution Un of the rc-particle system, (2.U) tf»=Jiw Then Un is a random variable with values in &{W\ and we can show that Un converges as n->cc to the McKean measure \L. In order to discuss a central limit theorem for Un let us introduce some notations. From now on, we fix T>0. For any weWTi O ^ t ^ T and Ee&s set (2.12) AKt,w;EHX/(vv(s)e£,w(s)*vv(s-)), and (2.13) N(t^E)-\Qu(s)(w(s);E)ds, o where 7(B) stands for the indicator function of B. For each (w, w')eWT x WT set (2.14) N(t,w;E) = a{wM) = \ J fa((w'(s-),w(s-);j;)-l)JV(d5)d3';w') S [0,T] where u(t) = jgr(w(f);/i). Central Limit Theorem for a System of Markovian Particles 445 We note qt(x,x';y) is bounded by (2.4). Furthermore since a{w,w') is square integrable with respect to /i(x)/x, there corresponds a Hilbert-Schmidt operator A on I}c{WT,ii), (2.15) A<j){w)= j a(w,w')<j>{w')u{dw'). wr Note that l}c(WT,y) is a separable complex Hilbert space. Then it is shown that J — A is an invertible operator. Let us describe our main results. Let (2.16) U^)-Yn<.Un-^y^ for t*W *eL 2 (W r ,M) where S — 0—^fx,eL2{WT,u)} satisfying (2.17) Gaussian £K(*)«y)] = ((/-A)-1«,(/-^)-i#)L2(HrTirt in the sense of convergence of finite dimensional distributions. Next let us consider another interpretation of the operator (I — A)'1 as in Tanaka [15] and Sznitman [13]. For any fixed l^eW and e > 0 there exists a unique / i j j e ^ W ) such that (2-18) is a ^-martingale (2.19) *(wW)-Je ( B c w + ,* C U ) ) *(wW)^5 for any 0eIB(S), and J?(w(t);£) For any &el}(WT,fi) (2.20) = ul(t), J?(w(0);^) = n. and e > 0 nB${Q = \*> is well-defined, and we can show (2.21) n
exists in L2{WT,fi). e->0 Furthermore the following identity holds. Theorem 2.2 (2.22) I +n = {I-A)~1. We will prove these theorems by a series of lemmas. Lemma 2.3. Assume (2.1)-(2.3). Then for any UE^(S) there exists a unique McKean measure fj. corresponding to {Qv;ve0>(S)} with J^(w{0);//) = w. 446 T. Shiga and H. Tanaka Proof. Let v(t) be a ^(5)-valued function defined on [0, oo) such that (v{t),<j>) is Borel measurable in t for any <j>{w{t))-\Qv(s)<j>{Ms))ds o is a v-martingale for any (/>eIB(S), and (2.24) J2?(w(0);v) = i;(0). For a o\\q\\(X) = supq(x,x') define a probability kernel Pt by x,x' lj(i;E)^p(1/x;£\{x})+p^(')])^(E) (2.25) where xeS, Ee$s and ^[x, y] = jg(x, y)u(^y). Then s (2-26) e B W * W = c[JJ?(x;d>;)^(y)-0(x)]. Let X(0) be an S-valued randon variable with the distribution u(0), and Nt be a Poisson process on {0,1,2,...} with the intensity c independent of X(0). Denote by an the n-th jumping time of Nt. Let us construct an S-valued process X(t). Set X(t) = X(0) for 0^t{Sy valued solution t (2.27) {u{t\<j>y-{u,<j>y = \iu{slQu{s)4>yds for any eIB(S). o Let u°(t) = u. If un~1(t) is defined we have a unique solution / / of (2.23) and (2.24) with v(t) = un-1{t), and set un(t) = &{w(t);n"). Then we have (2.28) (u"{t\}-(u,<}>)^\(u»(s),Qun-1{s)yds o for any <j>eJB(S). Noting (2.3) it follows (2.29) \\u"+1(t)-u"(t)\\vaT ^2\\q\\„$(\\u"+Hs)-u»(s)\\var+\\u»{s)-u"-Hs)\\vJds, 0 where ||-||var stands for the total variation norm. This yields that there exists u(t)e0>(S) such that (2.30) lim sup \\un{t)-u(t)\\yai = 0 H-.00 0 for any T>0. 210 Central Limit Theorem for a System of Markovian Particles 447 Thus we get a solution u(t) of (2.27), and the uniqueness also is obvious. Finally, let fi be a unique solution of (2.23) with v{t) = u{t). Then u{t) = if(w(t);ju) is a solution of t (2.31) - } = J rfs ^elB(S). o Since the uniqueness of solutions to the linear equation (2.31) can also be proved easily, u{t) coincides with u(t) which is also a solution of (2.27). Therefore fj, is a McKean measure corresponding to {Qv;ve^(S)j with if(w(0);/i) = u which is also unique by virtue of the uniqueness result for (2.27) and (2.23). From now on, we fix the McKean measure /i obtained in Lemma 2.3. Lemma 2.4 (i). For any Ee&s (2.32) N(t,E;w) is a p-martingale, and (ii) for any Ex and E2 of $s (2.33) N{t,E^w)N{t,E2;w)-]Qu(s){w{sy,E^E2)ds o is a ii-martingale. Proof Noting that for any eIB(S) (2.34) 4>(w(t))-<j>(w(0))= J ((y)-(w(s-)))N(dsJy;w), and [O.rjxS t (2.35) (w{t)) — $Qu(s) we have (2.36) j {<j){y)-4>(w(s-)))N(ds,dy;w) is a fi-martingale. [0,1] x S Also it is easily seen that for i//eIB(5) (2.37) j" yj/(w(s-))({w{s-)))N(ds,dy;w) is a fi-martingale. [0,1] x S In particular, if <j)(x)\j/(x) = 0 for all xeS then (2.38) j" is a fi-martingale. [0,1] x S Furthermore noticing N((s,y);w(s~) = y, 0 ^ s ^ f ) = 0 for any t > 0 it is easy to see that for any /eIB(5 x S) (2.39) | f(y,w(s-))N(ds,dy;w) is a \i-martinga\e, [0,t]xS from which (i) follows, (ii) is immediate from (i) combined with the following identity obtained by integration by parts, 211 448 (2.40) T. Shiga and H. Tanaka S(t,£i;w)%£2;w)-iV(t,£in£2;w) J\N(s-,E1;w)dN(s,E2;w) 0 + \N(s-,E2;w)dN(s,E1;w). 0 Let {Xn(t) = {Xu1(t),...,XHH(t))) be the Markov process on S®" with the initial distribution w®", generated by Q(n) of (2.10). Let us denote by Pn the probability distribution induced by {XM(t)}0^t^T on the path space (WZ,&£), where W£ = Wrx...xWrand^t', = #rx...x#Ifor0^t^T. Lemma 2.5. For each n^l P" and ju®" are mutually absolutely continuous and (2.41) (2.42) ~ ^ ( w ) = exp/Tr(W), J j lo g- £ g > * ( * - W S - ^ W M ^ W J J*;(w)= t »=1 [0,1] xS L " j = l J - i £ 1 1 ^ ( W i ^ w j M ) - i^K(5),W(S)])rfS for W = ( W 1 , . . . , W I I ) G W ? and £>0. Proo/ Let M"(w) = expH"(w). Since {w ; (t)} 1 ^ i ^ n have no common jumps almost surely (^®n), it follows from Ito's formula (cf. Ikeda-Watanabe [4], p. 66) (2.43) MJ(w)-l=£ j Mat-(-t^^-\Wj(s-);y)'l) N{ds,dy;w^ \M j = l i= 1 [ 0 , f ] x S ' Hence M"(w) is a /i®"-martingale. Next, for eJB(S®n), set KM) = (yv(t))-\Qin)(yv(s))ds (2.44) (2.45) K^HMt))- Z JlQ I f W (w i ( S );dy)^0(wW)^. ;=i o s It is obvious that K0(t) is a //®"-martingale. We will show (2.46) M^K^t) is also a fi® "-martingale. By integration by parts (2.47) M;ic+w-jx+(s-)dM;+jM;.^M 0 + 0 \MUdK^s)-dK4,(s))+Yl(M:-M"s_)(K4S)-K4s-)). 212 Central Limit Theorem for a System of Markovian Particles 449 Also, (2.48) Im-MUiK^-K^s-)) = I(M;-M;J(^(W( S ))-0(W(S-))) = IjM:-(j S r~i)^*(w(s-))^s,dy;w 1 ) =t i M;_(itq>,.( S -),w,( s -);y)-l) i = 1 [0,1] x S V* j= 1 / •4f0(w(s))N(ds,dy; Wi ). On the other hand noting that (2.49) \M*8_(dK+{s)-dK+(s)) o = ~ t i$Mns- (- Z«,(w i (*w J W;y)-l)Q B W (w i (s);dy)^f*(w( S ))d S , i=l 0 S \"j=l / we have (2.50) M;X + (t) = J ^ ( S - ) d M ; + iM;_dK + (s) 0 0 +1 I M;_(it^( Wi (s-), Wj .( s -);y)-l) i - 1 [0,(]xS V j - l / Hence MJK 0 (r) is a ^® "-martingale. Finally for each Ae^j we set Pn{A) = \MnT{Yi)ii®n{dw). (2.51) A Then P" is well-defined as a probability measure on (W£t&£). Moreover it is easily seen that for any eB(S®B) K^(t) is a Pn-martin%ale (2.52) since MntK^) is a ju®"-martingale. Therefore P" = Pn, because of the uniqueness of the martingale problem (2.52) with if(w(0);F'I) = M®n, which completes the proof of Lemma 2.5. Next we discuss a limit of a functional Hj(wlt ...,w„). Since H " ^ wj can be regarded as a symmetric statistic of {wn}*= 1( which is a sequence of (Wp^yJ-valued independent random variables with common distribution p, Dynkin-Mandelbaum's theorem of § 1 is applicable. Lemma 2.6 \imHnT(w1,...,wn)=±I2(f)-±TriicQAA* (2.53) n-><x> 450 T. Shiga and H. Tanaka in the sense of convergence of probability distributions, where f{w,wl) = (2.54) a(w,w')-^a(w',w)-ja{w,w")a{w',w")p.(dw") and l2{f) is a multiple Wiener integral associated with {WT,p). Proof. Using Taylor's expansion of logx we have (2.55) H- r ( W l ,...,wJ i = l [OJlxS V j = l " i;=i t / I £ t *>*(*-)• w / s -);>0-l)/ N(dt,dy;WJ [o.rjxs '"j=i +i t I (" t^>i( s -X w i( s -)^)-0 N(dt,dy;wi -it f ^WiV(rft,rfy;wj ; = i [o,f]xs -/;+/;+/j+/a, where (2.56) 1^(01^ const. (- t «,(w ( (s-), W j ( 5 -),>)-l) . We first note (2.57) /I=-t t"K^)- Since it is easily seen that (2.58) a(w,w')eL2(^ (2.59) $a(w,w'){i(dw') = $a(w',w)ft(dw)=0 ^-a.s., it follows immediately from Theorem 1.1 (2.60) l i m J ^ / ^ a + a*). where a* is defined by a*(w,w') = a(w',w). Let (2.61) b(w;w',w") = I [0,T]xS toXs-),w'(s-);>)-l)te,(w(s-),w"(s-);y)-l)JV(ds,dy;w). 214 Central Limit Theorem for a System of Markovian Particles 451 Then (2.62) t ix*,;*;,**) /!=-AE •^« i=i ;= l fc=i distinct 1 n — f 1 Z i 4= j _L ln _(_ P _1_ J" " " i= 1 _L- /•" *~i2,l ^ J 2 , 2 ^ i 2 , 3 + J 2 , 4 ^ J 2 , 5 ' where 6[/i;w',w"] = jb(w;w',w")/i(dw). Note that (2.63) 6(w;w',w") = b(w;w",w') and \b(w;w',w")dfi{w") = 0. Also it follows from Lemma 2.4 (2.64) fc \ji\ W, w"] — ja(w', w) a(w", w) /i(d w). Accordingly, by virtue of Theorem 1.1 we see I2,i^2,4 anc ^ ^2.5 v a m s n and furthermore we get by (2.64) (2.65) limJ" 2 i 2 =-il 2 (&|>;-,-]) (2.66) limJ2i3«-£ f - —I as «-*co, b\jx\W,W\ii(dW) j a(w,w')2/x(<2w)^(dw') = —lTrace/1,4*. Consequently we obtain (2.67) limr2-i/2-i6[/i;%-]-iTrace^J4*. n-> 00 As for 1\ and I\, by making use of Theorem 1.1 repeatedly we can show that they both vanish as n->oo, and the proof of Lemma 2.6 is completed. Lemma 2.7. Traced" = 0 for all « ^ 2 . Proof. By (1.7) it suffices to show that for any n^2 (2.68) \ a{w1,w2)a{w2>wz)...a{wrt,w1)fi{dw1)...fi{dwn) = 0. Let (2.69) a(t;wM)= J [0,r]xs te,(w'(s-),w(s-);y)-l)JV(ds,rfj;;w'). 215 452 T. Shiga and H. Tanaka Since a(t;w1,w2), a(f;w 2 ,w 3 ),...,a(t; wn, wx) are square integrable n®n martingales of bounded variation, and any two of them have no common discontinuities with respect to ^®", Theorem d) of [2] p. 367 implies that a(£;w1,w2)a(£;w2,w3)...a(£;wII,vv1) is a (i®"-martingale, which yields (2.68). Now we are in position to prove Theorem 2.1. Let any el}(WT,[i) be fixed. By Lemma 2.5 (2.70) Er" [ e x p / ^ I (fn{Un -/i), 4>>] = £ ^ n [ e x p | i / ^ ^ X ^ ) + ^>i'---^n) It follows from Lemma 2.6 that expH^(w) converges as n->oo to exp|(/ 2 (/) — TraceAA*) in the law sense. Noting Lemma 2.7 we can apply Lemma 1.2, and we get (2.71) £[exp^(/ 2 (/)-TraceXA*)] = l. Hence {expH^-(w)} is uniformly integrable. Also, using Theorem 1.1 again f 1 l00 " e x p V - l - ^ j : * ( W | ) + H-r(w)^ ( ynui Jn=i is uniformly integrable and converges as n->cc to exp{y^T/ 1 ( lim E*®" [exp { / ^ - ^ = I *(*",)+ H"T(w)}l cxp{-^\\(I-A)-^\\lHWT^ which completes Theorem 2.1. Next we proceed to prove Theorem 2.2. We can show in the same way as Lemma2.3 that for any fixed £eWT and £>0 there exists a unique solution of the martingale problem (2.18) and (2.19), which we denote by JX\. Moreover the proof of Lemma 2.5 implies that n\e&(WT) and \ie^{W^) are mutually absolutely continuous and dLfi (2.73) ^P(w)-expH £ /(w), where (2.74) H^(w)= J {logg,[w( S -),uJ( S ) + £^ w ;y]}JV(d S ,dy;w) [0J]xS T -iq[yv(s),u\(s)-\-e6f.{s)-u(s)]ds 0 216 Central Limit Theorem for a System of Markovian Particles 453 where t^(s) = J5?(w(s);j$, u(s) = &{w{s);pi) and qs[x,v;y~\ is defined similarly to qlx,v\. Lemma 2.8. There exists C > 0 independent of (,tW and e > 0 such that (2.75) \\ul(t)-u(t)\\vaI£Cst for O^t^T, and \HBfHw)\SCe(l+N{T,S;w)) (2.76) for any weWT. Proof For any <£eIB(S) (2.77) || = J<«5(s)»erfW+J!*C(.)0>rfs-j<M(s),e«(,)0>ds J < t t { ( s ) - M ( 4 e « w 0 > r f s + J<Mj(s),G B c w+ «a«, ) _ IlW 0>ds JI|i4W-«(s)ll va JIG B (s)0IL^s+J|ie BCW+rfc( . ) - 8(J) 0IL^ o o < Const £ ^(jlliiiW-fiWIL.^ + et). Thus we see ,«|(t)-«(r)|| var ^Const, (f ||MJ(S)-w(s)|| var + et) (2.78) which yields (2.75). (2.76) follows immediately from (2.75), noting that qs(x,x';y) is bounded from above and from zero and that a trivial identity qs[x,u(s);y~\ = 1 holds. We give the proof of Theorem 2.2. We have for any $el}(WT,fi) (2.79) ii« 4KO=io4-* $> =-<M^-I)
e E e Using Taylor's expansion Hy is represented as follows: y*"0Kf) Hf{w)= J (qslw(s-),ul(s);yl-l)N(ds,dy;w) [0,T]xS +£ J (qMs-U(s)iy)-l)N(ds,dy;w) [o,r]xs (2.81) |/f5(w)|^Const. J to, n x s + JE/(w) (q3bv{s-)M(snedi{!l);y-]-l)2N{ds9dy;W). 217 454 T. Shiga and H. Tanaka Since (2.82) lg s [w(s-),^(s) + e( 5 C(s) ;3;]-l| = k 5 [w(s-),M^(s)-u(s);j]-e(? s (w(s-),C(s);y)| S\\qs\\JH(s)-u(s)\\VAr + s) it follows from Lemma 2.8 that |J^(w)|^Const.e 2 iV(T,5;w). (2.83) Furthermore we note by (2.14) and (2.80) (2.84) H£/(w) = ja(w',w)(^-/i)(rfw') + ea(C,w) + J ^ . Hence combining these it is easily checked (2.85) lim (-(ti,H¥ uniformly in (eW T . Also, using an inequality |e* —1 — x | ^ ~ - e | x | and (2.76), we have (2.86) -K(iMH^-l-H¥)>\ 2e 1 2lH ^-jn ^\H^\^ LHWT,^<^ which tends to 0 as EJ.0, since (n,eaN(T's''w)}< + co holds for any a > 0 . Consequently we obtain by (2.79), (2.85) and (2.86) (2.87) hm(nE{I-A)&{0-A =0 uniformly in £EWT. Noting that I —A is invertible and (I —A)'1 is a bounded operator on l}{WT,}x\ (2.88) s-\imIIE
A{I-A)-i ElO which completes the proof of Theorem 2.2. § 3. The Case of McKean's Model of Boltzmann's Equation Let us consider McKean's 2-velocity model of Maxwellian gases. For each i;e^({±l})set (3.1) Q>(x) = w ( - l ) ( 0 ( - x ) - 0 ( x ) ) x=±l. 218 Central Limit Theorem for a System of Markovian Particles 455 Let any 7 > 0 be fixed. WT denotes the set of all right continuous step functions defined on [0,T] taking values in {±1}. Let any ue0*{{±l}) be given. Then by Lemma 2.3 there exists a unique McKean's measure \i on WT corresponding to {Q0;ve&{{±l})}, and u(t) = JS?(w(t);/*) is a solution of the following Boltzmann's equation: (3.2) £ < « « , * > = <««®«*«»e*>, *elB({±l}), at where Q(j>{x,y) = (xy) — <j){x). To avoid triviality we henceforth assume 0 ^ u ( + l ) < l . The n-particle system X{n)(t) — (X"(r), ...,XJ(t)) is defined as a Markov process on the n-fold product space { + 1}" with generator <24>(*i,-,*„)=- (3.3) Z zw(..,)c;- *i,..)-#i ^j) where £ is the sum with respect to the two types of collisions c \X'J \ Xj ) VCiXj) ' It is assumed the initial distribution of X{n){t) is u®". Noting (3.4) QM<j>(x) = t uf(x){(x1,...,-xi,...,xn)-(x)} (x = (x 1 ,...,x n )) i=l 1 " with u{"){x)=- Y,'{-i}(*/)» w e s e e t n a t this model is essentially contained in the framework of the preceding section although the condition (2.4) fails. If we set for each weWT Nt{w)=Y4I(w(sHw{s-)) t Nt{w) = Nt{w)-\u{s,-l)ds o t Nt{w) and Nt{w)2 — \u{s, — X)ds are //-martingales as in Lemma 2.4. We note o that Lemma 2.5 should be modified in the following way: Lemma 2.5'. Let any T> 0 be fixed, and let P" be the probability measure on Wj = WT x ... x WT induced by {Xin)(t)}0^t^T. Then P" is absolutely continuous with respect to ft9*, and (3.5) where ^ ( W ) = expH"T(W) 219 T. Shiga and H. Tanaka 456 U H^)=t]\o ^P^dN g s(Wi i^t 0 U(S, - 1 ) - £ ](u^(s))-u(s,-\))ds. We notice that - oo ^ HnT(w) < + GO. For (w,w')eWT x WT we set a(w,w') = f/ { _ 1} (w( S -))rfiV>'), (3.6) and (3.7) /(w, w') = a(w, w') + a(W, w ) - Ja(w, w") a(w' w")/i(dw"). Then we have Lemma 2.6' limexpifnT(w) = e x p { i / 2 ( / ) - i T r a c e ^ A * } (3.8) n->co in the sense of convergence of probability distributions where I2(f) Wiener integral associated with {WT,n). Outline of the proof of the lemma. Let W; = -{w = (w1,...)w11); sup w(n,(w(r)) «(s,-l) -1 is a multiple 1 2 K Then we claim limn®n(Wa) = l. (3.9) Note IU(w,(t))-< M (t),0>-J M ( s ,-l)(^( Wi (s))-eIB({ + l}). Making use of the law of large numbers and a maximal inequality for martingales, it is easy to check sup Z*(w i (t))-<M(t),*> 0 vanishes as n~->cc in probability w.r.t. /j®", and (3.9) follows immediately from this. Accordingly we obtain Lemma 2.6' combining the proof of Lemma 2.6 with (3.9). Thus we see that Theorem 2.1 and Theorem 2.2 are valid in the case of McKean's 2-velocity model of Maxwellian gases since the remaining parts of the proof are quite same as the preceding section. Central Limit Theorem for a System of Markovian Particles 457 § 4. The Case of Diffusion Processes Let b(x,y) be an Revalued bounded measurable function on RdxRd. For each veP(R?) set (4.1) QM*)A X ^ T * W + Z^[x^]^-0(x) z i=it7xi i=i vxt where bi[x,v~\ = jbi{x,y)v{dy)(i = X,...id). Let any T > 0 be fixed. Denote by WT the set of all Revalued continuous functions defined on [0,T], equipped with the usual <x-fields &v Let any ue&{Rd) be given. Then we have Lemma 2.3". There exists a McKean measure fi on WT corresponding to {Qv;ve^(Rd)} with if(w(0);/i) = u. Namely \i is a solution of the following martingale problem: (4.2) *(W«)-|G,,W<«WM)^ 0 2 is a fi-martingale for any bounded C -function , and (4.3) &{w(t);fi) = u{t) {O^t^T) and u(0) = u. We omit the proof since it can be shown easily along the same line as in Krylov [5] concerning the existence theorem of SDE. But we think it is not easy to prove the uniqueness of the above martingale problem by some direct methods. However, the method of Sznitman [13] for proving the central limit theorem works even if we do not know the uniqueness in advance and we can easily obtain the uniqueness as a consequence of the central limit theorem. Now we take a solution \i of the above martingale problem. Then (4.4) B(t, w) = w(t) - w(0) -\b lw{s), u(sf] ds, 0 is a d-dimensional Brownian motion with respect to /1. The corresponding nparticle system Xln){t) = (Xn1(t),...,Xnn(t)) is an (Revalued diffusion process having the initial distribution w®" with generator (4.5) <2<">0(x1,...,xw) = i f z l X i 0 ( x 1 , . . . , x n ) i=l 1 n n "i=lj=l where Ax. and Vx. act on the i-th variable of 4>(xl,...ixn). Let Pn be the probability measure on W£= WT x ... x WT induced by {Xin)(t)}0^t dPn w = 7^r( ) dj? ex P^(w) 458 T. Shiga and H. Tanaka where tfW= I I (" t^i{s)^j(s))-blWi(s)Ms)li)dB{SyWi) 2 -\ t III- ib(Wt(s),Wj(s))-blWi(s)Ms)-] For(w s w')GWrX W r l e t (4.7) fl(w,w') = J»(w'(5Xw(5))-l>[w'(5XttW])dB(3,w') 0 and (4.8) /(w,w') = a(w,w') + a ( w » - J a(w,vv")a(w>'>(<*w''). Then we see that Theorem 2.1 also holds in the present case. In particular, for any bounded measurable function # on WT Iim Ep"[exp{y^T 0>}] = lim &9n [exp j ^ f ^(wj + H»Tw\] = exp{y^T}, / 1 " \ and hence ( i 5 " , - £ $(wf) >-»•> a s n-*°o- Since P" is unique, this implies that <#,//> is uniquely determined by the initial distribution w. Thus p is the unique solution of the martingale problem (4.2) and (4.3). Furthermore, by the same reason, for any (,eWT and e > 0 there exists a unique probability measure fi\ on WT such that (4.9) <Mw(OWG(rf W+ «„.,) e is a ^-martingale and (4.10) J?(w(ty,£) = ul(t) and «J(0) = «. Noting that /4 and u are mutually absolutely continuous JJE of (2.20) is welldefined, and by a similar argument to § 2 we see that Theorem 2.2 is also valid for the diffusion case. References 1. Dawson, D.A.: Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Statist. Phys. 31, 29-85 (1983) 2. Dellacherie, C , Meyer, P.A.: Probabilities et potentiel, Chapitres V a VIII. Herman: Paris 1980 3. Dynkin, E.B., Mandelbaum, A.: Symmetric statistics, Poisson point processes and multiple Wiener integrals. Ann. Statist. 11, 739-745 (1983) 4. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North-Holland/Kodansha 1981 222 Central Limit Theorem for a System of Markovian Particles 459 5. Krylov, N.V.: On Ito's stochastic differential equations. Theory Probab. Appl. 14, 330-336 (1969) 6. Kunita, H., Watanabe, S.: On square integrable martingales. Nagoya Math. J. 30, 209-245 (1967) 7. Kusuoka, S., Tamura, Y.: Gibbs measures for mean field potentials. J. Fac. Sci. Univ. Tokyo 31, 223-245 (1984) 8. Maruyama, G.: On the transition probability functions of the Markov processes. Nat. Sci. Rep. Ochanomizu Univ. 5, 10-20 (1954) 9. McKean, H.P.: Propagation of chaos for a class of non-linear parabolic equations. Lecture Series in Differential Equations, 7 Catholic Univ. 41-57 (1967) 10. McKean, H.P.: Fluctuations in the kinetic theory of gases. Commun. Pure Appl. Math., 28, 435-455 (1975) 11. Parthasarathy, K.R.: Probability measures on metric spaces. New York: Academic Press 1967 12. Simon, B.: Trace ideals and their applications. London Math. Soc. Lecture Note Series, 35. Cambridge: Univ. Press 1979 13. Sznitman, A.S.: Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56, 311-336 (1984) 14. Sznitman, A.S.: A fluctuation result for non-linear diffusions. [Preprint] 15. Tanaka, H.: Limit theorems for certain diffusion processes with interaction. Proceeding of the Taniguchi International Symposium on Stochastic Analysis (K. lto ed.), pp. 469-488. Tokyo: Kinokuniya 1984 16. Tanaka, H., Hitsuda, M.: Central limit theorem for a simple diffusion model of interacting particles. Hiroshima Math. J. 11, 415-423 (1981) Received October 16, 1984 223 Probab. Th. Rel. Fields 71, 69-83 (1986) PTObflDlllty Theory &— © Springer-Verlag 1986 Propagation of Chaos for Diffusing Particles of Two Types with Singular Mean Field Interaction Massao Nagasawa and Hiroshi Tanaka Institut fur Angewandte Mathematik der Universitat Zurich, Ramistrasse 74, CH-8001 Ziirich, Switzerland Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, 223 Japan Introduction In a system of diffusion processes with a pair interaction b(x, y) described by Xi{t) = Bi(t) + - £ n SdsbiX&lXjis)), i = l,2,...,n, j= l o where {B^t): i—1,2,.,.,«} is a family of mutually independent Brownian motions, if the number n of particles tends to infinity, then the distribution of (X1,X2, ...,Xm), for fixed m g n , converges to that of a collection of independent copies of a (non-linear) diffusion process X(-) with the so called mean field drift b[x,u(ty] = $b(x,y)u(t,dy) induced by the distribution were called "propagation of 1 law of large numbers: un=n u(t, *) of X(t) itself. Limit theorems of this kind chaos" by McKean [4]. One can regard it as a " £ $xt-*u- The propagation of chaos for a system ;=i of diffusion processes was proved by McKean [5] for Lipschitz continuous interaction b{x, y). Besides this Tanaka [13] discussed an associated fluctuation problem (central limit theorem) modifying a method of Braun and Hepp [1] for Vlasov equation. The same problem has been treated also by Kusuoka and Tamura [2] as a limit theorem of Gibbs measures for mean field potentials, and by Sznitman [11] with the help of (Cameron-Martin) Maruyama formula [3], which enabled him to handle the case of bounded measurable interaction b(x, y) (cf. also Shiga and Tanaka [10]). The present paper is concerned with the propagation of chaos for a system {(Xi, Yj): i = l , 2 , . . . , « } of particles of two types with a singular pair-interaction, which arises in a statistical model for segregation of interacting random motion (cf. Nagasawa [6, 7], Nagasawa-Yasue [8] and Nagasawa-Tanaka [9]). It 70 M. Nagasawa and H. Tanaka can be described by a system of stochastic differential equations: (i) Xi(t) = Xt{0) + Bt{t) + \ds\- 1 t f(Xi(s)o {J 1=1 < = l,2,...,n^ Yl{s)) + v{Xt{s))\ + 0i(t)i ) Yj{t) = Yj{0) + ffj(t) + ]ds\-±n t fiX^-YjW o I i=i j=l,2,...,n, + viYjWl-Vjitl J where {Bi(t),B'j(t);i,j = l,2,...,n} is a family of mutually independent Brownian motions, X^O) (resp. - 7,(0)) has a common distribution v (+) (resp. v<_)) whose support is in [0, oo), f(x) is a non-increasing continuous function on (0, oo) which may diverge at the origin, an example of which is f{x)=^, a>0, x v(x) is an odd function which is continuous and nonincreasing in R —{0}, and (pfi) (resp. Wj(t)) is a non-decreasing continuous function which makes the origin a reflecting boundary for Xt(t) (resp. Yfi)). That is, X^t) (resp. Yj(t)) is a reflecting diffusion process on [0, oo) (resp. ( — oo,0]) repelled by Yt(t) with the interaction -f{Xt— Yj) (resp. —f(X[—Yj)\ under the influence of an environ- ment potential v(Xf) (resp. v(Yj)). The propagation of chaos implies that (Xt, Y^ converges in law, as n->co, to (X, Y) which is a solution of a system of (non-linear) stochastic differential equations (2) X(t) = X(0) + B(t) + \ds{ J f(X(s)-y)uY(s,dy) « <-« Y(t)=Y(0) + B'{t) + \ds{{ f(x-Y(s))ux(s,dx) 0 + v(X(s))} + v(Y{s))}-Y(t), [0,oo> where wx(s>') and uY(s, •) stand for the distributions of X{s) and Y(s\ respectively, B(t) and B'(t) are Brownian motions, and X(0) and 7(0) are distributed by v(+> and v (_) , respectively. Because of singularity of f(x) (and possibly of v{x)) at the origin, it seems that the methods of [2, 5, 11] and [13] can not be immediately applied to the present case. However, thanks to the monotonicity of f(x) and v(x), one can find a rather direct way of proving the propagation of chaos based on the construction of unique solutions of Eqs. (1) and (2) by iteration (Theorem 1 in §1). In Sect. 2, preparing several lemmas, we will prove a limit theorem (Theorem 2). The propagation of chaos, which is a direct corollary of the limit theorem, will be discussed in Sect. 3. In the original model (cf. [8, 9]) the interaction f(x) (e.g. f{x) — l/xA) diverges at the origin so rapidly that the Y-particles lie always left to the Xparticles even without the additional terms Propagation of Chaos for Diffusing Particles 71 particles are separated from the X-particles by a moving boundary. The terms fpt and — ¥J are added in (1) artificially in order to avoid mathematical difficulties of handling the moving boundary. The propagation of chaos for the unmodified model (i.e. without 0- and — ¥J) is an open problem. 1. Existence and Uniqueness of Solutions In this section we will prove the existence and uniqueness of solutions of a system of equations ,° (3) n(ttz) W = z(t) + $d$\ br,(s,z),Z{s,w))ti{+)(dw) + Ht,z\ 0 W where b(xty) is the sum (4) b(xiy)=f(x + y)+v(x), o f / a n d v with the properties (5) f{x) is a non-increasing continuous function on (0, oo) which may diverge at the origin, and v(x) is an odd function which is nonincreasing and continuous in IR —{0}. As will be seen in §3, solutions of the system of Eqs. (3) will provide those for the Eqs. (1) and (2) at the same time. In Eq. (3), w and z are elements of W=C([0,oo)-+Rn{w:w(0)^0}, /i ( + ) and fi(~i are probability measures on ^ ( and rj are elements of W+ = C([0,oo)->IR+), <£e?=C([O,oo)->]R+)n{0: nondecreasing, <j>{0) = 0 and constant on each connected component of {t>0; £(t)>0}}, andtye C(t,w) = w(0)vl+(w(f)-w(0)) + ct + ^(t,w), where c ^ / ( l ) + i>(l). The solution (£) can be given explicitly (cf. §1 of [9]). Because c^f{x + y) + v{x) for x ^ l and y^O, Lemma 5 of [9] implies that for any (£, q) satisfying (3) (8) 0g«t,w)SC(t,w), 0gfj(t,z)£Cfe4 We impose, in addition, integrability conditions on /i (+) , fi{~\f and v: (9) (10) JH0I/^H>) w p = i£+\pi-\ j J/-(C(5,w) + C(s,z))M(+)(dw)^->(dz), ww 72 M. Nagasawa and H. Tanaka and (11) /i = fii+),/f-\ Sv-(as,w))ii(dw\ w are bounded on each finite time interval, where / ~ = ( — / ) vO and v~ — (-i?)vO. Theorem l. 1 Under the conditions (5), (9), (10) and (11) on f, v, /i ( + > and ii(~\ there exists a unique solution {(£(£, w), fit, w)), (n(t, z), il/(t,z))} of the system of Eq. (3), where %e\V+, fe&^, neW+ and i ^ e ^ . Proof. Denoting />"=( — b)v0 and a^(s,x)=-ib-(x,asj))^-\dz), ur (12) w we define (£ (13) ?0)(t,w) = W{t) + $dsai+)(s,^0)(s,W)) o + i0)(t,W), ^ 0 ) ( ( , z ) - z ( r ) + jrfsa ( -»(s,V 0) (s,z)) + iA(0)(^2), £*>(£, W) = w(t) + lds j fc(f(«(Sj 0 (14) W ), ,,<*- D( S , 2)) M *->(dz) W +* ( k ) (t,w), 0 W + Ogf(0>few)^f(Wfew)^C(t,w), and for all fc^O, fw^f^^f5^...^f^^f(2^f0\ and the same inequalities for (?/(k), ^ (A) ). Therefore there exist monotone limits §{t9 w)= lim ?2k)(t, w), ^ ( t , w)= Urn f2k\t, l{t, w)= lim ?2k+ l\tt w), fit, w)= lim fi2k fc—'CO fc-»CO 1 This is a reformulation of Theorem 2 of [9] w), + 1) ((, w), 73 Propagation of Chaos for Diffusing Particles and correspondingly r), \j/, Y\ and ip. They satisfy (H) f(f, w)^f(t,w), 0(t,w)^ n(t,z)^t\(t,z\ i/>(t,z)^(t,z), o^M (+) Kfe-)]^M (+, Kfe-)]^^ (+) Kfe •)]<«>» (18) Og/i<->[i|(t,-)]^Ai ( - ) Wfe-)]^M ( - , Kfe •)]<<». t 0 (19) __ TV r 0 W t](t,z) = z(t) + jds\b(rj(s,z)^{s,w))^+\dw) 0 (20) + \l/{t,z), W I 3fez) = 2(t) + Jd5jfe(g(5,z),f(s,w))/i (+) (dw)+^fez). To get (19) and (20) from (14) notice, for example, & « ( 2 * + 1 , , g ) ^ W 2 * + 2 , ) rj )^b{^ff2k)), where the first term (resp. the third) is monotone nondecreasing (resp. nonincreasing). (2k) Then, Lemma 1 below completes the proof of the existence of solutions. Lemma 1. 7/(17-19) and (20) are fulfilled, then ? = £, <j> = <j>, rj=ij and \j/ = \j/. Proof Because of the assumptions (5), (10) and (11) it holds that JrfsJ j/-(f(s,w)+^(s,z))^-»(dz)M ( + Vw) 0 WW 0 1 WW _ t JdsJi)-(^(s,w))/i ( + )(dw)^J(dw) 0 0 W and corresponding inequalities for fj. Therefore, each term of the right hand side of (19) (resp. (20)) is jU(+) (resp. ^(~') integrable because (9) is assumed, and 0^/i<+>K(f, • ) - « * , •)] + * ( - ) R f c -)-3(t, •)] ^\ds J {p(J(s,w))-t)({(s,w))} /* ( + ^w) + M (+) [£fe *)-*(*,•)] 0 W I + JdsJWjf(s,z))-^,z))}^-»(dz) + ^ ) [ | f ( t , - ) - * ( t ) ' ) l o tr ^M ( + ) [*(^-)-*(*,-)] + ^ - ) [ * ( t , - ) - * f e - ) ] < a > . 74 M. Nagasawa and H. Tanaka The inequality above combined with 0 — 0 ^ 0 and ijj — \j/^0 implies that (t,w) = (f>(t,w) for ^ ( + ) -a.e. weW^resp. ij/(t,z) = \//(t,z) for /x (_) -a.e. zeW) and hence J(t, w) = £(£, w) for /i t + ) -a.e. w e W ( r e s p . fj(t,z)=ij(t,z) for /i ( _ ) -a.e. zeVP). To show the equalities for all w, zeW, denoting c(s, x)= J b(x, fj(st w z))ii(-\dz) we consider a Skorokhod equation on [0, oo) 0 Since the solution of the equation is unique by Theorem 1 of [9], £(£, w) = £(£,w) = J(£,w) and 0(£, w) = 0(i, w ) = 0 ( t , w) for all weW. The same argument applies to showing r\{t, z)~rj(t, z)=rj(t, z) for all zeW. This completes the proof of Lemma 1. To prove the uniqueness let (£i,0i,*/ 1 , and f (k) (t,w) = w(0 + i^Jft(f*>( S ,w), J/ < t -»(5,z)) /i ( ->(«iz) 0 W + 0 ( k , (t,w), ij (k) (t,z) = z(t) + J w))n{+)(dw) JP + *<"(*, z), for / c ^ l . Then it is easy to see that ?2k)£Zi,Z2£ti2k+1)> {(0) ^ {(2) ^ {(4) ^ for fc^O, t _? 0<2> ^ 0(4) ^ 0(6) ^ ^ and the same inequalities for (rjlk\ ij/ik)), which imply the existence of monotone limits {(t, w) = lim £ ( 2 k + ^(f, w), 0(t, w) = lim 0 ( 2 k + ^(f, w), Jc-»oo {(t, w)= lim £<2k,(f, w), ft-*oo k-»co 0(r, w) = lim 0<2k>(t, w), k-»oo and correspondingly (fj,\[/,rj,}j/). It is clear that they satisfy (17-19) and (20). Therefore, Lemma 1 implies ? = £ , 0 = 0, *? = */, ^ = ^ , and hence £1 = £2> 0 i = 0 2 , rj1 = r}2 and iA 1 =iA 2 ' completing the proof of Theorem 1. Propagation of Chaos for Diffusing Particles 75 2. A Limit Theorem Given a function b(t,x) defined on [0, oo) x(0, oo) with the property (21) fo(i, x) is continuous in (t, x)e[0, oo) x(0, oo) and nonincreasing in x for each r^O, we consider a Skorokhod equation on [0, oo) (22) «t,w) = w(0 + Jdsb(5,{(s,w)) + *(t,w). 0 Theorem 1 of [9] guarantees the existence of a unique solution (£(£, w), 0(£, w)) for weW, where £eW+ and 0 e ^ . We will denote it as (£(t), (£)) without w, if there will be no confusion. Lemma 2. Let b{t,x) satisfy the condition (21) and let {£","), a > 0 , foe the solution of a Skorokhod equation on [a, oo): Za(t) = a + W(t) + \dsb(s,Za(s)) + <>(t). (23) o Then £"(r)\£(t), as a\0. Proof See Lemma 5 and what follows in Sect. 3 of [9]. Lemma 3. Under the assumption (21) define for a > 0 (24) ba{t, x) =fo(f,x v a), /or (t, x)e [0, oo) x (0, oo). Let (&,,0fl) foe f^e solution of (22) witnfofl(t,x) in p/ace ofb{t,x). 77ien U 0 / « t ) , «s a\0. Proof Theorem 1 of [9] implies that £a(t)/' and <j>a(t)\, as a \ 0 . Therefore $dsba{s^a(s)) = Ut)-w{t)~a(t)S, o as a\0, and hence the limit i lim $dsba{s>£a{s)) exists. Denote i(t) = lim (M Since ba(s,£a{s))l{E(a)(£a(s)) decreases to fo(s, £(s)) 1(E ^ ( s ) ) , as a\0, we have for e>0, 76 M. Nagasawa and H. Tanaka i lim ldsba(st£a(s)) a\0 o = ]dsb{s,Z(S))l{,iai)(£(s)) + \\m 0 Idsb^U^ko^M "NO 0 = Jdsfcfc|(s))l (0ia)) (fW) + lim lim Jdsi fl (5,^(5))l [0 , Bl (^( S )). 0 e\0 a\0 o Therefore £(t) satisfies (25) f(t) = w(O-ct + Jds{b(s,|(s)) + c} + 0W, o where a constant c is chosen to be b(t, l) + c>0, for Vte[0, T], and $ is defined by # ) = lim lim Jds{fcflfcUs)) + c}l[0te]Kfl(s)) + Hm^ffl(t). It is easy to see that lj(t) is continuous in t (cf. §3 of [9]), and hence <j>(t) is also continuous in t by (25). Moreover, f(t) is monotone nondecreasing and stays constant on each connected open interval in which |(f)>0. Because of the uniqueness of solutions for a Skorokhod equation (22) which is the same as (25), the (£, <j>) must coincide with (£, ), completing the proof. We consider Wand W+ as metric spaces with the metric of the uniform convergence on each finite time interval. Lemma 4. Let b(t,x) be bounded and Lipschitz continuous: \b(t, x) -b{t,y)\^c{T)-\x-y\,for Vte[0,T], and Vx,.ye[0, oo). Then the solution £(£,w) of Eq. (22) is continuous in (f, w)e[0, T~\y.W. Proof. The lemma is an easy consequence of Lemma 2 of [9]. In fact, as is shown there, the solution £(t,w) is given as the limit of {£(k){t,w)} defined by (26) <S(Wfew)-w(r) + Jdsfc(s,f(k-1)(s,w)) + ^k>(t,w), 0 and the convergence is uniform in (r, w)e [0, T] x W. Since it is easy to see by induction that £ik){t,w) is continuous in (r, w)e[0, T] x W, so is £(t, w), completing the proof. Lemma 5. Under the condition (21) on b{t,x), the solution f(r, w) 0 / £ g . (22) is continuous in (t, w)e[0, T] x JV,/or any VT>0. Proof Let iC be an arbitrary compact subset of W and denote M = sup weK sup £(r, w) Skorokhod equation on [1, oo) ((t,w) = w{0)vl+{w{t)-w(0)) + ct + \j/(t,w) 231 Propagation of Chaos for Diffusing Particles 77 with c^ sup b(t,l). It is to be noted that ba{t,x) (resp. b(t, x)) can be approxile[0, T] mated from below (resp. from above) on [0, Af + 1] (resp. [a, M + l]) by Lipschitz continuous ones. Therefore, %a(t, w) (resp. £a(t, w)) is the increasing (resp. decreasing) limit of continuous functions on [0, T] x K, and hence lower (resp. upper) semi-continuous in ( t , w ) e [ 0 J ] x X . Now, let a \ 0 . Then £a{t,w) (resp. £"(*>w)) increases (resp. decreases) to £(?, w) by Lemma 2 and 3. Therefore £(£, w) is lower and upper semi-continuous, and hence continuous in {t, w)e[0, T] x K. This completes the proof of Lemma 5. Lemma 6. Let b(t,x) and bn(t,x), « = 1 , 2 , 3 , . . . , satisfy the condition (21), and for Va>0 and V T > 0 lim bn(t,x) = b(t, x), uniformly in (t, x)e[0, T] x a,- , (27) n-*oo L a] sup max 6 n (t,a) Let £n(r, w) denote the solution of Eq. (22) wfcfcfc„(£,x) m p/ace ofb(t,x). Then for any compact subset K of W (28) lim £„(*, w) = £(*, w), uniformly in (t, w)e[0, T] x K. Proo/. We first prove the assertion assuming that b(t, x) is bounded above and bn(t,x) converges uniformly on [0, T ] x 0 , - . With the help of Lemma 1 of L aJ [9] we have \Ut)-m\2^2ids(Us)-m{bn(s,Us))-h{s,Us))} 0 where the first integral is nonpositive and the second is, by Schwartz inequality, dominated by lds\Zn(s)-^2 0 }ds\bH(s,tts))-b{s,Z(sy)\2. + 0 Therefore, applying Gronwall's lemma, we have (29) \UtM~atM\2<eT$ds\bn(s,Z(S,w))-b(s,Us,™))\2- Since sup sup £(s, w)^sup sup £(s, w) weK se[0, T\ for any compact subset K of W, the integrand of the right hand side of (29) is bounded and converges to zero uniformly in (s, >v)e[0, T~\ xK, as n->00. Therefore (29) implies that £n(t,w) converges to £((, w) uniformly on [ 0 , T ] x K . 78 M. Nagasawa and H. Tanaka We apply what we have proved above to ba(t, x) and (£?„)„(£, x) defined by (24) (resp. b(t,x) and bn(t,x) for x^a) and obtain, using the notations in Lemma 2 and 3, (30) KJfl(t,w)-> £.(*,*), (W(t,w)-+?{t,w), as n-oo, uniformly on [0, T] x K. On the other hand £a(t, w)/ f (t, w) and ffl(t, w ) \ <^(t, w) as a \ 0 uniformy on [0, T ] x K by Lemma 1 and Dini's theorem, and hence there exists a>0 such that 0 ^ T ( t , w)-£,(t, w)< £ , uniformly in (t, w)e[0, T ] x X . Because of (30) it holds that for this a > 0 IKJ.fc w) - y r , w)| < e, \(Qa(t, w) - £»(*, w)| < e, uniformly on [0, T ] x X, for sufficiently large n. Therefore we have If (t, w)-{„(*, w)| ^ 3 e , uniformly in (t, w)e[0, T ] x K, where made is use of the inequality (£„)a = £n^(£nf- This completes the proof of Lemma 6. Theorem 2. Besides (5) assume that f is bounded below.2 Let p ( + > , /i < _ ) , /^ + ) , ^ _ ) , n—l,2,..., be probability measures on W satisfying conditions (9) and (11) for given v. Let (^n) (resp. (£„,rjn)) be solution of (3) with (/i ( + ) ,//"') [resp. (/iJ,+),//J,_))). If fi[+) (resp. //„ _) ) converges to pi{+) (resp. / / _ ) ) weakly, then for any compact subsets Kt, K2 of Wand V T > 0 (31) lim fB(t, w) = f (t, w), n-"X> lim nn(t, z) = r,(t, z) n-*<x> uniformly in (t, w, z)e[0, T] x K^ x K2. Proof. Let (f ( k U ( k ) ) and (& t ) ,rf ) ), ^ 0 , be defined by (12), (13) and (14) with (fii+\fi{~)) and (ju(n+),/i(n_)), respectively. We will prove first of all that £{®(t,w) (resp. rj{^(ttz)) converges to f(k)(£, w) (resp. >/(k)(£, z)), as n-*OO, uniformly on [0, T ] x K, where K is a compact subset of W. We show it by induction in k. Since filn+) (resp. / ^ - ) ) converges weakly to ^ ( + ) (resp. /* (_) ) by the assumption, (ai+)(syxl fl(-}(js,x)) and (fli +, (s,x),fli->(s,x)) defined by (12) with ( M ( + V ~ ) ) and (ju^juj, - *), respectively, satisfy the conditions of Lemma 6. Therefore, fj,0)(t,w) (resp. r}^](t, z)) converges to f (0, (t, w) (resp. n{0){t, z)) uniformly on [0, T ] xK. Assume that ^\t,w) (resp. rj^faz)) converges to foo, uniformly on [0, T ] x K . We will only prove that fS,k+1)(£, w) converges to f ( k + 1 ) (r, w) uniformly in ( t , w ) e [ 0 J ] x K , since the same argument can be applied to n%+1)(t,z) and ^ k + 1 , ( f , z ) . Define l^+1\stx)^if(x w + riik\stz))ti^(dz) + v(x)> fc(s,x)= j / ( x + ftf>(s,z)W->( Then / satisfies the condition (10) automatically Propagation of Chaos for Diffusing Particles 79 It is clear that £ and £j, k + 1 ) satisfy Skorokhod equations ?k+1)(t,w) = w(t) + \dsb{k+1){s,?k+l)(s,w)) o + 4>ik+1)(t,w), respectively. If we prove that b(k+1){t,x) and blk+1)(t,x) satisfy the conditions of Lemma 6, it implies that £(k+1){t,w) converges to £{k+l){t,w) uniformly on [ 0 , T ] x K , a s n-*oo. The condition (21) is clearly satisfied. The condition (27) can be verified as follows: l*? +1) (s,x)-& (k+1) (s,x)| (32) converges weakly to / i ( _ ) by the assumption, there exists a compact subset K0 of Wsuch that 2f(a) Keeping this in mind, we estimate (33) The first term of the right hand side of (32) ^ J l/Cx + i ^ f e z B - Z ^ + ^ f c z B l r i - ' ^ + a, K0 where the integral is smaller than e for sufficiently large n uniformly in (s, x)e[0, T] x a,- , because the integrand converges to zero uniformly in (s, x, z)e[0, T] x \a,-\ x i t 0 ; and (34) The second term of the right hand side of (32) ^\$f(xWk)(s,z))(iii-\dz)-^-\dz))\ where f(x + rjw{s,z)) + E, is uniformly continuous in (s, x, z)e[0, T ] x o^- and the integral is smaller than 3e uniformly in (5, x)e[0, T ] x a , - f ficiently large n. In fact, 3 let r xX0, for suf- Hi (s, x)e[0, T ] x a , - > be a covering of 3 Thanks to a referee's remark one can invoke Theorem 6.8 (p. 51) in Probability measures on metric spaces by Parthasarathy 80 M. Nagasawa and H. Tanaka [ 0 , T ] x [ a , ~ l , where U{s,x) = {(t,y):sup\f(x + t1ik)(s,z))-f(y + r}W(t,z))\<s}. zeK0 Then, there exists a finite subcovering {U(sitx^\ i = l,2,..., /}. Let n be sufficiently large so that | J / ( x i + i^>(S()z))(A4->(dz)-/i'->(dz))| < £ for i= 1,2,...,/. For any (s, JC)G[0, 71] x I a , - | we can find U(sitx^ L (s,x)eU{si9x^ and hence such that | j / ( x + V k, (s,2))( M ( n - , (rfz)-^- ) (dz))| <2e + | J / ( x i + ijt*>(si,z))0i(1|-)(dz)-^-)(^))l <3e, for sufficiently large n. Thus we have finally \b(nk+1)(s,x)-b(k+1)(s,x)\^6e sup (s,x)e[0,T]x[a,-~\ for sufficiently large n. Hence b(fc+1)(s,x) and b{k+l)(s,x) satisfy the condition (27), the second condition of (27) being trivial. Now, the proof can be completed as follows: Because of Theorem 1, especially inequality (16), Lemma 5 and Dini's theorem we can find k0 for any e>0 such that (35) Q^?2k+1)(t,w)-?2k\t,w)Sz, for k^k0, uniformly in ((,w)e[0,T]xK. On the other hand there exists n0 such that for \&k)(t,W)-ek%w)\ n M Se, uniformly in (t,w)6[0,T]xX as is proved above. Therefore, combining (35) and (36) with (37) ii2k% w) ^ t(tt w) ^ ")* «.(t.w)^." + 1 ) (t,w), we obtain sup ((,w)e[0,T]xJi: |£(t,w)-£„(t,w)|^3£, for n ^ n 0 . Propagation of Chaos for Diffusing Particles 81 Thus £n(£, w) converges to %(t, w), as n-*oo, uniformly on [0, T ] x K for any T > 0 and compact subset K of W. This completes the proof of Theorem 2. 3. Propagation of Chaos We consider a system of Eq. (3) with / and v satisfying the condition (5). Moreover we assume that / is bounded below. Let Px be the Wiener measure on W starting from x e R , v ( + ) and v(~> be probability measures on R + with (38) v = v(+),v<->, J xv{dx) and define probability measures / / + ) and J J ( _ ) on W by (39) / / + > ( - ) = j vM(dx)Px(-), ,!<->(•)= f v^(dx)Px(-). Because of (38), the probability measures / j ( + ) and fx{~) satisfy the condition (9). In addition we assume the condition ( l l ) . 4 Then there exists a unique solution {£{t,w),yj{t,z)) of (3) with (^ +) ,/i<->) for all {w,z)eWxW by Theorem 1. A stochastic process (X, Y) on (W x W, fii+) ®/i ( _ ) ) defined by (40) (X{t),Y(t)) = {at),-r}(t)), coincides clearly with the process described by Eq. (2). Let (Q, P) be the infinite product of (W x W, ^ ( + ) (§)//<->) i.e. Q=W2xW2x..., W2 = WxW, (41) For a) = ((w 1 ) z 1 ),(w 2 ,z 2 ), ...)<E£ define X(r, a>) by Yi{t,(o)=-t](t,zi), i = l,2,3,..., and (43) X(£) = ((X 1 (t), YMKXM Y2(t))t...). By definition {(A^t), Yj(£)): i' = l , 2 , 3 , . . . } is a family of independent copies of the process described by (2). Now, define for (44) M(„-,(coJ-) = ^ i ^ ( - ) . 4 A sufficient condition for (11) is given in Proposition 1 of [9], i.e. if v~ satisfies (i) v~(x)^ c(l + |x|a) for some nonnegative constants c and a, and if v = v(+), v <-) satisfy (ii) j xav(dx)<(X), then E+ the condition (11) is fulfilled 82 M. Nagasawa and H. Tanaka They satisfy trivially the conditions (9) and (11). Therefore, by Theorem 1, there exists for each (w, z)eWxW a unique solution (£„(£, w, co), nn{t,z, co)) of (3) with (£+Ka>lpfc-\Q>)). Define, for co=((wlizll(w2,z2), ...)eQ, processes (X\n\Y™), i = l,2,...,n, on(fl,P)by Xi")(t,co) = ^(t,w I .,w), (45) and put (46) x«(0-(OT(t), yrW), ...,«>(4 y«(t))). n) Clearly {(X<- (t), Ytn)(t)): i = l,2, ...,n] is a family of particles of two types described by (1). Since /xj,+)(cu) (resp. p.(n~)(co)) converges to /i ( + ) (resp. /i (_) ) weakly for P-a.e. meQ by the strong law of large numbers, Theorem 2 implies that for P-a.e. coeQ the sequence (£„(£, w, co), */„(£, z, co)) converges to (£(t, w), ^(f, z)) uniformly in (*,w,z)e[0, T ] x X , x K 2 , for V T > 0 and V compact KltK2cW. In other words, the propagation of chaos (of Kac-McKean) holds for a family of diffusing particles of two types with singular interaction described by (1). Theorem 3. (Propagation of chaos.) Under the stated conditions5 let ((X(1">(t,co),yl(">(tJ))) be the system of 2n-particles of two types defined on (Q,P) by (45) and (46). Then, for P-a.e. cueQ, {(X^{t,co),Y<"\t,oj)),..., {X%\t,aj), Y£\t,a>))), for fixed mXn, converges uniformly in re[0, T] for VT>0, as n->oo, to ((Xt(t,cy), Y^co)), ...,Xm{t,co), Ym(t, co))) which is a family of independent copies of the non-linear diffusion process of two types defined by (42). That is, the diffusion process on (R 2 )" described by equation (1) converges in law to the infinite direct product of the diffusion process described by (2) as n tends to infinity. Remark. Px in (39) is not necessarily to be the Wiener measure, and Theorem 3 remains valid for arbitrary Markov processes on W, since our treatment of the problem is purely pathwise and does not depend on a special choice of probability measures on W. See the first paragraph of this section References 1. Braun, W., K. Hepp: The Vlasov dynamics and its fluctuations in 1/JV limit of interacting classical particles. Commun. Math. Phys. 56, 101-113 (1977) 2. Kusuoka, S., Y. Tamura: Gibbs measures for mean field potentials. J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 31, 223-245 (1984) 3. Maruyama, G.: On the transition probability functions of the Markov processes. Nat. Sci. Rep. Ochanomizu Univ. 5, 10-20 (1954) 4. McKean, H.P.: A class of Markov processes associated with non-linear parabolic equations. Proc. Natl. Acad. Sci. 56, 1907-1911 (1966) 5. McKean, H.P.: Propagation of chaos for a class of non-linear parabolic equations. Lecture Series in Differential Equations. Catholic Univ. 41-57 (1967), Washington D.C. Propagation of Chaos for Diffusing Particles 83 6. Nagasawa, M.: Segregation of a population in an environment. J. Math. Biol. 9, 213-235 (1980) 7. Nagasawa, M.: An application of segregation model for septation of Escherichia coli. J. Theor. Biol. 90, 445-455 (1981) 8. Nagasawa, M , K. Yasue: A statistical model of mesons. Publication de l'lnstitute recherche Mathematique Avancee (CNRS) 33, 1-48. Universite Louis Pasteur (Strasbourg) (1982/83) 9. Nagasawa, M., H. Tanaka: A diffusion process in a singular mean-drift-field. Z. Wahrschemlichkeitstheor. Verw. Geb. 68, 247-269 (1985) 10. Shiga, T., H. Tanaka: Central limit theorem for a system of Markovian particles with mean field interaction. Z. Wahrscheinlichkeitstheor. Verw. Geb. 69, 439-459 (1985) 11. Sznitman, A.S.: Non-linear reflecting diffusion processes, and propagation of chaos, and fluctuations associated. J. Funct. Anal. 56, 311-336 (1984) 12. Tanaka, H.: Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9, 163-177 (1979) 13. Tanaka, H.: Limit theorems for certain diffusion processes with interaction. In: Stochastic Analysis, pp. 469-488. ltd, K. (ed.). Tokyo: Kinokuniya Co. Ltd., Amsterdam-Oxford-New York: North-Holland Pub. Co. 1984 Received November 29, 1984 238 Saisho, Y. and Tanaka, H. Osaka J. Math. 23 (1986), 725-740 STOCHASTIC DIFFERENTIAL EQUATIONS FOR MUTUALLY REFLECTING BROWNIAN BALLS YASUMASA SAISHO AND HIROSHI TANAKA (Received August 1, 1985) Introduction In this paper we construct a random motion of mutually reflecting hard balls of diameter p in Rd by solving certain stochastic differential equation (abbreviated: SDE) with a kind of singular drift. For simplicity we first consider the motion of mutually reflecting Brownian balls of diameter p. In order to construct such a motion we pose the following problem. Let W denote the space of continuous paths in Rd. Given rol, •••, w„^. IV satisfying (1) l«tfO)-wXO)l£p, l£* solve the equation (2) « * ) = WiW+ 2 ('(«')-£/*))<**«(*). l&Zn, under the following conditions (3) and (4). (3) gi*=Wtlgi£n, and \^(t)-^(t)\^pt l£i<j£n, feO . (4) (j>i/& are continuous non-decreasing functions with ^,y(0)=0, *«(o = ('ip(iw*)-exoi)^iX'). Jo where lP(r) = 1 if r = p, and = 0 if r^p . A pair (f, <j>) of functions or simply a function If is called a solution of (2) provided that (2), (3) and (4) are satisfied. One of the main results in this paper is that there exists a unique solution of (2) for given tolt •*•, wn. By taking wly •••, wn to be independent •". £»(*))• This 1S w n a t w e ca^ t n e motion of mutually reflecting Brownian balls. £,-(*) denotes the center of the i-th Brownian ball at time t. In analogy with Skorohod's equation for a 1-dimensional reflecting Brownian motion ([3] [5] [7]), the equation (2) may be regarded as Skorohod's 239 726 Y. SAISHO AND H. TANAKA equation for the motion of mutually reflecting Brownian balls. We next consider the SDE on a probability space (Cl, 3?y P): (5) dX§(i) = a{Xi{t)) dBM+biXiit)) dt where b: Rd -> Rd are given, A",-(0)'s are 2 r 0 _ m e a s urable initial values satisfying |X,(0)—-X"/0)| ^/>, l£i<j^*nt and £,-(£), l^i^n, are independent ^-dimensional £? r adapted Brownian motion with £ , ( 0 ) = 0 . Here i3ft}ti/} is a right continuous nitration on (fl, U , P) such that each £F, contains all P-negligible sets. As in (2), Xt(t) and <&ij{t) should be found under the following conditions (6) and (7). (6) Xi(ty& are ^ - a d a p t e d continuous processes with \Xt(t)—Xj(t)\ (7) >p, l^i< Oij(tys are £F*-adapted continuous non-decreasing processes with 0 ^ ( 0 ) = 0, 0(7(*)=<M') and *iX0 ^ T M | A , « - X , M I ) <**„(*) • Jo The equation (5) may be considered as Skorohod's SDE for mutually reflecting diffusion balls with coefficients D = {(*i, - , xn)GR«: | * r - * , | > p , l£i<j£n} . So the crucial point of our discussions is to prove that the domain D of (8) satisfies Conditions (A) and (B). In solving Skorohod's SDE for D we make 240 MUTUALLY REFLECTING BROWNIAN BALLS 727 use of Theorem 5.1 of Saisho [6]. In 1 we state briefly the results of Lions and Sznitman [4] and Saisho [6] concerning Skorohod's equation for a general domain. We prove that the domain D of (8) satisfies Condition (B) in 2 and (A) in 3. We solve Skorohod's equation (2) in 4 and SDE (5) in 5. 1. Some known results on Skorohod's equation for an iV-dimensional domain with reflecting boundary Let D be a domain in RN and define the set 1RX of inward normal unit vectors at JCGE9Z> by ULX — mx,r= \J "*-x r > r>0 {ns=RN: |n|=l,£(x-w,r)nZ)=*h where i?(z, r)={y^RN: \y—z\. In what follows <-, •> denotes the usual inner product in RN. We introduce two conditions on the domain D. Condition (A) (uniform exterior sphere condition). rQ>0 such that 32* = Jtx.r^fy for any There exists a constant x<=dD . Condition (B), There exist constants 5 > 0 and j3(l^/3<<=>°) with the following property: for any x^dD there exists a unit vector lx such that >llP for any nG (J 32g. j/eB(jc,3)nai> 1.1. For any fixed r > 0 and a unit vector n the following two statements are equivalent. REMARK (i) B(x-rn,r)nZ>=*. (ii) +——Iif—x| 2 ^0 for any 2r REMARK 1.2. y^D. D satisfies Condition (B) if it satisfies the following condi- tion. Condition (B'). There exist 8 > 0 and a ( 0 = g a < l ) with the following property: for any x^QD there exists a unit vector lx such that C(y, lx, a)nB(x, 8)c5, Vy£EB{xt 8)f]dD, where C(y, lx, a) is the convex cone with vertex y, defined by 241 728 Y. C{yylxya) D). SAISHO AND H . = \z^RN: TANAKA ^a\z-y\} . Denote by W{RN) (resp. W{D)) the space of continuous paths in RN (resp. Skorohod's equation for D with reflecting boundary is written in the form (i.i) «o=w(t)+r«(o^w, Jo where zo^W(RN) is given and satisfies W ( 0 ) G 5 ; a solution (£, <j>) of (1.1) should be found under the following conditions. (1.2) feW(S). (1.3) <j> is a continuous non-decreasing function such that (0) = 0 and Jo (1.4) «(5)e3Z e(l , if f(j)e3D. The following theorem was proved by Lions and Sznitman [4] under the additional condition that D is admissible. Frankowska [2] and Saisho [6] removed this additional condition. Frankowska's result is of a general type but contains what we need only in a less explicit form, so we state the theorem in the form of Saisho [6], Theorem 1.1. If the domain D satisfies Conditions (A) and (B), then there exists a unique solution of (1.1) for any given w^W(RN) with w(0)^.D. Next, given RN®RN} b;D-+RN, we consider Skorohod's SDE (1.5) dX(t) = dt+n(t) d®(t), where the initial value X(0)^D is assumed to be ^-measurable and B(i) is an ^-dimensional £F,-adapted Browman motion with 5 ( 0 ) = 0 . Here {3?t} is a right continuous filtration on (fl, S% P) such that £F0 contains all P-negligible sets. A solution (X(i), (£)) should be found under the following conditions (1.6H1.8)(1.6) X(t) is a 5-valued ^ - a d a p t e d continuous process. (1.7) 3>(i) is a continuous non-decreasing process with (0)=0 and *(/) = \'hD(X(s))d 242 MUTUALLY REFLECTING BROWNIAN BALLS (1.8) n{s)(E3lx(s) if 729 X(s)GdD. In addition to Conditions (A) and (B), Lions and Sznitman [4] introduced the following Condition (C) and discussed the existence and uniqueness of the solution of (1.5). Condition (C). There exists a function / in C2(RN) which is bounded together with its first and second partial derivatives such that 3 y > 0 , V J C ^ Q D , + ^ - < V / ( j c ) , 7 ny\y-x\2>0. The following theorem was proved by Lions and Sznitman ([4: Theorem 3.1]) under Condition (C) and the admissibility of D; however, recently Saisho ([6: Theorem 5.1]) removed these additional conditions. Theorem 1.2 ([6]). Let D satisfy Conditions (A), (B) and assume that D satisfies Condition (B') Let D be the domain in Rnd defined by (8). We are going to prove the following proposition. Proposition 2.1. The domain D satisfies Condition (B')y that is, there exist constants 8>0 and a(Q^a<\) with the following property: for any xGQD there exists a unit vector lx such that (2.1) C{y,lx,a)nB{x,$)aD holds for any y^B(x, 8) 0 3D. For a non-empty subset / of {1, 2, •••, n} and x~(xlt •••, x„)^D we set x(2) = {x{: s ' e l } . (i) (2.2) DEFINITION 2.1. Let I, V be non-empty subsets of {1, 2, •••, n}. x(I) and x(F) are said to be separated if i n / ' = 0 and |*(— Xj\ >2p, V i e / , V / e / ' . (ii) x(I) is called a cluster if (2.3) for any i,j(=I with / #=/ there exist j 0 (= i), iu ••-, ip_lt ip(= j) in / such that \xik_l—xik\<2p9 XSk^p. 243 730 Y. SAISHO AND H. TANAKA REMARK 2.1. If x(I) is a cluster, then (2.4) \Xi-Xj\<2(n~l)p, Vijel, (2.5) <2(n~\)Pi Vte/ ( where Xj=(#J) _ 1 S y s / #,• and §7 is the number of elements in I. Let x=(xly •••,#„). Then {*lt •••,*„} can be represented as the sum of mutually separated clusters: (2.6) W •••,#„} = U x(Ik). Here l^tn^n. In what follows we assume that x^dD and keep it fixed, so Iu •••, Im appearing in (2.6) are also fixed. Let c>l be a constant which will be determined later and set —,yn)^R"d Also we set for y=(ylt v i = yik+c(y*—yrk)> v= *e**» l^k^m, nd (vu—,vn)e-R . We are going to prove Proposition 2.1 with (2.7) s- e 4(c-l)+2 ' a — cos 0 , where c, £ and 6 are constants determined by , 2S . (2 8) J (e-l) (»-l) = 1/8 , U = (c-l) P /4, ' (2.9) sin | - = — - L - -—, O <(0 ^ T T / 2 . From now on let c, 5, S be constants determined by (2.7) and (2.8). B(xt 8) fl dD and set z = y+(u—x). Let y& 244 MUTUALLY REFLECTING BROWNIAN BALLS Lemma 2.1. 731 I«—x\ = \z—y\<2(c— 1) \/~n~(«—1) p . Proof. Since i'G/A for some k, we have (2.10) k - * , | = (c-l)\Xi-xIk\<2(c-l) (n~l)p by (2.5), and hence we obtain the lemma. Lemma 2.2. (i) For any w ' e B(u, 6), v'&B(vy 6) and 0 < r ^ l , u" = ( 1 - * ) X + , ' B ' 6 D , (ii) B(uy6) xf' = (l-t)y+tv'^D . B(vy6) Proof. Since (ii) can be proved by setting 2=1 in (i), it is enough to prove (i), that is, \uV-u'/\>p, Case (I): i,j^Ikfor uV-u'/ some k. \vy-v'/\>Pt l^i<j^n. Since = ( 1 - 0 (*,-*,)+*{(«,_ M . ) + K - M , - ) - W — u,)} = {1+(C-I)i} (*|-*y)+*{(tt{-tt*)-(llj-tf,-)} , we have I«{'-«J' I > { l + ( c - l ) t}p-2£t>p and a similar inequality for \vfif—v'/ |. CVwe(II): i^Ik,j^It(k3=l). Since Mi'—#,• = *(«'—#,) = t(ut—xt)-\-t(u'i—Ui) , , we have Wt— Xi\ ^ |w,—*f| + |tt{—!*,-! (setting f = 1) < 2 ( f i - l ) ( « - l ) p + f i (by (2.10)), and hence, making use of the inequality \x{—x$\ ^2p which is a consequence of the assumption that x(Ik) and x(Ii) are separated, we have I«{'-«}' I ^ \*i-xs\ - | « { ' - * , I - I «}'-*/1 >2p-4(c-l) («-l) p-2£>p. Next, to prove the inequality for | a"—w}71, we notice that^ \{yi-yik)~{Xi-xIk)\ ^ 2 max = $yt-Xt)—±r \yp-xp\^2\y-x\<2B> E0v-*,)1 245 732 Y. SAISHO AND H. TANAKA from which it follows that \yi-yik\ < ! * , - * / J + 2 5 < 2 ( * - l ) P+2S (use (2.5)). Therefore \vi-yA= (c-l)\yi-yIk\<2(c-l)(n-l) p+2(c-l)8, and hence \vY~yi\^t\vi~yA^t\v\-vi\<2{c-${n-\) 9^2{c-\)8^B. Thus we have ^ I**—*/1 -1yi-Xi| -1yj-xj\-\vV~yA-\vy-yj\ >2p-28-4(c-l) (n-l) p-4(c-l) 8-26 The proof of the lemma is finished. Lemma 2.3. | z—v \ <€ \2. Proof. Taking Ik such that f €/>, we have I *<—«>( I = I ( £ - ! ) (*<—#)—fc—l) (*/s—^/*) I <(c-\)\Xi-yi\±{c-\)\xIk-yIh\ , and hence (2.11) \x-v\t = ±\z,-v,\t i=l ^2(c~\f\x-y\^2(c-lf±fiIk\xrk-yIk\2. Because |ar /t —^/J*^(#/*)" l 53 l*j—»I*» we have S#i'»l*/ t -y/J , ^l*-Fl , > and (2.11) yields |*-p|^2(c-l)|x-ffI<2(<:-l)8<e/2. The proof of the lemma is finished. Proof of Proposition 2.1. By (ii) of Lemma 2.2 and Lemma 2.3 we have B(z,£/2)C5(P,£)CZ), and hence by (i) of Lemma 2.2 246 MUTUALLY REFLECTING BROWNIAN BALLS (2.12) 733 .Dothe convex hull V of the set B(z, £/2) U {y} . We have also (2.13) \z-y\ = |M-X|^£>2S, because x^B(u, €) by (ii) of Lemma 2.2. Therefore, if 6 is defined by (2.9), then (2.12) combined with Lemma 2.1 and (2.13) implies that (2.14) C(y, lx, cos 0)V\B{y> 2 5 ) c F c 5 . Since B(x, 8)dB(yt 2S), (2.14) implies (2.1) with a = c o s 0 . Proposition 2.1 is finished. 3. The proof of D satisfies Condition (A) For l ^ / < y ^ w let Dtj be the domain defined by Di$ ^{x = (xlt - , xn)(=Rnd: \xi-xi\>p} . The domain D of (8) is expressed as D = n D„ i£» and x^dD implies x^ L^LX f] dDiit where = i(ij): Obviously, if x^D then x^dD a unit vector it^ in 72"'' by l£i<j£n, \xi~x}\ = p} . is equivalent to Lx =N(£. For (i,j)^Lx we define »,,. = {o,.... o, M , o,.... o, x>=^, o,.... 0}, (f-th) 0-th) let a ( 0 ^ a < l ) be the constant appearing in Proposition 2.1 and set Proposition 3.1. and for any x^dD (3.1) The domain D satisfies Condition (A) with rQ=p J Ulx= { n = 2 , c w «i,: ^ 0 , | n | - 1} . L e m m a 3.1. (i) B ( J C - 2 " ^ pii f/l 2'^ p) 0 A v = * , x<=QDti. (ii) B(x~2-^ pniJt 2 - ^ p ) n Z ) = * /or any (i,j)(=Lx> x<=3D, that is, « i 7 e 247 734 Y. SAISHO AND H. TANAKA Wxa-if*Pfor any Proof. We set {i,j)e.Lx. Since (ii) follows immediately from (i), it is enough to prove (i). y=x-2-^pniJt that is, Then for any z^B(y, f (Xi-{-Xj)l2 for k = \xk k =N i,j . for i,j, 2~1/2 />) we have I * , - * , I ^ I sti-yg I +1 yt-yj I -f I yj—zj I = \*i-yi\ and hence z^Dijy L e m m a 3.2. + \Zj-yj\ completing the proof. Let Jl'x be the right-hand side of (3.1). mx Then x<=dD. By Remark 1.1 it is enough to prove that for any n^.7lx 2r0 Let l=lx tion 2.1 be the unit vector appearing in Proposition 2.1. Then by Proposi- 0(xt/,a)nB(x, S)c5 from which it follows that (3.2) x-mx where C(x, I, a)* is the dual cone of C(JC, I, CK), that is, C(x, h ay = iz<=R»d: y - x > ^ 0 , V # e C ( x , /, a)} = {zejR"': < z - x , — 0 ^ \ Z T = G ? * I * — * I > • From (ii) of Lemma 3.1 and (3.2) x—ntj^Cfa h a)*t \f{i,j)^Lx , and hence (3.3) ^vT^?, M(i,j)eLx. 248 MUTUALLY REFLECTING BROWNIAN BALLS 735 Now let n^.Jlfx be expressed as »== 2 Ci*nih Cfj^O. Then by (3.3) and hence 1 2r0 1 ^ 1 > 2 C,y. 2 V 2 9 V1 - a V 2 p «.» e i Therefore, for any y^D 2 c,/if-*,nlV>+—L- c,vl*-irl2 2 by (ii) of Lemma 3.1 and Remark 1.1. T h e proof of Lemma 3.2 is finished. L e m m a 3.3. For any £ ( 0 < £ < 1 ) and x^dD C(V(JC, €)} D B(x, 8f)dD { n there exists 5 ' > 0 such that U {*} , 0',DsL where Cij(xt€)= Proof. Let y^C^y, iy^Rnd: £), y^x. y = x+z, (ij)^L. Then y can be expressed as ^£|*|>0, and hence I yt—yj 12 = I **-*> 12+2<*,—*,-, *,--»,•>+ |arf~*j 12 ^p 2 +2<Sf i ~2f i , Xi-Xj>>p2 , because V2p V^ \/2p 249 736 Y. SAISHO AND H. TANAKA Therefore C i ; (x, £)cD ( -yU {x}, and hence (3.4) n C0{X,G)C{ CI, DSL Since \xi—xi\>p B(x, S1) n Ai>u{*>. <.i,D<=L for any {i,j)^LXi \yi—yj\ >P there exists S ' > 0 such that for any # e f o r any (i,j)$Lx . This combined with (3.4) implies { n Cu(xt 6)} (1 B{xt 8') The proof of the lemma is finished. Proof of Proposition 3.1. We make use of Lemma 3.3 and then Corollary in [1: p. 11] to obtain x-mxc{ n cti(x,€)}*= E c,v(*,e)*, where 2 means the vector sum. Thus x-mxa n {WCilx~;sf}. But the right-hand side of the above is the convex cone with vertex x spanned by {x—niit (i,j)G:Lx}t so we have mx which combined with Lemma 3.2 implies This means that Proposition 3.1 holds. 4. Mutually reflecting Brownian balls Since the domain D satisfies Conditions (A) and (B') (and hence (B) by Remark 1.2), Theorem 1.1 guarantees the existence and uniqueness of the solution of Skorohod's equation for D: (4.1) %(t) = w(t)+\tn(s)d(s)y Jo where zv=(w1} ••-, zon), zok^W, l^k^ny and |z^,(0)—Wj(0)\ j ^ p , l^i<j^n. A solution of (4.1) is a pair (£, ) of functions satisfying (4.1), (1.2), (1.3) and (1.4). The component-wise expression of (4.1) is 250 MUTUALLY REFLECTING BROWNIAN BALLS (4.2) £>(*) = wk(t)+^nk(s) 737 <**(*), l ^ A ^ n . In this section we prove the following theorem by showing that (4.2) is equivalent to the equation (2) ?*(*) = *"*(*)+ 2 ['toM-gyM) «**«<*). i£*;Si». Theorem 4.1. There exists a unique solution of Skcrohod's equation (2) for the motion of mutually reflecting Brownian balls. Proof. By Proposition 3.1 we have (4.3a) n(,) = (4.3b) «*,•(£($)) - (0, - , 0, (4.3c) £ Cli(s) ^ _ — , 0, - , 0, ^——, (z-th) 0'™th) c«W^0, (4.3d) n„(g(*)), 0, »•, 0 ) , l^i<j^nt cfi(*) = 0 for & / ) $ £ « , > . The component-wise expression of n(s) is .. g.(*)-gX*) , „ nt(s) = . ^cti(s) V T p , , gi(«)-gi(«) +f S^,) V T p • Therefore if we define cfj(s) for * > / by cij(s)=cil(s)t then we have nk(s) — 2 cw(*) >: /** ' w V2p and hence gtW-g/*) V LP y: ,=£* Jo So if we set 1 f« *«(') = TV/ 2Tp7 Jo ^ we have (4.4) <M0 = 4,ik{t), J *W > 251 738 Y. SAISHO AND H. TANAKA (4.5) Mt) = (' 1P( | & ( f ) - E / i ) |) <%,(*), Jo because hj(t) can increase only when \^k(t)—^i(t)\=p by (4.3d). Thus (4.1) finally yields (2). Recalling (3.1), we see easily that (2) also implies (4.1) with n(t)= 2 where $(£)= S AiiW«*i(fW)/l 2 ^X')^ff«)l. ;;(*) anc * ^oW is the Radon-Nikodym derivative of d(f>ij(t) with respect to d(t). The proof of Theorem 4.1 is finished. 5. Skorohod's SDE for mutually reflecting diffusion balls Let Bi(t), l^i^n, be independent ^-dimensional Sv^rownian motions with B{(0)=0 defined on a probability space (fl, 2% P) with a filtration {S^J-^oWe assume that each £F, contains all P-nuIl sets and 2 r ( = 0 £?*+*. Given the coefficients b: Rd ^ Rd y we consider the following Skorohod's SDE for mutually reflecting diffusion balls: (5.1) dXt{t) = dt where the initial values are assumed to be ^-measurable random variables satisfying \Xi(0)-Xj(0)\>py \^i<j^n. The solution Xt(t)t l ^ i ^ n , should be found under the following conditions. (5.2) Xi(tys are £F(-adapted continuous processes with (5.3) <3>,v (/)'s are £F r adapted continuous non-decreasing processes with Otj{0) = 0, <£,-,.(*) = <£,,•(*) \Xi(t)—Xj(i)\^p> and %j(t) - T 1P( |*,«-*/(*) I )<**«X0 • Jo In this section we prove the following theorem. Theorem 5.1. Suppose a and b are bounded and Lipschitz Then there exists a unique strong solution of (5.1). Proof. Let continuous. 252 739 MUTUALLY REFLECTING BROWNIAN BALLS D = {x = (xly -,xH)(=Rnd: and for x — (xly •••, xM)^Rnd a(x) \Xi-Xj\>p, l^i<j , set 0 = , b(x) lb(x„\ T h e n as in 4 t h e S D E (5.1) can be regarded as Skorohod's S D E for D (5.4) dX(t) = a{X(t)) dB(t)+b(X{t)) dt+n(t) where B(t) is an M^-dimensional £F,-Brownian motion. t o b e found u n d e r t h e following conditions. (5.5) X(t) (5.6) n(t)^mx(t) (5.7) <£>(£) is a continuous non-decreasing process and d®(t), T h e solution X(t) is is a 5 - v a l u e d continuous process. if X(t)<=LdD. 4>(t)=\'uD{X(s))d®(s). JO But since D satisfies Conditions (A) and (B), it follows immediately from T h e o r e m 5.1 in [6] t h a t t h e Skorohod's S D E (5.4) has a u n i q u e strong solution. T h e proof is finished. References [1] [2] [3] [4] [5] [6] [7] [8] W. Fenchel: Convex cones, sets, and functions, Lecture notes, Princeton Univ., 1953. H . Frankowska: A viability approach to the Skorohod problem, Stochastics 14 (1985), 227-244. N . Ikeda and S. Watanabe: Stochastic differential equations and diffusion processes, North Holland-Kodansha, Amsterdam-Tokyo, 1981. P.L. Lions and A.S. Sznitman: Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. 37 (1984), 511-537. H.P. McKean: Stochastic integrals, Academic Press, New York, 1969. Y. Saisho: Stochastic differential equations for multi-dimensional domain with reflecting boundary, to appear in Probab. T h . Rel. Fields. A.V. Skorohod: Stochastic equations for diffusion processes in a bounded region 1, 2, Theor. Veroyatnost. i Primenen. 6 (1961), 264-274; 7 (1962), 3-23. H. Tanaka: Stochastic differential equations with reflecting boundary condition in 253 740 Y. SAISHO AND H. TANAKA convex regions, Hiroshima Math. J. 9 (1979), 163-177. Department of Mathematics Faculty of Science and Technology Keio University Kohokuku, Yokohama 223 Japan 254 LIMIT DISTRIBUTION FOR 1-DIMENSIONAL DIFFUSION IN A REFLECTED BROWNIAN MEDIUM By H. Tanaka Introduction In analogy with Sinai's problem C8] on a random walk in a random medium, Brox C1J considered the diffusion process X(t) described by the stochastic differential equation CD dX(t) = dB(t) - -g-W'(X(t))dt , X(0) = 0 , where (W(x), x £ R } is a Brownian medium independent of another Brownian motion B(t), and proved that (log t ) ~ X(t) converges in distribution as t -*• °° . Similar results in the case of a considerably wider class of self-similar random media were obtained by Schumacher CVj , Recently Kesten C53 obtained the exact form of the limit distribution for Sinai's random walk as well as for a diffusion in a Brownian medium. See also C2J for a related problem. In this paper we substitute W(x) in (1) by a nonnegative reflected Brownian medium and find the corresponding limit distribution. The result was already anounced in C9J without proof but the Laplace transform of the limit distribution given in C9: §3) is not correct. We give here a full proof to the whole result of t9: §3} with a correction (see Theorem 1 and 2 below). Our method is similar to that of C U . Theorem 1. Let X(t) be a solution of (1) where W + = {W(x),x >. 0} and W_= {W(-x), x > 0} are independent reflected Brownian motions on the half line CO, °°) starting from 0 which are also independent of the Brownian motion B(t). Then the distribution of (log t ) ~ X(t) converges as t + «> to the distribution \i defined by (2) where u = /mliTQ(d¥) m^ is the probability measure on ]R defined by (3.1) and the probability measure on the space of media W = C0R such that Q y has a symmetric density and for ^x e" Xx u(dx) / J2\ 0 sinhV2T coshJZk + t sinh/zx is 0,°°)) /MW:W( 0) =0} W + are independent reflected Brownian motions on CO, °°) . Theorem 2. (3) + A > 0 dt (1 + t ) 2 255 247 The present case is not contained in the framework ofC73 since the nonnegative reflected medium ing its minimum. The case of was discussed in C93 • Acknowledgment. W(x) has (uncountably) many points giv- a nonpositive reflected Brownian medium Some generalizations will be discussed in C53 . I wish to thank Professor H.Kesten for pointing out mistakes of the first version of this paper. §1, Preliminaries and exit times from valleys Let W and Q be defined as in Theorem 1. For each W C 1W solu^ tions of the stochastic differential equation (1) define a diffusion process in E with generator (1 1) W -,J JL e W(x)_d_( e -W(x)_d_) 2 e dxVe dx ; * 1) Such a diffusion can be constructed from a Brownian motion B(t) as follows (C43)• Let 12 be the space of continuous functions w •* CO, «>)•*• H with w(0) = 0, and denote by P the Wiener measure on fi, Denote the value of a) at time t by u>(t) or by B(t) and put / • * L = limJIlTi(B(s))ds v. e+0 e / L X , x + e ; (local time), 0 ¥( S(x) =J e ^dy , •I A(t) - f o S~ , A" e-^^ S ' 1 ^s^^s « / e- 2W f S_1(x ^L(t, X )dx , t > 0 m = the inverse functions Then the process X(t, W) - S" (B(A~ (t))) defined on the probability space (£2, P) is a diffusion process with generator (1.1) starting at 0. If we set (W x )(*) = W(- + x) , then X x (t, W) = x + X(t, W x ) is a diffusion process with generator (1.1) starting at x . Let T(xv 1) x 2 ) = inf {t > 0 : B(t)£(x.,, x 2 )} , The Brownian motion here is not the same as the one in (1) but we use the same notation B(t) . 248 Lfx-j, x 2 , x) = L(T(x 1( x2)» x) , x€lR , x,(t) = x(t, xw) , x*(t) = x + x(t, xw x ) Next we define a valley. is called a valley of W Given W € \W » a quartet (i) a < b (ii) W(b 1 ) = W(b 2 ) = 0, (iii) 0 < W(x) < W(a) for a < x < b1 , 0 < W(x) < W(c) for b2< x < c , (iv) V .= (a,b 1 ,b 2 ,c if < 0 < b2 < c , W(a) = W(c) = D , A_ = s u p {W(y) - W(x) : a < x < y < D , < x < y < c } < D . 2) We call D (resp. A = A V A, ascent) of V. ) the depth (resp. the inner directed It is clear that there exist valleys of W with A < 1 < D for almost all reflected Brownian media W . In what follows let W € TW be given and V = (a, b 1 , b ? , c) be a valley of W with the depth D and the inner directed ascent A . We put T x = T x (a, c) = inf {t i 0 : X*(t)£ (a, c)} . The f o l l o w i n g t h r e e lemmas were proved in C O Lemma 1. For . a < x < c c Tx (a, c) = | L ( S x ( a ) , § A ( c ) , S^ty) J e ^ ^ d y •J a where Sx(y) -j and d = means t h e e q u a l i t y i n Lemma 2. For each {L(xxv 2) a V b = max {a, b} , e»Wd2 distribution, A > 0 Ax 2 , Xx), d x£]R} = U L ( x v a Ab = min {a, b) X 2 , X)„ X€IR} 249 Lemma 3 . For (1.2) X > 0 W. (e tW) W € tW d 7 XWX), t >. 0} = {X~^X(X 4 t, W), t > 0) {X(tt where and , is defined by W,(x) = X" 1 W(X 2 x) , x€ H . The following lemma plays an essential role in our discussions. Lemma A. For any X > 0 and Cu, v ] C ( a , c) inf p{e*< D - 6) < T* < e X C D + 6 H + 1 , u<x X* The proof is similar to that of the corresponding lemm; [1) but even much simpler. x €C U » Let v 3 ^e fixed. y Setting Ac ;„1 = = ^(yJ/S^e) £ (^ /.^ tn\ = / .j*.™l*h.j/j\™Mi, (y) and applying Lemma 1 and 2, we have c T X = S X(c) / L •/.' <sX 1 ' sx(y))e -XW(y) dy . Since { 3) S,(c) <. (c - x)exp X max W > [x.c] J ( T, < (c - x)(c - a)exp X max W - X min W > L' <. (c - a ) 2 L ' [x »c} Ca, cJ J L' = max L(-°°, 1 , y) , we have P Tx > e X(D + 6) A D > eX(D+6)} e A6 /(c - a ) 2 j •* 0 To obtain an estimate from below first we notice that (1.3) 3) lim X" log C, = D , A X+" max W = max{W(x), x £ 1} , I min W = min{W(x), x f l ) I 258 250 where Cx = |SA(a)| A|SA(c)I , x £ (u, and the convergence is uniform in v~) . Next, for given 6 > 0 we set ^ S/O = sup{x C, ^(y^e-^^dy XW(y) dy L(-1, 1, s,(y)) > e A ^ D ~4^(b 1 - a 1 )L x expJ-X max W = (b., - a 1 ) L A e A C D " 2 ) . Since A log|s.(a 1 )| X* 1 and X log[s,(b-)| converges to max W - D Xx 1 QcAa^ ,xva^] max W - D , respectively, which are both negative, we have CXAD.| .xVb-p lim s,(a-) = lim s.(b.) = 0 the convergence being uniform in x £ £u, v~\ . >* ^ cex(:D-fi)J ^ P {L x T;< formly in x e[u, vj , because Therefore < (^ - a i ) - 1 e - A 6 / 2 j * 0 , X + lim L, = L(-1, 1, 0) > 0 X->-°° §2. The limit distribution of X(e Xr , AW) In this section we change the notation slightly. a valley Given We W and V = (a, b., b2» c) of W , we set n = c(to, «) -*IR) , Q = C(C0, «) •* Ca, O ) , and denote by P* , x € B (resp. P^ , ye(a, c)) the probability measure 259 251 on fi (resp. ft) i n d u c e d by t h e d i f f u s i o n <2'1J 2 e dP e process with generator dxJ (resp. the diffusion process on [>» c } with (local) generator (2.1) and with reflecting barriers at a and c). The latter diffusion has the invariant probability measure m, given by *x(dy) = e - ^ ' d y For any i n t e r v a l (u, vjC[a, c] | /0 e_AC K([u, v], C)d€ e " A ? K(Ca, c}, 0 where, for an interval I in M , K(l, £) the reflected Brownian medium, i.e., K(I, $ ) = lim-i- \% (2.2) Therefore is the local time at £ for f+_v(W(s))di r°° e " C K(Cu, v ] , J (2.3) mx(£u, VJ) " ° 7 / e " C K(Ca, 0 K(Cu, v l , K(Ca, c ) , N e x t we A~1Od£ cj, X" 1 C)dC 0) _ w < ; o , r 0) v3) f A + set P X = / "X^H • **'* = P X® P I ' R = R(to f u) = i n f { t > 0 : u ( t ) Lemma 5 . F o r any P X = P>PX = w(t)} . 6 > 0 limP?(R< eX xProof. (2.^) Fitfst it we we pprove r o v e that tha 11HI3P*(R < eX{A+5)\ 1 holds for x = b. and b_ 252 Without loss of generality we nay consider the case b instead of we define b2 for simplicity. For any £>0 x = b„ . such that a^ € (a, b^) , a 2 e (a, b 1 ) , c 2 £ ( b 2 , c) L1 = m a x j x 0 : w(t) = aA , T1 = T.,(u>) = i n f l t ^ O : w ( t ) ^ ( a v c2)| T 2 = T 2 (co ) = i n f { t > 0 : w(t) $ ( a 2 , c2)j Then we can prove e a s i l y t h a t (2.5) , T { o< 0 0 ]^ P S A (c 2 ) sA(b) S ( c 2 ) - S ps (a 1 AVO 1 , A—>oo , and hence (2.6) R^T0} > l P ^ { ^ ( 0 ) e (a, b ) , w ( T Q ) € & b > P ^ { ^ ( 0 ) e (a, )) + P A{£v ^{u)(t)e LaV = m A ([a, b}) +J 0) v c c )} - 1 j}PAfToedt} " 1 —>1 , A-^oo , by (2.3) because m ( / x e ( a , c) : W(x) = o|) = 1 . On the other hand Lemma A a p p l i e d t o the v a l l e y ( a p» ^1» b ? ' c 2^ whose depth i s A+(S/2) implies (2.7) P* T1 }^{ A(A+S)' T2<e' A->OT , and so P A x { R <e^ A + 5 ) } > P >x ; ( T 0 < e — ">'\ - o(1 > P ^ { T 1 < e A ( A + 5 ) , T1 = T 0 ) - o ( D (by ( 2 . 6 ) ) 261 253 ^ { T ^ e ^ ^ } _» 1 , - 0(1) (by (2.5)) as A ^ o o (by (2.7)) . Next , to consider the case where the diffusion starts at 0 wi shall consider three diffusion processes starting at 0 , b1 and b_ respectively. By making use of the comparison theorem in onedimensional diffusion processes (for example, see [3 : p.352] ) we can construct, on a suitable probability space (Qx» %.) * "three processes X^Ct) , "x". (t) and "X^C*) such that the probability measure on Q , induced by X*Q(t) (resp. X^(t), X2(t)) coincides with P A (resp. P^ , P A 2 ) and lC^(t) (2.8) , ^t^O , P^-a.s. Put R± = i n f j t > 0 Since "RQCR^V^ : X"±(t) = ( 0 ( t ) } , i = 0, 1, 2 . by ( 2 . 8 ) , we have S^A(RI V R ^ < e A(A+S) >P^{R<eA(A+"}+pb2{R<e^(A+S'j - 1 by ( 2 . 4 ) , completing t h e proof of Lemma 5. Lemma 6 . (u, v l in R lim A - ^ o° For any r- , r« with A- P A ° L ( e * r ) e Cu, v)l = *<(>, v ) n l > , . b , } ) *• uniformly in J r€ (r* , r^y , where m is defined in (2.3), Proof. Denote by T (resp. T) the exit time of (a, c) for co(t) (resp. to(t)), and by ^ (resp. TR) the exit time of (a, c) for w(t) (resp. uj(t)) after the collision time R . Since mA(U) -» 1 as A-»oo for any open set U containing {xc(a, c) : W(x) = 0}- , it follows from Lemma 4 that P A (e*< D - S ><W D+ S>} 262 254 rmA(dx)p;{e^D-5' a 1 , A - » co T h i s c o m b i n e d w i t h Lemma 5 i m p l i e s PA Ar f)f 1 X r Q /^ I 1 2 : =P°(R<e <e < TR } nr Ar- >IPj;[R<e Arp '< e > 1 , A->*> Therefore for r£(v^f ^i * ^\ ^ CvT . r2} P^fco(e A r )eru, v]} (2.9) >PA°[R<e ri , w ( e A r ) € : £>, v) , e = P ° { R < e A r i , 0(e*r)€ >PA fr, ^ C ^ } v j , e**2 + mA(Cu, vj) - 1 — » m ( [u , vj A Cb., , b2J ) , A -> oo ; as for the above equality we used the strong Markov property. we have Similarly lim P ° f w ( e A r ) £ [ u , v 3 c } > m ( [ u , vj c ^ fb- , b j ) , which combined with (2.9) implies P ° [ w ( e A r ) £ [ut v]}-»m([u, vj A £b., , b 2 J) , A-> co . The uniform convergence in §3. r£ £r-, r 2 3 is also clear. Proof of Theorem 1 Let V = (a, b. , b 2 > c) be a valley of W such that A < 1 < D . Such a valley exists with Q-probability 1. In fact, b. and b ? are taken as b 2 = the largest root of where a T = s u p [ x < 0 : W(x) = l} endpoints a and c and W(x) = 0 in (0, c') c' = infix > 0 : W(x) = l] . can be chosen suitably so that The a < a ' , c > c ' and 255 V = ( a , b 1 , b 2 , c) is a V = ( a , b 1 , b 2 , c) d e n o t e s such a v a l l e y p r o b a b i l i t y m e a s u r e on / -i -, \ JR d e f i n e d tr (3 1} valley T\ with A<1 . W . In what follows We d e n o t e by m thf by K( C u f , v Q , 0 U - VC > *y = K < c v b j | o f = where C u ' , v J C", v 3 ^ 0 > i • b 2 ^ • T h e n , i n t h e n o t a t i o n reads as follows: For any i n t e r v a l I i n JR a n d {r( A ), A > o l satisfying lim r ( A ) = 1 , C3.2) p f x ( e A r ( A ) , AW) e l l lim with respect to Q . o f §1 Lemma 6 for any family = mw(l) for almost all W Now we define IP = P ® Q yU = Jm.,Q(dW) . Integrating both sides of (3.2) with respect to and Q we have lim ff>lx(eAr(A), A W ) e ij = Ai(l) . (3,3) Next, define W^ a s i n Lemma 3 . Then | w A ( x ) , x e l R J r e f l e c t e d B r o w n i a n medium. Therefore (3.3) yields (3.4.) is again a p { x ( e A r ( A ) , A W A ) £ i ] = yW(i) . lim We now apply the scaling relation (1.2) to (_3.4)» the result is lim IpJA- 2 X(AV r ( A ) , W)€ll =yU(I) . -i Taking r ( A ) = 1 - 4A~ * l o g A i n t h e a b o v e , we o b t a i n p [ A ~ 2 X ( e A , W ) C l ] = /U(I) . lim This completes the proof of Theorem 1 §4- .Proof of Theorem 2 The absolute continuity of W /U is the measure in E n /*n ( l ) " then M E l can be proved easily. In fact, if defined by fK(ln[b1( b-)) K([b1> b a3 ) . I ; K(Cb is absolutely continuous because ^ n (l)£nE Q {K(I^Cb v b2))} 2ntj p(/x|, 0, 0)dx , V V»>iJ > 264 256 where p(t, $ , 7 ) i s the transition density of the Brownian motion with absorbing barriers at +1 . Thus kX is absolutely continuous because Ain f pL as n ^oo . time %= 0 We proceed to the proof of (3). Let K(l) = K(I, 0) be the local at 0 for the reflected Brownian medium as defined by (2,2) with and consider the number of times d^(t) that the reflected Brownian path t , {w(u) : u > 0 ]• crosses down from Then as found in CA= Qjlim 2fcdf(t) = K(CO, t}) , t > 0 (4.1) \U0 a' , c 1 , b.. Let Lemma 7 . For f Q and b- V— 1 JW -^ J (|9) - K(CO, b^}) =1 2/2/5" e + r^ _ e-/3r f -cXK(C0,b23)l 2 W . «-/?" (f3) is exponentially EQ e • distributed: 1 V= L- . L J 2c/ + 1 Proof. Since c((5)~1 as p I 0 , (4.3) follows letting p i> 0 . To prove (4.2) we first apply (4.1) 2 EQ|e (4.4) = E QL-* t ' Qf = where E^ - /^'l - 2 c * £ dfc, ( c < ) - | 3 c ' | r £ e - - - ' E « { e - ^ } n denotes the expectation > - ^ with i T o < T l respect } ^ e - to measure of the reflected Brownian motion starting at £ T x = inf ju S O : W(u) = x | . If we set from (4.2) by to write down ' j K( 0 c,:i) = lim EQje before time 2oC + e 727 (4.3) 0 (X , p > 0 where In P a r t i c u l a r , to be defined as in the beginning of |3. -«K(C0,b2J)-(ac'J E^je (4.2) £>o p.48) P the and V T i < T o probability 257 then (4-4) yields u.5) 4 } -li-^z^ n=0 lira -I _ a Next we make use of the well-known formula f -<XT x l I a b J ^725T(b-x) ey2S(b-a) _ -,/2tf(b-x) e-j2#(b-a) where E denotes the expectation with respect to the probability sure of a standard Brownian motion starting at x . We then have (4.6) /3T 1 f -pT E P if£ E«{e-^J=2E ;T £ 2£/2^ _ -2S/2P 1 + 0(£2) , £^0 J PTn (4.7) 1 -^»(1-£) W^<^}=^^ (4 .8) E«{e~ 1; T^Tj,} = IM. From (4.6), (4.7) and (4.8) we obtain „-/9(1-£) 72F P • ^ ° 266 258 £. 1 1 - k ^ 2j2p 2ot + c (p) ' em w h i c h combined w i t h (4-.5) p r o v e s t h e G i v e n x > 0 we s e t = K(Cbv0]), 1 Lemma 8 . For x >0 (2t Proof. K3 = K ( C x , b 2 3 . t>0 •t(K-+K0+K,) EQ-i| KK33 ee ' ; (4-9) lemma. K2 = K ( [ 0 , x 3 ) , and x ^—^{(l-WUne"^0^; The left hand side of (A.9) K tKl )4: t 3 £ J,0 _ e-/2p x equal: -t(K0+K 2 ' " 3 \; x < b. ' 2 } 1 Since E Q je | = (2t + 1 ) " 1 by Lemma 7, for the proof of the lemma it is enough to show f •riKje (4.10) - t ( K +K ) | * * : x 2 ; x< c'j . (2t + 1)' To prove this we introduce the smallest C-field ^p on W which makes W(s) , 0<"s^x , measurable and consider the event f that the shifted trajectory W( • i x) hits 0 before it hits 1 . Then first using the strong Markov property of the reflected Brownian motion and then (4.3), we have •tK. {^'"Wz} = (l - W{X)]E Q {K(C0, b2))e = 2 —-5(1 - W ( x ) | , (2t + 1 ) Since 2 <• and r -t{K +K3) EyJK3e * ^ ; {x<< } i x «< c ' l €" ^ » a.s. i j x -tK„ 2 n x • } 4/ t K 3 vM} we have 267 259 -? (2t proving (4.10) EEQy JC (1 l ( and h e n c e t h e Lemma 9 . C4.11) + 1) ^* For oo A>0 "- WW< (x x>>) )ee" • tK. 2 i * lemma, and t>0 e"* x E«{(1 - W ( x ) ) e - t K ( C 0 ' x 3 ) ; | x < B ,J 0 -J-fi " A I (2t + 1)S1 C + 2tS J , where C = cosh/2A (4.12) in S = s i n h/^ '2 A Proof. Let , (-1, 0) w(Q, 1) J -j{f'(0+) - f'(0-)} = 2tf(0) f(-D = f(D = 0 . in To solve (4.12) we first find the solution g^ of Af - -|f" =9 (-1,1) with boundary condition f(-1) - f(1 ) = 0 and then express f\ as follows: f,(x) =i A g.(x) + c sinhy"2X(1+x) X for x d - 1 , 0) g (x) + c sinhv/jA (1-x) A for x<=(0, 1) . If we determine c so that the above fj. satisfies the second condition of (4.12), then the f^ is a solution of (4.12). Thus *\(0) can be computed and we obtain (4.11). Now Theorem 2 can be proved as follows. By Lemma 8 we have .((x.oo)) = E Q EQ fK((x, xvb2l)] ( K(Cbv b^)} {KI + \\ + K3 : *< b 2 } -t 1 x 268 260 2 (2t + 1 K EQ[(1 _ w(x))e-tKCC0' x ^ ; x < c'ldt I and hence by Lemma 9 e^^Cx^JJdx = , 3—v4fl J ' 0 (2t + 1)^ * L J rCO 0 - i 2 t + 1)S ] dt C + 2tS J §_d 2 2tS (2t + 1)* ' C + t Thus integration by parts yields e'>XyW(dx) =-^- *j e_Axy«((x, oo))dx I 0 §§ v2t (notice that U((0,oO)) =-1-) at + 1) (C + 2tS) 00 f Sdt Jn (t + 1)2(C + ts) 0 and this proves (3). REFERENCES £1} T.Brox, A one-dimensional diffusion process in a Wiener medium. to appear in Ann. Probab. C2J A.O.Golosov, The limit distributions for random walks in random environments, Soviet Math. Dokl., 28(1983), 18-22. £3} N.Ikeda and S.Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, 1981. £4} K.IfS and H.P.McKean, Diffusion Processes and Their Sample Paths, Springer-Verlag, 1965. £5} K.Kawazu, Y.Tamura and H.Tanaka, One-dimensional diffusions and random walks in random environments, in preparation. (_6j. H.Kesten, The limit distribution of Sinai's random walk in random environment, to appear in Physica. (jO S.Schumacher, Diffusions with random coefficients, Contemporary Math. (Particle Systems, Random Media and Large Deviations, ed. by R. Durrett), 41(1985), 351-356. 269 261 [8J Y.G.Sinai, The limiting behavior of a one-dimensional random walk in a random medium, Theory of Probab. and its Appl. 27(1982), 256-268. [9] H.Tanaka, Limit distributions for one-dimensional diffusion processes in self-similar random environments, to appear in the Proc. of the workshop on Hydrodynamic behavior and interacting particle systems and applications held at IMA, University of Minnesota, March 1986. Department of Mathematics Faculty of Science and Technology Keio University Yokohama, Japan Reprinted from Seminaire de Probabilites X X I , 246-261, Lecture Notes in Math., 1247, Springer-Verlag, 1987. LIMIT DISTRIBUTIONS FOR ONE-DIMENSIONAL DIFFUSION PROCESSES IN SELF-SIMILAR RANDOM ENVIRONMENTS H. Tanaka Department o f Mathematics F a c u l t y of Science and Technology Keio U n i v e r s i t y , Yokohama, JAPAN Introduction Let X ( t ) be t h e one-dimensional d i f f u s i o n process described by t h e stochastic d i f f e r e n t i a l (1) where dX(t) = dB(t) - | - W ' ( X ( t ) ) d t , B(t) (W(x), x£|R} motion equation i s a one-dimensional Brownian X(0) = 0 , motion s t a r t i n g at 0 and i s a random environment which i s independent of t h e Brownian B(t). We are i n t e r e s t e d i n t h e asymptotic behavior of Under what s c a l i n g does X ( t ) have a l i m i t d i s t r i b u t i o n ? X ( t ) as t + ~: S i m i l a r problems random walks were considered by Kesten, Kozlov and S p i t z e r [ 5 ] and Sinai for [8]. The problem we discuss here i s a d i f f u s i o n analogue of S i n a i ' s random walk problem [ 8 ] . In the case of a Brownian environment Brox [ 1 ] proved t h a t d i s t r i b u t i o n of (log t ) X ( t ) i s convergent as t + » . the S i m i l a r r e s u l t s were o b t a i n e d by Schumacher [ 7 ] f o r a c o n s i d e r a b l y wider c l a s s of s e l f - s i m i l a r environments. As was seen by these works t h e assumption o f the self-similarity of the random environment i s important and t h e n o t i o n of s u i t a b l y v a l l e y s of t h e environment plays a c e n t r a l role in the proof. random defined However, i t was assumed t h a t t h e environment has only one p o i n t which gives t h e same value o f l o c a l minima or maxima ( t h e bottom of a v a l l e y c o n s i s t s of a s i n g l e p o i n t ) , and t h e e x p l i c i t form o f t h e l i m i t d i s t r i b u t i o n was unknown u n t i l a recent discovery by Kesten ( [ 6 ] ) f o r S i n a i ' s random walk which corresponds t o t h e case of a Brownian environment i n our d i f f u s i o n setup (Golosov a l s o obtained the same r e s u l t as K e s t e n ' s ; see also Golosov [ 2 ] f o r t h e corresponding r e s u l t i n another different model). 271 190 In this paper we discuss the following three typical examples of random environments with emphasis on finding the l i m i t d i s t r i b u t i o n s : (i) Nonpositive reflected Brownian environment, (ii) Nonnegative reflected Brownian environment, ( i i i ) Symmetric stable environment. In the f i r s t two examples the environment has (uncountably) many points giving the same value of local maxima or minima. For details see [ 9 ] . The proof in ( i i ) is only sketched. The result in ( i i i ) on the l i m i t d i s t r i b u t i o n is an exten- sion of Kesten's result [ 6 ] . In [4] a unified d e f i n i t i o n of valleys of an environment is given in a general setup containing the above examples and some results similar to Brox's and Schumacher's are obtained, but here we l i m i t ourselves to the above examples because we are interested mainly in the form of the l i m i t distributions and we have e x p l i c i t results concerning this only in some special cases. The author wishes to thank K. Kawazu and Y. Tamura; his frequent discussions with them were very valuable. 1. Preliminaries and the Result of Brox 1.1. real line Given a real valued right continuous function W(x) defined on the IR and having l e f t l i m i t s , we consider the stochastic d i f f e r e n t i a l equation (1) with environment W(x). Since W(x) is not d i f f e r e n t i a t e in general, what is meant by a solution of (1) w i l l need explanation. without considering a solution i t s e l f for a given Brownian motion However, B ( t ) , we just interprete the diffusion described by (1) as a diffusion process s t a r t i n g at with generator (1 1) [iml) Ie«(x)i.(e-M(x)L, 2 e dx {e d x >• Such a diffusion can be obtained from a Brownian motion through a scale change and a time change ([3]). To be precise let 0 272 191 a = c([o,«) + iR) 1 'flMo) = o>» P = the Wiener measure on B{t) = oi(t) = the value of n, at time w L(t X) - - l l ! - r fo W e ) ( B ( s ) ) d s S(x) - f* eW<*>dy t, 'loca1 time >- , A(t) - t > fli = the inverse functions. X(t,W) = S~ (B(A~ ( t ) ) defined on "the probability space (n,P) is a diffusion process with generator (1.1) starting at 0. The Brownian motion B(t) used here is not the same as the one in (1) but we use the same notation. Let (W x )(.) = W(-+x). For a fixed x € IR we replace W in X(t,W) by Wx and then consider X x (t,W) = x + X ( t , W x ) . Then X (t,W) is a diffusion process with generator (1.1) starting at x. In X(t) = X(t,W) and X x (t) = XX(t,W) and the this paper we shall deal with following notation will be used throughout. S x (x) - f* e xW du, W : a function (environment) defined by W (x) = x~ W(x 2 x) for V x € I R , x > 0 being fixed. In the following lemma due to Brox the medium Lemma 1.1 ([!]). For each 1) is fixed. For the proof see [1], x>0 {X(t,xwx), t>o> = i\~h{\h, where W w), t>o> d = means the equality in d i s t r i b u t i o n . For a topological space R the notation C([0,<») + R) stands for the space of R-valued continuous functions defined on [ 0 , » ) . 192 When we consider the environment to be random, we denote by Q the proba- b i l i t y d i s t r i b u t i o n (on the space of environments) of the random environment. Since we are assuming that the random environment and the Brownian motion are independent, the f u l l d i s t r i b u t i o n is p ment is fixed X(t) i s governed by = P x Q. B(t) Thus, when the environ- P, and when the environment is random X(t) is governed b y p . 1.2. In this subsection we l i m i t ourselves to the case of continuous environment and state a result of Brox [1] in a form which is convenient for our use in §2. Given a continuous function W on IR which is supposed to be the environ- ment in (1), we define a valley of (1.2) \ y W following [ 1 ] . For x #y = sup (W(y') - W(x')> where the supremum is taken over all pairs of x' and y 1 x < x' < y' < y V = (a,b,c) such that or y < y' < x' < x according as x < y or y < x. A triple is called a valley of W if (i) a b < W(a) - W(b). V = ( a , b , c ) , D = (W(a) - W(b)) A («(c) - W(b)) A = H .VH , a,b c,b are called the depth —'— V, where (resp. uVv) denotes min {u,v> (resp. max (u,v>). u/\v of and V and the inner directed ascent of I t is obvious that A < D. Theorem 1.1 (Brox [ 1 ] ) . depth (1) (i) Let D and the inner directed ascent Let T x be the exit time of V = (a.b.c) be a valley of W with the A. (a,c) for the process X x (t) = x + X(t,xW x ). A A Then for any 5 > 0 and any closed interval lC(a,c) lim inf P { e ^ D " « ) < T X < e x ( D x X+- x € I + s) > = 1. 274 193 (ii) For any e>0, any closed interval I C (a,c) and for any closed interval J C (A,D) lim sup P { | X x ( e x r \ xW) - b| > e } = 0. x+» x£l re J First proving the above result and then making use of the scaling relation (Lemma 1.1), Brox derived his main theorem: Theorem 1.2 (Brox [ 1 ] ) . For any e>0 P{|x" 2 X(e" X , W) - b x | > e> * 0, X+» in probability with respect to the probability measure environment, where b Q of the Brownian is the unique bottom of a valley A w A A A A such that (1.3) 2. V = (a , b . c ) of a < 0 , A < 1< D. Nonpositive Reflected Brownian Environment As a simple example in which the environment has (uncountably) many points giving the same value of local maxima we consider the case of a nonpositive reflected Brownian environment. Main ideas of this section grew out from the conversation with Y. Tamura. We f i r s t introduce the space the probability measure on (W(-u) : u>0} tj y = C(IR+(-<», 0 ] ) f l (W(0) - 0} and denote by Q with respect to which (W(u) : u>0) are independent reflected Brownian motions on and (—,0]. An essen- t i a l difference between the present case and Brox's one Is that there is no valley of W satisfying (1.3) in the present case. For W£ W we put 275 194 W*(u) = W(u) - inf W, u>0 , [0,u] = W(u) - inf W, u<0 , [u,0] and define a + = a + (W), b+ - b (W) and c 1. d + = + min {u > 0 : W*(±u) = 1} , 2. b + = + ntfn (a > 0 : W(+u) - M+J where 3. c M = min W and M C0.d + ] " - min {u > b = c (W) as follows. = min W, [d_,0] : W(u) = 0 ) , a + = max {u 4. e + = + min {u > d + : W(±u) = W(b ) } , 5. a = max {u , Ce_,b_] c = min {u > b, : W(u) = max W}. [b+,e+] Then a V+ = (a + , b + , c ) are valleys of < 0 < a ascent we have Q-a.s. < c , Q-a.s. W with As for the depth and the inner directed A+ < 1 < D+ , Q-a.s., because Substracting a suitable null set from ments hold for a l l W{a_) > W(d_) and W(c+) > W(d+), W we may assume that the state- W in the subtracted space and so we often omit the phrase "Q-a.s." d 1 b \J \ i n \ ' / 1 195 Our aim is to know the asymptotic behavior of c i s e l y , to find the l i m i t d i s t r i b u t i o n of observe the process X(t,xW). If \ X(t,W) as (log t ) X(t,W) t-»-», or more pre- as t+». First we becomes large and time goes on, the pro- cess eventually f a l l s into one of the valleys w i l l t e l l about the asymptotic behavior of V+ and thereafter Theorem 1.1 Ar Then a X(e , xW), x + «, r ~ 1. use of the scaling relation (Lemma 1.1) w i l l give the result for the l i m i t d i s t r i b u t i o n of x" X(ex,W) as To be precise choose (2.1) x+«. a such that -min W< a < 1 [c_,a + ] and put x + = ± min {u > 0: W(±u) = - a } , T* = min {t > 0: X(t,xW) = x + >, f x = T~ Denote by L_(I,0 (W(u), u e i R } , the local time at 5 for the reflected Browm'an environment i.e., L (I,e) = l i m e e+0 Lemma 2 . 1 . T* . (i) fT 1-,(W(u))du, I = interval in |R. l ^-e,?J There exists 5 - $(W) > 0 such that lim P{f < e x ( 1 " ^ > = 1. \+« A + lim P{T" < T \ = L_([0,a+], 0)/L_([c_,a+], 0 ) . (ii) X-t-» a Proof. The condition (2.1) implies the existence of a modification W of with the following properties (2.2) and (2.3). (2.2) W # e C(tR *|R) and W # (x) = W(x) for (2.3) There exists a valley J Vx€[x_,x+]. V # = (a # , b # , c # ) of w # with depth and a # < x_ < x + < c . Let D # T* = min {t > 0: X(t,xW#)€^ (x_, x + )>. Then for any + 26 < 1 we have «>0 such that D# < 1 196 < e*< 1 -«>> - P f T* < e ^ P A by ( i ) of Theorem 1.1. 1 > P { T* < e » < D # + a > } • 1, X+» -^, A The proof of ( i i ) is easy; in fact - + s (x x +> - V0) e A * L ( [ 0 , x ] , Ode 0 » e X 5 L ([x , x j , Ode f ' _ j i e L j C 0 , x + ] , x _1 c)dc 0 f r 1 e 5 L_([x_,x + ], A ^ J d ? L_([0,x + ], 0 ) / L _ ( [ x _ , x + L 0), x * = L_([0,a + ], 0)/L_([c_, a + ] , 0). Lemma 2.2. There exist e > 0 and r, and r with rl < 1 < r„ such that for any r€[r1,r2] 11m P{X(e Xr \ xW)€U£(b)} = p x+» holds with p_ = 1 - p+ b = b+ and and U (b) p = p+ r The above convergence I t is easy to see that there exist valleys tf+ = (a"+ , ff , c ) a < a_ < b_ < x_ < c_ < c_ < c" < c of and , respectively. Denote by T* the exit time of x+ «* T - (a" , c.) for the process X ( t , xW) and by T*~ the exit time of x * t = (a" , c" ) for the process X ~(t,xW). Then using the strong Markov property of < a < x b. on [ r \ , r „ ] . W with depth ff+ > 1 and satisfying a p + = L ([c_,0],0)/L ([c , a + ] , 0 ) , denotes the e-neighborhood of i s uniform with respect to Proof. where X(t,xW) we have P { X ( e x t l " 6 ) , XW)£ I + } (2 "2) >P{J+^Jx<^l-^} P|T^+ > e ^ 1 - * ^ 197 and a similar inequality with $ = s(W) of + replaced by - . We now take Lemma 2.1 and then let x+» in (2.2). Then the assertion (i) of Theorem 1.1 combined with Lemma 2.1 implies Vim P{X(e x ( 1 " 5 ) , x W ) £ I+> > P + (2.3a) X+™ and a similar inequality (= (2.3b)) with + replaced by - . (2.3a) and (2.3b clearly imply that lim P{X(e x ( 1 _ 6 ) , xW) €. T+> = p + . (2.4) X+o. r. We now take and r„ such that A _ V A+S/(l-s) < rx < 1 < r 2 < D _ A D + and prove the lemma with these t =e xr X US - e ( K r. and r_. Let r- < r < r„ and Then we have P{X{e x r , xW)£ u G( b +)> P { X { e x ( 1 " 5 ) , xW)£ dx}P{X X (t,xW)eU £ (b + )} = f +o(l) + which tends to p valley uniformly in r as V + because x+» by Theorem 1.1 as applied to the xr' t can be expressed as t = e with r'+r as x*«. The other case (b = b_ , p = p_) can be proved similarly and so the proof is finished. Since measure y b + and p + are Borel functions of W we can define a probability on R by /*U)u_(dx) = f{p_ *(b_) + P+ *(b + )>Q(dW), Theorem 2.1. The full distribution of t •*• » •€CbOR). -2 (log t) X(t) converges to M_ as . Proof. converges to Lemma 2.2 p as implies that the f u l l d i s t r i b u t i o n of x-»"» provided that r(x)-»-l as x-*-«. Brownian environment is s e l f - s i m i l a r , i . e . , {W ( x ) , x £ I R } X ( e > r ^ x \ xW) Since our reflected is again a reflected 198 X(e* r '*\ xW ) Brownian environment, we see that the full distribution of converges to . v also Applying the scaling relation {Lemma 1.1) we see that the x ~ 2 X ( x V r ( x \ w ) converges to full distribution of p_ . If we take r(x) = 1 - 4 A ~ log*, the last statement is nothing but the assertion of the theorem. Theorem 1.2. y has a symmetric density and 1 M ux) = = r j"r—r2——J "..d-5/%-xx• XX y_Cdx) 2 (2.5) 0 Proof. " If and U t > 0, Q-a.s. = L([0,a+],0) (2.6) /2T d ( t ) denotes t h e number of times t h a t the r e f l e c t e d Brownian path {W(u) : u>0} crosses down from = L_([0,tL0), _ — r ~ z r . * > o. 0 U + l ) { ( j + /2T coth /2x)cosh 0 to (see [ 3 ] ) . -e before t , then Therefore, putting we can w r i t e f o r l i m 2cd ( t ) f = mi"n{u>0 : W(u) = - l > or, X > 0 EV^'S +0 n=0 n EJj.fe where (-»,0] H a denotes the h i t t i n g time t o and t h e s u f f i x x i n E;* Using t h e e x p l i c i t form of be computed. The r e s u l t E"{e" c(x) = ( e ^ E Q {e g + = b+ - f + . x. 2/21 Let _ e-/2~x - e"*^)" n e n t i a l l y d i s t r i b u t e d w i t h mean where position is - e } , e t c . , t h e r i g h t hand side of ( 2 . 6 ) can ^ + e~^)(e^ (2.8) f o r t h e r e f l e c t e d Brownian n o t i o n on indicates that the i n i t i a l x e/2"x where ; H_! < H Q , is q -tfL 2 ~xf E > (2.7) a ~xH li 2. + > 1 /^ . 2a + c(x) ' In p a r t i c u l a r l_2 In a s i m i l a r s p i r i t we see e a s i l y e e L x = L_([c_ , 0 ] , 0 ) . Then i s expothat 280 199 ;fl e p_(dx) - E { i + i 1 ' "2 • e + ) = J E {L,e i 0 }da , and making use of (2.7), (2.8) and the fact that L. , g + and {L„ , f+> are independent we can compute the right hand side of the above. We thus obtain (2.5). 3. Nonnegative Reflected Brownian Environment In this section we consider the case of a nonnegative reflected Brownian environment. This is a typical case where the bottom of a valley consists of (uncountably) many points. Let y be the space C(|R + [0,-))/"! (W(0) = 0} and consider the probability measure Q on W with respect to which {W(u):u > 0> and {W(-u):u > 0) are independent reflected Brownian motions on [0,«). The study of asymptotic behaviors of X(t,W) as t + » can be done by a method similar to that of Brox [1] as will be sketched here. The definition of a valley given in 1.2 must be slightly modified. Given V = (a, b-, b „ , c) is called a valley of U if W £ y, a quartet (i) a 0, (iii) 0 < W(x) < W(a) 0 < W(x) < W(c) D and tively. 1. A = H for 0 < x < b, , for b2 < x < c, a,b2VHCtbi<°. A are called the depth and the inner directed ascent of There exists a valley of W such that A < 1 < D with V , respecQ-probability In f a c t , l e t a' = max {u < 0 : W(u) = 1} • 1 b. = min {u > a : W(u) = 0} , c' = min {u > 0 : W(u) = 1} , b„ = max {u < c ' : W(u) = 0} . 200 Then with a suitable choice of V = (a, b . , b 2> c) follows becomes a valley of As in [ 1 ] we can prove that [ b j , b2] as in the l i m i t ? A < 1 < 0, Q-a.s. X { e , xW) f a l l s into an e-neighborhood xr X ( e , AW") distributed on [ b , , b „ ] This l i m i t d i s t r i b u t i o n can be i d e n t i f i e d with the l i m i t , as x-*-», (local) generator c. I f In what W. We f i r s t observe xr x+» , r ~ 1. Then how is the invariant probability measure and W with a < a1 , c1 < c, the quartet V = (a, b , , b „ , c) denotes such a valley of X(t,xW). of a and c with j x W e ^ ^ m (e~ xW of the diffusion process on [ a , c ] with W ^ ) and with reflecting barriers at a L ( I , z) denotes the local time at environment (W(x), x £ | R > , then f o r an interval mx(I) - / I e ' ^ d y / jj e for the reflected Brownian IC[a,c] e'^dy = f - e _ x c L + ( I , 5 ) d 5 / r"e _ X 5 L ++(Ca,c], Ode. 0 0 • L + { i n C b r b 2 ] f 0 ) / L + ( [ b r b z ] , 0). Define a probability measure mu in (R by m w ([u,v]) = L + ( I ' , 0 ) / L + ( [ b r b 2 ] , 0) where I ' = lu.v] Olb,, b_]. Then for any interval family {r(x), x > 0} satisfying I in 1R and for any lim r(x) = 1 we have X+™ lim P{X(e x r t x ) , x W ) C l > = m w ( D for almost all U with respect to Q. Now we define l i m p { X ( e x r ( x ) , x W ) £ 1} = Substituting W in the above by the scaled W v = \ nuQCdW). Then V+(I). and then using Lemma 1.1 we obtain the following result. Theorem 3.1 {[9]). to y as t •*•-> . (i) The full distribution of (log t)"2X(t) converges (ii) u has a symmetric density and for \ > 0 201 e"w j " 0 v + (dx) - /" 0 cosh/27-1 ( 0 + l) 3 { cosh/2x + a H ! M X } A /2i dp > x > 0> 4. Symmetric Stable Environment In t h i s section the space y of the environments is D{|R + |R) ' f) (W(0) = 0} and Q is the probability measure on W with respect to which and {W(-u) : u>0} fW(u) : u>0} are independent and symmetric stable processes with exponent ct(0 < a < 2) such that E Q {e /-nw(u), a . e. | u | | T | » > u e i R i R- This case is contained in the frame of Schumachers work [ 7 ] (see also Kawazu-Tamura-Tanaka [4] for additional information) and so the f u l l d i s t r i b u t i o n of (log t ) " a X(t,W) is convergent as t + « . The purpose this section is to give a simple probabilistic description of the l i m i t d i s t r i b u t i o n . In the case of a Brownian environment Kesten [ 6 ] proved that the density of the l i m i t d i s t r i b u t i o n is given by (4.1) *- ^ e - C 2 k + 1 ) V | x | / 8 j X e ( R i We begin by stating some known results (see [ 7 ] , [ 4 ] ) . First we give the d e f i n i t i o n of a valley, which is a slight modification of the one given in §1. Let W€ y. By d e f i n i t i o n U is said to be o s c i l l a t i n g at x (€(R) i f sup W > W(x±) , inf W < W(x±) for any e > 0, where have a local minimum at I = (x-e, x + e). I + = (x, x + e) x if and I_ - ( x - e , x ) . inf W = W(x)AW(x-) f o r some A local maximum is defined s i m i l a r l y . W is said to e>0 where Denote by u W the set of W£ W with the following four properties. T) This is the space of IR-valued right continuous functions on |R with l e f t limits. 202 (4.2a) Tm {W(x) - i n f W} = Tm x+» [0,x] (4.2b) If W i s d i s c o n t i n u o u s at (4.2c) For any open set {x£G G in IR {W(x) - i n f W} = » . x+-~ [x,0] x , then W i s o s c i l l a t i n g at x. both t h e sets ; W( x ) = sup W}» { x £ G G : W(x) = inf G U) c o n t a i n at most one p o i n t . (4.2d) W does not have a l o c a l maximum at Let HgW1 , Then x = 0. V = (a»b,c) i s c a l l e d a val 1 ey o f (i) a b < W(c) - W(b), H c > b Here the n o t a t i o n d i r e c t e d ascent continuous at proved t h a t H A x if and < tf(a) - W(b) . are d e f i n e d as i n § 1 . The depth Notice t h a t 0 and t h e i n n e r implies that x. W is I t can be Moreover, making use of t h e s e l f - s i m i l a r i t y of a sym- W such t h a t a < 0 < c such a v a l l e y i s u n i q u e l y determined by A > 0 (ii) W has a l o c a l minimum (or maximum} a t m e t r i c s t a b l e environment we can prove t h e f o l l o w i n g : V = ( a , b , c ) of if c, i s d e f i n e d as i n ( 1 . 2 ) . Q(Vr) = 1 . W we d e f i n e the scaled environment and There e x i s t s a v a l l e y A < 1 < D, Q - a . s . The bottom of W and so i s denoted by w" by b = b(W). a W?(x) = \~ W(A x), x £ l R . as a s p e c i a l case of [ 6 ] (see also [ 4 ] ) we have t h e f o l l o w i n g r e s u l t : For Then For any e >0 P { | x " ° X ( e \ W) - b(W*)| > e > in probability with respect to Q as a +0 x-»-» and consequently the full distribution of (log t)" X(t) converges to the distribution of b = b{W) as t * «.. 203 Our task is now to compute the d i s t r i b u t i o n of b = b(W). Although we are unable to give an analytic representation l i k e (4.1), we have the following simple p r o b a b i l i s t i c representation of the l i m i t d i s t r i b u t i o n . The f u l l d i s t r i b u t i o n of (log t ) " a X(t) converges, as Theorem 4 . 1 . t -»• », to the d i s t r i b u t i o n with density *(x) = { - & £ U . } 2 . Q { r(f ) SUp W* < 1} , x £ | R , [0,|x|] where W*(u) = W(u) - inf W, u > 0. [0,u] The proof w i l l be based on the following description of For b(W) due to Kesten. V l £ W* we set d + = ± inf{u > 0 : W*(±u)>l} where W*(-u), u>0, is defined in a way similar to W(b.) = inf W, W(b ) = C0,d + ] " W*(u), and define b by inf W. [d_,0] We also set V = W(b ), * * Lemma 4.1 (Kesten [ 6 ] ) . a < 0 and A < 1 < D. M. = Let (a, b, c) be a valley of b =b W such that Then b = b+ and the equality sup W, M = sup W. [0,b+] " [b_,0] or b holds i f and only i f one of the following conditions holds: V > V+ and M+ < (V_ + 1 ) V M „ » (11) V_ < V+ and M_ > (V+ + 1 ) V M + • (i) Since we deal only with the probability measure simply E { . } instead of E {•} Q in the sequel, we write for the expectation with respect to Q. Again we 204 introduce notation: F x (x) = E{e + ; V+ < x - 1 , M+ < x> = E{e ; V < x - 1 , M < x} . •Ab, Clearly F (x) vanishes identically on (-», 0] and equals Lemma 4.2. Proof. E{e E{e~ x b ; b > 0} = fg F x (x)dF Q (x) , \ > 0. Let lx = {V_ > V+ , M+ < (V_ + 1 ) V M J . E2 = {V: < V+ , M_ > (V+ + 1 ) V M + } . Then i t is easy to see that ( E j l ) E2)C = {V_ < V+N/{M+ - 1), M_ < (V+ + l ) V M + > and hence E{e"* b ; b > 0} = E{e + E{e = E {e -xb + + •Xb, + ; V_ > V+ , M+ < (V_ + 1 ) \ / M _ } ; V_ < V+ ; E 1 U E2> , M_ > (V+ + 1)N/M+) } on [!,«). 205 + = E{e ; ( E ^ E2)c> } - E{e _ X b _Ab + + = E{e + / } - E{e = F x ( l ) - E{e * v ; V_ < V + V ( M + - 1 ) , + I M_ < (V+ + 1 ) V M + > . F Q ((V + + 1 ) V M + ) } ' 'o V x ) d F x ( x ) • ^ - To F x < * > d F o ( x ) Now t h e r e s t of the proof of Theorem 4 . 1 i s d i v i d e d i n t o two p a r t s . 1. P r o p e r t i e s of a symmetric s t a b l e p r o c e s s . s t a b l e process {x + W ( t ) , t > 0 , x £ t R , Q> our l a t e r use. We deal w i t h t h e symmetric and prepare some of i t s p r o p e r t i e s For the proof o m i t t e d h e r e , see [ 1 0 ] . Sx = i n f a (t > 0: Tx = i n f {t > 0 : x + W(t) x + W(t) For for x , a £ 1R we set > a), < a} , a Tx = S X A T X Let g ( x . y ) be t h e Green f u n c t i o n of order w i t h absorbing b a r r i e r s at 0 and 1. x > 0 o f the symmetric s t a b l e process We thus have x T E{ / 0 It i s known t h a t e " x t f ( x + w(t))dt> = J 1 g . ( x , y ) f ( y ) d y , 0 A g ( x , y ) i s symmetric i n notation: x and y.. ^X$X p"(x) = E{e A o < x < 1. We s t i l l need t h e -xTx ; S^ < TJ} , p~(x) = E{e P x (x) = P+{x) + p" x {x) = E { e _ A T X } — q + (x) = Zahvia/l)}'2 , — -1 X2 ( 1 - x ) 2 a a q ' ( x ) = 2 a _ 1 { r ( a / 2 ) } ' 2 (1 - x ) 7 x 7 u ; VQ < S^l following 206 q* (x) = q*(x) - \ /J g x (x,y)q ± (y)dy 1 £ -1 £ r(x) = x f -1 (i - %r * Notice that 9 x (x,y) * g x (l - x, 1 - y), P x (x) = p~(l - x ) , p x (x) = p x ( l - x ) , q+(x) = q"(l - x) , q*(x) = q~(l - x), We also use the notation: c(«) = r ( o + l)w" sin(oTT/2): = / Q P(x)q(x)dx . Lemma 4.3. Proof. a { r ( a / 2 ) } 2 = . The l e f t hand side equals 2
= 2
= the right hand side; the second equality is obtained by addir*g the preceding two terms and then dividing the sum by 2. Lemma 4.4 (Watanabe [ 1 0 ] ) . (1) For any f € C([0,1J) lim x " a / 2 r 1 g (x,y)f(y)dy = < f , q V x+0 0 * * lim x _ c t / 2 J1 gAl x+0 ) - x, y)f{y)dy = p*{x) = « - 1 c( 0 ) / J g x (x,y)(l -yf'dy, pTfx) = a _ 1 c( 0 ) j \ g.fx.yjy - 0 dy , pj(x) = 2l-aT(a){?(a/2)}'2 f ^ * (1 - y 2 ) 7 1 dy 207 Lemma 4 . 5 . We have (1) Hm p ! ( x ) x " o / Z = lim p"(l - x ) x _ a / 2 = K - x q + >, (ii) lim {1 - p 7 ( x ) } x _ a / 2 = lim {1 - p*(l - x ) } x _ o t / 2 = K + x q+> where K = a_1c(a) Proof. f1 q + (x)x" a dx = 2 r ( l - f )c( a ) / r ( a / 2 ) , c 0 We give the proof of ( i i ) . Since J 1 g.(x,y)dy = x _ 1 {l - p . ( x ) } n A A we have lim {1 - p : ( x ) } x " a / 2 = lim {1 - p , ( x ) > x " a / 2 + lim p * ( x ) x ~ a / 2 A A x+0 x+0 x+0 A A lim x _ a / 2 f1 g (x,y)dy + lim p*(x)x x+0 0 A x+0 K <X + a ' ^ C a ) . x " a , q*> = K + \ - X = K + X 2. Computation of Lemma 4 . 6 . For -Xb E(e" x ; b > 0>. \, v >0 - u(d -b ) E{e } = {K - + + -l , q >}{K + x
} . M
A In particular + E{e -xd E{e Proof. + } = K{K + x} _ 1 , + + + -l } = {K - x}{K + X> The proof is similar to that in [ 6 ] . For E>0 we set 208 T<°> = J{0) e = 0, T = T - Inf { t >T« n - X > : W(t) - W ^ " S<"> = S [ n ) = i m W " - 1 * 1 ') < - e> : W(t) - W(T { n - a ) ) > 1 - e>, n = 1,2, Then -Xb + - u (d + -b + ) E{e > {n) = lim y E , -xT {e e+0 n=l - , ( S ( n + 1 ) - T ( n ' ) T (k) Jk) MV ' ; P ' < Sv ' f for : < n s' n + 1 > < T ( n + 1 »} and 11. I <E{e"XTe40 n=l , „ „ 1 T° e < S?. e „ " E { e " p S ° - ; S° £ < T° £ , e i e l e e = lim I {P"x(€)>n - P X U). e+0 n=l I f we set Kx = K + x, K2 = K - x then 1 - p"x(e) - K l £ a / 2 , p+( e ) ~ Kgc^ 2 as e+ 0 . Therefore we have -xb + - M {d + -b_) E{e . } = lim J (1 - l ^ e * ) K e+0 n=l = K 2 /K : = fK - x}{K + x } _ 1 completing the proof of the lemma. For the proof of the theorem it is enough to prove the following lemma. Lemma 4.7. For x > 0 E { e " x b ; b>0> = {r(cc + l ) / r ( a / 2 ) } 2 / " e~ xt Q{d., > t } d t . 0 209 Proof. We f i r s t notice that -Xb^ Fx(x) = E{e -xT = E{ ; V+ < x - 1 , M+ < x } x l ~ , n -xb n ; T x °_ : < T°}E<e ,1-0 / w - / ,^."2 } = p*(x)E{e r-l+2x , , F0(x) = PQ(X) = 2A a r(a){r( a /2)} dF Q (x) = r { a ) { r ( a / 2 ) } _ 2 r ( x ) d x -xb + + } .2*7 " / , J T " (1 - y T dy, . We thus have E { e " A b ; b > 0} = / * (4.3) = E{e + Fx(x)dFQ(x) > J * p * ( x ) dF Q (x) = r ( a ) { r ( « / 2 ) ) ~ 2 E{e = 2_1r(a){r(a/2)}"2E{e + + } }
r> . On t h e o t h e r hand, using Lemma 4.3 we have f0°°e"xtQ{d+ > t } d t = x _ 1 { l - E(e + = o T ^ r ^ ) } " ' = (aK) ' { r ( a / 2 ) } ) } = q + >{K + x q + > > " !
{K + x
> E{e >
• Combining this with (4.3) we obtain Lemma 4.7. This completes also the proof of Theorem 4.1. Remark. The density <j, in Theorem 4.1 is symmetric and (n e"*%(x)dx = * t t ^ 4 . 0 {r(a/2)} + • * > °. K + x
291 210 References [ 1 ] T. Brox, A one-dimensional diffusion process in a Wiener medium, to appear in Ann. Probab. [ 2 ] A.O. Golosov, The l i m i t distributions for random walks in random environments, Soviet Math. Dokl., 28 (1983), 19-22. [ 3 ] K. Ito and H.P. McKean, Diffusion Processes and Their Sample Paths, Springer-Verlag, B e r l i n , 1965. [ 4 ] K. Kawazu, Y. Tamura and H. Tanaka, One-dimensional diffusions and random walks in random media, in preparation. [ 5 ] H. Kesten, M.V. Kozlov and F. Spitzer, A l i m i t law for random walk in a random environment, Compositio Math., 30 (1975), 145-168. [ 6 ] H. Kesten, The l i m i t d i s t r i b u t i o n of Sinai's random walk in random environment, to appear in Physica. [ 7 ] S. Schumacher, Dtffusions with random c o e f f i c i e n t s , Contemporary Mathematics (Particle Systems, Random Media and Large Deviations, ed. by R. Durrett), 41 (1985), 351-356. [ 8 ] Y.G. Sinai, The l i m i t i n g behavior of a one-dimensional random walk in a random medium, Theory of Probab. and i t s Appl., 27 (1982), 256-268. [ 9 ] H. Tanaka, Limit d i s t r i b u t i o n for 1-dimensional diffusion in a reflected Brownian medium, preprint, 1986. [10] S. Watanabe, On stable processes with boundary conditions, J. Math. Soc. Japan, 14(1962), 170-198. 292 J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 34 (1987), 351-369. Stochastic spatially differential equation homogeneous Maxwellian corresponding Boltzmann and non-cutoff to the equation of type Dedicated to Professor Seiz6 Ito on his 60th birthday By Hiroshi T A N A K A Introduction It is known that a Markov process can be associated with a certain nonlinear equation of Boltzmann type ([2][3][4][5][7]). In the case of the spatially homogeneous Boltzmann equation of Maxwellian molecules, the associated Markov process was constructed by solving certain stochastic differential equation (abbreviated: SDE) based on a Poisson random measure ([7], see also [5][6]). The purposes of this paper are to simplify the proof of existence of solutions of the SDE of [7] by modifying the form of the SDE and also to give some remarks concerning the uniqueness of solutions. We consider the Boltzmann equation of Maxwellian molecules: (1) d ™-=[ {u'u[-uu1)Q(6)dededxl, t>0, xeR\ where u=u(t, x), u x =u(£, a^), u ' = ^(£, a'). u[=u{t,x[) and de = de/27r. Q(0), O<0 (non-cut- off) but (I) [*dQ(d)dd However, the special form of Q(6) is not important in our methods and hence in this paper we assume that Q{6) is an arbitrary nonnegative function satisfying only the condition (I) or even the following weaker one: 293 352 Hiroshi TANAKA [*02Q[d)dd < o o . (II) The two cases (I) and (II) are discussed separately. A molecule with velocity x collides with a similar test molecule with velocity xx; the post-collision velocities are denoted by x' and x[t respectively. If S[x,Xi) denotes the 2-dimensional sphere with center (x+Xi)j2 and radius \x—Xi\j2t then x' and x{ are always on S(x,%i), or move precisely, S(x', xi) =S(a, a^). Taking a spherical coordinate system on S(x,Xi) with north pole x, denote by 6 (resp. e) the colatitude (resp. longitude) of x'. Then xf and x[ can be regarded as functions of x, xu 0 and e. We set a{x, xlt0, €)=x,—x. A probability measure valued function u(t), £>0, is called a weak solution of (1) if where (K { J(0,lt)X«>,2!E) In [7] the following SDE was considered in connection with the Boltzmann equation (1) under the assumption (I); (2) X(t,a>)=X(0,a>) + f a(X(8-, ft)), F ( s - , a), 6, e)N(dsdddeda). Here, iV(-) is a Poisson random measure on (0, oo) x (0, K) X (0, 2n) x (0,1) with intensity measure dsQ{0)deda, and the solution process {X{t, a>), t>0] is to be found on a basic probability space {Q, P} under the condition that the process {Y{t, a), t>0], defined on the probability space {(0,1), da} and describing the motion of a test molecule, is equivalent in law to {X(t, a>), t>0}. The relation between the Boltzmann equation (1) and the SDE (2) is that the probability distribution of X(t) is a weak solution of (1) (general theory of SDE's including jump parts goes back to K. Ito mi. The modification we are making for the SDE (2) in proving existence theorem is as follows: 294 The spatially homogeneous Boltzmann equation 353 ( i ) N( •) is replaced by a Poisson random measure (again denoted by N(-)) on (0, oo) x (0, x) XS 2 xQ x with intensity measure dsQ{6)dddodPlt where {Qu Pi} is a copy of the basic probability space {Q, P] and do is the uniform probability distribution on the 2-dimensional unit sphere S2. (ii) Y{s — ,a) is replaced by X{s —, wj. (iii) a($,xlfd,e) is replaced by b(x,xlf0,a) (the definition is given in §1). Thus in the case (I) the modified SDE can be written as (3 ) X(t, (o) =X{0, a>) + [ b(X{s-, a>), X{s-, a>J, 6, ^NidsdOdad^), where St = (0, t] x (0, TT)XS2X QI (in the case (II) the modified SDE is given by (3.3) in §3). Advantage of the modified SDE (3) is that the new coefficient b[x, xu 6, a) is Lipschitz continuous in (x, xj as an Ll(cio)-valued function for each fixed 6 (see Lemma 1) and that the process describing the motion of a test molecule is exactly a copy of the solution process X{t, a)), and in fact, by virtue of these, (3) can be solved easily by using a routine iteration method. The proof of pathwise uniqueness for (3) is also easy. Most of the discussions on the uniqueness in the law sense are essentially the same as the proof of Theorem 4.1 of [7] but they are somewhat simplified. In formulating the uniqueness in the law sense we further modify the SDE (3) as follows: (V) {Qt, Pi} is replaced by a probability space {D, P} which need not be a copy of {Q, P). (ii') X(s —, a),,) is replaced by X(s, a>) which is an arbitrary measurable process defined on {Q, P) such that it has the same distribution as the solution X{s, ft)) for each s. The uniqueness in the law sense is proved for this modified SDE so (in the case (I)) the solution process has the same law as the solution process of (3) (and also (2)). Similar discussions in the case (II) are also given. § 1. LVLipschitz continuity of b(x, xu 0, a) Think of S(x, xt) as a celestial globe with north pole x and let C{x, xu 6) denote the circle on S(x, a?x) with constant colatitude 6. Given 295 354 Hiroshi TANAKA oeS\ let s($ixu<j)=2rl\%—xl\o+2-x(xJrx^, let M(xtxu the b9(x, xu 6, a)=C(x, xlt 6) f\ M{x, xl9 6) = a point on S{x, xj, b{x, xu 6, o) =b0[x, xu 6, o)~x. Then for fixed x,x1eR\x^=x1 and 0€(O,JT), b0(xt xlt 6, a) is uniformly distributed on C(x, xx, 6) as a random variable defined on the probability space {S2, da}. When x=xlt we set b{x,x1,0,a)=0. LEMMA 1. (1.1) For any x, xlt y, yx € R* and 6 € (0, it), [ , \b(x, xu 6, o)—b(y, ylt 6, G)\(Lo Js 2 where const, is independent of x, xu y, yx and 6. PROOF. (i) First we consider a special case. Special case: S(x, xx) =S{y, yi) =S 2 . In this case the integral on the left in (1.1) depends only on 8 and the angle f ( O ^ I ^ T T ) between x and y. Therefore, it is enough to consider the case (1.2) z = ( 0 , 0 , l ) , #=(0, sin£, cosf), xx=-x, yl=—yt and prove that the integral on the left in (1.1) is dominated by const. 0£. Let A be the rotation in Rs around the a^-axis by the angle £. Then in the case (1.2) we have (1.3) b[y, ylt 6, a) =A~xb{x, xlt 6, Aa). A point aeS2 is expressed as a = {r,Vl~rzcos(p,Vl—rzsm 296 The spatially homogeneous Boltzmann equation Then b{x, xu 6, G) = (sin 6 cos a, sin 6 sin a, cos 0 — 1) b [x, xx, 0, Aa) = (sin 0 cos a, sin 0 sin a, cos 0—1), and hence from (1.3) b {y, ylt 0, a) = (sin 0 cos a, sin 0 sin a cos £ + (cos 0 — 1) sin £, —sin 0 sin 5 sin £ -f- (cos 0—1) cos £). Therefore (1.4) 6(3, xu 6f a)—b(yt ylt d, a) — (sin 0 (cos a—cos a), sin 6 (sin a—sin a cos £) + (1 —cos 0) sin £, sin 0 sin a sin f— (1—cos 0) (1 —cos £)), and hence f \b(x, xu 0, ~^—\ \ \HX> %\,0,o)~b(y,yi,d,o)\drd const.0£ 2ff Jo Jo follows from (1.4) once we prove the following estimates. (1.5) I I |cosa—cos #|drcfy> <; const. £. (1.6) I I |sinor — sinacos£|drc^<;const.£. The proof of (1.5) is as follows. fi(rt Setting /(r, I |cosor—cosa\drd because \ft{rr,
y)Wd}~ 8/2 |cos^sin^|drd^ =4^ [V(l-^){r*+(l-rV}~w^da! < 4 l I rx(ra;)"8/2drda; (use r"+(l—r*)af^ra;). 355 297 356 Hiroshi TANAKA As for (1.6), it is enough to prove i ran \sina—sin dt\drd o Jo and for this it is also enough to prove that (1.7) ft \gt(r, where g(r, /^rpvi^pf^+fi-^cos2^}-^^^^^^ <4f 1 f 1 {r 2 +(l-r 2 )a; 2 }- I / 2 dr^ r2+{l-r*)x2>rx). J^rfjl-^^^+tl-^cos^J-^cos^lsin^ldr^ < 4 P f 1 {r 2 +(l-r 2 )x 2 }- 3/ Vdrda; jc-x^ l»-aJi| _ y~y1 |0-Vi| we have [ 216(», »1( 0, o)-b(yt yu d, o)\ + -eudta)-b(e2, -e2,d,tr)\d<j |i^_Jl^|.j s 2 | 6 ( e 2 ,^,, f f ) 1 ^ + ^-^l-^-^l k where we have used the result of case (i). Now (1.1) follows from the 298 The spatially homogeneous Boltzmann equation 357 following trivial inequalities. |&—3i| — | y — # i | \x-v\ + \3h—Vi\. (x-x1)-\^~^(y~yl) ^2(1^-1/1 + 1^-^1 \V-Vi\ § 2. Stochastic differential equation—I In this section we assume that Q(0) satisfies the condition (I). 2.1. Existence theorem We assume that a basic probability space {Q, 3, P], equipped with a filtration {EFJ of increasing sub-<;-fields of 9% satisfies the following conditions. (2.1) The tr-field ffo contains all P-negligible sets and is rich enough in the sense that, for any probability distribution (i in i?3, there exists an EFo-measurable Unvalued random variable with distribution j«. (2.2) There exists an Svadapted Poisson random measure N(-) on (0, co)x(0, Trjx&xQi with intensity measure dsQ{6)d$SiadP1 where [Qlt P J is a copy of {Q, P}. A Poisson random measure JV(-) on (0, co)x(0,7t)xS 2 xQ 1 is said to be Si-adapted if, for each t>0, EFScS-t and 3t is indepedent of 2*1, where 3:* (resp. 3L) is the smallest c-field on Q with respect to which the random variables N{A), AeJll (resp. A&JLI,), are measurable; here The SDE we are going to discuss is the following (=(3)): (2.3) X{t, a>) =X(Q, <&) + f b{X(s-, to), X{8-f o)l), 6, a)N(dsdddad(o1). By a solution of (2.3) we mean an EF(-adapted process X(t, a>)> t>0, which is right continuous and has left limits for almost all to. X(t,to), t>0, 299 358 Hiroshi TANAKA T ! E\X{t,co)\dt 0 THEOREM 1. Let the condition (I) be satisfied and let X{0, co) be a given 3'^measurable random variable with E\X(§, PROOF. We successively by (2.4) set X0{t, ft>)=X(0, a>), t^O, Xn(t, oi)=X(Of w) + f biX^is-, and define Xn(t,a>), n^l, o))t Xn_x(s-, ft>J, 6, o)N[dsdedada>1). The stochastic integral is well-defined for each n by virtue of the estimate \b(x,xltd, a)\<\x—£i|0/2. By Lemma 1 we have E{sup\Xn+1(s)-Xn(s)\) < ^ ( j s |&(X.(s- t a>) f X.(s- ( ©J,*?, a) -biXn-tfa-, a>), X n _ ! ( s - , coj, dt a)\N{dsdddad(o^ < c o n s t . S [ j s {\Xn{s-, w)-Xn_1(s~, a>)\ + \Xn(s-,a>1)-Xn^s-,1)] < const. 1 E\Xn{s, a>) —X n _i(s, o>)\ds, and hence E E\ sup |xn+1(s)-xM(s)|)<; f; cfc'i)' with some constants c and c'. Therefore X(t,(o)=limXn(t,Q>) exists as a uniform convergence on each finite ^-interval (a.s.); clearly X{t, (o) is an integrable solution of (2.3). To prove the uniqueness, let X(t,(o) and Y(t,a)) be any integrable solutions of (2.3). Then we have E\X(t)-Y(t)\< const. [*E\X{8)-Y{a)\d8 and hence X{t)^Y{t), t>0, a.s. 300 The spatially homogeneous Boltzmann 2.2. equation 359 Uniqueness in the law sense The uniqueness in Theorem 1 asserts that there is only one solution of (2.3) so far as the basic probability space, the initial value and the Poisson random measure are fixed. Different choices of the basic probability space etc. yield different solutions, but we can prove that their probability laws in the path space are the same provided that their initial distributions are the same. We prove this uniqueness in the law sense for a slightly modified SDE (Theorem 2). Let {Q, 3 \ P} be a probability space with a filtration {EFJ satisfying (2.1) and (2.2) as before. But now we replace [Qlt P J by {Q, P) which need not be a copy of [Q, P}. A process Y{t, a>) defined on [Q, P} is said to be integrable if it is jointly measurable and if Tf \Y(t,a>)\dtdP 0 Let W denote the space of i?8-valued right continuous paths with left limits. Given an integrable process Y(t, a>) defined on {Q, P], we consider the SDE (2.5) X(t, a>)=X{0, ), Y(s, ©), 0, a)N[dsdddadd>) where St = (0, t] X (0, ic) X S2 X Q. PROPOSITION 1. Let the condition (I) be satisfied and let X(0, co) be a given 9'^-measurable random variable with E\X{§, a>)|) there exists an integrable solution of (2.5). Also the law uniqueness holds in the following sense: The probability measure on W induced by a solution of (2.5) is uniquely determined by uQ and u{t), t>0, where u0 is the probability distribution of X(Q, a>) and u(t) is that of Y(-,a>) at time t. The existence of a solution is proved by a routine iteration method as in the proof of Theorem 1. The law uniqueness is proved as follows. First we choose a sequence {hn(t),t>0} of step functions such that PROOF. (2.6) each hn(t) is expressed as h m=(° nl } \tnk for for i = 0 tnk (fc=0,1, • • •), 301 360 Hiroshi TANAKA where {tnk} satisfies 0 = £ B0 <^i<-*', limiBt = oo, limsup(£ni+1-£nfc) = 0; fc-KO (2.7) »-•«> k lim f* f \Y(s,a)-Y(hn(s),Q>)\dsdP=Q, 0 Let X(t) be the solution of (2.5) and let Xn(t) be the solution of (2.8) Xn{t)=X(0) + [ b{Xn{K(s)), Y{hn{s),&),d,o)N(dsd0d<jdo>). Jst Then Xn{t) is obtained as follows: (2.9) Xn(t) =Xn(U + [ b(Xn(tnk), Y{tnk, <5), 0, o)dN, (fc^O). Making use of the estimate (2.10) \b(xtx1)dya)\<\x-x1\6j2t and also (2.7), we can easily prove that (2.11) E\Xn(s)\^ const., (2.12) snpiElX^-XMl-.OKt^t^t, 0<s where const, may depend on t but not on n. Then, making use of Lemma 1 and then (2.7), (2.12), we have E\Xn(t)-X{t}\< const. [E\Xn(hn{s))-X(s)\ds + const. f'f |r(/i n (s))-r(s)|dsdP < const. [E\Xn{s)-X{s)\ds+o(l), and hence by Gronwall's inequality (2.13) \imE\Xn{t)-X(t)\=Q. W-K30 On the other hand, by (2.9) we have for any geK* and tnk 302 The spatially homogeneous Boltzmann equation {ev^-b^Y{t^^8^~l)Q(6)ddd,adp] = expl L J(0,K)XS XJ3 I = exp(V-Tf-x+(*-u[ L 361 ,a J J }ev^-K'^°>-l)Q(d)dddou(t„t,dy)] J(o,*)xs xjiH ) J where we put x=Xn[tnk). This conditional expectation formula implies that the probability measure on W induced by the process Xn(t) is uniquely determined by u0 and u(t), £!>0. Therefore, by (2.13) the probability measure on W induced by X(t) is also uniquely determined by u0 and u(t), t>0. This completes the proof of the proposition. Now we consider the following SDE for which we are going to prove the law uniqueness: (2.14a) X{t)=X{0) + [ b(X(s-)f X(s, &), 6, <j)N{dsdddadcb). Here, an 9^-adapted integrable solution X{t) is found under the condition that (2.14b) X(t,6>) is a measurable process denned on the probability space {Q, P) such that X(t, <w) has the same distribution as X(t) for each t. REMARK 2. When {Q, P} = {Q1,P1}, a solution of (2.3) is also a solution of (2.14). THEOREM 2. Let the condition (I) be satisfied and let X(0, a>) be any Revalued and ^^-measurable random variable with E\X{0, ft>)|). Let Q=[0,1], EF^the afield of Borel subsets of [0,1], P(A) = the Lebesgue measure of A (€ 9 ) and let {Qu Px} be a copy of {Q, P). As in 2.1 we construct, on the probability space {Q, P], a ^-distributed random variable Jt and a Poisson random measure N(-) on (0, oo) x (0, it) X S2xQi with intensity measure dtQ(0)dddadPx so that £ and N(-) are independent. We then consider the SDE of the type (2.3) PROOF. (2.15) £(*)=.£+f b(t(s~)1Jt(s-,co1),e,a)dN, 303 362 Hiroshi TANAKA where St = (0, t]x (0, ^)xS z xQ x . We are going to prove that for any solution X{t) of (2.14) there exists a solution of (2.15) which (as a process) is equivalent in law to X(t). Once this has been proved, the law uniqueness of solutions of (2.14) follows immediately from the pathwise uniqueness of solutions of (2.15). On the probability space {Qlr P J we can find an ^-valued right continuous process X(£, a>i) having left limits which is equivalent in law to a solution X(t) of (2.14). Given such a process 3LQ(t, co,), we consider the SDE (2.16) X(t) =%+ [A b(X(s~), XQ(s-, co,), 0, o)dN. Since the both (test) processes X(t,co) and Jt0{t —, a>0 in (2.14) and (2.16) have the same marginal distribution at each time t, Proposition 1 implies that the unique solution Jt^t) of (2.16) is equivalent in law to a solution process X(t) of (2.14). Next we construct £,(*) for n>2 by .£.(*)= the solution of (2.16) with X(s—, a>0 replaced by XM-i(s—, o>i). Then as in the proof of Theorem 1 we can prove that Jtn(t) converges to a solution X{t) of (2.15) as n->oo. Since each process £%{t) is equivalent in law to X(t), so is £{t). This completes the proof of Theorem 2. § 3. Stochastic differential equation—II In this section we assume (3.1) [* 6Q{0)d0 = ao, [%d2Q(6)d0 Let [Q, P], {Qu P J and N(-) be the same as in 2.1 and set M{A)=N(A)~ 2(A) for a measurable subset A of (0, oo) x (0, K) XS 2 xQ x with Z{A)~ \ dtQ{0)dOdadPl Then the stochastic integral on the right of (2.3) can be written as b{X(s-, (o), X(s~, co,), dy o)dM+ [ b(X(s-, w), X(s-, o)x),6, a)dX. The first integral in the above will make sense under the condition (3.1) while the second integral equals -c[t{X(s,co)~X(s^co)}ds 304 The spatially homogeneous Boltzmann equation 363 where X{s, w)=E{X{s, co)} and c=\* 2~1(1-cos 0)Q(0)d0. So we are led to the following SDE: (3.2) X(t, co)=X{0, co) + [ b(X(s-, co), X{s~, cox),6, a)dM -c ['{X{8, co) -X(s,co)}ds. If I \b{x,xlt8,a)—b(y,yi16,(x)\ida were dominated by a constant JS2 multiple of {|a;—2/|2+|a?i—Vil2}^2» we could solve (3.2) easily. But this is not likely to be true. So we further modify the SDE (3.2) so that it can be solved in an easier way. First we introduce the predictable <7-field $ on [0, oo) x Q X Qi; it is defined as the smallest <7-field on [0, oo) x Q X Qi with respect to which all functions a(t, co, co:) satisfying the following conditions (i) and (ii) are measurable. (i) For each fixed il>0, a(t, co, co^ is 2 r ( ®2 r -measurable where {Qlt 3, Px] is a copy of {Q, 2 \ P}. (ii) For fixed co and colf a{t,co,w1) is left continuous in t. Let 31 denote the class of predictable processes (i.e., immeasurable functions on [0, co)x£?X&i) with values in the space 0(3) of orthogonal matrices of degree 3. Then our modified SDE can be written as (3.3) X{t, co)=X(0,fl>)+ ( b{X{s~, o>), X{s-, co,), 0, R(s, co, co,)a)dM -c[t{X{s,w)-X(s,a))}ds. By a solution of (3.3) we mean an S^-adapted process X{t, co), t>0, which is right continuous in t, has left limits for almost all co and satisfies (3.3) with some R=R(t, co, a>0 € 31. X{t, co) is said to be square integrable if [TE{\X{t, co)\2}dt If we N(A) = f set Js T O M{A) = lA(s, d, R(s, ft>, co^NidsdOdada),), R(A)-Z[A), then N{-) is also an Svadapted Poisson random measure with the same 305 364 Hiroshi TANAKA intensity measure I and (3.3) becomes (3.2) with M replaced by M. In this sense (3.2) and (3.3) may be regarded as equivalent. The corresponding martingale problems are the same. For a, o' € S2 we denote by R(a, a') the rotation (orthogonal matrix) in /?3 which sends a to a' along the geodesic connecting a with af, and (y-yij\y—yx\~1) set R(x,xl,y,yl)=R{{x-xl)\x-x1\'\ = t h e identity matrix (otherwise). (see Lemma 3.1 of [7]). LEMMA 2. For any (for x^x^yj-yj, Then we have the following lemma x, xlt y, 2/i€ i?3 \b{x, xlf 0, ff)—b(y, yu 0, R(x, xlf y, y,)o)\< const. {\x-y\ + K-2/i|}0. THEOREM 3. Let the condition (II) be satisfied and let X{0, co) be a given 9\-measurable random variable with E{\X(0,a))\2} Define Xn(t,co), n>0, by XQ(t,co)=X(0,co), Xn(t, co) =X(0, co) + f HX^s-, <w), X^s-, co,), 6, Rn_,a)dM — c\ {Xn^(s, ft>)-Xn_!(s, o>)}dst where Rn=Rn(s,co,(o1) = J[R(Xk_1(s~,(o), X^s n>l, — ,^,), Xk{s-,co), X*(s —,ft>J). Then, making use of Lemma 2 and the convergence of the second integral of (3.1) we have E{mp\Xn+l(s)-Xn(s)A < 8 # [ j s \b(Xn(s-, ft>), Xn(s-, co,), 6, Rno) -6(X n _ x (s-, co), Xn_,(s-, co,), 6, R.-!a)\*dx} + 2cH^E{\Xn(s, co)~XJ^)~Xn_1(s, < const. (l + t)[E{\Xn(s)-XnMs)\2}ds, co)+Xn„1(s, co)\*}ds 306 The spatially homogeneous Boltzmann equation 365 and hence by a routine argument we can prove that (3.4) E snp\Xn{s,co)-Xn_1(s,o>)\ is convergent for all t>0 with probability 1. We denote by Q the set of 0 and also by Qt (c£?0 the copy of Q. We set (limX(£, a;), o)^0 { 0 , otherwise, r = {(t, a>, a)J G [0,cx>)xQxQi : X(t-, a>)*X(t~, ft>0}Then r^S*. We first claim that Rn(t, o>, o>0 is convergent as w->oo for each fixed (t,a),?}— Rn-l{t, OJ, 0)011 = ||{JB[X..1(*-, o>), X ^ i - , OH), X„(£-, a>), X „ ( i - , © J ) - ! } ! * , . . ^ ,fl>,©OH < v /y X , . . ^ —, o))-X w _ 1 (t- > o)Q _ \Xn_1(t-,a>)-Xn_1(t-,a>1)\ Xn(t-,oj)-Xn(t~,w1) \Xn(t-,a>)-Xn(t-,wl)\ < 2VT{|X w _ 1 (t- > o))-X w (t- > o))+X w - 1 (t-,o>0-X w (^- > o)01} |X M (i-,o>)-X f t (i-,o)0| If (£, o), o>0 G A then |X„(£ —, o>)— Xn{t —, o>i)|^e for some e>0, and for all sufficiently large n (say, for n>n9) and hence CO £ < 2 ||-R„(t, o>, ^0~i2 B _i(t, OJ, o>0|| ^ 1 f; {|X- B («- f a))-X.. 1 (t- I fl))| + |X.(i-,a> 1 )-X._ 1 (t- f fi> 1 )|}«x>. Next we define R € 51 by (lim i2w(i, o), o)0, (t, a), o)0 G r R(tf co, 0)^ = 1— {identity, otherwise, and claim that {X(t,a>),R} is a solution of (3.3). For this it is enough to prove that (3.5) E^s{b(Xn(s-,w),Xn(s-tw1)tdfRno) - 6 ( X ( s - , o)), X ( s - , o)0,0, ifcr)}di!f H 307 366 Hiroshi TANAKA tends to 0 as w - x » . If Rn = R{Xn{s~, then (3.5) is dominated by 2 £ { j g \b(Xn(s-, a>), Xm{8-, a>), Xn{s-, a>) -X(s~, ©J), a>), X(s-, wj, 6, RnRn<j)\*dx} a>J, 6, RnRna) -b(X(s-, < const, f E{\Xn(s)-X(s)\2}ds w), X{s~, w,), 0, Rna) -b(X(s~, + %E{\S \b(X(s~, ©J, X(s-, + 2El[ to), X(s~, m), 6t bn(s, a>, a>x, 6, Ra)\2dx\ a)di\, where 6,(8, o), o)u 6, a) = \b(X(B-t (o), X(s~, ©J, 6, R'%o) ~b(X(s~, a>), X{8-, o),), 6, o)\\ Ri = RnRnR~\ Moreover, we can easily prove t h a t R£{s, a>, O J J - * / as w->co for each fixed (s, a), (Oi) 6 r, and hence for each fixed (s, co, a)=0 n-*oo for almost all a with respect to da. Since we also have t h e bound bn(s,co, a>lt 6, v)<\X{s —, to) — X{s —, coJ^-O2, an application of Lebesgue's dominated convergence theorem yields lim El I bn(s, co, coi, which implies t h a t (3.5) tends to 0 as n-*co. Thus t h e existence proof is finished. To prove t h e law uniqueness let hn(t)=k2~* for &2~*<J^(fc-f 1)2 - " and h(0)=0. L e t X(t,co) be any square integrable solution of (3.3) with auxiliary process R=R{t,o),coi)e3l and consider t h e SDE (3.6) Yn(t,a>)=X(0, <*>) + [ b(Yn(hn(s),a>), Y.(M*) f a>J,0, R*v)dM Jst -cj]{y.(*.(8))-y^w)}d8f where Rn—Rn(s, co, a)i) =R(X(s — ,co),X(s-,wi)t Yn(hn(s),a>), Yn(hH(s), (o^Rfe, a>, o j . 308 The spatially homogeneous Boltzmann equation 367 (3.6) can be solved easily; in fact, once we know Yn(t,(o) for 0<1t<1k2~*, we can define Yn(t,a>) for k2-n const. \t-s\, 0^s and also, by making use of Lemma 2, that E{\ Y«(t) -X(t}\2}< const. [ # { | Yn{s)-X(s)\2}ds+const. An(t), where A%(t)~ sup{E(|Y n (u)-Y n {s)| 2 ) const, being independent of n. (3.7) : 0 < s < w < £ , ^ - s < 2 " n } < const.2"W, Therefore E{\Yn(t)-X(t)\2} •<), w->oo. On the other hand, let k2-n) +\ (eix-b~l-ix-b)d* JSt-Sk2-n -ic(t-k2-n)x-{Yn(k2-*,Q))-Yn{k2-n,a>)} (where b=b(Yn(k2~\ a>), Y%{k2-*t <*),$, a)). Then, for k2~n a.s., and hence the probability measure on W induced by the process Yn{tt w), t>0, is uniquely determined by u0. This combined with (3.7) proves the law uniqueness of square integrable solutions of (3.3). In the rest of this section let {Q, P] and {Q, P] be the same as in 2.2 and consider the SDE (3.8) X{t)=X{0) + [ b{X(8-), Y(s,a>),0,R{s,a>,d>)<j)dM ~c[{X{s)-Y&tij}ds where Y(t, <&) is a given square integrable process defined on {Q, P] and R=R(t,co,d>) is similar to one in (3.3). PROPOSITION 2. Let the condition (II) be satisfied and X(0, w) be a given 3'0-measurable random variable with E{\X(0, a))]2} 309 368 Hiroshi TANAKA any given square integrahle process Y(t, d>) there exists a square integrahle solution of (3.8). Also the law uniqueness holds in the same sense as in Proposition 1. The above proposition can be proved by a method similar to Proposition 1. Only point one has to be careful is that the SDE (2.8) is now replaced by Xn(t)=X{0) + [ b(Xn(hn(s))f Y(hn(s),d>),e,Rna)dM -c^{Xn(hn(s))-Y{hn(s))}ds, where Rn—Rn(s, o),w) = R(X(s-,a>), Y(8,a>)tXn(hm(8),a>), Y(h%(8),w))R{s,w,&). Next we consider the SDE b(X(s~),X(s,S)^,R{s,co,w) (3.9) X(t)=X(Q)+[ Jo Jst where X{t,w) satisfies the same conditions as stated in (2.14b). Then the following theorem can be proved in the same spirit as in Theorem 2. THEOREM 4. Under the condition (II) the probability measure on W induced by any square integrahle solution of (3.9) is uniquely determined by the probability distribution u0 of X{0, a>). REMARK 4. The martingale problems corresponding to the SDE's (2), (3), (2.14), (3.2), (3.3) and (3.9) have the same form and the probability distribution u(t), at time t, of a solution to any one of these SDE's is a weak solution of the Boltzmann equation (1). References [ 1 ] Ito, K., On stochastic differential equations, Mem. Amer. Math. Soc. 4 (1951). [ 2 ] McKean, H. P., A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. USA 56 (1966), 1907-1911. [ 3 ] Murata, H., Propagation of chaos for Boltzmann-like equation of non-cutoff type in the plane, Hiroshima Math. J. 7 (1977), 479-515. [ 4 ] Sznitman, A. S., Equations de type de Boltzmann, spatialement homogenes, Z. Wahrsch. Verw. Gebiete 66 (1984), 559-592. 310 The spatially homogeneous Boltzmann equation 369 [ 5 ] Tanaka, H., On Markov process corresponding to Boltzmann's equation of Maxwellian gas, Proc. 2nd Japan-USSR Symp. on Probab. Theory, Lecture Notes in Math. vol. 330, Springer-Verlag, Berlin-Heidelberg-New York, 1973, 478-489. [ 6 ] Tanaka, H., Stochastic differential equation associated with the Boltzmann equation of Maxwellian molecules in several dimensions, Proc. of Conf. on Stochastic Analysis (edited by A. Friedman and M. Pinsky), Academic Press, New York-San FanciscoLondon, 1978, 301-314. [ 7 ] Tanaka, H., Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Gebiete 46 (1978), 67-105. (Received September 19, 1986) Department of Mathematics Faculty of Science and Technology Keio University Yokohama 223 Japan 311 LIMIT THEOREM FOR ONE-DIMENSIONAL DIFFUSION PROCESS IN BROWNIAN ENVIRONMENT Miroshi TANAKA Department of Mathematics. Faculty of Science and Technology Keio University, Yokohama, 223 (Japan) INTRODUCTION Let 1W be the space of continuous functions W: IR •*• IR with WiO) = 0. In this paper an element of W is called an environment. Given an environment W, Brox[l] considered a diffusion process starting at 0 with generator (1) U ' f = -L- e W(x), d ^W 2 dx te -W(x) d } 5bT' * Such a diffusion, denoted by X(t, W) , is constructed from a onedimensional Brownian motion B(t) by a scale-change and a timechange. The probability measure governing B(t) is denoted by P. We consider the Wiener measure Q on W. Thus {W(x), x £ 0, Q> and {W(-x), x £ 0, Q} are independent one-dimensional Brownian motions starting at 0. We assume that B(t) and W(x) are independent so the full distribution governing X(t, W) is $> = P © Q. If W(") were smooth, X(t, W) would satisfy 1 I/rtW' (X(s) X(t) = a Brownian motion - —•*— z ) ds . J Q Although our W(-) is never smooth, the above remark will explain that X(t, W) is regarded as a diffusion analogue of Sinai's random walk in a random environment([11]). X(t, W) is called a diffusion process in a Brownian environment. The problem is to study the limiting behavior of X(t, W) as t •* °>. Broxfl] obtained the following result which is analogous to that of Sinaifll]: For any e > 0 P{(log t)"2X(t, W)6U £ (t, W) } , which is regarded as a W-random variable, converges to 1 in probability as t -*• » , where U (t, W) is the e-neighborhood of b t < w ) which is defined suitably in terms of "valleys" of the environment. The distribution of b (W) is independent of t so the full distri—2 bution of (log t) X(t, W) converges to that of bj(W) . Kesten[7] obtained the exact form of the limit distribution. Kesten's result was then extended to the case of symmetric stable environments([12]) . 312 157 The purpose of this paper is to elaborate Brox's result. We prove that, without scaling but only by centering, X(t,») has a limit distribution as t -»• °°. To state the result more precisely, put b(t, W) = (log t) bt(W) and let § be the probability measure on W such that (W(x), x £ 0, 0} and {W(-x), x £ 0, 5} are independent Bessel processes of index 3 starting at 0. Note that e—W G L 1 (IB) with ^-measure 1. Let fi be the space of continuous paths u>:t0, ») -W 1 •* R and, for each W with e £ L (R), denote by P w the probability measure on fi such that {to(t), t £ 0, P w ) is a diffusion process with generator (1) and with initial distribution yw(dx) = e - W < x ) d x / | Finally put y = fp 0(dW) stated as follows. and e-W(y»dy . P = JPwQ(dW) . Then our result is Theorem. The process iX(tff + t3 V) - "b(tQ3 WJ3 t £ 0> p} converges as *„-*•<*> to the stationary process {u(t)t t £ 03 P] in the sense of weak convergence of probability measures on ft. In particular the distribution of X(t3') - b(tt>) converges to p as t -*• ~. Similar results were also obtained by Golosov[2] for a reflecting random walk in random environment. Our method is on the extension line of Brox's and uses fine results on one-dimensional Brownian motion obtained by Levy[8], It6 and McKean(4] and others. SI. OUTLINE OF BROX'S METHOD 1.1. Let ft0 be the space of continuous paths to: [0, «>) -*- IR with to(0) - 0 and let P be the Wiener measure on ftQ. We write B(t) for w(t), the value of to at time t. Thus (B(t), t £ 0, P> is a Brownian motion starting at 0. For a fixed W £ W we set S (x) = r x « w ( y»dy , S~ (y) - the inverse function of S(x), A~ (t) - the inverse function of A(s). Then X(t, W) = s" (B(A~ (t))) (1) starting at 0. If we set is a diffusion process with generator (WX°)('J = W(- + xn) - W(x n ), then 313 158 X °(t, W) = x- + X(t, f»°) starting at is a diffusion process with generator (1) x„. Regarding W as a random element as well as process X(t,>) P © Q}- This full process is denoted by {U x 1W, J9 = defined on the product probability space X > 0 and (X(t,-)) {X(t, W ) , t Z 0, P} from the diffusion process For w, we have a W£W we define W^^W to distinguish it with a fixed by W A (x) = A xeR. The following scaling relation is important ( [1 ]) : fixed X > 0 and -1 W. W(A 2 x) , For any W6W {X(t, AW X ) , t > 0, P} = U ~ 2 X ( X 4 t , W) , t 2 0, P> (1.1) where = means the equality in distribution. 1.2. We give the definition of a valley. {W(x), a < x £ c} of (i) a < c , (ii) there exists (iii) b WGW. such that x £ (a, b) , W(c) > W(x) > W(b) for every x £ ( b , c) , b as above H_ = sup{W(y) - W(x) : a < x < y £ b } < W(c) - W(b) , H + E sup{W(x) - W(y) : b £ x < y < c } < W(a) - W(b) . b and also for simplicity, we write A = H (abbreviated to i.d.a.) and of the valley. (a, b, c) (a,b,c) see that if containing (a, b, c) 0 and positive constant), then and for any W in some exists a valley 0 (see[l]). (a, b, c) b = b*. (a, b, c) for if A < r < D, subset We denote by 0 A and a < 0 < c. (a', b', c') and satisfying of b{W) WMx) = with W A < r < D with the unique b(W) A It is easy to (r W both is a r > 0 Q-measure 1 there b for x i 0, for x < 0, and containing of such a valley we put t W(x) - min W :x,0] are used for (a,b,c). It is known that for any r = 1. W(x) - min W [0,x] D are valleys of A' < r <: D' W (c^W) To give another description of directed D = (W{a) - W(b)) A (W(C) - W(b)) When the letters is said to contain in- is called the inner vH_ notation they always mean the i.d.a. and the depth of valley A part if for every {W(x), a < x < c } . is the depth W in the above definition is particularly important and so, to stress ascent b 6 (a, c) Let of W(a) > W(x) > W(b) for the same The value stead of is called a valley W 314 159 d + = inf{x > 0 : W # ( x ) = 1} , d_ = sup{x < 0 : W # ( x ) = 1} , V = min W , [0,d + ] and define mined with V_ = min W , [d_,0] b + and b_ by W(b+ ) = V + (such Q-measure 1 ) . We also set M = max W , [0,b + ] are uniquely deter- M_ = max W. [b_,0] Then another description of b(W) is given as follows(see [7]) : if M + v (V + + 1) < M_ v (V_ + 1 ) , "b+ (1.2) b+ b(W) if Moreover, if we define a(W) M + v (v+ + 1 ) > M_v (v_ + 1). and c(W) by a(W) = the infimum of the set of a's (a < b(W)) such that W(a) > W(x) > W(b(W)) for every x £ (a, b(W)), sup{W(y) - W(x) : a < x < y < b(W)} < 1, c(W) = the supremun of the set of c's (c > b(W)) such that W(c) > W(x) > W(b(W)) for every x£(b(W), c) , sup{W(x) - W(y) : b(W) < x < y £ c} < 1. Then (a(W), b(W), c(W)) is the maximum valley of W containing and satisfying A(W) < 1 < D(W), and is called the standard valley W. Note that a(W), b(W), etc., are Borel functions on W. 0 of and, for X > 0, let T x x be the exit time from (a, c) for the diffusion process X (t, A W ) . The following lemma is due to Brox[l], 1.3. Lemma (1.3) Let 2 ([!]). (a, b , c) For any lim m f Pie be a valley of W 6 > 0 and \ a closed e interval I <=• (at c) i = 1 . 1.4. Brox([1]) employs a coupling technique. To explain it we had better adopt the path space representation of X(t, A W ) . So let Si = C([0, «) •*• R) and denote by P^ w the probability measure on fl induced by the diffusion process X(t, A W ) . Moreover, we use the following notation. For an arbitrary interval [a, c] and an environment W we denote by P„r , the probability measure on the path space n.La, c,j = C([0, «») -*• [a, c]) induced by the diffusion process on at [a, c] with (local) generator ^xw * with reflecting barriers a and c and with initial distribution 315 160 y• * /J "W/. M „ I= =e « - * W * xd )x. W[a,c] ( d x ) /e r -Aw(y> J ^ " ' a T h i s r e f l e c t i n g d i f f u s i o n is stationary since invariant measure. valley o f ^ W, P„ In p a r t i c u l a r , in case P*[a#c], (a, b , c) are "[a,cj is t h e standard abbreviated to P*, and ft, respectively. We now assume that w and ^fa,cl y„r , is its w [a , c j d and and 0(t) at time {w(t), t S 0} space (a, b, c) is the standard valley of stand for generic elements of and ft W. with values Let w(t) t, respectively, and consider two processes iCS(t) , t £ 0} and {ft x ft, P w > independent. ft where defined on the product probability B?w = P^ w © P Thus the two processes are Put R = inf (t £ 0 : w(t) = fi(t)} , T R = inf(t Z R : w(t) fc {a, c) } , T R = inf(t £ R : 6(t) $ (a, c)} . Notice that these are random variables defined on (to1 (t) , t £ 0) define a process u(t) {ft x ft, P }. If we by for 0 < t £ R , for t > R , o>» (t) . fo(t) then (1.4) {«(t), 0 £ t £ T R , P*l i {ui'(t), 0 £ t S T R , P*} . The following lemma is also due to Brox[l]; the equality is a consequence of Lemma 2 (1.4). ([1]). For Xr7 x (1. 5) P,lR W any < e = PW{R r„ such that A < r7 < r ? < D Xr9 < e < e and 2» * < < e rJ R < TR) •*• 1, X •+ » . Using Lemma 2 and the scaling relation (1.1), Brox([l]) obtained his main results: (1.6) For any e > 0 p{|X" 2 x(e X , W) - b ( W x ) | > E } -> o in probability with respect to Q as X •* °°. By the same argument as Brox's we can obtain a refinement of his result as will be discussed in the next subsection. 316 161 1.5. et We keep the notation of 1.4 and, in addition, we denote by (resp. §fc) the shift on fi (resp. ft) defined by (e.uH-) = = a(t + • ) ) . For X > 0, 7. denotes the 2 - 4 W ^ w ) (t) = X w(X t) , t £ 0. $ t denotes the u(t + .) (resp. ( e J K ' l map: U -+ fi defined by o-field on o, generated by the sets of the form 0 £ s < t, x G R , and J6=\/tf3t- For is u £ Ji and {to: to(s) < x } , x£IR, to - x o>(t) - x ; fl - x (for denotes Q £ Q) the path whose value at time t also denotes a similar path. The following notational convention is used: <1"7B> P W[a,cJ - P W[a,c]' d.7b) p£{fi£r} = pjirnfi}/ T * $> re£. Note that the right hand sides of the above make sense since both Q o,. . and la, c j are measurable subsets of ft. For any family {r(X)} such that r(x) + 1 (X -*• «•) Lemma 2 implies (1.8) as e A <W) E l - 1P*{R < s(x) < s(X) + t(x) < T R } - 0 x •+ °° tox any Wtii1? and the same is true with T replaced by T_, where s(X) = X _ 4 e X , (1.9) t(x) = x " 4 e X r ( X ) . We are now in position to state Refinement of Tx&J3u(x), Brox's u(\) result. = eXr(X), For X > 03 i P*{fi - h(W)^y-K1(Yx)}A) where b(-) is defined by (1.2) — random variable (1.22) \ex(W3 defined on TX)\ 1 zx(W) The convention + Zx(W3 and and for any rAJ , £,(*.» Y-.) is a suitable A satisfying . Since the scaling relation implies *) as above have A ( W3 Q) The proof is as follows. {r(\}} we (1.7b) is used. (1.1) 317 162 { ( e exPAM)(t)' S ' °' 3 V { ( e ^ (X) S u ) ( t ) ' * S P °' XW > A using the notation V e (1.9) expX u we h a v e x 2 b t W - X,GrX} x2b(w ,er -PAWX{VBUx = PXW{VS(X)*- A2b(W)er,> x} (since W^fl W) 1 = BP*{R < s ( X ) , 6 s ( x ) w - b ( W ) G Y x < r A ) , s ( X ) + t ( X ) < T R } + (by *{R < s ( A ) , 8 s U ) o ) - b(W) £ 1 Tx = P W { @ s(X) fi-b(W)€YJ1(rx)} = p*{d - M w j e T ^ d y } where e = (W, I\) E A A and + ex + E ° (1.4)) (by ( 1 . 8 ) f o r TR) + ex, e. = e. (W, T.) A A ° (1.8)) (rx) , sU)+tUXTR} (by e A are suitable random vari- A ables satisfying (1.11) . Before ending this section we state one more lemma. Lemma 3. Let (a3 pfXJj X > 0, (1.12) p(\) then for r - and for £ 0 any (1 131 bt c) Y and standard a' = o(eeX) p(\) X > 03 xe^Pa;> any be the as oil) condition c' with is uniform and (1.15) with Wh(*) Since for any \i , {i~6,-&) WD[a'-b,c'-b] 1 and (1.12) that X •+ °° for a < a ' 0 , } respect 6 > 0 P*{T > p(x)} * 1, to both 1 have x- ~ , A = f/C + b) tend to we { r , } + o(i), WD[a'~bJc'~bJ A (1.13) Proof. W (£1W). x » o . and W where of satisfying ph& - 2>er,} = Pxh (i.i4) valley satisfies as the - choice of r, under the W(b). yA{(b - 6, b - 6)} and x -*• «», it follows from Lemma PXb {T» > p(X)} •* 1 VTfa'-bjC'-b] (X * ->, 318 163 where T and T' are the exit times of (a*, c') and (a'-b, c'-b), respectively, for the processes under consideration. Moreover, we see that [a*, c1] , (1.16a) u^(dx) = K (1.16b) K(A) = | " e - X W ( x , d x / / V A W ( x , d x Therefore, we have as on + 1, X * • . X •*• °° = P^{fi - b £ rx, T" > p(A) } + o(l) = K(A)P A , [I\ n {T* (by (1.15)) > p(X) }] + o(l) = PK [I\ fl (T' > p(X) }] + o(l) VTIa'-b^'-b] A = ?\ (by 1.16a)') A W^ta'-bfC'-b] (I*,) + o(l) (by 1.16b)) (by (1.15)) , A V^ta'-bjc'-b] completing the proof of the lemma. §2. THE LAW OF THE STANDARD VALLEY. Recalling the notation of 1.2 we put W + = {W(b+ + t) - W(b + ), - b + £ t S d + - b + , Q} , W_ = {W(b_ - t) - W(b_), b_ < t £ - (d_ - b _ ) , Q} . On a suitable probability space such that (x+(t), t £ 0> and reflecting Brownian motions on RBM ) . Let {Jt+(t), t € R } be *-X *+«t) = i ™ 2T-J x where I = put 3+(t) t 2 0} is BES° (3)) . (2.1a) (2.1b) x we consider a process {x (t), t £ IR} (x+(-t), t £ 0} are independent [0, °°) starting at 0 (abbreviation: the local timeat 0 of x + (t), that [o, E ] ( x + ( E »» d s • [0, t] or [t, 0] according as t 2 0 or t < 0. Also = x+(t) + t+{t). Then by Pitman's theorem([9]) (S + (t), a Bessel process of index 3 starting at 0 (abbreviation: We put a+ = the smallest zero of x,(t) the t < 00 w i t h t h e maximum oof f t < T+ = min(t > 0 : 3 (t) = l) . in x,(t) (z, 0] = 1 where z is 319 164 Proposition. (2.2) W and W_ £ W+i Proof. W are {$+(t)> independent Q+ < t < The independence of equality in (2.2) are obvious. and T+} W_ . and W+ and the first law For the proof of the second law equal- ity it is convenient to use the construction of an equivalent of by means of the excursions of a RBM°([4][5][3]). space of w: [0, °°) -*• TR VJ* W the satisfying (i) w(t) > 0 (ii) w(0) = w(t) = 0 We consider a Denote by for 0 < t < c(w) = min{s > 0 : w(s) = 0} , t > i; (w) . for a-finite measure n tO' on defined by + n [{w(t 1 ) £ A l f w{t 2 ) £ A 2 ,..., w ( t n ) G A n } ] - I K + (t , x.JdXj / p (t 2 -t 1 ,x : , f x 2 )dx 2 / a A l ' " where 0 < t± JAA P A 2 °{Vtn-l'xn-l'xn,axn n < t2 < • • • < t P°(t,x,y) = ^ { A^^flR f I 2 K + (t, x) = l - ~ e" X / r irtJ 2 t e-^» 2 , / 2 t ), t > 0, -e-< x + 1 ^ 2 t Let p(t) be a stationary Poisson point process on teristic measure n and set x(t) X(t) where at w(t) ftf with charac- a(-) . by p{s) (t - a(s-)) ={o s = l(t). 0 {[3]). , ^ 1 _ c(p(s)) , 0<s£t £(t) = the inverse function of We define } x, y e R + - t > 0, a(t) = 3 Then For x(t) w Q. Vfr if a(s-) S t < a(s) if a(s-) = t = a(s) , is a RBM and we denote by and put £ = min{s > 0 : h(p(s)) > 1} . T = min{t. > 0 : p<£) £(t) h(w) is its local time the maximum value of 320 165 Note t h a t K < °° a . s . because n [{h(w) > 1}J = 1 < °°. W~ = ( W ( b + + t ) - W(b+), - b+ < t W+ = {W(b + + t ) - W(b + ) , 0 <, t £ d + - b + , Q} , W° = ( W ( t ) , 0 < t < 0 , Q) We a l s o put , S b + , Q} . Since <W*(t), t £ 0, Q} is a RBM° and -min{W(s):0<s£t} is its local time at 0, we see that the joint distribution of W and W* is the same as the joint distribution of the following processes (2.3) and (2.4): (2.3) (x(t) - M t ) , 0 £ t <, a(C-)} , (2.4) {pit) (t>, 0 S t S T} . On the other hand, {p(s), 0 £ s < £} and (p(£)(t), 0 £ t < x} are independent and hence the processes (2.3) and (2.4) are also independent. Therefore, W + is independent of W. and consequently of W + . Moreover, since (p(£)(t), 0 £ t £ x}is a part of a BES (3) ([13], see also [10]), it follows that w t - <e + (t), 0 £ t £ x+> . It remains to prove W~ = {3 + (t), a+ $ t $ 0} . (2.5) To prove this we must consider the time reversal of (2.3), or more precisely, of (x(t) - M t ) + Z, (2.6) 0 £ t £ aU-)} . Put VX\ = {w£Zl> + : h(w) < 1} , V$-\ = (w£&T + : h(w) Z 1} , and define two point processes pi(t) = p(t) where D p0 on U^Q and t£D , (the domain of definition of Dp Then for p„ and = (te (0, ~ ) : p ( t ) £ ^ } , p. and 2£K p.. by i = 0, 1 , p.) is given by i = 0, 1 . are independent stationary Poisson point processes with characteristic measures n.= the restriction 321 166 of n tf 6 UT+ to W-, i - 0, 1, r e s p e c t i v e l y . For w£^D we d e f i n e by |"w(£(w) - s) f o r 0 < s < r (w) w(s) = I (. 0 for s > c; (w) . Then the measure n_ is invariant under the map: w -*- w. for t <£ for t £ £ . We define It can -be proved that p Q is again a stationary Poisson point process on Wn with characteristic measure n- and is independent of p.. Therefore, the point process p defined by f S> 0 ( t) P(t) = | ( for t6D ° PT (t) is equivalent in law to for p. t£ D p l Therefore, if we set 0<s£t £(t) = the inverse function of and if we define p x(t) a(«) , by p(s)(t - a(s-)) if a(s-) £ t < a(s) x(t) = 0 if a(s-) = t = a(s) where s = %(t), then x(t) is a RBM and £). Moreover, it is seen that (drawing a picture is helpful) 5(t) = x(a(e-) - t) , 0 < t £ a(£-) , £(t) = M a U - ) ) - Ma(£-) - t) = 5 - i(a(£-) - t) , 0 < t < a(£-) . Therefore, the time reversal of (2.6) is (2.7) {x(a(£-J - t) - Ua<£-) - t) + £, ^ {x(t) + X(t), Since W~ 0 < t £ a(M) 0 < t S a(C-)} . is equivalent in law to {x(aU-) + t) - M a ( H + t) + £, (2.7) implies (2.5) as was to be proved. -a<£-) £ t < 0} , 322 167 §3. PROOF OF THE THEOREM We are going to prove the theorem announced in the introduction with b(t,W) = (log t ) 2 b ( W l Q g (3.1) Let P t t > 1 . be the probability measure on ft P (du)Q(dW). t Z 0, converges in law to the process x (t) {ft, P} and to P T_ and x (t) x_(t) 0 of p(\) t as X -*- °°. of §2, we need another process { x + ( t ) , t £ R} . We are defined on a common probability x_(t), put Denote by g_(t) = x_{t) + £_(t) in a way similar to (2.1). (1.12) and let IP (dtodW) = The process P} and that they are independent. is denote by Let defined by which is equivalent in law to local time at o_ w {w(t), t > 0, P} In addition to the process (x_(t), t £ R } assume that x Then the theorem is rephased as follows: {o)(eA + t) - X 2 b(W x ) , space ), H_(t) the and define The expectation with respect E. X > 0, be a given function satisfying the condition I\e^g ,,,, X > 0. Then the condition for refinement of Brox's result is automatically satisfied. I\ in the Therefore, by (1.10) and (1.2) we have (3.2) VV e «px w - A 2 b<w x >er x H = EQ[p£{a - b(w) £ YA_1(rx)}] + o(i) = I x + I I X + o(l) , where h = V p w< f f l - b + £ V 1 ( V } ; nx and E0 b = b } b= b ] + = E Q I P * { A - b_eTx-x(rx>>j b - b j , denotes the expectation with respect to denotes the integral over the set noted that in the above (and also in what follows) which tends to 0 as Q while E _ { — ; {b = b }. It is to be o(l) means a term \ •* *> uniformly with respect to the choice of {T,} so far as it satisfies the condition (1.13). A use (2.2) to compute I and II,. A A We put b+ = - c t b_ = a_ , We are going to 168 d - + H T+ - a+f 3_ - - (T_ - a_) , = max 6, - 0,(a,), H = max B_ - 6 (a ) , [0+,O] + " [a_,0] " " = - e + (o + ), V_ = - B_(o_) , V+ J+(a+ + t) - B+(o+) for t £ 0 , 3_(a_ - t) - B_(a_) for t < 0 , Then the proposition of §2 implies (3.3) {W(t), d_ < t £ d + } ^ (B As in (1.2) we define b by b+ if M + v (V+ + 1) < M_v (y_ + 1) , b_ if M + v (V+ + 1) > M^v{V_ + 1) . b Using Lemma 3 and then (3.3) we have (3.4) = E [PX {y"1 Q V^+t-b^ d + -b + ] X I X where to P^ , , p+1 '' *J P *, , . For e > 0 1+, Also let u,0 e X + is the probability measure defined in a way similar put = max {t < 0 : x+(t) = 0 o (T )}; b = b ] + o(l) t < ?s < 0 s.t. and x+(s) = e}-, e + [o £ ,T + 3 bi be the probability measure on R: A0 + (dx) = e " X 6 + { x ) d p. x/pe-*Mt) dt , Ap+ and l e t P be t h e p r o b a b i l i t y measure on ft d e f i n e d i n a way Ap + similar to (3.5) P „ (see the introduction). Then the scaling relation { A - 1 0 + U 2 t ) # t e R) & { 0 + < t ) , t e i m p l i e s t h a t f o r any m} 6 >0 iJ xp {(-6,6) } = j \ e~Mx,dxyY V B + ( t ) d t •* 1, A •* « 324 169 and hence E [ y . R { ( - 6 , 6 ) } ] + 1. Similarly E[p* . , { ( - 6 . 6 ) }] * 1 and E[u Qp -l a s A -*• » . T h e r e f o r e , j u s t a s i n t h e T ,{(-6,6)}] + 1 + °e'T+J c a s e of Lemma 3 , we can p r o v e t h e f o l l o w i n g : L e t I \ , A > 0, b e t h e same a s b e f o r e . Then (3.6) B I P ^ V 1 = E[PS + [a + f T + 3 { V 1 < r X>>] ^ ) ) ] = E[P* #e{YA" From (3.4) and (3.6) we have for any (3.7) I x = E[P* EWx" 1 1 + °<1J ( r A ) }] + o ( l ) , A -*- « . e >0 (r x )}? b = b + ] + o(l), A - co , and a similar formula for II,. We put B+(t) = B + (a £ + t) - B+(ae) , a = the smallest zero of x (t) in (z, 0] where the maximum of t < 0 such that x (t) = 1 , z is K = » a x n l s + - s+(&+> m Then m < m m = [o+,0] max 6,» [a+/0] + me = max f} . [a£,0] for all sufficiently small e > 0 (P-a.s.) and < m + <==> a+ < a =^» B+(o+) = B+(o+) + 3 + (o £ ), M + = M+. Therefore {b = b + } n {m£ < m + } = {M+V(l - 6+(5+) - 6+(oe)) < M_v(l - 8_(o_))} n {m£ < m + } and hence < 3 - 8) ^[P6+,e{V1 /e + Me,A) , _1 B = B ] + ( r A ) > ; M + v(i-e + (d + )-e + (a £ )) < M _ v u - M a j n 325 170 A( e ,M where P 6 Y e* A~ 1(I tends to 0 as e is a measurable V* since the process +0 uniformly in function of (B + (t), S+£tS0} is equivalent in law to A. Since (x + (t), aESt^T+ } and conditioned by (x + (t), o£St£T+} {0 (t), a+5t£0}, by using the strong Markov property of x (t) we see that the right hand side of (3.8) equals (3.9) where as E[Pg e iMx) = P(M + V ( l - 6 + ( o + ) - x ) e-i-0 and \ iMx) -*• 1 / 2 E[Pg fe as + A(E,A) , < M^v ( l - 6 _ ( a _ ) ) } . x -+ 0 , {Yx_1(rx)}] since (3.9) i s egual &+(ae) •* 0 to + A'(e,M which, by (3.6), is again equal to § E[P A0 tY A _1 (r A )}] + A'(e,A) + A"(e,A) , (3.10) where A'(e,A) -»• 0 as A •*• <*• for each fixed (3.11) lA= e + 0 uniformly in A and A" (e,A) -»• o as e > 0. Therefore, from (3.7) ^ (3,10) we have ^ E ( P i g ( Y ^ V ^ ) } ] +o(l)y A -*• « f and a similar formula for II, . To complete the proof of the theorem we need the following simple lemma. Lemma 4. For any A > 0 and r£ $ we have Proof. For each W with e~ W £ L 1 (R) , X(t,W) denotes the stationary ^ -diffusion process with initial distribution p . Fixing a sample path &,{*), we put X(t) = X(t,B,(')) and X^(t) = X(t,B+(A 2 •)). Since the scaling relation (3.5) implies J A B + { V 1 « r > } ^ g S + |A 2 ., { ^" 1 « r » ) ' for the proof of the lemma it is enough to show 3,(*). In what follows the notation P 326 171 stands for the image measure of where S (x) = / e ^+ y dy. p . . Similarly under the map f} stands for the image 0 2 measure obtained by replacing can be easily verified: (3.12) S: R ->- \Rf |3+(-) by 6+(A •). The following fact If B(0) is a random variable with distribution 2 A B(0) is distributed according to f) . As in 1.1, * (t) can be constructed from a Brownian motion with initial distribution — 2~ 4 B. ( t ) = B ( 0 ) + X B(X t ) . p . Write Then B ( t ) A ' A with initial distribution fl . X (t) , we.use process B (t) X (t) p then B(t) B(t) = B(0) + £{t) and put i s again a Brownian m o t i o n Therefore if, in the construction of instead of B ( t ) , we still have a diffusion which is equivalent in law to X (t) - A By an easy A calculation we see that X x (t) = x~ 2 S _ 1 (x 2 B(0) + B(A~ 1 (x 4 t))) , = x"V1(fi(X~1(X4t))) , where B(t) = X B(0) + Bit) bution p 2 is a Brownian motion with initial distrik~ (by (3.12)) and (t) is the inverse function of _1 &(.» = rv e + <s tB(u))) d u m Therefore, law to {y X (t)} XX {X(t)}. and consequently {y X (t)} AX is equivalent in The proof of the theorem is now completed as follows. arbitrary (3.2). TBjQtr t > 0 being arbitrarily fixed, and put Take an r = r in Then from (3.2), (3.11) and Lemma 4 we have lim E Q [P w {e expx oj - x 2 b(w x >er}} = E [ P B m i , and this prove the theorem. REFERENCES [1] Th. Brox, A one-dimensional diffusion process in a Wiener medium, Ann. Probab., 14(1986), 1206-1218. [2] A. O. Golosov, Localization of random walks in one-dimensional random environments, Commun. Math Phys., 92(1984), 491-506. [3] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, 1981. 172 4] K. Ito and H. P. McKean, Diffusion Processes and Their Sample Paths, Springer-Verlag, 1965. 5] K. Ito, Poisson point processes attached to Markov Processes, Proc. 6th Berkeley Symp. Math. Statist. Probab, III, 225-239, Univ. California Press, Berkeley, 1972. 6] K. Kawazu, Y. Tamura and H. Tanaka, One-dimensional diffusions and random walks in random environments, to appear in Proc. 5th Japan-USSR Symp. Probab. Th. 7} H. Kesten, The limit distribution of Sinai's random walk in random environment, Physica, 138A(1986), 299-309. 8] P. Levy, Processus stochastiques et mouvement brownien, Gauthier-Villars, Paris, 1948. 9] J. W. Pitman, One-dimensional Brownian motion and the threedimensional Bessel process. Adv. Appl. Probab., 7(1975), 511526. 10] J. W. Pitman and M, Yor, A decomposition of Bessel bridges, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 59(1982), 425-457. 11] Y. G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium. Theory of Probab. Appl., 27(1982), 256-268. 12] H. Tanaka, Limit distributions for one-dimensional diffusion processes in self-similar random environments, to appear in Hydrodynamic Behavior and Interacting Particle Systems, the IMA Volumes in Math, and its Appl., Vol. 9, 1987, Springer-Verlag. 13] D. Williams, Path decomposition and continuity of local time for one-dimensional diffusions, I, Proc. London Math. Soc., (3)28(1974), 738-768. Reprinted from Stochastic Analysis, 156-172, Lecture Notes in Math., 1322, Springer-Verlag, 1988. 328 On the maximum of a diffusion process in a drifted Brownian environment KIYOSHI KAWAZU AND HIROSHI TANAKA 1. Introduction In this paper we investigate asymptotic behavior of the tail of the distribution, of the maximum of a diffusion process in a drifted Brownian environment. This problem is a diffusion analogue of the Afanas'ev problem([l]). Our result is naturally compatible with that of Afanas'ev[l]. Let {W(x),x € R, P} be a Brownian environment, namely, let {W(<),< > 0, P} and {W(—t),t > 0,P} be independent Brownian motions in one-dimension with W(0) = 0. We consider a diffusion process X(tj W) defined formally by X(t, W) = Brownian motion - i (t{W\X(st W)) + c}ds, 2 Jo where c is a positive constant. The precise meaning of X(t} W) is simply a diffusion process with generator 2C dx{€ dxh starting at 0. Such a diffusion process can be constructed from a Brownian motion through changes of scale and time. For a fixed environment W — (W(x), x € R) we denote by Pw the probability law of the process {X{tj W)} and put 7> = Jp(dW)Pw. Thus V is the full law of {X{t} -)} . We often write X{t) = X(t, •) . Since c > 0, max t > 0 X(i) is finite ("P-a.s.). The problem is the following : How fast does V{mzxt>o X(t) > x} decay a s i - » o o ? Since (1.1) P{maxA"(0 > x) = E{A(A + B ) " 1 } , 329 79 where A = f° ewW+«dt, (1.2) B= f J—oo ewW+«dt, Jo the problem is nothing but to find the asymptotics of E{A(A + B)'1} result varies according a s c > 2 , c = l , 0 < c < l , a s as x —*• oo . The will be stated in the following theorem. T H E O R E M . ( i ) / / o l , then 7>{maxX(t) C«y Ifc-l, 2c — 2 1 > x} ~ ^ — j - e x p { - ( c - -)x}} x -> oo. then 1>{igxX(t) > x] ~ ( 2 A ) 1 / 2 I - 1 / 2 e x p { - x / 2 } ) x - oo. (»i> / / 0 < c < 1 , *Aen 7>{maxX(t) > s } ~ const.x'312 e x p { - c 2 s / 2 } , x - * oo, u/Aere cons*. = 2 5 / 2 - 2 c r(2c)~ 1 / " / ~ r f°° z(a + Jo Jo Jo Jo zyia^e-'l^e-^usinyiudadydzdu, A = (l + j/ 2 )/2 + ycoshu . 2. P r o o f of t h e t h e o r e m Since A and B are independent, the right hand side of (1.1) equals E{Af(A)} f(a) = E{(a -f 5 ) _1 where } , a > 0 . Fixing x > 0 , we consider the time reversal W(t) = W(x — t)~- W{x)t 0 < t < x . Since {W(t)> 0 < t < x} is also a Brownian motion, we have /(a) = E{{a + / * exp{Wfy) + ct} dt)'1} = E{(a + e " ^ j * exp{iy(i - i ) + ct} (2-1) h = c^^-^^^e^f'J-" + | dt)'1} J ' e H 'M- r f Jo In deriving the fifth equality in the above we used the formula of Cameron-Martin-MarayamaGirsanov; the last equality was derived by using W(t) as in the case of the first equality. From the fifth equality of (2.1) we obtain the following lemma. 330 80 LEMMA 1. (2.2) For any c> Q and x > 0 >x} = e^2-^xE{A{Atw^'{^> V{m*xX{t) + j \ w ^ - ^ - ^ dt)~1} , where A is given by (1.2). The following lemma due to Yor will also be used. LEMMA 2(Yor[2]). For any v > 0 we have (2.3) r ex P (W(i) - ~)dt I 2/Z„ , Jo 2, where = means equality in distribution and Zv is a gamma variable of index v , that is, P{Z„ € dt} = T(v)-H"-1e-tdt {i > 0). 2.1. Proof of (i) When c > 1 , Lemma 1 implies Jim e-W7-c>?{maxX(t) > x} = E{A(j°° e™^"1* dt)'1} . It is easy to see that the above expectation is finite. To obtain its exact value we use Lemma 2. We thus obtain (i). 2.2. Proof of (ii) For x > 0 we put ew<0 dt}} ^(x) = ~ Then it is easy to see that yj{x) = E{{ (' ew& dtr'eW} Jo = E{( f ew& di)"*} • Jo in fact, the second equality is a consequence of the last equality of (2.1) with o = 0 and c = 1 . Thus ip{x) is monotone decreasing in x . LEMMA 3 . ElAife^^dty^^^x'^e^7 (2.4) Proof. When c = 1 , we have as x -> oo Since E{A} = 2 in case c = 1, the left hand side of (2.4) equals 2£{(/ 0 * ew^M x 2 w which also equals 2e- f E{(JQ*e & w dt)-*t &} E{A(f dt)' by virtue of (2.1) with a = 0 and c = 1 . Thus we have (2.5) . e W(t > +t dt)~1} = 2e-*t2i>{x). 331 81 On the other hand, using the scaling property {W(t)} = {•y/xWtyfx)} we have dt) + logs, and hence = 5{maxo yfe/i, which combined with the monotonicity of i>(x) = f'{x) implies -4>(x) ~ (2TTX)"1/2 a s z - » o o (2.6) . This together with (2.5) proves the lemma. LEMMA 4. For x > 0 we have E{( f tw& dt)-2ew<*>} < tf (x/2) 2 . (2.7) Jo Proof. The left hand side of (2.7) is dominated by E{( f " ew<*> *)-»( f Jo ew«> dt)-^*)} Jx/2 ew^-w^^dt)-ltw^-wixt2)} = EUf^e^Utr^r J0 Jar/2 = E{( T / 3 tw™ dt)~x}E{{ r / 2 ew& d t ) - 1 ^ ' } JO = TK*/W JO ; in deriving the second equality in the above we used the fact that {^(*+|))—M^(f),* > 0} is a Brownian motion independent of {W(<), 0 < t < x/2} . The proof of (ii) is now given as follows. By (1.1) we have 0 < E{A([" ew&+t dt)-1} - P{maxX(t) > i } (2-8) = J5{AB-' - A{A + B)- 3 } < ^-M3/^-3'2} = 2-1E{A*l2}E{B-*l2}. We prove (2.9) £{A 3 ' 2 } < o o , (2.10) £?{£- 3 ' 2 } < const, i" 3 /^"*' 2 . 332 82 (2.9) follows immediately from Lemma 2 ; a direct proof can also be given as follows. Using Holder's inequality we have < ( 5 / 3 ) " ^ { j H exp{|(W(<) - j)}dt} = (5/3) 1 /* - (40/3). (2.10) can be proved by making use of the CMMG formula, the Schwarz inequality, Lemma 4 and then (2.6) ; in fact, putting B0 = J? e W ( 0 A we have E{B~*I2} E{BoS,7ew^^t2} = t-^E{B^ew^yi^E{B^ew^Yl2 < e-xl2i>{xyt7j>(x/2) < < const, e - ' / 2 ! " 1 / 4 - 1 " 1 / 2 . The assertion (ii) of our theorem follows from Lemma 3, (2.8), (2.9) and (2.10). 2.3. Proof of (iii) The proof of (iii) relies essentially on the following Yor's formula. Y o r ' s f o r m u l a ( [ 3 : the formula(6.e)]). For any bounded Borel junctions f and g we have E{f(Jo e™Mds)g(ewM)} = ct Hay f°° dzg(y)f(l/z)exp{-z(l Jo Jo + y3y2}TPst{t), where ct = W (t) = (2x 2 i) _ 1 / 2 «p{7rV20 I exp{-u 2 /2f}e- r ( c o s l 1 *)(sinh ti) S in(™/*) du. r Jo T o proceed to the proof of (iii) we put f(a,z) = a(a + 4z)-\ g(y) = y2e, B^(i)= fe^^ds. JO Using first the CMMG formula and then Yor's formula we have E{a(a + r e w ( 0 + e i rft)"1} = E{a(a + 4B ( 2 c ) (i/4))" 1 } Jo = E{a(a + 4B< 0 >(*/4))- 1 exp(2c^(i/4) - &)} = exp(^x/2)E{f(a^\x^))g(ew^)} = exp(-cai/2)c,/4 r JO dy H dzg(y)f{a,l/z)exp{-z(l JQ + y2)/2}t^(x/4). 83 Since Lemma 2 implies P{A G da} = 2 2 c r(2c)- 1 a- 2 c - 1 c- 2 /Va (a > 0 ) , we have 7?{max(>o^(t)>i} (2.11) =2 2 c + 1 / 2 r(2c)-%- 1 exp(27r 2 /a:)a:- 1 / 2 exp(-c 2 i/2) roo ^oo roo x / dy / dz / Jo Jo Jo where ^ / azftt + ^ - ' a - ^ ' e - ^ - d a , Jo /i(z) = X LEMMA 5 . duy 2c A(z)e~ A *exp(~2u 2 /x)(sinhu)sin(4 (l + y 2 )/2 + ycoshu. = Let 0 < c < 1 andpui F(y, z, u) = y 5e A(z)e -Ax u sinh u. Then we have roo M Proof. too roo = / / / F(V> z>u) dydzduKoo, Jo Jo Jo By a change of variable coshu = v , we have M= dv y^h{z)e-Xz f°° dy f°° dz r Jo Jo J\ log(u + y/v*^l) , where A = (1 + y 2 ) ^ + yv . Since h(z) = T ^ z r u^h-^u Jo + 2 l)-ldu, it is easy to see that h(z) —> 2 _2c r(2c) (2.12) (2.13) M*)~«,|0 as2-»oo , 2~2c~lV(2c-\)z if c > 1/2, 2"2zlogl/z if c=l/2, 2-4eJ«a3c-l(a+1)-l(fa.z2c if 0 Therefore for any e > 0 and a> 0 we have Mj » y°° dy J~ dz j°° dv y2ch(z)e-Xl < const, r fW y^v'X^e-* log(v + V ^ ^ T ) dydv < const. / ~ /°° y 2c u e A-° tfydu < const. / ~ / " y2Q~€-\\ Jo Jo + y 2 p + 1 + V ( l + z)"° dy dz 334 84 (by putting v = (2y)~ 1 (l + y2)z with y fixed ), which is finite if e > 0 is sufficiently small and a > 0 sufficiently large. Note that const, in the above may vary from place to place and depend on e and a . Next we prove that (2.14) r dy f1 dz / " dv y2ch(z)e~Xl Jo JO Ji M2= log(v + v V - 1) < oo . Assume 1/2 < c < 1 . Then by (2.13) M2 < const. / dy f dz I dv < const. r / ° ° A - J j / V < const, r Jo ri?*-l-{\ y2cze~x'vc dydv + ( we used tf)-l*f(\ + / ze~x'dz < A"2 ) z)-*dydz Jo (by putting v = ( 2 y ) - 1 ( l + y2)z with y fixed ) which is finite for sufficiently small e> 0 by virtue of 1/2 < c < 1 . When c-l/2 , (2.13) implies M 2 < const. /°° rfy / dz /°° dv t/2 1_e e~ A2 v' < const.A~ 2+ ' , we have for 0 < e < 1 . Since /„' z^'e^'dz M2 < const./ / < const, r r JO JO X"2+'yv'dydv y-'(l + y2)-1+2'z'{l + z)" 2 + e dy dz < oo provided that c > 0 is small enough. Finally assume 0 < c < 1/2 . Then by (2.13) M2 < const, r dy j dz T dv^z^e'^v' < const. /°° < const. f ° / ~ y 2 c ~ ' - J ( l + y 2 ) ' 2 c + V ( l + z^'dydz JO r V ^ V V d y d v < oo JO provided that e > 0 is small enough. Thus (2.14) is proved. We can now complete the proof of (iii) as follows. From (2.11) we have (2.15) > z} = 22c+bf2T(2Cy1exp(2w2/x)x'zf2exp(-c2x/2)M(x) V{maxX(i) where rCO M(i) = / Jo /OO / Jo fOO / Jo F(y, z, u) s i n ( 4 W s ) / ( 4 W x ) exp(-2u 2 /:r) dy dz du t 335 85 By Lemma 5 we have l i m * . ^ M(x) — M which equals 2_4c + /* r r r ^° ^^^y^^u ^^«& ^ & <** • Thus the assertion (iii) follows from (2.15). Acknowledgment. We wish to thank Prof.S.Kotani and Prof. M.Yor for giving us valuable information ; Prof. M.Yor Idndly sent us preprints including [2] and [3], without which the result (in) would not have been obtained. References [1] V.I.Afanas'ev, On a maximum of a transient random walk in random environment, Theor.Probab.Appl.35(1990), 205 - 215. [2] M.Yor, Sur certaines fonctionelles exponentielles du mouvement brownien reel, J.Appl.Probab.29(1992) , 202 - 208. [3] M.Yor, On some exponential functionals of Brownian motion, to appear in Adv.Appl. Probab. (September 1992). Kiyoshi Kawazu Hiroshi Tanaka Department of Mathematics Department of Mathematics Faculty of Education Faculty of Science and Technology Yamaguchi University Keio University Yosida, Yamaguchi 753 Hiyoshi, Yokohama 223 Japan Japan Reprinted from Seminaire de Probabilites X X V I I , 78-85, Lecture Notes in M a t h . , 1 5 5 7 , Springer-Verlag, 1993. No. 9] 86, 377 Proc. Japan Acad., 69, Ser. A (1993) Recurrence of a Diffusion Brownian Process in a Multidimensional Environment*^ By Hiroshi TANAKA Department of Mathematics, Faculty of Science and Technology, Keio University (Communicated by Kiyosi ITO, M.J. A., Nov. 12, 1993) Introduction. Let W be the space of continuous functions on R vanishing at the origin. In this paper an element of W is called an environment. Given an environment W, we consider a diffusion process Xw = {X(t)t l > 0 , ? ^ e Rd) with generator When W is bounded, the result of Nash [8] for fundamental solutions of parabolic equations guarantees the existence of a diffusion process Xw with generator t-i d*k v dxj' For a general W we still have a nice diffusion process Xw (e.g. see [4]) and hence Xw can be constructed from Xw through a random time change. Without any assumption on the behavior of W{x) for large | x | the process Xw may explode within a finite time, but such a case is excluded automatically since we are interested in the recurrence of Xw. We consider the probability measure P on W with respect to which iW(x), x e R , P] is a Levy's Brownian motion with a rf-dimensional time. The collection of diffusion processes X — iXw) in which W is allowed to vary as a random element in (W, P) is called a diffusion in a tf-dimensional Brownian environment. W h e n d = 1 this was considered by Brox [1] and Schumacher [9] as a diffusion model exhibiting the same asymptotic behavior as Sinai's random walk in a random environment ([10]); see also [11] for some refined results. Recently Mathieu [7] obtained some very interesting results concerning a long time asymptotic problem for X in the case d > 2. Motivated by [7] the present paper was written. In this paper we prove that Xw is r e c u r r e n t for almost all Brownian environments W in any dimension d, namely, for any nonnegative Borel function f on R such that / > 0 on a set of positive Lebesgue measure the equality Pxw{j f(X(t))dt = ooj = i, x e R\ holds for almost all W with respect to P. In [3] Fukushima, Nakao and Takeda discussed the same problem but with the replacement of W{x) by J ^ ( | x | ) , ' This research was partially supported by Grant-in-Aid for Science Research No. 0 4 4 5 2 0 1 1 , the Ministry of Education, Science and Culture, Japan. 378 [Vol. 69(A), H. TANAKA where W{t) is a Brownian motion with a 1-dimensional time. To obtain our result we employ Ichihara's recurrence criterion ([4]) which, in the present special case, a s s e r t s that Xw ( W i s fixed) is r e c u r r e n t where d6 is the uniform distribution on S . We can also employ Fukushima's recurrence criterion ([2]) which, in the present special case, asserts that Xw { W i s fixed) is recurrent if there exists a sequence iu„} such that 0 ^ un ^ 1, lim u„ = 1 a.e. and lim#(w„, un) = 0, where 8(ut v) is the Dirichlet form associated with Xw, namely, §(u, v) = -^ I Vu- Vu e~wdx. Since it is obvious that Xw is r e c u r r e n t if and only if Xw is recurrent, either criterion yields our result. But for the verification of these criteria we need some information on the asymptotic behavior of W{x) for large | . r | . A key point in obtaining this informaiton is to consider the o n e - p a r a m e t e r family {Tt, t e R} of measure preserving transformations on (W, P) defined by (1. 3) and then to use its ergodicity. §1. Brownian motion with a {/-dimensional time. Let d > 2 and as before let P be the probability measure on W such that {W{x), x ^ R , P} is a Brownian motion with a d-dimensional time ([6: p. 277]), that is, a Gaussian system with (1.1) EiWix)) = 0, W(0) = 0, (1.2) E{W(x)W{y)} = j i \ x \ + \ y \ - \ x - y \ ) . For each f £ R and W ^ W we define an element TtW of Why (1.3) (T,W(x) = e~t/2W(ex), x e Rd. Then {Tt, t e R} is a one-parameter family of measure preserving transformations on the probalility space (W, P). Using (1. 2) we can easily compute the covariance matrix of e~tnW(exx)t e-tnW(exd,---,e-t/2W(exJ, W(xO, W(x$,• • •,W{x' n ) for fixed t ^ R and x l t . . . ,xm, x[,. , . ,x'n ^ R , and the following lemma can be proved in the same way as in Ito [5J. Lemma 1. {Tt, t ^ R} is mixing and hence ergodic. Next let 0 < a < b, put K = {x e Rd ; a < \ x | <= b) and consider the Banach space B = C(K), the space of real valued continuous functions on K, and the real Hilbert space H — L (K t dx). The inner product in H is denoted by ( ' , " > . Regarding WK = iW(x), x ^ K) as an H-valued random variable, we denote by y the probability distribution of WK. Since every Borel set in the space B is also a Borel set in the space H and since WK is regarded as a B-valued random variable, we have y{B) — 1. y is a Gaussian measure on H with (1.4a) J e yidg) = ^ { e x p J fix) W{x)dx] = e x p { ^ } , / € H, No. 9] (1.4b) Recurrence of a Diffusion Process Af(x) = j \ i \ x \ + \y\~ 379 \x~y\) f(y)dy. For 0 = Af0 with / 0 ^ H we define the 0 - t r a n s f o r m j ^ by 7^ CO — ?"({£ : g + 0 ^ / } ) . Then the following Cameron-Martin formula is easily verified by using ( 1 . 4). (1.5) rMs) = e x p { < / 0 , g>~\ f0>}r(dg). Lemma 2. Any nonempty open set in the space B has a positive ^-measure. Proof. We first prove that the range R = (Af :f^H] is dense in B. If this were not true, there exists a finite signed measure pt # 0 on K such that (1.6) f Afix)fi(dx) = 0 for all Since the left hand side of ( 1 . 6) equals if, g(x) = fK\i\x\ / e H. g} where + \ y \ - \x-y\} fi{dy) e J/, (1.6) implies g = 0. Therefore, regarding |/ as a signed measure in R have (L7) we J l i l l (UI + l*/|-|.r-2/|} fi(dy)fi(dy) = 0. In the same way as in the proof of Theoreme 5 8 of [6: p. 2 7 6 ] we can prove that the left hand side of ( 1 . 7) equals const, f |?r_1|£(?) -/2(0)|2rfff where /5(f) is the Fourier transform of p.. Therefore fi must be concentrated on {0}. But this is impossible because 0 ^ K and hence R must be dense in B. Next we notice that the whole space B, which has f - m e a s u r e 1, can be expresses as a union of a countable number of open balls of the form Be( 0 being a r b i t r a r y but fixed. On the other hand by the Cameron-Martin formula (1.5) y{BB{<j>)) = r-(Be{0)) > 0 if and only if r(Be(Q)) > 0 provided that 0 e R. Therefore we must have ?(B£( 0 for all 0 e R. This implies the assertion of the lemma. §2. Recurrence of Xw. Since our result in the 1-dimensional case is easily obtained from a general theory of 1-dimensional diffusion processes, we assume d > 2. Theorem 1. Xw is recurrent for almost all Brownian environments W. Proof. It is enough to prove t h a t Xw is r e c u r r e n t for almost all Brownian environments W and, according to Ichihara's criterion ([4: Theorem A]) it is also enough to prove that (2.1) J^lfst_ie-w ~, If we p u t M(t) = mm{(TtW)(6) : 6 e S"" 1 }, then (2.2) the left hand side of (2. 1) P.a.s. 380 [Vol. 69(A), H. TANAKA f~ e(2~d)t { / ^ e x p C - et/2(TtW)(d))dd\ 00 i2 d)t exp{et/2M(t)}dt > f e ' > f°° ' dt liao0)(M(t))dt, •"0 provided that a > 0 is chosen so that (2 — d)t + ae ^ 0 holds for 0. Next take K = ix ^ Rd : 1 < | x \ <* 2} and consider B, H and the preceeding section. Since T = {<$ ^ B : min (0Cr) : | x | = 1) > open set in B, we have y(D > 0 by Lemma 2. The ergodicity of {Tt, now implies and lim T " 1 f l ( a > 0 0 ) ( M ( / ) ) ^ = £{l ( f l i T O ) (M(0))} = r(n hence liaoo)(M(t))dt J/ o = °° , P-a.s., which > 0, combined all t > y as in a) is an t e R} P-a,s., with (2.2) proves (2. 1). Remark 1. Xw is n u l l - r e c u r r e n t (P-a.s.) in the sense that mw{dx) = e dx is an invariant measure for Xw with mw(R ) = ° ° . Remark 2. Fukushima's criterion can also be used for proving Theorem 1; in fact, by virtue of Lemmas 1, 2 it is still easy to prove the existence of a sequence of radial functions un in C^(R ) such that 0 < un ^ 1, lim un = 1 a.e. and $(un, un) = 0. This argument also proves the recurrence of Xlwl for almost all Brownian environments W. Remark 3 . X_lw} is r e c u r r e n t for d = 1 and transient for d !> 2 for almost all Brownian environments W. The proof in the case d ^ 2 is as follows. According to Theorem B of [4] the transience of X_\w\ (and consequently of X_\Wl) follows if one proves that, for almost all Brownian environments W, /o n\ C (2.3) / -\W(r0)\ e -d+l r , ,. dr < °o for 6 belonging to some subset (which may depend on W) of 5 with a positive uniform measure. But this can be proved by showing that the expectation (with respect to P) of the left hand side of (2. 3) is finite for each fixed 6. References [ 1 ) Th. Brox: A one-dimensional diffusion process in a Wiener medium. Ann. Probata, 14, 1 2 0 6 - 1 2 1 8 (1986). [ 2 ] M. Fukushima : On recurrence criteria in the Dirichlet space theory. Local Times to Global Geometry, Control and Physics (ed. Elworthy). Research Notes in Math., Series 150, Longman (1987). ( 3 ] M. Fukushima, S. Nakao and M. Takeda: On Dirichlet forms with random data recurrence and homogenization. Stochastic Processes - Mathematics and Physics II (eds. S. Albeverio, Ph. Blanchard and L. Streit). Lect. Notes in Math., vol. 1250, Springer-Verlag (1987). [ 4 ] K. Ichihara: Some global properties of symmetric diffusion processes. Publ. RIMS, Kyoto Univ., 14, 4 4 1 - 4 8 6 (1978). 15] K. ltd : On the ergodicity of a certain stationary process. Proc. Imp. Acad., 20, 5 4 - 5 5 (1944). No. 9] [6] Recurrence of a Diffusion Process 381 P. Levy: Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris (1948). [ 7 ] P. Mathieu: Zero white noise limit through Dirichlet forms. Application to diffusions in a random medium (1992) (preprint). [ 8 ] J. Nash: Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. , 8 0 , 9 3 1 - 9 5 3 (1958). [ 9 ] S. Schumacher : Diffusions with random coefficients. Contemporary Math. Particle Systems, Random Media and Large Deviations (ed. Durrett). vol. 4 1 , pp. 3 5 1 - 3 5 6 (1985). (10] Y. G. Sinai: The limiting behavior of a one-dimensional random walk in a random medium. Theor. Probab. Appl., 27, 2 5 6 - 2 6 8 (1982). (11] H. T a n a k a : Limit theorem for one-dimensional diffusion process in Brownian environment. Stochastic Analysis (Proc. Japanese-French Seminar, Paris 1987). Lect. Notes in Math., vol. 1 3 2 2 , Springer-Verlag, pp. 1 5 6 - 1 7 2 . 341 Localization of a Diffusion Process in a One-Dimensional Brownian Environment HIROSHI TANAKA Keio University Dedicated to H. P. McKean. 1. Introduction Let W be the space of continuous functions W : U -* U with W(0) = 0 and Q be the Wiener measure on W, namely, let {W(x), x s 0, Q} and {W(-x), x ^ 0, Q} be xndependent one-dimensional Brownian motions starting at 0. An element of W is called an environment. Given an environment W we consider a diffusion process X(t, W) starting at 0 and with generator (1.1, Lw--\e™±(e-™±). X(t, W) can be constructed from a one-dimensional Brownian motion B{t) by a scale change and a time change; see [3]. We denote by {tt,P) the probability space on which B(t) is defined and put & = P ® Q. Thus B{t) and W(x) are independent. X{t, W) is then called a diffusion process in a Brownian environment. Sinai presented this model as a diffusion analogue of his random walk in a random environment; see [10], page 268. Brox (see [1]) and Schumacher (see [9]) proved that X(f, -) exhibits an asymptotic behavior as t — oo which is similar to that of Sinai's random walk of [10]. We consider anotherj>robability measure Q on W such that {W(x),x ^ 0,Q} and {W(-x),x ^ 0,Q} axe independent Bessel processes of index 3 starting at 0. Then e~w € LHR^Q-a.s. For each W with e~w e Ll{U) let fiw be the probability measure in IR of the form (1.2) ^(dx) = const. e~wix) dx . For each \ > 0 w e define b\(W), a function of the environment W alone, by (3.1). Then it was proved in [11] that the distribution of X ( e \ •) - W O converges to (j, as X. — oo, where \i = J pwQ(dW). A similar localization theorem had already been obtained by Golosov (see [2]) for reflecting random walks on Z + . In the present paper we restate the above localization theorem for X(ty •) in a modified form and prove it by a method different from that of [11]. THEOREM 1.1. For any /,, 1 ^ j' ^ k, with 0 < t\ < *•• < tk the joint distribution ofX{extj, •) - W 0 , 1 = j = K with respect to & converges to the mixture f fAwQ{dW) as \ — oo where pw is the k-fold product distribution of fj,w- Communications on Pure and Applied Mathematics, Vol. XLVII, 755-766 (1994) © 1994 John Wiley & Sons, Inc. CCC O01O-3640/94/050755-12 342 756 H. TANAKA Ogura (see [7], Example 7.2) gave another proof to Brox's Theorem 1.4 (see [1]) from which the present method was suggested. The notion of valleys of environments still plays an essential role as before. The main difference between the present proof and that of [11] is as follows: In [11] we used Brox's estimates for exit times from valleys and a coupling technique, but now, instead of these, we use a theorem of Ogura (see [7]) concerning the convergence of a sequence of one-dimensional (generalized) diffusion processes. 2. Convergence Theorem for Diffusion Processes in One-Dimension It is well known that the generator of a diffusion process in U can be expressed as a diffusion operator d/{m{dx)}d/{dS(x)}. The measure m(dx\ called the speed measure, is finite on compact subsets and positive on open subsets (* 0 ) of U, and the function 5(JC), called the canonical scale, is continuous and strictly increasing. Ogura (see [7]) discussed a more general case, namely, the case of "bi-generalized diffusion processes," but for our present purpose it is enough to consider a sequence of diffusion processes. Thus, suppose we are given a sequence of diffusion operators Ln with speed measure mn{dx) and canonical scale S„(x\ n = 1,2,... , and denote by X%(t) the diffusion process with generator Ln starting at x. We assume that the following conditions (2.1), (2.2), and (2.3) are satisfied: (2.1) For each n S„(0) = 0 and S„(x) tends to oo or - c o accordingly as x — oo or JC — — oo; for each x the canonical scale Sn{x) tends to 0 as n — oo. (2.2) For any / £ Co(R), the space of continuous functions with compact supports, Hm / fdmn = I f dm , where m is a nontrivial finite measure (namely, mn converges vaguely to m as n — oo). (2.3) The measure m„ = mn o S" 1 converges vaguely to C6Q as n -* oo, where 5^"' is the inverse function of 5„,c = m(U) > 0 and 6o denotes the 5-measure atO. The author learned the following theorem through the kindness of Y. Ogura. OGURA's THEOREM. (SEE [7]) Let e be an arbitrary constant such that 0 < e < 1, put Tk,e = {(tU---,tk)€Wk:e^tl < tk ^ 1/e , . S e ( l ^ V./^*)} (2.4) tj-tj-i and consider a sequence {x„} satisfying (2.5) |S„0cB)| s e / o r * = 1,2,... . DIFFUSION PROCESS 757 Then for any fj G C 0 (R), 1 ^ j'< ^ K 4n/;Nrn/ c ''^ m as n — oo uniformly in {x„} satisfying condition (2.5) and* in (t\,- • • , **) G r*iB. Proof: Since the case of Hm5*(jt) = 0 is excluded in Theorem 5.1 of [7], strictly speaking the above theorem is not a straightforward consequence of Theorem 5.1 of [7]. The proof can be done, however, in the same way. Denote by Qn(hx,y) the transition density with respect to mn of the diffusion process with speed measure rhn and canonical scale x. It is defined through formula (5.6) of [7] by taking m = m„ and Q = U. It is known that qn{t,x,y) is continuous in (t,x,y) G (0, oo) X R X U and symmetric in x andj>. Also we define q(t,x,y) through formula (5.6) of [7] by taking m = C6Q and Q — U. By an easy computation we see that q(t,x, y) = \/c for any (t,x, y) G (0, oo) x IR x U. Probabilistically (and formally) q(t,x,y) is the transition density with respect to C6Q of the "generalized diffusion process" which jumps to 0 immediately at t = 0 and remains there for any t > 0. Now Proposition 5.1 of [7] implies that q„(t,x,y) converges to q{t,x, y) uniformly on each compact subset of (0, oo) x U X U as n —• oo. Therefore, if pn{t,x,y) denotes the transition density with respect to mn of Xn(t), then for any / G C0(R) / pn(t,xn,y)f(y)mn{dy) = Jqn(t,Sn(xn),Sn(y))f(y)mn(dy) - J c-tfdm as n — oo uniformly in / ^ e and in {xn} satisfying condition (2.5), e > 0 being an arbitrary constant. This implies the theorem. 3. Valleys 3.1. We give the definition of a valley of an environment W following [1]. A part {W(x), x\ ^ x ^ X2} of W is called a valley of W if: (i) x\ < x2\ (ii) there exists XQ G {x\,x2) such that W{xi) > W(x) > W(xQ) W{x2) > W{x) > W(x0) for any x G (XI,JC0) , for any x G (xo,x2) ; (iii) for the same JCO as above H- = sup{W(y) - W(x) : x\ ^ x < y ^ xQ} < W{x2) H+ = sup{W(x) - W(y) :xo^x W(x0), W(x0) • 344 758 H. TANAKA For simplicity we write (xi,xo,x2) instead of {W(JC),XI ^ x ^ x2} and call xo the bottom of the valley. A = //+ V H- is called the inner directed ascent (abbreviated to i.d.a.) and D = flVfci) - W(jt0)) A {W(x2) - W(x0)) is the depth of the valley. A valley (xi,xo,x2) is said to contain 0 if x\ < 0 < x2. For any fixed r > 0, with Qmeasure 1, there exists a valley of W with i.d.a. < r < depth and containing 0 (see [1]); moreover, there are many such valleys for fixed W but it can be easily seen that their bottoms are the same. We denote by b = b(W) the unique (common) bottom of such valleys for r = 1. To give another description of a valley due to Kesten (see [6]) let \ > 0, W E W and put ( W(x)-minW forx^O I W(x)-minW for x < 0 dt = min{x > 0 : W*W = \} , Vx = min W , [0,d x + l Vx = min W . Wx,01 We define &x and 2>x hy W(£>x) = V\ (such fcjf are uniquely determined with ^-measure 1 for each fixed \ > 0). Let Mt = max W, + Jt = Ml v (Vx+ + M , Mx = maxW , A" = M [ V (Vx + M , [0A 1 and define &x = K W by (3.1) WW) = { Ux" ifA + >A". Moreover, we define ax = o\{W) and ex - c\{W) by (3.2) ax(W) = the infimum of the set of a's (a < &x) such that W{a) > W{x) > W{bx) for any x G (a, fcx), sup{W(y) - W(x) :a^x (3.3) C$W) = the supremum of the set of c's (c > 6X) such that W(c) > WW > W{bk) for any x e {bK, c) , sup{W<x) - W{y) :bx^x DIFFUSION PROCESS 759 Then we see that, with Q-mea.sur& l,(a\,b\,c$ is the maximum valley of W containing 0 and satisfying Ax < X < D\, where A\ and D\ are, respectively, the i.d.a. and the depth of the valley. Thus b\ coincides with b of the preceding paragraph. From now on we suppress the suffix 1 in a],b],c\,Di,d],d~[, etc. 3.2. For X > 0 and W 6 W we define i y x e W b y WM = \~*W(\2x),x G R. Then for each fixed X > 0 {W\,Q} is equivalent in law to {W,g} and flx(W) ='\ 2 fl(W\)A(W0 = \2b(Wx\cx(W) = \2c(Wk),Dk(W) = \D(WX). As stated in the Introduction we denote by X(u W) the diffusion process with generator Lw and starting at 0. Then the following scaling relation holds (see [1], Lemma 1.3): For any fixed X > 0 and W G W (3.4) {X{t,\Wx)9t ^ 0,P} = {\-2X(\4t,W),t ^ 0,P} where = means the equality in distribution (W is fixed). 3.3. The following is a result of Brox, proved in [1], Proposition 4.1 (see also [4], Theorem I-A-l): If (XI,XQ,X2) is a valley ofW with i.d.a. < r < depth and containing 0, then X(eXr, \W) converges to XQ in probability as X -* oo (W is fixed). 3.4. On a suitable probability space we consider a process {w+(x),x G R} such that {w + M,x g 0} and {w+{-x),x ^ 0} are independent reflecting Brownian motions on [0, oo) starting at 0, Let {€+(x),x G U} be the local time at 0 of w+(x), namely, <+W-Hm(£)/W"+W>*. where / = [0,JC] or [x,0] accordingly as x ^ 0 or x < 0. Now we put (3+{x) = w+(x) + €+(x). Then by Pitman's theorem (see [8]) {/3+(x\ x ^ 0} and {/3+{-x), x s 0} are independent Bessel processes of index 3 starting at 0. For X > 0 we put (3.5a) a\ = the smallest zero of w+{x) in (z,0] where z is the maximum of x < 0 with w+(x) = X . (3.5b) rit = min{x > 0 : /3+(x) = X}. We next introduce another process {w~(x),x G U} which is equivalent in law to {w+{x),x G R}. We assume that w+(x) and w~(x) are defined on a common probability space (Q,P) and that they are independent. Denote by £~{x) the local time at 0 of w~(x), put 0~(x) = w~(x) + f~(x) and define o\ and T\ in a way 760 H. TANAKA similar to equation (3.5). We put b\ = -at , bx = o\ , d\ =Tx - <Jx > Vjt = -p+(at) d\ = , = M , + V (vj - <Jx) , Vx = - / r ( a x ) , + Af£ = max£ - /3+(at), K+,o] A+ ~(TX Mx = max^~ K,o] + x), /Tfa), /x - Mx v (Vx + W • The following lemma for X = 1 was proved in [11]. The proof for a general X > 0 can be done in the same way. LEMMA 3.1. (SEE [11]) Let X > 0 and put Wt = {W(x + bt) - W(bt), -bt ^ x ^ dt - K } , Wx = {W(-x + bx) - W(bx) , bx ^ x =g -(dx - bx)} - Then, under the law Q (Wx, Jx) and {Wx, Jx) are independent and have the same distribution which is equal to the joint distribution of {/3+{x),ax ^ x ^ r ^ } and Jx • LEMMA 3.2. Let Wxix) = \{W{\~2x converges in law to {W,Q} as X —• oo. + b) - W(b)},x € R. 77ien { H \ , g } Proof: It is enough to prove that, for any positive constant K and for any bounded continuous function F on W depending only on {W(x\ \x\ ^ K}, (3.6) Hm JF{Wx) Q(dW) = J F'dQ . From now on we suppress the suffix 1 in J~\,Jt,W\, etc. We put T x == {J+ < J~}n {\~2K < b+ A (d+ - b+)} and notice that <2(rx) increases to Q{J+ < J~} = 1/2 as X T oo. Moreover, if W G Fx then the value of F(Wx) is determined by W+, namely, we can write F(WX) =Gx(W+) onr> DIFFUSION PROCESS 761 with a suitable function G\. Thus, using the notation applying Lemma 3.1, we can write EQ{F;T} = JrFdQ and EQ{F(wk);J+ =EQ{GK(W+); n}+o(l) I \ } + o(l) = E {Gx (p+(x\a+ = ^x^r+); f \ } + o(l) E{F(\/3+(\-2-));fk}+o(\) = E{F(\p+(\-2-)) ; / + < / " } + o(l) , where f x = {7 + < / " } fl { X 2 ^ < b+ A (d+ - S + )}, and o{\) represents a term which tends to 0 as k — oo. By an argument similar to that used for arriving at expression (3.9) of [11" we can prove that the last term in equation (3.7) is asymptotically equal to (3.8) E{F($3+(\-2-))}/2 as \ - oo. Since {\p+{\~2-),P} equal to (3.9) is equivalent in law to {£ + (-),P}, (3.8) is E{F(/3+(-))}/2 = fFdQ/2, and hence the first term of equation (3.7) tends to (3.9) as \ —• oo. Since > J} can be treated similarly, we obtain equality (3.6). EQ{F(W$,J+ 4. Proof of Theorem 1.1 The idea of the proof is to apply Ogura's theorem to study the limiting behavior ofX(e x /, . ) - M ' ) a s \ - oo. 4.1. We consider the diffusion processes (4.1) X%(t) = X(ext, W) - W W ) , (4.2) i f (f) = \ 2 {X(\~4e\ \W) - b(W)} , rgO, t i= 0 . The scaling relation (3.4) implies that, for each fixed \ > 0 and W, {tf 762 H. TANAKA Since the distributions of W and W\ under Q are the same, {Kr'(rt,/£o}i{yyw,rso}, where = means the equality in distribution {W is random). Thus the processes (4.1) and (4.2) are equivalent in law under {?. For a fixed W the generator of the diffusion process Y™(t) is expressed as the diffusion operator L™ with the following canonical scale and speed measure: (4.3) (4.4) S^{x) = Ze~x T e x p {X (W(\~2y + b) - W(b)) } dy . Jo ng(dx) = exp {-X (W(k~2x + b) - W(b)) } dx . LEMMA 4.1. (i) For any x the canonical scale S™{x) tends to 0 as X —> oo with Q-measure 1. (ii) We regard mx and rh\ = m^ ° (S\ ) ~ l as random variable taking values in the space of non-negative Radon measures in U equipped with the topology of vague convergence. Then the joint distribution {under Q) ofm™ and m™ converges to the joint distribution {under Q) of e~w^dx and const.8o{const — fe~wdx) as X -*• oo. Proof: The assertion (i) is obvious. To prove (ii) we first put x\ = max{x < 0 : W(x + b) - W{b) = 1} , x2 = min{x > 0 : W{x + b) - W{b) = 1} , and observe that, with Q-measure 1, for any s > 0 there exist x' E [x\ — e,*i) and x" G {x2,x2 + e) such that W{x' +b)- W{b) > 1 and W{x" + b) ~ W(b) > 1. Thus, if x varies with X in such a way that x > \2{x2 + e) or x < \2{x] — e), then iSjfC*)! -* oo as X ^ oo. On the other hand, if JC varies with X in such a way that \2{x\ + e) < x < X2(JC2 - e), then S$x) — 0 as X — oo, where e > 0 can be chosen arbitrarily under the condition e < (—*i) A x2. Therefore, for each fixed y =£ 0 X~2(SX ) _1 (y) tends as X — oo to x\ or x2 accordingly as y < 0 or y > 0. Thus, putting zi = (S$~l(yi), i = 1,2, for any given y\ and y2 with y} < 0 < y 2 , we see that X~2zi — x\ and X~2Z2 — x2 as X — oo. Now we write mf{\yuy2])=m^[zuz2]) j exp {-\(W{\~2x + b) -W(b))}dx J Zi where •l/e l/e / //—1/e —i/e , l/e h= y-\2e /*A e +/ ./l/e ,—\'e , /3 = / A, rz: +/ •/ * 2 < DIFFUSION PROCESS 763 and prove the following: (4.5) {m^,Q} converges in law to {e~WMdx,Q} as \ —• oo. In particular, for each e > 0{I\,Q} converges in law to {j\x\ 0 lim£i0 lim^_oc QUi > y} = 0, i = 2,3. (4.5) is an immediate consequence of Lemma 3.2. For the proof of (4.6) with i = 3 it is enough to note that h = \2[ + \Jk-ht Je u:*o ) exp{-\(W(* + b) - W{b))} dx exp{-MW(* + b) ~ W(b))} dx -0, \ - oo provided that W is in the event {(-xj) A X2 > e} whose ^-measure tends to 1 as e i 0. As for (4.6) with i = 2, by replacing W by Wx we see that I2 is equivalent in law to + (/k2 / jcxp{~(W(x + bx)-W(bx))}dx, which is dominated by (4.7) {J j ^ {~{WiX + h^ ' ^^ "Vl . +/ ^ 1 exp{-(W(-x + ft) - JV<0> dx , provided that W is in the event {dt-b£>\2e>l/e, (4.8) Hdx-bx)>\2ei -K <~\2s, b£<-\2e} . But by virtue of Lemma 3.1, (4.7) is equivalent in law to + (f +f * J exp{-0-{x)}dx , which is dominated by (4.9) f J\x\>i/e exp{-/3+(x)}dx + f exp{~f3~(x)}dx . J\x\>\/e H. TANAKA 764 (4.9) obviously tends to 0 as e i 0, P -a.s. Since g-measure of the event (4.8) equals ^-measure of the event {d+ - b+ > e,b+ > e,b~ - d~ > s,b~ < - e } , which clearly tends to 1 as e i 0, we obtain (4.6) for i = 2. The assertion (ii) of Lemma 4.1 now follows from (4.5) and (4.6). 4.2. We now proceed to the final part of the proof of Theorem 1.1. For any fj and tj, 1 ^ j' ^ k, with fj 6 CQ(U) and 0 < t\ < • • • < tk we are going to prove that (4.10) J E | n fj (**(',)) I Q(dW) - fu(W)Q(dW) as \ — oo, where um = flffjdfiW. By the equivalence in law of the processes (4.1) and (4.2), (4.10) is equivalent to (4.11) JE j n / y (y™Uj)) \ QW) - fu(W)Q(dW). Let {\»,n = 1,2,...} be any positive sequence such that \n — oo as n — oo. Lemma 4.1 enables us to use Skorohod's realization theorem of almost sure convergence. Thus, on a suitable probability space which we still denote by (fi, P\ we can choose a sequence {Wn} and W in such a way that the following hold: (4.12) {W„,P} is equivalent in law to {W,Q}. (4.13) {WyP} is equivalent in law to {W,Q}. (4.14) m„ converges vaguely to e~wdx while mn converges vaguely to const.6o (const. = / e~wdx) as n — oo,P-a.s., where m„ and m„ are defined below. The measure m„ is defined by equation (4.4) but with the replacement of W and * by Wn and \„. m„ = mn o S„l where S„ is defined by (4.3) with the replacement of W and \ by Wn and \n. Now, fixing the value of Wn, we denote by Yxn(t, Wn) the diffusion process with canonical scale Sn and speed measure mn and starting at x (the probability space on which Yxn{t, W„) is defined is still chosen as (ft,/*)). Then (4.14) and (i) of Lemma 4.1 imply that, with probability 1, the conditions (2.1), (2.2), and (2.3) are satisfied. (Strictly speaking, as for condition (2.1) we must take a suitable subsequence of {\„}, but this subsequence is still denoted by {\„}.) Thus Ogura's theorem is applicable to K£(f,MO: Putting 8n(tu--.^xyWn)^Elf[fj(Yxn(thWn))y DIFFUSION PROCESS 765 we have \imgn(tit...,tktx„tW„) =U(W) , P-a.s., n— oo provided that {Sn(xn)} is bounded. In proving (4.11), however, a difficulty arises since the initial value of the process (4.2) (with W and X. replaced by W„ and X„) grows so fast with X„ that the condition (2.5) is not satisfied. To overcome this difficulty we use Brox's result. Let A = A(W) be the i.d.a. of the valley (a,b,c) of W introduced in Section 3.1. Then A < 1,2-a.s. Let r be a constant such that 0 < r < 1 and suppose A{W) < r. Then there exists a valley (a\byc') of W with i.d.a. < r < depth, a ^ a',c' ^ c and containing 0. Therefore, by Brox's result stated in Section 3.3, X(eXr, \W) tends to b as X — oo in probability {W is fixed), and hence for each e > 0 / ' { K W l < X 2 e } - 1, (4.15) X^oo, where / is determined by \~4ext = ekr, namely, / = X4e"x*1-r). If we denote by vn(W,dx), the probability distribution of Y™(t) with X = kn,t as above, and W fixed, then (4.15) implies i/B(W, (-\2ne,\2e)) (4.16) - 1 , n - oo , if A(W) < r. Next we introduce £(W) = max lx < 0 : W{x + b) - W{b) = - } , rj(W) = min {*> 0 : W(x + b) - W(b) = - \ , and put £n = £,{Wn),rin = T){Wn). If £„ A fjn > e then it is easy to see that for 0 < e < 1/8, \Sn(±\2ne) | < 2\l f\xp{-\l/2} (4.17) dx < 1 . Jo Now, for any e > 0 small but fixed we choose r = r{e) € (0,1) so that Q{A < r} > 1 - e and then determine t by X~4exf = eXr as before. Putting r„, e = {A(W„) < r(e), e < £„ A 77J, we have the left-hand side of (4.11) = JdPJvn{Wn,dx)gn(tx -t,...ytk~t,x,Wn) (4.18) = -/r„,e dP uH(WB9dx) y(-x2e.x2e) x #„(>, - f , . . . , r * - * , * , # „ ) + tf„,£. H. TANAKA 766 Since P{r„, e } = Q{A < r(e), e < f A r?} — 1 as e 1 0, from (4.16) we have (4.19) lim lim/? n e = 0 . Moreover, Ogura's theorem implies that (4.20) sup \gn(t1 -t9...ttk-t9x,W„) -U{W)\~0, />-a.s., as n - oo. Therefore, it follows from (4.17M4.20) that the left-hand side of (4.11) tends to JU{W)dP = JU{W)Q(dW) as X - oo via {\ n }. Since { \ J is an arbitrary sequence with \n —• oo, this implies (4.11). The proof of our theorem is finished. Remark. In [5] a localization theorem was obtained for diffusion processes in a considerably wider class of random environments which are asymptotically self-similar. It will be possible to discuss this case by the present method but most of the essential arguments of [5] will not be simplified as much. Bibliography [1] Brox,T., A one-dimensional diffusion process in Wiener medium, Ann. Probab. 14, 1986, pp. 12061218. [2] Golosov, A. O., Localization of random walks in one-dimensional random environments, Comm. Math. Phys. 92, 1984, pp. 491-506. [3] Ito, K., and McKean, H. P., Diffusion Processes and Their Sample Paths, Springer-Verlag, New York, 1965. [4] Kawazu, K., Tamura, Y., and Tanaka, H., One-dimensional diffusions and random walks in random environments, pp. 170-184 in: Probab. Th. Math. Statist. 5th Japan-USSR Symposium Proceedings 1986, S. Watanabe and Yu. V. Prokhorov, eds., Lecture Notes in Mathematics No. 1299, SpringerVerlag, Berlin-New York, 1988. [5] Kawazu, K., Tamura, Y, and Tanaka, H., Localization of diffusion processes in one-dimensional random environment, J. Math. Soc. Japan 44, 1992, pp. 515-550. [6J Kesten, H., The limit distribution of Sinai's random walk in random environment, Physica 138A, 1986, pp. 299-309. [7] Ogura, Y, One-dimensional bi-generalized diffusion processes, J. Math. Soc. Japan 41, 1989, pp. 213-242. [8] Pitman, J. W., One-dimensional Brownian motion and the three-dimensional Bessel process, Adv. in Appl. Probab. 7, 1975, pp. 511-526. [9] Schumacher, S., Diffusions with random coefficients, pp. 351-356 in: Particle Systems, Random Media and Large Deviations, R. Durrett, ed., Contemporary Math. 41, 1985. [10] Sinai, Y. G., The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl. 27, 1982, pp. 256-268. [11] Tanaka, H., Limit theorem for one-dimensional diffusion process in Brownian environment, pp. 156172 in: Stochastic Analysis, Proc. Japanese-French Seminar, Paris, 1987, Lecture Notes in Mathematics No. 1322, Springer-Verlag, Berlin-New York, 1988. Received June 1992. 353 Diffusion Processes in Random Environments HlROSHI TANAKA Department of Mathematics Faculty of Science and Technology Keio University Yokohama 223, J a p a n 1 Introduction Problems concerning limiting behavior of random processes in r a n d o m environments have been discussed mostly in t h e framework of random walks (e.g., see [1], [4], [6], [15], [17], [23], [24]). Most of the problems, naturally, can also be treated in t h e framework of diffusion processes. We give here a survey of some recent results concerning diffusion processes in r a n d o m environments, mainly of one dimension, with emphasis on t h e following two examples of problems. T h e use of methods a n d results in theory of diffusion processes makes our argument transparent. (I) (IT) Localization by r a n d o m centering (depending only on t h e environment) of a diffusion in a one-dimensional Brownian environment. Limit theorems for a diffusion in a one-dimensional Brownian environment with drift. We also give a brief survey concerning (EI) results for a diffusion in a multidimensional Brownian environment. 2 A diffusion in a one-dimensional Brownian environment (with drift) Let P be t h e Wiener measure on W = C ( R ) n {W : W(0) = 0 } . T h e processes {W(t),t > 0, P} and {W(~t),t > 0, P] are thus independent Brownian motions. Let Q — C[0, oo) and denote by cj(t) t h e value of a function w(€ fi) at time t. For fixed W and a given constant K we consider a probability measure Pw on Q such t h a t {uj(t),t > 0 , P w } is a diffusion process with generator /;)r = VMA(e-"'Ml) 2 and starting at 0, where W(X,K) of X^/ = {cj(i),£ > 0,Pw} can scale change and a time change. t h e probability space (W x Q,V) X = {u>(t),t >,"P} is then called ax \ dxJ ~ W(x) — \KX. I t is well known t h a t a version be obtained from a Brownian motion through a We can regard W and {co(t),t > 0} as denned on where V{dWdw) = P(dW)Pw(duj). T h e process a diffusion in a Brownian environment (with drift Proceedings of the International Congress of Mathematicians, Zurich, Switzerland 1994 © Birkhauser Verlag, Basel, Switzerland 1995 354 1048 Hiroshi Tanaka if K ^ 0). Symbolically one may write duj(t) = dB(t) — ^Wf(oj(t), n)dt where B(t) is a Brownian motion independent of W{-)\ however, this stochastic differential equation has no rigorous meaning. When K — 0, X is a diffusion model of Sinai's random walk in a random environment [23]. In this case Schumacher [22] and Brox [3] showed that X exhibits the same asymptotic behavior as Sinai's random walk, namely, that the limit distribution of (\ogt)~2uj(t) as t —*• oo exists. Kesten [14] obtained the explicit form of the limit distribution. Golosov [7] also obtained a similar explicit form for a reflecting random walk model. Some generalizations of these results were done in [10] and [25]. The problem (I) stated in the introduction is to elaborate the result of [22] and [3] by taking account of a random centering that depends only on the environment W. This will be discussed in the next section. It is to be noted that a similar localization result was already obtained by Golosov [6] for reflecting random walks on Z + . The problem (H) is concerned with the case K ^ 0 and may be regarded as a diffusion analogue of what was discussed by Kesten-Kozlov-Spitzer [15], Solomon [24], and Afanas'ev [1]. Here we are mainly interested in limit theorems concerning the first passage time Tx = inf{£ > 0 : u(t) = x} as x —» oo. As will be seen in Section 4, the result varies with K and naturally is compatible with those of [15] and [1]. 3 Localization by random centering in the case K, = 0 The argument of [3] relies on the notion of a valley introduced in [23]; in order to state only the result, however, it is adequate to start simply with the definition of the "bottom" (denoted by b\) of a suitable valley around the origin. Given a Brownian environment W = {W(x),x 6 R}, let us define bx = b\(W) following [14] for each A > 0. Setting W*(x) = W(x) - min W, [xA0,xV0] d+ = minjo; > 0 : W*(x) = A}, V? - min W, [0,dj] dl = msix{x < 0 : W*(x) = A}, V~ = min W, we first determine b^ and b^ by W(b^) = V^ and W(b^) — V^", respectively (such bx are uniquely determined with P-measure 1 for each fixed A > 0), and then define 6A = bx (W) by X{ \b- ifM+V(V+ + A)>M-V(VA-+A), where Mf = max{W(x) : 0 < x < 6+} and M~ = max{VF(x) : 6~ < x < 0}. When A = 1 we write b — b(W) suppressing the suffix 1. We also define W$€ W) for each A > 0 and W e W by Wx(x) = \-1W(X2x),x e R. Then {WXlP} is equivalent in law to {W, P} and hence the distribution of b(Wx) is independent of A >0. 355 Diffusion Processes in Random Environments 1049 Let X = {o>(£),£ > 0,V} be a diffusion in a Brownian environment (K = 0) starting at 0. According to Schumacher [22] and Brox [3] X~2u(ex) ~ b(Wx) ^ 0 (3.1) in probability with respect toVas X —*• oo. Localization by random centering arises from the following question: Under what scaling does t h e left-hand side of (3.1) admit a nondegenerate limit distribution? T h e answer is simply t h a t uj(ex) — \2b(W$ does. To state t h e result more precisely we need to introduce another probability measure Q on W , defined in such a way that {W(x),x > 0, Qj and {W(—x),x > 0,Q} are independent Bessel processes of index 3 starting at 0. Let fiw be the probability measure in R of the form const. exp{—W(x)}dx; it is well defined for almost all W with respect to Q because exp(—W) e ^ ( R ) , Q-a.s. For an integer k > 1 we set ^ — u,w®- • -®u.w (the fc-fold product) and (xk — J u!^Q(dW). T H E O R E M 1 ([26], [28]). For any * i , . . . ,tk with 0 < t\ < • • • < tk the joint distribution of u(extj) - b\(W), 1 < j < k, with respect to V converges to /ifc as X —* oo. This theorem was proved in [26] for k = 1. T h e case k > 1 was proved in [28] by making use of Ogura's theorem stated below. Suppose we are given a sequence of diffusion operators d mn(dx) d dSri[x) n > 1, and denote by X*(t) the diffusion process with generator Ln starting at x. We assume t h a t t h e following conditions (i), (ii), and (iii) are satisfied. (i) 5 n ( 0 ) = 0 and Sn(x) tends to oo or —oo accordingly as x —• oo or x —* —oo; for each x, Sn(x) —• 0 as n —• oo. (ii) T h e measure mn converges vaguely to some nontrivial finite measure rn as n —> oo. (iii) T h e measure rrfn — rnn o 5 " 1 converges vaguely to C6Q as n —+ oo, where S^1 is t h e inverse function of Sn,c — m ( R ) > 0, and 6Q is the ^-measure at 0. O G U R A ' S T H E O R E M ([21]; see also [26]). For any e e (0,1) and an integer fc > 1 we set r M = { ( t i , . . . ( t f c ) € R f c : e < t i < t f c < l / e , t i - t j _ i >e (1 < Vj < k)} and consider a sequence {xn} satisfying \Sn(xn)\ Then for any continuous (3.2) functions n>l. fj in R with compact (3.3) supports, 1 < j < k, dmo as n —> oo uniformly in {xn} satisfying the condition T/fc e , where mo is the probability measure c~xm. (3.3) and in (*i,...,£&) G 356 1050 Hiroshi Tanaka It is known (see [3], Lemma 1.3) t h a t , for fixed W , t h e process {to(X4t),t > Q,Pw} is equivalent in law to {X2u)(t),t > 0 , P \ w x } . This combined with t h e fact t h a t bx(W) = X2b(Wx) implies t h a t the process {uj(ext) - bx(W),t > 0,PW} is equivalent in law to {X2(ui(X~4ext) — b(W\)),t > 0 , P X W x } \ in addition, W\ and W are identical in law. Therefore, for t h e proof of Theorem 1 it is enough to show JExw I n / i ( A 2 H A - 4 e A t O " b{W))) — j < n Jfjd^ \ ®(dw^ i P(dW) x (3-4) ^°°- For fixed W the generator of the diffusion process {X2(u;(X~4ext) 0 , P A w } is given by {d/mf\dx)}{d/dS^(x)}, where Sxv(x) mf(dx) = 2e~x [X exp{A(W(A- 2 2 / + b) - W(b))} Jo = exp{-X(W(X~2x + b) W{b))}dx. — b(W)),t > dy, L E M M A 1 ([28]). (i) S{v(x) tends to 0 as A -> oo with P -measure 1. (ii) If we regard m-Y and m^ = m™ o (S^)~l as random variables taking values in the space of Radon measures in R equipped with the topology of vague convergence, then the joint distribution (under P) of m^ and m^ converges to the joint distribution (under Q) of exp{—W(x)}dx and CW$Q as X —• oo where cw = Jexp{—W(x)} dx. Making use of L e m m a 1 and Ogura's theorem we can prove (3.4) and hence Theorem 1. For details see [28]. A similar localization problem was discussed in [11] when {M^(a;)} is a step process arising from a r a n d o m walk t h a t is assumed to converge in law, under a suitable scaling, to a strictly stable process. 4 Limit theorems in t h e case n ^ 0 Let X = {uj(t),t > 0,V} denote the diffusion in a Brownian environment with drift (K 7^ 0), and set Tx = inf{t > 0 : to(t) = x},ZJ(t) = max{o;(s) : 0 < s < t} and co(t) = inf {u(s) : s > t}. T H E O R E M 2 ([13]). (i) If K > I, then 4 lim Tx/x lim t^oo u(t)/t l converges to 4 in probability (ii) If K = 1, then (xlogx)~ Tx and each of t~x (log t)u(t), converges to 1/4 in probability K - l 4 t-1(logt)a;(t) and (w.r.t. V) as t —* oo. t"1 (log (w.r.t. V) as x —> oo t)u(t) 357 Diffusion Processes in Random Environments 1051 (iii) If 0 < K < 1, then lim V{x~^KTx t> 0, X—'OO K lim £>{£~ w(£) < x} = lim 7>{t~Kw(t) < x} t—*oo t—+oo = lim ? { t " V ( t ) < a;} = 1 - F j o T 1 / * ) , x > 0, where FK is the distribution function of a one-sided stable distribution with Laplace transform exp(—cXK). REMARK. The constant c in the Laplace transform exp(—c\K) is given by J21"T(K) J where (x) is the solution of ^ ^ • g - 2^,^(0) = l,'(0) = 0;M(x) is given by M(x) = 27(p _ 1 (x)), where 7(x) = J^ z~Ke~Azdz and p~1(x) is the inverse function of p(y) ~ j ^ zK~leAz dz. Theorem 2 is a diffusion analogue of (a part of) the results for random walks due to Kesten et al. [15]. We do not give a detailed proof here but we remark that our method of the proof, in particular, of (ii) and (iii) is different from that of [14] and is based on the following lemma due to Kotani. KOTANl'S LEMMA (1988, unpublished; see [13]). Let A > 0- Then for t > 0 Ew {e-XTt} = exp j - / Ux{s) ds\ , V - a.s., where U\{t) is the unique stationary positive solution of dUx(t) = Ux(t)dW(t) + J2A + ^-Ux{t) ~ Ux(t)2 j dt. By virtue of Kotani's lemma, for the proof of (iii) it is enough to show that, with X = C^1/K: lim £ \exp(-€Tx/xVK)\ = l i m £ I exp | - j Ux(s) ds)\ = expf-cf*), and the key point in proving the last equality is the use of Kasahara's continuity theorem [9] concerning Krein's correspondence (e.g. see [16]). A full proof is given in [13]. The following theorem is a diffusion analogue of the result of Afanas'ev [1]. 358 1052 THEOREM Hiroshi Tanaka 3 ([12]). (i) If-2 < K < 0, then P{TX < oo} ~ const.x~3/2exp(-K2x/8), x —> oo, where oo oo oo oo 0 0 0 0 (A = 2" 1 (l + y 2 ) + ycoshu). (ii) If K = - 2 , then V{TX < oo} - (2/7r) 1 / 2 x- 1 / 2 exp(-x/2), a: -> oo. (iii) If K < - 2 , then V{TX < oo} ~ —K. — 2 • exp{(« + l)z/2}, a: -> oo. — K, — 1 The proof of (i) relies on an explicit representation of the distribution of a certain Brownian functional due to Yor ([29], see the formula (6.e)). 5 A diffusion in a multidimensional Brownian environment One generalization of the model discussed in Section 3 to a multidimensional case is to take a Levy's Brownian motion with a multidimensional time as an environment. Let {W (x), x EHd, P} be a Levy's Brownian motion with a d-dimensional time that is supposed to be an environment. For a frozen Brownian environment W let X\y — {u)(t),t > 0,Pw} be a diffusion process with generator 2 _ 1 (A — VW - V) starting at 0. Existence of such a diffusion is guaranteed by the result of Nash ([20]). As in a one-dimensional case we call X — {cj(t),t > 0,"P} a diffusion in a ddimensional Brownian environment, where V(dWduj) — P(dW)P\y(duj). A similar diffusion model appeared in a heuristic argument of [18]. Durrett [4] obtained rigorous results on recurrence and localization for random walks on Zd described by a certain random potential having asymptotic self-similarity and stationary increments. The diffusion X may be regarded as the continuous time analogue of what was discussed in Example 2 (/3 = 1) of [4]. Recently Mathieu [19] considered the diffusion X itself and discussed its long time asymptotic behavior. THEOREM 4 ([27]; see also [4] for random walks). X ^ is recurrent for almost all Brownian environments W for any dimension d. This theorem can easily be proved by making use of Ichihara's recurrence test ([8], see Theorem A) concerning symmetric diffusions. We can also use Fukushima's recurrence criterion [5] in terms of the associated Dirichlet form. X|w| is also recurrent, P-a.s.; however, X iiv| 1S transient, P-a.s., for any d > 2 as can be proved by using Ichihara's transience test ([8], see Theorem B). From the argument of [27] it is also easy to see that Theorem 4 remains valid when {VF(a;)} is replaced by any continuous random field {V{x)} in R d satisfying the following conditions (i), (ii), and (iii). 359 Diffusion Processes in Random Environments 1053 Self-similarity: there exists a > 0 such that t h e law of {A~1V(\ax)} equals t h a t of {V(x)}, denoted by P , for each A > 0. (ii) {Tt,t € R } is ergodic, where Tt is a P-preserving transformation from C(Rd) onto itself defined by (TtV){x) = e~t^aV(etx),x £ Rd. (Hi) min-jV(x) : \x\ = 1} > 0 with positive probability. T h e argument of [19] entails the following theorem. (i) T H E O R E M 5 ([19]; see also [4] for r a n d o m walks). Localization in the sense that lim takes place for X Em P { A " 2 m a x ( | u j ( t ) | : 0 < t < e A ) > N} = 0. N—*oo A—t-oo It seems t h a t there is no proof of the existence of the limiting of A~ 2 w(e A ) as A —*• oo. It is to be noted, however, t h a t Mathieu existence proof together with an explicit representation of t h e limiting of \~2u(ex) in terms of t h e local time of \W\ at level 0 when W is distribution [19] gave t h e distribution replaced by \w\. T h e above results on recurrence and localization rely heavily on t h e (asymptotic) self-similarity of W as well as the symmetry of X ^ - W i t h o u t these conditions the situation will change much. In t h e case of random walks there is a profound work by Bricmont and Kupiainen [2]. References [I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [II] V. I. Afanas'ev, On a maximum of a transient random walk in random environment, Theor. Probab. Appl. 35 (1990), 205-215. J. Bricmont and A. Kupiainen, Random walks in asymmetric random environments, Comm. Math. Phys. 142 (1991), 345-420. T. Brox, A one-dimensional diffusion process in Wiener medium, Ann. Probab. 14 (1986), 1206-1218. R. Durrett, Multidimensional random walks in random environments with subclassical limiting behavior, Comm. Math. Phys. 104 (1986), 87-102. M. Fukushima, On recurrence criteria in the Dirichlet space theory, in Local Times to Global Geometry, Control and Physics (ed. Elworthy), Research Notes in Math., Longman Scientific & Technical, Essex 150 (1986), 100-110. A. O. Golosov, Localization of random walks in one-dimensional random environments, Comm. Math. Phys. 92 (1984), 491-506. A. O. Golosov, On limiting distributions for a random walk in a critical one-dimensional random environment, Russian Math. Surveys 41 (1986), 199—200. K. Ichihara, Some global properties of symmetric diffusion processes, Publ. Res. Inst. Math. Sci., Kyoto Univ. 14 (1978), 441-486. Y. Kasahara, Spectral theory of generalized second order differential operators and its applications to Markov processes, Japan. J. Math. (New Ser.) 1 (1975), 67-84. K. Kawazu, Y. Tamura, and H. Tanaka, One-dimensional diffusions and random walks in random environments, Lecture Notes in Math., Springer-Verlag, Berlin and New York, 1299 (1988), 170-184. K. Kawazu, Y. Tamura and H. Tanaka, Localization of diffusion processes in onedimensional random environment, J. Math. Soc. Japan 44 (1992), 515-550. 360 1054 Hiroshi Tanaka [12] K. Kawazu and H. Tanaka, On the maximum of a diffusion process in a drifted Brownian environment, Lecture Notes in Math. (Seminaire de Probabilites XXVII), Springer-Verlag, Berlin and New York, 1557 (1993), 78-85. [13] K. Kawazu and H. Tanaka, A diffusion process in a Brownian environment with drift, to appear in J. Math. Soc. Japan. [14] H. Kesten, The limit distribution of Sinai's random walk on random environment, Phys. A 138 (1986), 299-309. [15] H. Kesten, M. V. Kozlov and F. Spitzer, A limit law for random walk in a random environment, Compositio Math. 30 (1975), 145-168. [16] S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, Lecture Notes in Math., Springer-Verlag, Berlin and New York, 923 (1982), 235-259. [17] A. V. Letchikov, Localization of one-dimensional random walks in random environments, Soviet Sci. Rev. Sect. C: Math. Phys. Rev. 8 (1989), 173-220. [18] E. Marinari, G. Parisi, D. Ruelle, and P. Windey, On the interpretation ofl/f noise, Comm. Math. Phys. 89 (1983), 1-12. [19] P. Mathieu, Zero white noise limit through Dirichlet forms, with application to diffusions in a random medium, Probab. Theory Related Fields 99 (1994), 549-580. [20] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931-953. [21] Y. Ogura, One-dimensional hi-generalized diffusion processes, J. Math. Soc. Japan 41 (1989), 213-242. [22] S. Schumacher, Diffusions with random coefficients, Contemp. Math. 41 (1985), 351356. [23] Ya. G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium, Theor. Probab. Appl. 27 (1982), 256-268. [24] F. Solomon, Random walks in a random environment, Ann. Probab. 3 (1975), 1-31. [25] H. Tanaka, Limit distribution for 1-dimensional diffusion in a reflected Brownian medium, Lecture Notes in Math. (Seminaire de Probabilites XXI), Springer-Verlag, Berlin and New York, 1247 (1987), 246-261. [26] H. Tanaka, Limit theorem for one-dimensional diffusion process in Brownian environment, Lecture Notes in Math., Springer-Verlag, Berlin and New York, 1322 (1988), 156-172. [27] H. Tanaka, Recurrence of a diffusion process in a multidimensional Brownian environment, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 377-381. [28] H. Tanaka, Localization of a diffusion process in a one-dimensional Brownian environment, Comm. Pure Appl. Math. 47 (1994), 755-766. [29] M. Yor, On some exponential functional of Brownian motion, Adv. Appl. Probab. 24 (1992), 509-531. 361 Environment-wise central limit theorem for a diffusion in a Brownian environment with large drift Hiroshi Tanaka Department of Mathematics, Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama, Kanagawa 223, Japan Introduction General theory of one-dimensional diffusion process was established by Ito and McKean ([1]) more than thirty years ago. In this paper we discuss the central limit theorem concerning a diffusion process in one-dimensional Brownian environment with large drift. Many of our methods benefit from the theory by Ito and McKean. By a diffusion process in a Brownian environment with drift we mean a process X z ° = {u(t),t > 0, Vx°} which is denned in the following way. (i) W is the space of continuous functions on R vanishing at 0 and is endowed with the Wiener measure P. For an element w of W, wK denotes an element of W defined by wK(x) = w{x) — KX/2 where AC is a given positive constant, (ii) For w € W and xo € E, P£° denotes the probability measure on ft = C[0,oo) such that X*° = {u(t),t > 0,PX°) is a diffusion process with generator u ; ~2e dx{G dx} starting from x 0 ; w(t) denotes the value of a trajectory w(6 ft) at time t. (iii) {w(£), t > 0} is regarded as a process defined on the probability space (W x ft,VXo) where Vx°(dwdw) = P(dw)P^°(duj). This process is what we denote by X 1 0 = {<*>(£), t > 0,VXo}j x0 indicates the starting point of our process. The process X Xo is a diffusion analogue of what was considered by KestenKozlov-Spitzer [4] and Solomon [6] as a random walk in a random environment. Asymptotic behavior of the process X° as t becomes large was investigated by Kawazu-Tanaka [3]. The results are analogous to those of [4] [6] and vary with K. In particular, when K > 1, it was proved that lim Tx/x ~ m, lim u(t)/t x—*oo = 1/m, 7?°-a.s., t—*oo where m = 4(/v- 1 ) _ 1 and Tx is the hitting time: Tx = inf{t > 0 : w(t) = x}. The next problem is then to study a fluctuation. In this paper we assume K > 2 and discuss central limit theorems concerning Tx and u;(i). We p u t 374 H. Tanaka ew*M-w"Wdz, Mx = 2 / dy f JO J-oa fj,(t) = t h e inverse function of Mx, uJ(t) = max{w(s) : 0 < s < £}, u(t) = inf{u>(s) : s > t}-, U(K-1)'2(K-2)~1. A = T h e n it will b e seen t h a t E%{TX} = Mx, E{MX} = mx and E { V a r J , ( T a ) } = Ax for a; > 0 where J57JJ, and VarJ, denotes t h e expectation and variance u n d e r P £ , respectively. O u r results are t h e following where n > 2 is assumed. E n v i r o n m e n t - w i s e c e n t r a l limit t h e o r e m (I) For almost all w with respect to P the process 5 i ^-""-} converges in law to a Brownian motion as X —* oo (in the sense of of probability measures on the Skorohod space). (II) For almost all w the process u(Xt) - fi(Xt) convergence -} converges in law to a Brownian motion as X —* oo. The same is true uj(Xt) is replaced by either ofU(Xt) and tj(Ai). when Central limit theorem in r a n d o m environment, n a m e l y C L T under V° will b e discussed in a n o t h e r occasion. §1. Some preliminaries 1 . 1 . We c o m p u t e t h e first and t h e second m o m e n t s of various quantities related t o h i t t i n g times for X ° . Let T0i(, = inf{t > 0 : uj(t) = a or 6}, b < a. T h e n it is well-known t h a t u(x) = i=£{exp(-A / °'b Jo f(u,(t))dt)} satisfies (e.g. see [1]) , * J (Xf - £w)u = 0 in (6, a), { } \ * ( a ) = *(&) = 1. F o r n > Owe p u t un(x) = £*{(J 0 T o , b f (u(t))dt)n}. T h e n u(x) = ^=0un(x)x (~X)n/n\ a n d (1.1) imply E ^ o W ~ A i ^ M - * ) " / ™ * = °- Therefore we have uo = 1 and for n > 1 A diffusion in a Brownian environment J I <X"2) Cwun = -nun-if in (6,a), «n(a)=«n(6)-0. Let 5 w (x) = J0X exp{wK(y)}dy, (1.3) 375 the canonical scale of Cw. Then S™(-oo) = - c o , P-a.s., if K > 0. The following formulas (1.4) and (1.5), which hold for almost all w, can be proved under the assumption n > 0 by making use of (1.2) and (1.3): For x < a, (1.4) = 2 rew^dy K{Ta} (1.5) f £ { T * } = 8 ["ewMdy J — oo e~w^z)dz f i/X e-w«W f «/x ew*(u)du f J — OO e~w"(v)dv. f JZ */ —OO Making use of the above formulas we can compute VarJ,{Ts} for x > 0; the result is (1.6) Vful{Tx} = & f*dy J0 ew*M-w*Wdz I* J — oo [*dy [" «/z J—oo e ^ ^ - ^ W It will be useful for our later discussions to introduce a one-parameter family of measure preserving transformations 6f>t e R, on {W,P) defined by (0tw)(x) — w(x-ht) — w(t),x € R- It is easy to see that dt&a = &t+3 and {&t} is ergodic. If K > 0, then (1.7) e~w^dt /oW = / J ~ OO is finite (P-a.s.) and 0 t /o = /o(^t^) = J^e^^-^^ds. (1.6) we have for x > 0 (1.8) From (1.4) and [Xeyf0dy, E°W{TX} = MX = 2 Jo (1.9) V a r i { T . } = 8 f dy f JO = 8J*evgdy, Making use of E{exp(wK(x) e"-<»>—Mp./ofdz J—oo L{w) = J_ e—'l'Hetfofdt). — wK(y))} = exp{—7(3; — 2/)} for x > 2/ where 7 = («-l)/2, we have £ { / o } = 1/7 and hence ^ { M j } = m i , x > 0, if K > 1 (note that E{fo} < 00 iff K > 1 and E{f02} < 00 iff K > 2). From now on we 376 H. Tanaka assume K > 2. The constant A given in the introduction is expressed as A = 1 6 7 " 2 ( 2 7 - I ) " 1 . We put / = / 0 - 7 - 1 . Then we have the following: E{h2} (1-10) (1.11) E{fOxf} = 27-1(27 - I ) " 1 ; = 7~2(27 - l)-3e-7X E{VaT0w(Tx)} = Ax (1.12) for x > 0; for x > 0. 1.2. The process {0 x /o,£ G R, P } will often play an important role in our discussions. We make use of habitual notation t (instead of x) to indicate time. By an application of Ito's formula we have (1.13) dBth = Otfodw(t) - (76>t/o - l)dt, and hence $tfo is a stationary diffusion process obtained as the unique stationary positive solution of the stochastic differential equation. It can be also written as (1.14) 0tfo -fo=f Osfodw(s) - 7 / Jo 0sfds. Jo From now on we assume K > 2. Then we can easily prove that E{fo2 } < 00 if 0 < 6 < K/2. Therefore by the Burkholder-Davis-Gundy inequalities we have f (1.15) E < max 1 r "1 / esfQdw(s) \ < const.**, (t > 0,1 < 6 < K / 2 ) , where const, means a constant that may depend on 6 but not on i; such const, will also appear in later discussions and may vary from place to place. L e m m a 1. i - 1 / 2 m a x { 0 a / o : |s| < t} —> 0 as t —* 00, P-a.s. Proof. Since the stationary diffusion Offo is reversible, it is enough to prove that i" 1 / 2 max{0 a /o : 0 < s < t} —» 0 as £ —>oo, P-a.s. Define stopping times (Tn, 71 > 0, by <70 = inf{t > 0 : 9tfo = 2} and an — the time of first return of &tfo to 2 visiting 1 after 1) and then consider the random variables Xn = max{^/o : o"n_i < t < crn},71 > 1. Then Xn,n > 1, are i.i.d. random variables. Since an/n —+ const.> 0 (a.s.), Um t - 1 ' 2 max 83fo t—»oo = 0<5 lim o" n - 1 ' 2 max Xk n—>oo = l const. Um n~ ' n—»oo max X&-, a.s., l and the rightmost hand of the above equals to 0, a.s., because P{XX >x} = {5(2) - S(l)}{S(x) - S ( l ) } - 1 - const.x-*,z - • 00, (here S(x) is the canonical scale of the diffusion 0t/oJ it is given by S'(x) = xK~l exp(2/x),a; > 0). This proves the lemma. A diffusion in a Brownian environment 377 L e m m a 2. For any ci > 0 Mt+U -Mt = mu(l + o(l)) + o(VX), |i| < where o(l) represents a general term that tends to 0 as \ —> oo uniformly in (£,u) sxich that \t\ < cjA and u € M, /or almost all w; o(vA) ifi a /erm //ia£ can 6e expressed as o(l)vA. Proof. Stepl. For any positive constants ci and c 2 (i.i6) riM(+„-M( m :it|c2\/Ai sup < —»• 0 a s A - > oo, P-a.s. We give a proof in the case C\ = ci = 1 since the case of general Ci and c 2 can be treated without any essential change in the proof. We use the notation sup (x) to denote supremum taken over all t, u satisfying \t\ < A and u > yf\. Since M[/2 = &th 1S a stationary reversible process, it is enough to prove that (1.17) - / SUP (A) ejds 0 as A —* oo, a.s. By Lemma 1 we have sup ^x)^~1\^tfo — &t+ufo\ —» 0 as A —• oo, a.s. Combining this fact with (1.14) we see that, for the proof of (1*17), it is enough to show that r+ i (1.18) sup (A) - / \ u 93f0dw(s) 0 as A —• oo, a.s. Jt To prove (1.18) we take 6 and a such that (1.19) 1 < 6 < K/2, 0 < a < 1/2, 6/2 + a - 1 > 0, and prepare the estimate E{R{\)26} (1.20) < const.A'* 1 , where Si — 6/2 + a — 1 and ft+u x R(t,u) = -u / Oafodw(s), jR(A) = sup ( A ) |i?(t,t*)|. Jt Putting Ik = [&Aa,(fe + l)A a ] for an integer k and denoting by sup(jt^) the supremum over all t 6 h and u > A/A, we write suV{k,x)\R(t,u)-R(k\a,u)\ >t+u - suP(fc,A)~ ~ / Qsfodw(s)+ u I JkX" / 63fodw(s) Jk\a+u 378 H. Tanaka where a .k\ fie* U = \~1/2 V = sup max 0 max +r- j - r / Jk\a 0sfodw(s) - / u 0 O3f0dw(s) Jk\~+u Making use of (1.15) we have E{U26} (1.21) and E{V26} < const.A-* 1 - 0 ^, is dominated by 26 •JfcA°+ii+r sup VXn n=l - / max 0*fodw(s) 0 < A" 6 ] T n~26E{Rl6} = const.\~6 E{Rf}, n-l where Ogfodw M I If 0 , where lx = [ A * _ a ] . Therefore i2> < 2 max "~ and hence E{V26} 0 max / •JA"+i Osfodw(s) 0 is dominated by 26' const.A~*(JA + l ) £ < max / 9sf0dw(s) [0 < c o n s t . A - ( 1 - a ^ ^ D - ^ (usc(1.15)). This combined with (1.21) implies (1.22) # { s u p ( M ) \R(t,u) - R(k\",u)\26} ^conat-A^1-0^-1)-*, because (1 — a)6 > (1 — a)(S — 1) + | . Let fc(t) be the integer determined from t by k(t)XQ < t < (k(t) + l)A a and put Kx = {k(t) : |t| < A}. Then #KX < const.A 1 "" and (1.22) implies (1.23) # { s u p ( A ) \R{t,u)~ < co*st.#Kx R(k(t)\Q,u)\26} • \-b-«W-U-i < const.A"*1 A diffusion in a Browman environment 379 because (1 - a) (6 - 2) + \ > 6%. By a similar method we can prove t h a t E {sup ( A ) \R(k(t)Xa,u)\26^ (1.24) < const.A" 51 and hence (1.20) follows from (1.23) and (1.24). We finally prove (1.18). If j3 is a constant such that p6\ > 1, then E{R(nfi)26} < const.ra"^ 1 by (1.20) so by using the Chebyschev inequality and the Borel-Cantelli lemma we see that R(nP) - • 0 as n —* co, a.s. Likewise Re{nP) ~ s u p | | i 2 ( i , u ) | : \t\ < cn^.u > nP*2\ -> 0, a.s., for any constant c > 0. The assertion (1.18) now follows from R{X) < Rc(n(\)P) with c = 2^ where n(X) is the integer satisfying n(\)P < A < (n(A) + i y . Step 2 is to complete the proof of the lemma. The result (1.16) implies that (1.25) Mt+U -Mt = (m + o(l))u, if |t| < cx\ and |u| > \/A/2. When t and u are restricted to \t\ < c*A and \u\ < \/A/2, we write Mj+ U — Mt = M' + M" where Then applying (1.16) we have jtf' = (m + o(l))(VX + u) and M" = — (m + o(l))VA. Therefore M t + „ - Mt = mu(l + o(l)) + o(VX) if |t| < q A and |u| < \/A/2. This combined with (1.25) proves the lemma. §2. Environment-wise central limit theorem 2 . 1 . P r o o f of ( I ) . We put rk = Tfc - Tfc_a,Jfc > 1. Then rfc,yfc > 1, are independent under the probability measure J * . Since T n = £ " T>., the central limit theorem for Tn with fixed w can be obtained by verifying the Lindeberg condition. We have Var°„{T„} = £ Vaiifa} = £ V^-ito) where V(w) = Var£,{Ti}. We now assume K > 2 to ensure the existence of E{V}7 which is equal to A by (1.3). Since {6t} is ergodic, we have (2.1) Urn Var " { r " } = A, P-a.s. 380 H. Tanaka P u t t i n g fk = rfc - El{rk} and VN(w) = £ ° { | T i | 2 ; | r i | > N} ( t h e integral 2 of | f i | over {|fi| > N} with respect to P £ ) , we have for any e > 0 a n d for almost all w (2 2) vax^} g £" { N 2 ; N > 'A"^")} - < ^ X X { N 2 ; N > ^ } (by (2.i)) const. n ^ J Vjv(^fc-i^) —• const.^?{V^}» n —> oo. Since -E-fVjy} | 0 as N | oo because of E{V} < oo, t h e left h a n d side of (2.2) t e n d s t o 0 as n f oo ( P - a . s . ) , which shows t h a t t h e Lindeberg condition is satisfied for {ffc}. Therefore for almost all w t h e central limit theorem holds for T n with respect t o P £ . B y (1.4) E^{Tn} = Mn and by (2.1) V a r ^ { T n } ~ An,n —* oo ( P - a . s . ) . Therefore t h e Lindeberg condition t o g e t h e r w i t h an application of Theorem 3.1 of [5] implies t h e assertion of (I). F u r t h e r details will be given in a joint p a p e r with K. Kawazu. 2 . 2 . P r o o f o f ( I I ) : T h e probability measure we consider in this proof is P £ where w is t a k e n from some subset of W t h a t has P - m e a s u r e 1. T h u s events we consider here can b e regarded as subsets of !2o = {<*> € Q : w(0) — 0 } . For any s > 0, p € R and v > 0 we have, with convention &(oo) = oo7 (2.3) {T p + < *} = {&{s) >p}D D {T(p+„)+ {u>{s) >p}D {u{s) > p) <s}n{u(Tp+l,)>p}, where p+ = m a x { p , 0} and Tx denotes t h e hitting time to x 6 R. Similarly (2.4) {Tp+ > s} = {o;( 5 ) < p) C {«(*) < p} C {«(*) < p} C {T(p+W+ >s}U{o;(r/H.|/) L e m m a 3 . For any fixed t\ > 0 Zei ( functions (p and ip are absolutely continuous, (pr,tp' € L2[Q, ti], (t) for all t 6 [0, h] and Then for almost all w (2.6) lim^co PZLMK^&Lj^KtWforaUteKtA I Vm-3AX J P{v?(i) < w(i) < V(t) / o r ai/ t G [to,*i]}- A diffusion in a Brownian environment 381 Proof. For the proof it is enough to consider the case y(0) < 0 < tp(0). We put p(t) - fi(Xt) + y(t)y/m-3AX and p^t) = p,(Xt) -f i>(t)Vm-3AX. Then pi(t) > 0 for all t € [0,*i] if A is sufficiently large, but p(t) < 0 if t is sufficiently small for each fixed A. Since {y>(i) < (ra~ 3 AA) _1 / 2 (u;(At) //(At)) < ip(t)} = {/>(«) < w(Xt) < Pi(t)}, an application of (2.3) and (2.4) yields {T(p(t)+v)+ < Xt,Tpi{t) 3 > Xt,ui(Tp(t)+v) > p(t)} 1 2 C W(t) < (m" ^A)- / (o;(A*) - fi(Xt)) {t)} C {Tp{t)+ < Xt] n [{TpM+1/ > Xt] U fe(!Tft(t)+J < tuit)}]. Therefore if we put rx F x = = + A ^A -#A = = = {tp(t) < (m-3AXy^2(u;(Xt) T i (p(t)+»)+ < ^TpM T + - fi(Xt)) < j,(t) for all t € [0, t ^ } , > Xt for all t € [O,^]}, At r { ,(*) < ' M*)+* > A i f o r ^ * € [0, *!]}, {w(r p ( t ) + I / ) > />(*) for all t e [0, t!]}, {u(Tpi{t)+J/} < P\{t) for some t 6 [0, ^ ] } , then (2.7) rA-nAAcrAcr+u5A. Put 1/ = 6 log A and let us prove (2.8) Urn P^{A\} = lim P^{BX} A—*oo Since {u(Tp(t)+l/) = 0, P-a.s. A—*oo < p(*)} C { w ( r W t ) + v ] ) < [p(t) + v]-{v- 4c[J where p~ = min{[p (t)+v] :0 1)}, we have fe(r*)<*-(*-i)}, and p+ = max{[p(<)+i/] : 0 < t < t-i}. By an ergodic theorem Mx/x —* m as a: —• co (P-a.s.) and hence {i(t)/t —> m _ 1 as i —»• oo (P-a.s.), which implies p\* ~ A?7i_1ti and p\~ = o(A) as A —* oo (P-a.s.). Therefore for the proof of (2.8) it is enough to show that for any constant c > 0 lim Y, J2{ttPl)<*-("-l)} = 0, P-a.s. 382 H. Tanaka Taking the expectation we have El J2 [\k\ P°w{yi(Tk) \k\ < const.Aexp{—(/v — l)(v — l)/2} < const.A -2 , where we used the result of [2]. The above estimate holds for v = 6 log A + O(l). Therefore an application of Borel-Cantelli lemma implies lim J2 PH{uL(Tk) = 0, P-a.s., \k\ for any given sequence {vn} thus have sucn that i/n = 6logn + O(l) as n —* oo. We |fcj |fc| -» 0 as A -* 0 (P-a.s.), which implies the assertion (2.8) concerning Acx. The assertion for B\ can be proved similarly. To proceed let £ = Am" 1 and put a{t) = CXPW = A _ 1 ™ {**(**) + tp{t)Vm~3A\\ , P(t) = r V i W = A _ 1 m {//(At) + V'WVm-MA} . Then for almost all w both a(t) and /?(£) tend to t as A —» oo, the convergence being uniform on each finite t-interval. From now on we write T(x) and M(x) for Tx and Mxy respectively. Since //(At) = £a(t) - y?(t)V'm~3J4A, an application of Lemma 2 yields (2.9) At = M(£a{t) - = M(£a(t)) - + o(v/A)) where o(vA) is a term that, when divided by vA, tends to 0 uniformly in t £ [0, ii] as A —* oo for almost all w. Similarly (2.10) At = M(£/3(t)) - ^ ( t ) V m - M A + o(\/\). In what follows the notation o(l) represents a term that tends to 0 uniformly in t € [0,tx] as A —• oo for almost all w. Using (2.9) and (2.10) we can obtain the following (2.11) ~ (2-14) that hold for almost all w: 371 A diffusion in a Brownian environment (2.11, T ( ^ ) < A ^ (2.12) r ( ^ ) + ) ^ W + ) T(*(*» > At o nW))-£M)) (2.13) r((p(0 + i')+) < - , ( > f _m ) + 383 o(l); + o(1); < At * » r ((/>(*)+ ») + )--M(teW) < -(p(t)Vm- 1 A\ + o(\/A) r(eHt) + f) + )-M($( a (i) + |)+) <- T ( ( ^ ( t ) + «/))> At «• ^ ^ ^ ^ > -m+o(i). In deriving the second equivalence in (2.13) as well as the equivalence (2.14) we again used Lemma 2. From (2.11) ~ (2-14) we have the following: For almost all w, P^i^x} is equal to pw(£) where pw{0 is the probability, evaluated by P£, of the event ( {AQ-W 1 {T{ta(t)+) - M « a ( t ) ) + } < - 1 1 ] { (^)- ^{T(fo?(*) + v r ))--^«w*) + ^- ))}>-*(t)+o(i), [. ( for all t e [0,ti] J But by (I), for almost all w, pw(£) tends to = P{-0(t) < w(i) < -y?(t) for all t e [0,«!]} P{v>W < w(t) (t) for all i G [O,^]} a s ^ - K X ) (here we used the assumption (2.5)). We also have a similar statement for P^ir^}. Therefore from (2.7) and (2.8) we finally obtain Lemma 3. It is also to be noted that, by virtue of (2.3) and (2.4), Lemma 3 remains valid when w(-) is replaced by either of <«;(•) and w(-). For fixed ti > 0 we denote by F the set of all pairs ((p, ift) of functions on [0, ti] of the form (p = p\ Vp2 V • • • Vp n (maximum) and ip = qx A q2 A • • • A qn (minimum) where pj and qj are polynomials with rational coefficients satisfying pj(t) < qj(t) for all t € [0,ti],pj(Q) # 0,gy(0) # 0(1 < j < n). Since F is countable, Lemma 3 implies that for almost all w (2.6) holds for all ((p, ip) 6 F. Fixing an arbitrary w for which (2.6) holds for all ( 384 H. Tanaka admissible if Q\{A) —* Q(A) as A —*• oo. T h e n for each pair ( y , ^ ) G ¥ t h e open set U() = {w 6 C[0,*i] : y>(t) < w(t) < ^ ( i ) for all t e [0,*i]} is admissible. On t h e other h a n d any open set G in C[0, t\] can b e expressed as U^=lUn where each Un is of t h e form U(ip^tp) with (y?,^) 6 F . T h e intersection of any finite n u m b e r of U^s is admissible because it is still of t h e form £7(^,-0), so Gn = U j = i y f c is admissible because of t h e inclusion-exclusion formula n QA(G„) = X ; ( - 1 ) * " 1 k=l £ Qx(uhnui2n---nuih). l Since (?„ | G we have lim A tcc Q\(G) > Q(G)y which implies t h a t Q\ —> Q as A —* oo. Since t x > 0 is arbitrary, this completes t h e proof of (II). References [1] Ito, K., and McKean, H.P., Diffusion Processes and Their Sample Paths, Springer-Verlag, New York, 1965. [2] Kawazu, K., and.Tanaka,H., On the maximum of a diffusion process in a drifted Brownian environment, Seminaire de Probabilites, LMN 1557, 78-85. [3] Kawazu, K., and Tanaka,H., A diffusion process in a Brownian environment with drift, to appear in J.Math.SocJapan. [4] Kesten, H., Kozlov, M. V., and Spitzer, F., A limit law for random walk in a random environment, Composito Math. 30 (1975), 145-168. [5] Prohorov, Yu. V., Convergence of random processes and limit theorems in probability theory, Theor. Probab. AppL 1 (1956), 157-214. [6] Solomon, F., Random walks in a random environment, Ann. Probab. 3 (1975),131. Reprinted from Ito's Stochastic Calculus and P r o b a b i l i t y T h e o r y , 373-384, Springer-Verlag, 1996. J. Math. Soc. Japan Vol. 49, No. 2, 1997 A diffusion process in a Brownian environment with drift* By Kiyoshi KAWAZU and Hiroshi TANAKA (Received Feb. 8, 1995) Introduction. Let W be the space of continuous functions W defined in R and vanishing at the origin. Let P be the Wiener measure on W, namely, the probability measure on W such that \W(t), t>0, P\ and {W(-t\ t^O, P\ are independent one-dimensional Brownian motions. Let Q=C[0, oo) and denote by co(t) the value of a function a> (ei?) at time t Given a sample function W (eJF) and a nonnegative constant ic we consider a probability measure Pfv on Q such that \(o(t), t^O, Pw) is a diffusion process with generator 1 ,„ ,,, d / _„, ,_, d \ dx dx mw(dx) dSw(x) starting from x, where (2) WK(X) = W(X)-JKX, (3) Sw(x) = [*ew' mw(dx) = 2e'w^x>dx. It is well-known that a version of { dX(t) - dB(t)~W'JiX{t))dt, * This work was partially supported by Grant-in Aid for Science (No. 07454034), Ministry of Education of Japan. 190 K. K A W A Z U and H. T A N A K A where B{t) is a Brownian motion independent of W(-) (however, note that (4) has no rigorous meaning). When JC>0 X° may be regarded as a diffusion model of a random walk in a random environment discussed by Kozlov [14], Solomon [18] and KestenKozlov-Spitzer [10]. When /c=0 X° is a diffusion model of Sinai's random walk in a random environment ([17]). The asymptotic behavior of {a>(f), (t), 2>0} exhibits the same asymptotic behavior as Sinai's random walk, namely, that the limiting distribution of (log f)~ao>(0 as t—<• °o exists (see also [2], [3], [8], [11], [19] for related works). When K>0 (in particular when 0<£<1) it was an open problem to obtain results for X° which are (or at least expected to be) similar to those of Kesten-Kozlov-Spitzer [10]. The purpose of the present paper is to give some answer to this problem. Let Tx=mf{t>0:(s): 0^s(0=inf {0 is the following. THEOREM 1. (i) If 0 then l\m2»{x-i»Tx£t}=Ft(t), (5) t > 0, 2-*oo \im&°{rK(d(t)<x} = limS>°{rea>(0^x} (6) = \im&° {rra>(t)£x} = l-F^x'1"), x > 0, t-oo where FK is the distribution function of a one-sided stable distribution with Laplace transform exp(— cXK); the constant c is given by H^MTiigr}where u(x) is the solution of d dM{x) d u = 2u, dx u(0) = l , w'(0) = 0 the function M{x) being given in Lemma 1. (ii) / / JC=1, then (7) (xlogx) _ I T x converges to 4 in probability with respect to ^ (8) each of r\\ogt)B{t), rl(\ogt)(o{t) and rl(\ogt), (iii) If K>1, then (9) \imTJx = - ^ r , 2>°~a.s., as x ^ o o ; A diffusion process in a Brownian environment (10) lima>(f)/* = ^-r—, 191 4 t-oo The assertion (i) can be slightly strengthened as follows. 2. Let 0 < * < 1 . (i) The process \X~1,KTxx, x^O, iP0} converges to {L(x), x^O} as ^->°o in the sense of convergence of finite dimensional distributions, where \L{x), x^O] is an increasing stable process with Laplace transform THEOREM £{exp(-£L(l))} - e x p ( - c ^ ) ( ^ 0 . (ii) The process {X~Ka>(Xt), t^O, £P0} converges to {L~Kt), t^O} as Z^<™ in the sense of convergence of finite dimensional distributions, where L-Kt) = mf{x>0:L(x)>t}. In the case of random walks, results similar to (5) and (6) were obtained by Kesten-Kozlov-Spitzer [10] and results similar to (7), (8), (9) and (10) by Solomon [18]. Our method of proving (9) and (10) is similar to that of [18] but as for (5), (6), (7) and (8) our method is different from either of [10] and [18]; it is based on Kotani's formula (see § 1) which reduces our problem to the study of limiting behavior of another diffusion process described by a certain stochastic differential equation with non-random coefficients. In proving (5) and (6) we must also use Kasahara's continuity theorem ([7]) concerning Krein's correspondence ([6]) between the m-measure and the spectral measure (or more precisely the /i-function) of a one-dimensional diffusion operator. ACKNOWLEDGMENT. The authors wish to thank S. Kotani for valuable discussions on the subject and for permitting us to contain his formula (1.1) here. § 1. Kotani's formula. The following formula was obtained by S. Kotani in 1988 in his study of the limiting distribution of (\og ty2a>(t) in the case K—0 (unpublished). KOTANI'S FORMULA. (1.1) Let X>0. Then for ;;>0 E%r\e-XTt\ = e x p { - j V a ( s ) r f s } , P-a.s., where Ux(t) is the unique stationary positive solution of (1.2) dUS) = Ux{t)dW{t)+[2X+^Ux{t)-Ux{ty)dt. 192 K. K A W A Z U and PROOF. H. T A N A K A Taking an arbitrary but fixed c > 0 we put for 0 v(t) = E'w{e-*T«}. Then EQw{e-iT°\ =E°w{e-irt}Etw{e-iT"\ = v(t)/u(t), P-a.s.. Since £wv(t)=Av(t)t t (1.3) d{e-w<^u'{t)} =2Xe-w*^u(t)dt, t>0, or equivalently t>0. If we put UA(t)= {logu(t)}'=u'(t)/u(t), then Ux(t)>0, P-a.s.. Since u{t) and U2(t) are adapted to the filtration generated by \W(t)}, we can apply Ito's formula to compute the stochastic differential dUx{t). Using (1.3) we have dux(t) = d(u'(t)u(ty1) = die^^u'i^^uit)-') = ew^u(t)-ld(e-w^u'(t)) +e-w<^uf(t)u(tr1dew*^-e-w^l)u'{t)ew*{t)u{ty*du(t) = 2Xdt+u'(t)u(t)-1dWK(t)+2-1u'(t)u(ty1dt- {u'(t)u(f)-l}*dt = ui(t)dW(t)^2x+^ux(t)-ux(ty}dt. For A>0 we can write u(t-\-h)=u(t)u(h) where ff(A) = l/£fr{e- iP "-*} =u(h); in the above " = " means the equality in distribution. Ux(t) = u\t)u(trl d .. u{h)—\ = hm——t Therefore m \"'1 = lim ,, . A = M'(0) ftlO ft = tt'(0)tt(0ri = £7i(0). This implies that £/;(?) is a stationary solution of (1.2). The uniqueness of such a solution follows from Theorem 18 of Ito-Nisio [5]. § 2. Kasahara's continuity theorem for Krein's correspondence. Krein's theory of strings ([6]) has many applications to diffusion processes (e.g., see [12], [13], [20]). In this section we do not give the general theory but list some of the results of Kasahara [7] on Krein's correspondence that will be useful for our later discussions. For a general statement of Krein's A diffusion process in a Brownian environment m correspondence theory it is convenient to consider inextensible measures (e.g., see [20]). From the view point of its application to the present paper, however, it is enough to consider simply Radon measures in [0, oo). Thus suppose we are given a Radon measure m{dx) in [0, oo). We exclude the trivial case where m(dx)=0. The associated function M(x) is defined by M(x)=m(l0, x)) for x>0 and M(0)=0. Consider the generalized differential operator X=d/m(dx)-d/dx and let «(0) = 1 , »'(0) = 0 , (2.2) M(0) = 0 , w'(0) = l , respectively. For x>0 (2.3) a)m{dz)dy, JJoSz (2.4) Mx, a) = x+a\\ The pair { 0 h(a) = lim^fx, a)/ (2.6) The correspondence between m(dx) and h{a) is one to one ([6], see also [13]). (2.7) a~1h(ca) is the characteristic function of acM(ax) for arbitrary positive constants a and c ([7]). i Assume that oo is not regular for S, namely, that at least one of the integrals \\ (2.8) m(dy)dx, ( JJo \\ dym{dx) JJo diverges. Then for each a > 0 a positive decreasing solution u of Xu=au with w(0)=l is unique and expressed \ as u(x)~(x, a)/h(a) ([4]). We now state 194 K. K A W A Z U and H. T A N A K A KASAHARA'S CONTINUITY THEOREM ([7]). Let mn{dx), rc=0, 1, •••, be Radon measures in [0, oo) with associated functions Mn(x), characteristic functions hn{a) and systems of fundamental solutions { «)» n(x, «)} • Then the following statements are equivalent to each other. ( i ) Mn(x)—>M(i(x) asrc—>ooat each continuity point x of M0(-). (ii) For x^O and a>0 a)-*co, (iii) For a>0 hn{a)^h{a) as n—>-co, §3. Proof of Theorem 1 in the case 0 < « < 1 . Let Ux{i) be the diffusion process appearing in Kotani's formula and put Vx(t) = ~Ux(t). Since Vx(t) satisfies the stochastic differential equation dVx{t) = Vx(t)dW(t)+(l+^Vx(t)-2XVx(t)*)dt, the generator of the diffusion process Vx(t) is Xx = d/mx(dx)-d/dSx(x) where ^K_1exp(y+4Xy)dy, (3.1) Sx(x) = (3.2) mx{dx) = 2x~K~l exp( Ux\dx We also put Vz(t)=VMxl(t))> Yx(t) = Sx(Px(t)), where ^4jl(0 is the inverse function of -4;(s)=\ Vx(u)du. 9x(t) and Yx(t) are denoted by Ix X\=d/mx\dx)'d/dx with (3.3) where 0x{x) is the inverse of the diffusion processes |(t), t^O, tions with respect to Pf, respectively. We begin by For any X\ is then given by 2dx(x)-*<+1exp{~j^-SXdx(x)\dx, m\{dx) - LEMMA 1. and £\ respectively. The generators of function of Sx(-). The path space representations with generators Xx, %i and X\ are denoted by Pf] and {a>(t), tzO, Px'*}, respectively. The expectaPf and P\-x will be denoted by Ef, Ef and E\-x, proving the following lemma. x>0 A diffusion process in a Browntan environment 195 \imXl-'Mn(X-<x) = M{x), (3.4) where Mx{x) is the associated function of m\(dx) and M{x) = 2y{p-\x)), y{x) = ^y-Ke~iydy, p"\x) PROOF. is the inverse function of p(x) = I yK~le*ydy. By an easy computation we have for x>X Sx(X-1x) = X-*px(x), where px ^ ~ uyK lex 2X p ( —y + 4 > ) r f 3 ' It is also easy to see that px(x)-+ p(x) as X {0 and hence Sx(X~1x)^X~'p(x) as X | 0. Therefore for any se(0, x) Sx(X~1(x — e))<X~'p(x) holds for all sufficiently small ^>0. In other words (3.5) X'\x-e) < dx(X~Kp(x)) for all sufficiently small X > 0. Similarly (3.6) X-\x + s) > dx(X-Kp(x)) for all sufficiently small X > 0. Next we note that Mi QO = 2 ^ t * ^ - t e + 1 e x p ( - ~ a u ) S K « ) d « , and hence •sx^~sp^x)) _ / 2 Mx{X~'p{x)) = 2\ z-*exp{~--4Xz)dz This combined with (3.5) and (3.6) yields (3.7) 2^ < Mx{X~Kp{x)) *"' z-'exp(-j-Mz)dz < 2y z-Kexp{—^—4Xz)dz for all sufficiently small X>0. Since 2^ V*exp( 4Xz}dz = 2X'~1yi(x) where rM) = 2X \xyKexp(—y—*y)d 380 196 K. KAWAZU and H. TANAKA (3.7) yields < Mx(rKp(x))/{2^-i) rx(x-e) < r*(x + s), which again yields \imMx{X^p{x))/{2X^)^y{x)f because j^OO—•?(*) as X j 0. Taking p~1(x) instead of x we obtain limM1(X-'x)/(2X'-1) = rip'Kx)), ,110 which proves Lemma 1. By virtue of Kasahara's continuity theorem and (2.7) Lemma 1 yields the following lemma concerning the characteristic functions hx{a) and h(a) of Mx(x) and M(x) respectively. LEMMA 2. WmX* hi(Xa)=h{a), a>0. -J 1 0 Let r=inf{f>0:fl)(0=l}(3.8) Then for x>0 and a^O = Ef{e-*ir} E${exp(-aX[w(s)ds^ = E\*sx*>{e-"u)t where a—inf {?>0: 0 determine fl^(>l) by Also let Sx{ax)~a. S(*) = £V~lexp(—)dy, and determine fl0(>l) by S(a 0 )=fl. LEMMA 3. (3.9) / / 0 < / c < l , then for Then G^ | a 0 as ^ | 0. a>0 l - £ ^ | e x p ( - a y i r f t j ( s ) r f s ) | ~ c(a, a)X* as X 10, where (3.10) c(a, a) = a/h(a). PROOF. Since °° is not regular for X\, (2.8) implies that for each a:>0 a positive decreasing solution ux{-, a) of X\u—au with M ( 0 ) = 1 is unique and expressed as (3.11) ux{x, a) = x(x, a)/hx{a), where the pair { £ i a ; { e x p ( - a y t f V s ) r f s ) j = ux(a, <xX). 381 A diffusion process in a Brownian environment 197 By making use of (2.3) and (2.4) with m{dz) replaced by m%dz) we can easily prove that (3.13) 2ak\adx[XlX)y-<exp(~—-Uy^dy+oQ), (3.14) as X i 0. Denoting by 0{x) the inverse function of S(-), we have (3.15) Hrnl^t± = 2 ^ d x ^^ e x p („l)^ = comL . From (3.12) and (3.11) we have l - £ ^ j e x p ( - a ^ \ V s ) r f s ) | = l- aX)+17V«()} which, by virtue of Kotani's formula, equals (3.16) Hm£Jexp(-f ; *'c/jt(s)ds)} = l i m £ { e x p ( - 2 4 " *Vi(s)ds)} = lim£fi{exp(-24i *Vs)ds)|, where Ep denotes the expectation with respect to Px* = \px(dx)Pf, fix being the invariant probability measure of the diffusion process with generator Xx. We first compute Um£^jexp(-2vtP ' V s ) d s ) } , with the starting point ax defined by Sx(ax)=a, by showing that \ a)(s)ds o)(s)ds where oh and rk are defined as follows: <J0 = 0, zk = inf{f>ff*_l:Tb: k^l, k S 1. 198 K. K A W A Z U and H. T A N A K A LEMMA 4. (i) mx = El\{al}=ami(R+)0 and X^>0 (3.17) ^||^L_ O T i | > e |_ 0 X^O, where R+=(Q, oo). as ?z-*oo, and the convergence is uniform in *3e[0, 1], PROOF. For b>0 and x>0 let I~{x, (v, oo) according as 0<x Ef{a) b) and / = ( 0 , y) or I={b, Let a='mf{t>0:o)(t)—b}. x) and /= Then it is =\dSx{y)\mx{dz) holds (note that — Sx(0)=Sx(<x>)=°° is also taken into account in deriving the above formula). By virtue of (3.18) we can easily compute mx — E$i{Tl} + E^lffi—Tt}, obtaining (i). The assertion (3.17) is nothing but the law of large numbers for i.i.d. random variables. Only the uniform convergence needs proof for which it is enough to verify the uniform integrability of Wu P*\ 0 £ ^ l } t namely, (3.19) lim supf arfPi^Q. We use the fact that the diffusion process { dV(t) = V(t)dW(t)-^{l+-^~-V(t)-2XV(t)^dt with V(Q)=ax. Then a comparison theorem in stochastic differential equations implies that the solution of (3.20) lies below the solution of (3.20) with X=Q. From this observation we see that (3.21) PJMri>iV} S Pfir^N}, and by a similar argument (3.22) P*Ai*i~-Ti>N\ ^Pi{iV}, where o=inf{t>Q:0 and small s>0 we put »iW = M~**(l-e)/mo] , nt(X) = M^(l+e)/»i 0 ] . Then the uniform convergence of (3.17) implies 383 A diffusion process in a Brownian environment 199 l i m P W | ^ p - m J > m o s l = Of i = 1, 2. xio Since mz^>m0 \\ nt{A.) I J as A | 0 we have (3.23) l i m P j 4 | - ^ f - m 0 >m 0 sl = 0 , i = 1, 2, and in particular IimPJMff»1«)<«itf)mo(l+e)} = 1 . But n^wioU + eXJT'f for all sufficiently small X>0 and hence (3.24) limPJMff B l «) Similarly we have limP;aVn2m>r*n = 1 . (3.25) Next we put a)(s)ds, Then on the event ^ 4 i = Zk = I k ^ 1. {trniti)\ »i<*) 2 (r*+Z*) < ft=l r^-*t JO na(J) o>{s)ds < S (K* + Z*) fc=l holds and hence £}*{exp(-2J ^S ( r 4 + Z 4 ) ) ; ^ } £ £ ; a ; { e x p ( - 2 ^ V ^ ^ s ) ; ,4,} £ £j'{exp(-2A S x ( r f t + Z f i ) ) ; / I ; } , where the notation Elx{X)A] stands for the integral of X over A with respect to PIK Since P?\AX\^1 by (3.24) and (3.25), £";{exp(-2/|i)(yA+ZA))} (3.26) ^ Eaxx{exp(-2X^~Kta>(s)ds)}+o(l) where o(l) indicates a term which tends to 0 as X \ 0. We now compute 384 200 K. KAWAZU and H. TANAKA Urn £ ^ { e x p ( - 2 i Z! (*%+£*))} \imEV{Qxp(-2XY1)\ni^.Eaix{exp(-2XZl)}niiXK = By Lemma 3 we have lim[£?''{exp(-2;r I )}] ra i^> = lim{l-c(a, 2)XVl (3.27) = exp{—c(a, 2)f(l — e)/m0}. Similarly we have (3.28) lim£rtexp( - 2 * 2 K*H = exp{-c(a, 2)t(l+e)/m0\. On the other hand, we have 2X S Z* < 2ia(i<7n{u, < 2ajlai-1Ef(l + e)mS'1ff!.4<*>/«(W) which tends to 0 in probability as X | 0 by virtue of (3.29) 0 and (3.23). Therefore I i m £ W e x p ( - 2 / s Zk)\ = 1. From (3.26), (3.27), (3.28) and (3.29) we have for any e>0 exp{-c(a, 2)*(l+e)/m0} ^ lim inf £j i jexp(-2^f 'ft>(s) ^ lim sup Efx\exp( —2X\ 2)t(l-e)/m0}, which implies lim£j;jexp(-2^ (3.30) LEMMA 5. (3.31) PROOF. For any t (o{s)ds)j = exp{-c(a, 2)t/m0}. t>0 U m £ | e x p ( - ( ; "u^ds}} = exp{-c(a, 2)t/m0]. Note that by (3.16) the left hand side of (3.31) equals lim £ ^ { e x p ( - 2 ; ( Let tr0='mf{t>0:a>(t)=ax\ property we see that t and put ril={aa Then using the strong Markov 385 .-1 diffusion process in a Brownian environment 201 is bounded from below by £5 i {exp(-2^*Vs)rfs) ; T u | . EV{exp(-2kV *\»(s)ds)} and is bounded from above by £j i |exp(-2Ji\ Taking into account of the fact that P^x\ru}-^1 as w ^ c o uniformly in Ae(0, 1) and also of (3.30), we first left X { 0 and then M T °°. As a result we obtain (3.31). The proof of (5) in Theorem 1 is now completed as follows. By Kotani's formula we have for £>0 SMexpC-er,/* 1 '*)} = E{exp(-j*£/ f;B -i/.(s)ds)} H = £{exp(-£ where we put t~^ and X=£x~1,K. Letting x^co Ux(s)ds^, (so X I 0) we obtain lim^Mexpt-^T*/* 1 /*)} = exp{-c(a, 2)t/m0) = e"" = e~c^, X-+BO where c={2x-KV{K)h{2))'1, because c(a, 2)=a/h(2) and m0 = am0(R+) = 2a\"'x^e^'dx = 2l~ by (3.10) and (i) of Lemma 4. Finally we prove (6). Clearly we see that for any u>0, y>0 and t>0 d { T u + ^ t } u { inf o ) ( s ) - ( u + > ) ^ - 4 . We notice that (3.32) lim2>0{ inf y-oo fi*(s)-(M+^)^-yl U*Tu+y = limS»°jinffl)(s)^->i = 0 J y-<» la£0 J since 0 (3.33) &*{Tt*x^t\ S ^{ai{t)^tKx} ^ S*{SK0^FK} ^ ^ &o\ i?M^**+^}+<Winfft>(s)^-4. Uso J 202 K. K A W A Z U and H. T A N A K A For any sufficiently large fixed y, the result (5) ensures = lim&°{v-l,KTv^x-l"\ l-Fcix-1"), = t)-»oo which combined with (3.32) and (3.33) proves (6). § 4. Proof of Theorem 2. For the proof of (i) it is enough to show that (4.1) Um^jexp^S^r1'*^ for any £,, £2, •••, £n^0 and 0=x0<x1< ••• <xn. Take an s such that 0 < e < minfx* — xk_x: l<:k^n} and let us prove first that (4.2) Hm£°{exp(-S^r1^ In what follows Tx=Tx(a>) denotes the first passage time inf {/>0: to(t)=x] where x may lie either to the right or to the left of a>(0). If we put Fx.k = ^ ^ M e x p C - W ' T i t ^ . , , ) ; r J U , 4 . „ < T i U 4 . l . i ) } , Gz,k = E&*-i{expt-^r^T;(,,..,); Tx{Xk_E)>T,(Ii_1.„}, then ^{exp(-s i $*r , "(r i ,u 4 .. ) -r i » Jk . l ))} (4.3) = n ^ ft -Mex P (-^r i "T, ( ^., ) )} = n(Fa.»+Ga.*). k=i Making use of a trivial inequality 0^n?=i(fl*+^)-II"=i tf*^2"=i&* which holds under the assumption that ak, bk>0 and a * + 6 * g l ( l ^ & ^ n ) , we have (4.4) 0^£|n(Fu+G ft = l = 2 E\\ u )}-£|nFu} A=l ew'<*>dx/\ * * * ew*™dx\ 387 A diffusion process in a Brownian environment 203 (since VK* has stationary increments) On the other hand it is easy to see that (4.5) (4.6) E\llFi.i\ = TLEiFi,k}t WmE{Fx,k} = lim E{Fx,k+Gx.k] = Umfi 0 (exp(-e*r 1 '-r J ( , 4 _, t , I _,,)} = exp{-c(xk-xk_1~e)&}, the last equality being a consequence of (5). From (4.3)^(4.6) we obtain (4.2). To derive (4.1) from (4.2) it is enough to notice that 0 S £°{exp(~ I! ^ r ^ T ^ - o - T ^ ^ ) ) } - gi€**-l/'(TiXk--Tx*h„1j)} g S ^0{l-exp(-|Arj^(T;^-T;(Xfe_E)))} S^0{l-exp(-^-^r;e)} = -+ S {1—exp(-ce£|)} as ^-* co — 0 as s | 0. The proof of (ii) can be done in a way similar to (6). In fact, as in (3.33) we have for any tk>0, xk>0 (l^k£n) and ;y>0 &«{TXKXk^Xtk, ty K S^ {a- a)Utk)<xk, £5>°{7W+I,^a»f l^k^n} l-£k0\infa>(s)£-y\. 204 K. KAWAZU and H. TANAKA Letting X \ oo in the above we obtain \im£»{X-*<»{Xtk)^xk, l^k^n) X-*aa = P{L(xk)^tkl = P{L~l{tk)^xk, l^k^n] l^k^n}. §5. Proof of Theorem 1 in the case JC=1. Assume K=1 and recall Ek{e-XT*\ = exp|-2>i(V;(s)rfsl, (5.1) a.s., where V x(t)=(2X)~1U x(t) is a stationary diffusion process with generator d x x d mx(dx) dSx(x)' wherein Sx(x)~ \ exp(—\-4Xyjdy, m x(d x) = 2x~2 exp(— 4Xxjdx. Once the following proposition is proved, (7) of Theorem 1 follows immediately from (5.1). PROPOSITION 1. (5.2) For any X>0 (xlogx)~M Vuxl0gx}-i(f)dt->2 in probability as x -* oo. Before proving this proposition we prepare three lemmas. We put for f >0 a$ = mf(i2+)_1l us{x) = LEMMA 6. xm^dx), ^dS^y)^%-a$)m$(dz). a^ ~ —21og£ as £ \ 0. af can be expressed as af—2m^{R+)~l(I^+II^) PROOF. 7f=\ Jo x~1exp{—2x~1—4£x}dx, where 11$ — \ x"1exp{—2x"1—4£x}dx . JJV For fixed A/>0 Is remains bounded as £ | 0 and e~i/NII^£lI$^IIi II{ = ( V ' e - ^ d x = r y-le~"dy ~ - l o g £ where A diffusion process in a Brownian environment 205 as £ | 0. Thus the lemma is proved since mf(iJ+)_I—>1 as £ j 0. ^2\°u$(xyms(dx)^0 LEMMA 7. as £{ 0. PROOF. In what follows const, means a constant which is independent of £ but may vary from place to place. First we prove (5.3) 0 < —us(x) ^ const, x logy The restriction 0<x^as for 0 < x ^ a e . implies 0 < -u£x) = 2J X e xp{2y- I +4£y} dy\\a$-z)z'2exp{-2z~1-4^z} dz ^2fl^ X exp{2y" 1 +4?3'}rf3'^V 8 exp{--2^- 1 -^}^. Since 0 as £ j 0, we have 0<£v<£a f —0 as £ 1 0. There- i r* cy l — u*(x) ^ consMog-z-l exp{2jy-1}tf;yl z 2exp{—2z^\ dz ^ const, x log— £Jo Jo £ as was to be proved. (5.4) Next we prove 1 1 x 0 < — ue(x) S const, x log-z-+const.-r\og— ? £ a* for x > a*. Since - J o ( j ? - a f ) w e ( ^ ) = I (z—a$)me(dz), — wf(x) can be expressed as — ui(x)=2(I+II), where 2^\xp{2y-l+4£y}dy^\a$-z)z-2exp{-2z~l--4$z}dz Q S const, x log-£- (by (5.3)), exp{2y-l+4Sy}dy[°°(z-ae)z-2exp{-2z~1-4Sz}dz 0 < / / = 2(* ^ 2[X exp{2v _1 +4£v}dy\~z- l expl-4$z}dz. Jy J if If we put g(y)—\ l Jj iU z~ e~ dz, then 206 K. K A W A Z U and H. T A N A K A u-le~udu^<MyYle-* g(y) = [" and hence II ^ z[' exp{2y-l+4£y} .(4gyrie-<**dy 1 . dy 1. x 2 If < const.—\ —— = cons?.—log—. This proves (5.4). We can now complete the proof of Lemma 7 as follows. From (5.3) and (5.4) we have £2\ u$(x)2m^(dx) < const.(g\og-£-) I exp { —2x~x—&£x\ dx +const.I £ log-j- j \ exp{—2A:"1—4£x}dx -\-const.\ ( l o g ^ l x"3exp{—2x~*—4£x}dx g const.(% l o g i y f Y 4 ^ ; * ; +cons*.\ (log — ) .\-~2d* Ja$\ 1 \2 Of/ C"° ^ ~ j l (.iosy)*y~2dy ( logyj + ~const - 0 LEMMA 8. PROOF. as I i 0. (log—)" £\ \u'$(x)\sx*ms(dx)->Q as £ j 0. First we prove 0 < —urz(x) £ const, log-r- for 0 < x (5.5) Under the restriction 0 < x g a 5 we have 0< -M^)^2exp{2^-1+4|x}(%f~3;)^-2exp{-2j'1-4^}rf3; <2asexp{2x-1+4£x}\*y-2exp{-2y-1-4£y}dy 1 rx ^ const, log—evx\ y 2e £ Next we prove Jo 1 dy — const, log—. Vv £ A diffusion process in a Brownian environment 0 < — wK*) ^ const.eifx^(4£x) (5.6) 207 for x > a^, where \y~le~ydy. a$ then 0 < - ^ ( x ) = 2exp{2x- 1 +4^}f"(^-^)3'- 2 exp{-23;- 1 -4^}rf3 J y-le~**vdy <, const. e^A Now the proof of Lemma 8 is completed as follows. (togj)~ltfj \u&x)\'x*m£dx) af 1 < const.£\og—( exp{-2x" -4^x}dx (by (5.5)) t Jo ^ const. | log— • a5 —> 0 as £ | 0. To estimate the integral over (a$, co) we notice that 1 x 2 l and hence e e~x/x X 1 . A log— x Therefore (logy)"1^ as x —> co, as x | 0, \u&x)\%x*m£dx) c o n r f . ^ l o g j ) - 1 ^ e4**9>(46x)»dx = c072s£.Uogy) I (by (5.6)) ex ^ 0 as | | 0 . We now proceed to the proof of Proposition 1. Since X^u^x application of Ito's formula yields — a^, an 208 K. K A W A Z U and Putting t—x, £=Z(x\ogx) (x log x)'1 we have l (5.7) H. TANAKA and multiplying the both sides of the above by (x\ogx)-^Xo(V^s)-a$)ds = (x log x)" 1 {u$(V$(x))-Ui{V€(0))} - ( x log x)-^\'e(Vs(sWt(s)dW(s). The distribution of V^s) is c^m^dx) where cf is the normalizing constant which tends to a finite value as £ j 0. Therefore the second moment of the left hand side of (5.7) is dominated by i poo oo u^yYm^dy)-^const.(x logx)~2xl \u^(y)\zyzms(dy), which tends to 0 as x—> co by virtue of Lemma 7 and Lemma 8 because 1/ 1\-J (x logx) 2x ~ "TltoS-*-) £ as x -» co. This combined with (x log x)~M a$ds —> 2 as x —* oo proves Proposition 1. The assertion of (7) of Theorem 1 follows immediately from Proposition 1. The assertion (8) can be derived from (7) by a method similar to that used for deriving (6) in § 3. §6. Proof of Theorem 1 in the case JC>1. For any integer &;>1 we put r* = T* — 7 V , . Then for any Xu X2t ••-, Xn>0 we have (6.1) £Sr{exp(- g ;,r*)} = ft Efcx {exp(-XkTk)\ = fif(rk_1w,xk), where f(W, X)=E°w{exp(-XT1)\ and rs:W^W (for fixed x) is defined by (rxW)(y)=W(x+y)-W(x) for any y<=R. From (6.1) and the ergodicity of \rx} it follows that \zk, k^l, 2*°} is stationary and ergodic. Therefore (6.2) l i m — = l i m T i + ' " + r * - = <5?Bfo}, 5>°-a.s.. n-co n n-oo fl A diffusion process in a Brownian environment 209 The condition K>1 is equivalent to the finiteness of €*{c\) as will be seen below. First we compute Ew\Ti) ; the result is £H7\} =^dSw(x)^_mw{dy). Therefore e'fa} =E{W°w{Tl)} = 2[dx\* Jo E{ei&W.W-Wie(y)\\dy J-™ ^-^r. K—1 Thus (9) follows from (6.2). Next we prove (10). Clearly aS(f)-*°° as I f w (@°~a,s.) and for any e>0 Taut) < ^ ^ a»(0 - di(0 T&u>+t a»(f) Letting f| °° in the above we obtain t lim—- = 4 (-~ Q){t) K~l -, &°-a.s., which means that the left hand side of (10) equals (c—1)/4, a.s.. To prove that the second and the third terms of (10) equal (*—1)/4, a.s., we put 0(n)=(l—e) '(*—l)n/4 for an integer n ^ l and for 0 < £ < 1 . Then W inf ©(sJ-flCnX-VTl £{ptf»>(inf©(s)-tf(w)<—v/n")} a ,0 {inffl)(s)<-V"n" r . The last term in the above is a general term of a convergent series by the result of [9]. Therefore an application of Borel-Cantelli's lemma yields (6.3) inf a>(s)-0(n)S -V~n for all sufficiently large n, &°-a.s.. Since Tgin)/n—»-l—s as n —* co (tp°-a.s.), (6.3) implies (6.4) infw(s)-0(tt) ^ - Vn" for all sufficiently large n, 3»°-a.s.. For t>0 we now take n = n(t) such that « ^ f < n + l. Then (6.4) implies 210 (6.5) K. KAWAZU and H. TANAKA tf(n)-V» £ a»(w) < a>{t) £ (t) for all sufficiently large t, <2°-a.s.. Dividing (6.5) by t and then letting f f °° we finally see that the second and the third terms of (10) equal (K—l)/4, £B°-a.s.. % 7. Remark to the case K—0. A limit theorem concerning Qj(f)=max{0 and put W#(x) = WOO- min W, x e R, dJ = min{jE>0: W#(x)=t\, Vt = min W, di = max{x<0: W#{x)=t\, Vj = min W, and define # and bi by W(&(±) = Vf (such &f are uniquely determined P-a.s. for each fixed f>0). Let Mt=maxW, Jt = MtV{Vt+t), Wr = maxw, /r^Mrv(^+0, and finally define 6f by tf = min{x>0:W # (*)=n if Jt < Ji, mm{x>0:W(x)=Jt\ if / ? > / F - Then the process {X~2a>(eXt), t>0, £P0} converges to {bf, t>0, P) as A-^tx> in the sense of convergence of finite dimensional distributions. In particular, (\ogt)~zw(t) converges in law to bf as t-^cv and E{e-^\ =^E%{e-&i\dx, f ^ 0, where E+ and Tx denote the expectation and the first hitting time to 1, respectively, for the reflecting Brownian motion on [0, oo) starting at x. The corresponding result in the case of random walks on {0, 1, 2, •••} with reflecting barrier at 0 was obtained by Golosov [2]. References [ 1 ] T. Brox, A one-dimensional diffusion process in a Wiener medium, Ann. Probab., 14 (1986), 1206-1218. 395 A diffusion process in a Brownian environment [ 2] 211 A. O. Golosov, Limit distributions for random walks in random environments, Soviet Math. Dokl., 28 (1983), 18-22. [ 3 ] A . 0 . Golosov, Localization of random walks in one-dimensional random environments, Comm. Math. Phys., 92 (1984), 491-506. [ 4 ] K. Ito, Stochastic Processes II, Iwanami, 1958, (in Japanese). [ 5 ] K. Ito and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4-1 (1964), 1-75. [ 6 ] I.S. Kac and M.G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl. (2), 103 (1974), 19-102. [ 7 ] Y. Kasahara, Spectral theory of generalized second order differential operators and its applications to Markov processes, Japan J. Math., 1 (1975), 67-84. [ 8 ] K. Kawazu, Y. Tamura and H. Tanaka, Localization of diffusion processes in onedimensional random environment, J. Math. Soc. Japan, 44 (1992), 515-550. [ 9 ] K. Kawazu and H. Tanaka, On the maximum of a diffusion process in a drifted Brownian environment, In Seminaire de Probabilites, (eds. J. Azema et al.), Lecture Notes in Math., 1557, Springer-Verlag, 1993, 78-85. [10] H. Kesten, M.V. Kozlov and F. Spitzer, A limit law for random walk in a random environment, Compositio Math., 30 (1975), 145-168. [11] H. Kesten, T h e limit distribution of Sinai's random walk in random environment, Physica, 138A (1986), 299-309. [12] F. B. Knight, Characterization of the Levy measure of inverse local times of gap diffusions, In Seminar on Stochastic Processes, Birkhauser, 1981, 53-78. [13] S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, In Functional Analysis in Markov Processes, (ed. M. Fukushima), Lecture Notes in Math., 923, Springer-Verlag, 1982, 235-259. [14] M.V. Kozlov, A random walk on the line with stochastic structure, Theory Probab. Appl., 18 (1973), 406-408. [15] Y. Ogura, One-dimensional bi-generalized diffusion processes, J. Math. Soc. Japan, 41 (1989), 213-242. [16] S. Schumacher, Diffusions with random coefficients, Particle Systems, Random Media and Large Deviations, Contemp. Math., 41 (1985), 351-356. [17J Ya. G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl., 27 (1982), 256-268. [18] F. Solomon, Random walks in a random environment, Ann. Probab., 3 (1975), 1-31. [19] H. Tanaka, Localization of a diffusion process in a one-dimensional Brownian environment, Comm. Pure Appl. Math., XLVII (1994), 755-766. [20] S. Watanabe, On time inversion of one-dimensional diffusion processes, Z. Wahrscheinlichtkeitstheorie verw. Gebiete, 31 (1975), 115-124. Kiyoshi KAWAZU Hiroshi TANAKA Department of Mathematics Faculty of Education Yamaguchi University Yoshida, Yamaguchi 753 Japan Department of Mathematics Faculty of Science and Technology Keio University Hiyoshi, Yokohama 223 Japan Chaos, Solitons & Fractals, Vol. 8, No. 11, pp. 1807-1816, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960-0779/97 $17.00 + 0.00 Pergamon PII: S0960-0779(97)00029-5 Limit Theorems for a Brownian Morion with Drift in a White Noise Environment HIROSHI TANAKA Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 233, Japan Abstract—This paper discusses limit theorems for a diffusion analogue of Kesten-Kozlov-Spitzer's random walk in a random environment. The results obtained are similar to theirs but can be presented in a more explicit form by the use of Krein's spectral theory for one-dimensional generalized second-order differential operators of the form d/dA/d/dc. © 1997 Elsevier Science Ltd 1. INTRODUCTION The model of a random process in a random environment that we discuss in this paper has its origin on physical grounds in the works by Chernov [1] and Temkin [2]. Mathematical investigation of this model was then done by Kozlov [3], Kesten-Kozlov-Spitzer [4], Solomon [5] and Sinai [6], all in the framework of random walks. A diffusion analogue of Sinai's random walk was discussed by Brox [7] and Schumacher [8]. In this paper we discuss a diffusion model, namely, a model continuous in time and space (still in one dimension), which corresponds to what was discussed by Kesten-Kozlov-Spitzer in the framework of random walks. A rough description of our model is the following. Consider the heat or electric flow on a one-dimensional conductor such as a thin wire. Suppose the flow has a drift (or a bias) so that the movement takes place more likely to the right than to the left. In addition, the conductor contains some impurities located irregularly that cause fluctuation of the drift. We thus consider a particle performing Brownian motion with its own constant drift plus a random drift caused by impurities, or in a more mathematically idealized form, we consider the random process X{t) described by the stochastic differential equation jw'(X(t)) dt, X(0) = 0, (1) where B(t) is a standard one-dimensional Brownian motion, {w'(x), x e R} is a white noise independent of B(t) and K is a non-negative constant. The problem is to know how the long-time behavior of X(t) changes under the perturbation of the white noise term. The answer for this will vary with K as in Kesten-Kozlov-Spitzer [4]. The case K = 0 was treated by Brox [7] and Schumacher [8]; the limiting behavior of X(t) as r—»«= is the same as that of Sinai's random walk, namely, the limit distribution of (log t)~2X(t) as f-» co exists. Kesten [9] proved that the limit distribution has a density 7Tk=02k + 1 1807 397 1808 H. TANAKA The purpose of this paper is to report detailed results in the case K > 0. In this case one may anticipate the result for X(t) from that of Kesten-Kozlov-Spitzer. However, to obtain exact and more elaborated results such as the determination of constants appearing in the limit, we need rigorous arguments. The result for 0 < K < 1 and part of the results for K > 2 were obtained by Kawazu-Tanaka [10,11]. For the case K = 0 , cf. Tanaka [12] and references therein. The proof of the results for 1 < K < 2 is new but only its outline will be given (see Section 3). Full proof will appear elsewhere. It is to be noted that, our problem being formulated in the framework of diffusion processes, we can make much use of methods in differential equations, in particular, Krein's spectral theory. Our results (see Section 1) naturally correspond to those of Kesten-Kozlov-Spitzer, but some new light is shed from the viewpoint of methods. In Section 4 we give an explicit (but complicated) representation of the constant C(K) appearing in the limit distributions. 2. MAIN RESULTS We begin with giving a precise definition of X(t) since the stochastic differential eqn (1) has no rigorous meaning because of the term w'(X(t))dt. By a formal argument we see that a "symbolic solution X(t) of eqn (1)" is a diffusion process with the generator ^ = l ^ + ^l_lwWl=feMx,-(„V2l(e-^) 2 d x 2 4 & t d x djcV + («V2lY dxJ The latter expression of %w has a rigorous meaning since it is of the form of Feller's canonical representation of a generator (for a quick view cf., for example, Section 2.10 of Nagasawa [13]). The process X(t) is therefore defined as the diffusion process with generator £w (defined by the latter expression)—a Markov process with transition probability exp(tJ£w)(x,dy). To be more precise we introduce the space W = {w e C(R): w(6) = 0} and the Wiener measure P on W. Then the formal derivative w'(x), x e R, is a white noise defined on the probability space (W,P). For each H-eWwe denote by Pw the probability law of the diffusion process with generator S£w starting from 0. Therefore, if ft = C[0,cc) and if w(t) denotes the value of w{ e ft) at time t ^ 0 , then {w(t), t > 0 , Pw} is a diffusion process with generator !£w starting from 0. Let &(dwdw) = P(dw)Pw(dw). We then regard the process {w(t), t > 0} as defined on the probability space (W xft,).This is the process X(t) in the introduction and is called a Brownian motion with drift in a white noise environment in this paper. We are interested in the long-time behavior of w(t) as f —> °°, or more precisely speaking, the limit distributions of r(x) and w(t) under the suitable centering and scaling as x, r—»°o5 where T(X) = inf{s > 0: 0, Our main results are the following in which C(K) denotes a suitable constant depending on K. (I) (Kawazu-Tanaka [10]) If 0 < K < 1, then = l - FK(x-l/K), hm@\^p-^x} l—*x K t x>0, J where [ e-*div(0 = exp{-ci>)r}, £ ^ 0 . Jo 398 Limit theorems for a Brownian motion 1809 (II) If K = l , t h e n fT(x)-4xl02^ lim^j-^ ] 2 _ < r } = F,(r), l i m y ( m y ~ f ^2 < j | ,-,* I r(log/) J teR, = 1 - FJ-16X), X<EU, where V [ log* e-*d/?,(0 = exp{c(l)f + 4flog|} > £ > 0 . J —5C (III) If 1 < K < 2, then lim 0> (t(x)-mx \/K X—f^ I. lim#4^ 1 =£ t = FK{t), t 6 R, X ) m 1/K f < x | = l - F , c ( - m 1 + 1/'tJc), J t e R , where m = 4(K - 1) _ 1 and | e-f'dF.(0-exp{c(K)r}, £ ^ 0 . (IV) If K = 2, then l i m f f l ( r ( / x ) " 4 x < f U ( 2 ^ ) l / 2 r e~ 2 ~ v d5, r ^ R , *_»* LVjrlog* J J_x (V) Assume K > 2. (i) Environment-wise invariance principle [14]. (a) For almost all w (with respect to P) the process rr(A,)-M(A,) | converges in law to a Brownian motion as A-» oo (in the sense of convergence of probability measures on the Skorohod space) where A = 64(K — 1)~ 2 (K — 2) _ 1 and M(x) = Ew{r(x)} = 2 j dy j " exp{w(y) - w(z) - ^ ~ l)] dz, * > 0. (In this theorem the definition of x{x) is modified slightly so that T(X) becomes right continuous in x, namely, x{x) = \ni{s > 0 : (o(s) >x\). 399 1810 H. TANAKA (b) For almost all w the process converges in law to a Brownian motion as A-» °° where A is the same as in (a), m = 4(K - l ) " 1 and ix(t) is the inverse function of M(x). (ii) Invariance principle in random environment [11]. Each of the processes \r(\x)-\mx VCA ^0,*},1 |ffi>(A/)-Am -1 * '•*-"'"/' 1 VnT^cx t>o,0>\ converges in law to a Brownian motion as A —> °° where m is the same as in (i) and C = 64K(K-1)-3(K-2)-'>A Let E denote the expectation with respect to &. Then it is easy to see that E{t(x)}< oo iff K > 1 and E{r(x)2} < °° iff K > 2 for x > 0. This is a simple reason why the result is divided into the five cases (I)-(V). 3. KEY METHODS Here are two key methods in our arguments. One is to use Kotani's formula which reduces our problem to the study of another diffusion process Xx(t) described by a certain stochastic differential equation with non-random coefficients. The other is to use Krein's spectral theory with which we can obtain some asymptotic properties of the Laplace transform of a certain hitting time of Xx(t). 3.1. Kotani's formula (unpublished; see Ref. [10] for a proof) For A >: 0 and t ^ 0 (we use t instead of x) ^ { e " A r ( , ) } - e x p [ - | t/ A (s)ds}, P - a . s . , (2) where Ux(t) is the unique stationary positive solution of dUx(t) = Ux(t) (MO + J2A + — ^ Uk(t) - (/A(r)2} dr. For our later arguments it is convenient to consider Xx(t) = (2\y]Ux(t), formula eqn (2) then yields iUe-Ar<'>} = exp[-2A j'Xx(s) A > 0 . Kotani's ds}, P - a.s., (3) and Xx(t) is a stationary diffusion process on U+ = (0,°°) obtained as the unique stationary positive solution of dXx(t) = XA(t) dw(t) + ( l + ~^Xx(t) - 2A*A(f)2) dl. Note that eqns (3) and (4) hold for A > 0. The generator ££x of Xx(t) is given by ^ d mx(dx) d dsx(x) (4) 400 Limit theorems for a Brownian motion 1811 where H(x) = J V ~ ' exp(- + 4A^) dy, mx(dx) = 2x~K~l expf 4AJCJ dx, x>0. (5) (6) 3.2. Krein's spectral theory (Ref. [15], see also Refs [16-18]) Here we sketch its main part. Let M be the class of functions M:[0,°°)^[0,°°] that are right continuous and non-decreasing. Note that M contains the function =co. We put M(0 — ) = 0 and l = lM = sup{x: M(x) < °°}. When / > 0 (i.e. Ji(x) # °°) we consider Jo dy l (p(z,a)dM{z), Jo- 0<x dx S 77- « > 0 , (<«> if M(x)&0). Jo (p(x,ay When / = 0 (i.e. M(x) = °°) we put h(a) = 0. The function h(a) is called the characteristic function of M(x) and the correspondence M(x)^>h(a) is called Krein's correspondence. When M(x) ^ 0, h{a) can be represented as Ka) = (' h(a) = c+\ ^ ^ , a>0, (7) where c ^ 0 and o-(d£) is a measure on [0,°°) such that Q Jn ; 0++ au + T £ Then h*{a)<^M*(x) where -«-*(r-'«)/(i-^). 401 H. TANAKA 1812 y l(x) is the inverse function of 7(0 IV^)"-<•={"(-»• 4. OUTLINE OF PROOF We give here an outline of the proof of (II) since (III) and (IV) can be proved in the same spirit as (II). The key points in the proof are to make use of Kotani's formula and fact (iii) in Section 2. From a technical viewpoint of our proof, it is convenient to regard x in the first limit theorem of (II) as time so we use habitual notation t instead of x. From now on we assume K = 1 and put A = £t~l where £ > 0 and t> 0; £ is fixed but later we let r-» °° so A [ 0. By Kotani's formula we have £ { exp ( _ €«o-«**y A}=E{trw. AW( ,oS, (8) for any event A in the probability space {W,P), where V(t) = 2\fxx(s)6s h and the notation E{X; A] (resp. E{X; A}) stands for the integral of X over A with respect to 9 (resp. P). Given a > 0 we define ax by sx(ax) = a where sx(x) is given by eqn (5) with K - 1 . Obviously ax > 1 and aK f a0 as A J, 0. We define stopping times /„, T^, n ~ 0, 1,..., based on the process A^fs) as follows: T0 = inf{s > 0:Xx(s) - aA}f T„ = mf{s >T^:Xx(s)=ax}, r„' - inf{s > T0:Xx(s) = 1], T„ = inf{s> Tn:Xx(s) = l } , n > 1. Then Tn - T„-u n - 1, 2,..., are i.i.d. random variables with E{Tn~ Tn^} = amx, Var{r„-r„_,} 1/2 >e const. (e>0). (9) (log t)m and put «i(0 Then n,-(0 ~ t(am0) ' as f for i = l and 2, l [I(Lt£(0)l J L f(l-c(0)- H 2 ( 0 = tf/M> amx J oo because mA—»m0 as A J, 0. Therefore inequality (9) implies «/(0 awA > e ( 0 [ ^ 0 as f-»«>. (10) We now put A,(£) = {T„i(l) < t < TnM}. Then, by making use of fact (10) we can prove that 402 Limit theorems for a Brownian motion 1813 P{A,(t;)}—»1 as t—*oo for each fixed £ > 0 . On the other hand, by the definition of A(£), writing T{n) instead of Tn we have E{e-V A ( t y e4f log t < E{e-V«). A f ^ e 4 ^ log , < E j e - ^ . M ) ) ; At(f)} e4* lo*'. (11) Next we are going to prove, for i = 1 and 2, that lim £{e-"v(t)))} e 4 a f l o g ' - exp{2aA:(a)£ + 4a£ log £}, a > 0, (12) with some constant fe(a)>0. Making use of the strong Markov property of Xx(s) we see that £ { e - « ^ W » ) } = £ 0 (£ 1 £ 2 )"W f where £o = £ { e x p ( - 2 A a [ * A (j) 6s H, £, - £[exp(-2Aa: J Xx(s) ds H, £ 2 = £ J e x p ( - 2 A a | ' Xx(s) ds H, and the suffix i in «,(*) is suppressed to simplify the notation. Let Xx(s) = sx(Xx(x\s))) where <J>_,A(s) is the inverse function of &x(r) = Jr0Xx(u)du. Then Xx(s) is a stationary diffusion process in |R with generator where /*A(dr) = 20 A (x)~' exp{ - - y - - 8A0A(*)i dx, 0A(*) — the inverse function of sx(-). Denote by \ to(t), t^O, Pyi the path-space representation of the diffusion process with generator i? A starting from x. Then we can prove that E, = EA{t~2""} = ux(a,2\a), x where f = inf{s >0:w(s) = 0} and ux(x,a) = E x{e~ar} is the unique positive decreasing solution oiSExu = au, x>Q with «(0) = 1. The solution ux(x,a) can be expressed as Ux(x,a) = , a>0, hk{a) where dy i}/x(x,a) = x + a dy Jo Jo Jo Jo respectively, and hx(a) 403 1814 H. TANAKA 1 - Eu namely, of 1 - uk{a,2\a) as A J 0. For this it is enough to know the behavior of hK{2ka) as A J, 0. By fact (iii) of Section 2 we see that ht(a)=hK{a) ——-r+*Mt(x) «MA([0,«=)) where the explicit form of Mf(x) is given in terms of MA(x) = jiA([0,x]). Now an essential step is the following Lemma 1, which is stated in general (1 < K < 2 ) . We put e « W = f y-"e-*ydy, g«(x) = 4 {e«(J«\y))}2dy> fK(x)= [V_1e4ydv, £ ' = the inverse function, MA = AA([0,=»)) = 2 J Jc-*exp(---4A;t)ch:. x (13) (14) (15) 4.1. Lemma 1 (i) If 1 < K < 2, then lim M;2AK_I]M$(Ml2kK-2x) = M*(x), x>0, x^l* A |0 where A/*(JC)= • 2(eK°/:,°gK,)(x) =° for 0 < x < /*, for (16) x > /*, /*^gK(oo) Therefore Mm M^2X-(,<-2)ht{Aa)^h*{a), a >0, (17) A JO where h*(a)0, io A \ A/ A *#/*, where /* = 4/M2,, and M*(x) = 0 for 0 < A: < /* and =«> for x > /*. Therefore lim ( l o g - ) A|0 \ hf(\a) = h*(a), a>0, A/ where /z*(a) = /* <-» M*(JC). Coming back to the case K = 1, we now know the behavior of hf(2\a) as A J, 0 from eqn (17) with K = 1. Therefore we know the behavior of 1 - £, as A J, 0 and similarly of 1 - E2. Thus we can obtain the following result: £ , £ 2 ^ l - 4 f l a A l o g - + 2flaA:(a)A(l + o(l)) A rl dx J o as A J, 0; (2 - e~4x - e"2* - e" 4 " - e" 2 ") — + 2afc*(2a). X (18) (19) 404 Limit theorems for a Brownian motion 1815 Noting that E0^> 1 as t—> c0, we finally obtain £fQ-aV{1\n(tm\ Q4a£ log ' ~ (£ £ \"(') 4 e <"£ '°g ' - (1 + 2aotk{a)X)nit) e 4< * log '-*""*«* <°g "A —> exp{2afc(a)£ + 4a£ log £} as t^> «=, proving eqn (12). Combining eqn (12) with the fact that P{A t (f)}-»l as t->oo, we have lim E{c~V('\ A,(g)} = lim E{^Vi'\ At{£) f) A& + 1)} = exp{2fc(l)f + 4flogf}, which again combined with eqn (8) and inequality (11) implies lim E { e x p ( - f « > 7 " * ' ) ; A ( £ ) } = lim £{exp(- g « L ^ ' ) ; ^(fl n M€ + 1)} = exp{2A(l)f + 4flog^}. This proves the first limit theorem of (II) with C(K) = 2A:(1) in virtue of the following technical lemma. 4.2. Lemma 2 Suppose we are given a sequence of random variables {X„, n — 1, 2,...} and a sequence of events {A„(£), n = 1, 2,...} for each £ > 0 . We put *>„(£) = £{e-**-; >!„(£)}, *„(f) = £ { e - ^ ; ^ „ ( ^ ) n / l „ ( f + 1)}, and assume that the following conditions are satisfied: (i) \imn^P{An({)} = lfort>Q. (ii) There exists a continuous function for£>0. Then Ai, converges in law as n —» oo to some random variable X whose law is uniquely determined by E{e~*x} = 0. To close this section we remark that the second limit theorem of (II) can be obtained from the first one with a small amount of effort. 5. REMARKS 1. The limit distribution dF„ is strictly stable if K # 1 and stable if K = 1. The limit distribution concerning 405 1816 H. TANAKA (II) If K = 1, then c{\) = 4 (III) If 1 f1 Jo dx (2 - e'4x - G'2* - e~4/x - e~2/x) — + 4A*(2). x then C(K) = 2 J - 3 - [ jr* + V*dx + 2A*(2)J/m0 = { ^ J r ( 2 - *) + 4/i*(2)}/{2t-' HIROSHIMA MATH. J. 28 (1998), 129-137 Invariance principle for a Brownian motion with large drift in a white noise environment Kiyoshi K A W A Z U and Hiroshi TANAKA (Received November 14, 1996) ABSTRACT. This paper discusses an invariance principle for a Brownian motion with drift coefficient K/4 in a white noise environment under the assumption that K is large. Our method clarifies the relation between the environment-wise invariance principle discussed in [7] and the present result (the invariance principle in random environment). Introduction Let W be the space of continuous functions on R vanishing at 0 that is equipped with the Wiener measure P. F o r an element w e W let us denote by wK the element of W defined by wK(x) = w(x) — (KX/2) where K is a given positive constant. F o r w e W, Pw denotes the probability measure on Q = C[0, oo) such that X x = {&>(/), t > 0, Pw} is a diffusion process with generator starting at 0, where co(t) is the value of a function ca(e Q) at time t. W e regard co(t) as a process defined on the probability space {W x Q,0>} where @>(dwdco) = P(dw)Pw(dco). Then symbolically K dco(t) = dB{t) -\--dt 1 --w'(m{t)) dt, where B(t) is a standard Brownian motion independent of t h e white noise {\tf(x)}. W e call the process X = {co(t),t> 0,0*} a Brownian motion with drift in a white noise environment; in [2] [6] [7] it is called a diffusion process in a Brownian environment with drift. T h e present authors obtained some limit theorems for X in [2] (see [8] for further results; see also [6] for a brief survey on related problems), which are analogous to those of [3] and [5]; however, some problems remain open. The present paper is a continuation of [7] and 1991 Mathematics Subject Classification. 60J60 Key words and phrases. Invariance principle, Brownian motion, Random environment. 130 Kiyoshi KAWAZU and Hiroshi TANAKA discusses the central limit theorem, or more precisely speaking, invariance principle in random environment in the case K > 2, We set Mx(=M(x))=2 eWK{y)-w«&dz,xeR, f dy f JO J-OO fi(t) = the inverse function of {Mx, x e R}, t e R, Tx{= T(x)) = M{t > 0 : co(t) > x),x > 0. We use also the following notation: (d(t) — max{co(s) : 0 < s < t},co(t) = inf{a>(s) : s> t},(coe Q), y=(K-\)/2}m = 4/(K-\) A = 64(/r - \)-2{K = 2/y, - 2)" 1 = 16y~2(2y - l ) " 1 , B = 64(K - 1)~\K - 2)" 1 = 8y"3(2y - 1 ) " \ C = A + B = 64K(K - \)~\K - 2)~l = 8(2y + l)y" 3 (2y - 1)"'. The following theorem was proved in [7]. A {Environment-wise invariance principle, see [1]). we have the following: (i) For almost all w e W with respect to P, the process THEOREM [T**-M»-x x > 0 When K > 2, p \ converges in law to a Brownian motion as X —> oo {in the sense of convergence of probability measures on the Skorohod space). (ii) For almost all w, the process co(Xt) ~ y(Xt) Vm~*AX t > ' 0,J>„} converges in law to a Brownian motion as X —> oo {in the sense of convergence of probability measures on C[0, oo)). The same is true when co{Xt) is replaced by either of ca(Xt) and co(Xt). Our main theorems are the following (K > 2 is assumed throughout). THEOREM 1. (i) The process (M^-Xmx \ VBX | J converges in law to a Brownian motion as X —• oo. Brownian motion with large drift in a white noise environment (ii) The process I Vm^m converges in law to a Brownian THEOREM 131 2 (Invariance motion principle J as X —> oo. in random environment), (i) 7%e process {^••"'J converges in law to a Brownian (ii) 77ie process motion as X —• oo. {*££!..*<.} converges in law to a Brownian motion as X —• oo. 7%e rawie is true when co(Xt) is replaced by either of co(Xt) and m(Xt). A s in [7] w e introduce a o n e p a r a m e t e r family of measure preserving transformations 0t, teR, on (W,P) defined by (6tw)(x) = w(x +1) — w(t), x e R . Clearly 9t9s = 9t+s a n d {0t} is ergodic. Set (0.1) Mw)= f e-w^dt J—00 w Then Btf0 =fo(Otw) = J ^ e *«-w*W EW{TX} = MX = 2 f 0yfody; Jo the first equality of (0,2) hclds for x > 0 and the second one holds for x e R. (0.3) Var w {7i} = 8 f 8ygdy for x > 0 ( JO (0.4) (0.5) (0.6) E{fQ} = y-\E{/S} J-oo = l y " 1 ^ - l)"1. E{Vzrw(Tx)} = Ax for x £ 0. VarjAfJ = fix + 0(1), x -> oo, (Var = variance). It was also observed in [7] that (0.7) e~w^(0tfo)2dt). d0J0 = ej0 dw(t) - (Y0,fo -l)dt, 132 Kiyoshi KAWAZU and Hiroshi TANAKA so 8tfo is a stationary diffusion process obtained as the unique stationary positive solution of the stochastic differential equation (0.7). Therefore (0.8) 0,/ o - / o = [' Osfodw(s) - A f Jo Jo dsfds, where / =fo — y" 1 . 1. Proof of Theorem 1 For the proof of Theorem 1 we need some lemmas. LEMMA 1 ([7]). r 1 / 2 m a x { f t / 0 : \s\ < t} -> 0 as t —> oo. LEMMA 2 ([7]). For any positive constants c\9 Ml+U -Mt~ mu{\ + o{\)) + o(y/l), \t\ < aX, u e R, where o(l) represents a general term that tends to 0 as X —• oo uniformly in (t,u) such that \t\ < c\X and w e R , for almost all w; o(\fX) is a term that can be expressed as o(\)yfk. To prove (i) of Theorem 1 it is enough to consider Jjj* 6yfdy (0.2). Making habitual use of t to indicate time we write by virtue of 4 ? ( ' 6>fds = - 7 f r fl'/o«*"(') ~ 7 ? ( ^ / o - / o ) v/. Jo yVX Jo yvA By the ergodicity of {6t} and also by (0.4) we see that the quadratic variation of the stochastic integral term in (1.1) tends to St/4 as A-+ oo, a.s., so the stochastic integral term itself converges in law to {(B/4) ' w(t),t >0,P} as X —> oo. The second term of the right hand side of (1.1) is negligible by Lemma 1. Therefore X+ = {A"1/2 J^ 6sfds, t > 0} converges in law to {{B/4)1/2w{t),t>0}, so does Xj; = {X~1/2 £** 0sfds, t > 0} because of the reversibility of the diffusion &tf. Now the assertion (i) of Theorem 1 follows from the fact that X£ and X^ are asymptotically independent as X —> oo. To proceed let £ = Xm and put (1.1) (1.2) ft(») = (BX)-l/2(MXl - Xmt), fat) = (M~3«)-1/2(A««0 - Zm~lt). Then the assertion of (ii) of Theorem 1 follows immediately from the following Lemma. LEMMA 3. For any to > 0 and e > 0 lim i>f sup |A(0+A(/)|>4=°- *-*°° \\t\ J 410 Brownian motion with large drift in a white noise environment PROOF. From the second M(At + fJx(t)m-l\fBX) and hence (BXy1/2{M(Xt equality + px{t)nrl>/Bk) of (1.2) - M(Xt)} = we 133 have Xmt = -fix{t), so an application of Lemma 2 yields px(t)(l + o(l)) + o(l) =—0x(t), where o(\) is a term tending to 0 uniformly on each finite f-interval as X —> oo, a.s. This implies the lemma. The following observation will lead to another proof of (ii) of Theorem 1. If v(/) denotes the inverse function of ^ Oyfody, then v(t) = pi(2t) and the derivative v^/), which equals to l/0v(t)fo, is a stationary diffusion process obtained from 8tf0 by changing time and scale. 2. The proof of Theorem 2 We give the proof of the part (i). Taking an arbitrary positive sequence {Xn,n = 1,2,...} tending to oo, we denote by pW the probability law of the process {Xnl/2(TXnX - Xnmx),x > 0 , ^ } . Note that P ( n ) is a probability measure on the Skorohod space D = D[0, GO). For the proof of the part (i) it is enough to show that P ^ converges to the probability law of the process {y/Cw(x)tx > 0,P} as n —• oo. We first prove that the sequence {p(n\n > 1} is tight. If QV denotes the probability law of the process {X~l^2(TXnX - MxnX), x > 0,P„}, then 0$ -> Si(P-a.s.) by Theorem A and hence <2(n) =](0P{dw) also converges to Q\ as n —• oo where Q\ is the probability law (on D) of the process {y/Aw{x),x>0,P}. All the convergence here is to be understood as the convergence of probability measures on D. Therefore for any e > 0 there exists a compact set K\ <= D such that Q^n\K\) < e2 for all n > 1. We then have P{Ln} < e where Ln = {w : Q{^(KCX) > e} = {w : $\KX) < 1 - e}. We also introduce, for each fixed w, an element 0, P}. We can thus find a compact set K2 <= Q such that P o ^ - ^ ^ l ) < £ for all « > 1. We now put K — {w\ + w2 : w\ e K\, w2 e K2}. Then K is a compact subset of D. Since \x ~ Kmx yXn we have = TknX - Mx„x \[Xn MXnX - X„mx y/Xn 411 134 Kiyoshi KAWAZU and Hiroshi TANAKA FW(K) = J J ljr(wi +9H(w))Q$(dwl)P{dw) Z JJ ljc,(wi)U(^(w))fiJ>(^i)P(^) = f el"'W(Ari)/»(iftv) > [ (l-e)P(^)^(l-a)(l-.2e), which proves that {p("\n > 1} is tight. it is enough to show that (2.1) Jim \f(w)P{n\dw) Therefore, for the proof of Theorem 2 = J J / ( W , + w 2 )ei(^i)fi2(rfw2), for any function / of the form f(w) = expi v ^ T ^ where a,- e R and */ > 0, 1 < j < k. equals Mm J J / ( W l = = ccjw(tj) I, For such an / the left hand side of (2.1) + which also equals the right hand side of (2.1). This completes the proof of (i) of Theorem 2. The part (ii) of Theorem 2 can be proved in a way similar to the above by making use of Theorem A and Theorem 1. 3. Supplement to the proof of (i) of Theorem A The proof of Theorem A was given in [7]; however, some details in the proof of the part (i) were omitted. It will be worth supplementing them. The proof of Theorem A given in [7] proceeds as follows. Let x^ = 412 Brownian motion with large drift in a white noise environment 135 Tk — 7&_i, fk = Tic — Ew{Tk}, k>\. Then it was proved that, for almost all w, {tk,k> l,Pw} is a sequence of independent r a n d o m variables satisfying the Lindeberg condition. Therefore the central limit theorem holds for Tn with respect to Pw, for almost all w. N o t e that Ew{Tn} = Mn and \axw{T„} ~ An as n —* oo (P-a.s.). N o w the rest of the proof, whose detail was omitted in [7], is given as follows. Let t* = Vaiw{Tk}/Vaxw{Tn}, (nk = (An)~l/2{Tk - Mk}, 1 < k < n, and tno — CnO = 0. For each fixed w we construct a piece-wise linear function £H(x), 0 < x < 1, with vertexes (XJ/=o k/>C«t)> 0 oo. We now modify £„(x) slightly, namely, we consider a piece-wise linear function n„(x) with vertexes (k/n,(„k), 0 < k x uniformly as n —* oo (P-a.s.). Therefore ,- n the process {//„(*),0 < x < \,PW] converges in law to a Brownian motion as n ~* oo for almost all w. We finally prove that the process {(AX)~l/2(T^ - MAx),x e [0,1],P„,} converges in law to a Brownian motion as X —> oo for almost all w; the time interval [0,1] can be replaced by an arbitrary interval [0, *o] with a minor modification of the proof. Given x e (0,1] and an integer n > 1 we take the integer k such that (k - \)/n < x < k/n. Then Tnx - Mnx > Tk-\ — Mk > \fAnnn(x) - Tk - ink where mk = Mk - Mk-i. Similarly Tnx~Mnx< ^J~Am\n (x) -\-Tk-\-mk and hence ^*Vn(x) ~ (T* + mk) < Tnx - Mnx < VAnrjn(x) + [xk + mk). This implies that for x e [0,1] (3.2) VAnnn(x) - (f„ + m») < Tnx - Mnx < VAnnn(x) + (f„ + m„), where fn = m a x { ^ : 1 < k 0 let n = n(X) be the integer such that n - 1 < X < n. Then T^^Mix < Tix — Mix ^ Tnx — Mix, which combined with (3.2) implies 413 136 Kiyoshi KAWAZU and Hiroshi TANAKA ^ ( " - I k - i H - (r n -i +m„_!) - (MXx M{n_l)x) VAiifjn(x) + (T„ + m„) + (Mnx - MXx). Since M& - M(n-^x and J / ^ - Mix are dominated by 2mn> we obtain (3.3) (lz!) rj^ix) - (AXyWfo + 3m„) < {AXf^T^ < (jr) - M*) ^ W + (^A)- l/2 (f n + 3m n ). On the other hand we can prove that for almost all w (3.4) p J l i m f B /vfi = 0 ) = l, (3.5) lim mniy/n — 0. In fact, it is easy to see that {T^,A: > 1,^} is stationary and ergodic. Since x T| is integrable we have n~ J ] T ^ ^ const, as n —> oo (^-a.s.) and hence «~'TJ —> 0, namely, rCll2Tn —*• 0 (^-a.s.). This implies n~l^2f„ —* 0 (0*-a.s.) and hence (3.4). (3.5) can be proved in a similar manner. By virtue of (3.1), (3.4) and (3.5) the processes of the leftmost and rightmost hands of (3.3) converge in law to a Brownian motion as X —* oo. Therefore (3.3) implies the assertion for (AA)~l/z(Txx - Mix) that we wanted to prove. References [1] K. Ito and H. P. McKean, Diffusion Processes and Their Sample Paths, Springer-Verlag, New York, 1965. [2] K. Kawazu and H. Tanaka, A diffusion process in a Brownian environment with drift, J. Math. Soc. Japan 49 (1997), 189-211. [3] H. Kesten, M. V. Kozlov and F. Spitzer, A limit law for random walk in a random environment, Composito Math. 30 (1975), 145-168. [4] Yu. V. Prohorov, Convergence of random processes and limit theorems in probability theory, Theor. Probab. Appl. 1 (1956), 157-214. [5] F. Solomon, Random walks in a random environment, Ann. Probab. 3 (1975), 1131. [6] H. Tanaka, Diffusion processes in random environments, Proc. International Congress of Mathematicians, Zurich, 1994, 1047-1054, Birkhauser Verlag, Basel, 1995. [7] H. Tanaka, Environment-wise central limit theorem for a diffusion in a Brownian environment with large drift, Ito's Stochastic Calculus and Probability Theory, ed. by N. Ikeda et al., pp. 373-384, Springer, 1996. 414 Brownian motion with large drift in a white noise environment 137 [8] H. Tanaka, Limit theorems for a Brownian motion with drift in a white noise environment, Chaos, Solitons & Fractals, 8 (1997), 1807-1816. K. Kawazu Department of Mathematics Faculty of Education Yamaguchi University Yosida, Yamaguchi 753-8513 Japan H. Tanaka Department of Mathematics Faculty of Science and Technology Keio University Yokohama 223-0061 Japan 415 Tanaka, H. Osaka J. Math. 38 (2001), 369-377 SOME THEOREMS CONCERNING EXTREMA OF BROWNIAN MOTION WITH d-DIMENSIONAL TIME Dedicated to Professor N. Ikeda on his 70th birthday HIROSHI TANAKA (Received October 16, 1999) Introduction Let X = {X(x), x G R^} be a Levy's Brownian motion with d-dimensional time ([2]) defined on a certain probability space (Q, P ) ; thus X is a centered Gaussian system with continuous sample functions satisfying X(0) = 0 and E{X(x)X(y)} = (\x\ + \y\ — \x — y\)/2. For a nonempty subset A of Rd we put X(A) = inf{X(x) :x e A}, X(A) = sup{X(x) : x e A}. We often use the notation X{A) to denote either X_(A) or X(A). For example, X(A) X(B) denotes any one of X(A)~X_(B), X ( A ) - X ( £ ) , X ( A ) - X ( B ) and X(A)-1((B). A point x in R^ is called a point of local minimum (resp. local maximum) of a sample function X if there exists a neighborhood V of x such that X{x) = Kill) (resp. X(x) = X(U)). A point of either local minimum or local maximum is called an extreme-point. The following are typical of those problems and theorems we discuss in this paper. (I) Under what condition on A does the probability distribution of X(A) admit a strictly positive C°°-density? (II) Under what condition on A and B does the joint probability distribution of X(A) and X(B) admit a strictly positive C°°-density? (III) Almost all sample functions X have the following property: There are no distinct extreme-points x and y with X(x) = X(y). We give some sufficient conditions that will give positive answers to the problems (I) and (II) and then give a proof of (III). Formulating the problems somewhat generally we state our main results in the following theorems. Theorem 1. Let A^, \ < k < n, be nonempty bounded closed sets not containing the origin 0. Then for any constants Ck, \ < k < n, such that c\ +C2 + • • • + cn ^0, the probability distribution of c1X(A1) + c2X(A2) + ••• + cnX{An) 416 370 H. TANAKA can be expressed as a convolution 7 * p where 7 is a nondegenerate Gaussian distribution with mean 0 and p is some probability distribution in R. In particular, the distribution of each of X_{A) and X(A) has a strictly positive C°°-density provided that A is a nonempty bounded closed set not containing 0. Theorem 2. Let Aj, Bk, 1 < j < m, 1 < k < n, be nonempty bounded closed sets such that Uj=lAj is separated from Uk=iBk by a certain (d — \)-dimensional hyperplane Y\ passing through the origin 0, Then for any constants Cj, c'k, 1 < j < m, 1 < k < n, such that X]T=i cj ^ 0 and YTk=i c'k ^ ^> tne J°^nt distribution of m (1) MX) = 5^C;X(Ay), j=\ n f2(X) = J24X(Bk) k=l has a form (71 & 72) * v where each 7, is a nondegenerate Gaussian distribution with mean 0 and v is some 2-dimensional probability distribution. In particular, the joint distribution of X{A) and X(B) has a strictly positive C°°-density provided that A and B are nonempty bounded closed sets separated from each other by a certain (d — 1)dimensional hyperplane passing through 0. Theorem 3. Let A-}, Bk, 1 < j < m, 1 < k < n, be nonempty bounded closed sets such that Uj=lAj is separated from Uk=iBk by a certain (d — \)-dimensional hyperplane. Then for any constants Cj, c'k, 1 < j < m, 1 < k < n, such that £7=i c j = YH=\c'k ^ °' tne probability distribution of f\{X) - fi{X), with f\ and fi given by (1), has a form 7 * p where 7 is a nondegenerate Gaussian distribution with mean 0 and \x is some distribution in R. In particular, the distribution of each of K(A) - K(B), X(A) - X(B) and X(A) - X(B) has a strictly positive C00-density provided that A and B are nonempty bounded closed sets separated from each other by a certain (d — \)-dimensional hyperplane. Theorem 4. Almost all sample functions X have the following property: There, are no distinct extreme-points x and y of X such that X(x) = X(y). An example of the applicability (or our motivation) of Theorem 4 will be given in the final section. 1. A lemma Given a centered Gaussian system {X\, A G A} defined on a certain probability space (Q, P), we denote by H the real Hilbert space spanned by {Xx, A G A} and by H0 the closed linear span (abbreviation: c.l.s.) of {Xx - X^, A, p G A}. Clearly HQ C H C L2(Q, P). We now introduce the following conditions. Condition (A). There exists a nondegenerate Gaussian random variable YQ inde- 417 EXTREMA OF BROWNIAN MOTION 371 pendent of {Xx - Y0, AG A}. Condition (B). There exists A € A such that X\ £ // 0 . It is easy to see that the condition (B) implies that X\ £ H$ for all A G A. Denote by RA the space of real valued functions on A; it has a Borel structure defined in a natural way. Then we can regard XA = {X\, A G A} as a random variable taking values in R A . The following lemma is rather trivial; nevertheless, it plays a fundamental role in this paper. Lemma 1. (i) Let f be a Borel function from RA to R such that (1-1) f(w + tl) = ftw) + ct for any w G RA and t € R where c is some nonzero constant and 1 denotes the function on A that identically equals 1. Then under the condition (A) we have f(XA) = CYQ + Y with a suitable random variable Y independent of YQ; in particular, the probability distribution of f(XA) has a strictly positive C°°-density. (ii) Suppose A is a locally compact space with a countable open base and assume that X\ is continuous in A with probability 1. We regard XA = {X\, A G A} as a random variable taking values in the space C(A) of continuous functions on A, which is equipped with the compact uniform topology. Then, under the condition (A), the conclusion of (i) remains valid for any Borel function f from C(A) to R satisfying (lA)for w G C(A) and t G R. (iii) The condition (B) implies the condition (A). REMARK 1. Let A*, 1 < k < n, be subsets of A and let ck, 1 < k < n, be constants such that c\ + • • • + cn / 0. Let w(Ak) indicate either inf{w(A) : A G A*} or sup{u;(A) : A G A,t}; the choice may depend on k but not on w. Then (1.2) f(w) = cltw(Ai) + .-- + cIIi0(A,I) is a typical example of / satisfying (1.1) with c = c\ + - • - + cn provided that / can be defined to be a Borel function. REMARK 2. Let F be a class of functions defined on [0, 1] and taking values in A (an example of such an F is the space of continuous paths in A connecting two given points of A). Then the function / defined by f(w) = \nf{g(w, u) : u 6 F} with g(w, u) - sup{w(u{t)) : 0 < t < 1} satisfies (1.1). REMARK 3. If {X\, A G A} satisfies (A) (resp. (B)) and if A\ is a nonempty subset of A, then the sub-system {X\, A G Aj} also satisfies (A) (resp. (B)). Proof of Lemma 1. (i) Under the condition (A) XA~Y0l and YQ are independent so f(XA) — cY0 = f(XA - YQ1) and Y0 are independent. If we put Y = f(XA) - CYQ, 418 372 H. TANAKA then we have the expression f(XA) = cYo + Y in which YQ and Y are independent and Ko is a nondegenerate Gaussian random variable. The assertion (ii) follows from (i). (iii) It is easy to see that X\ + Ho = {Xx + Y : Y e HQ} does not depend on A. The condition (B) means that X\ + H0 $ 0. Since Xx + Ho is a closed convex set, there exists a unique YQ e Xx + H0 such that yJE{Y*} = min UE{\XX + Y\2}: Y € H0\ > 0. Then clearly Y0 _L H0. Since Xx - Y0 £ H0, Xx - Y0 ± Y0 for all A. This implies that Yo is independent of{X A — /(), A G A}. • 2. Proof of Theorem 1 As stated in Introduction let X = {X(x), x £ Kd] be a Brownian motion with ^-dimensional time. For any fixed pair of real numbers t\ and t2 such that 0 < t\ < h we put A = {x e Rd :?i < |x| 3. Denoting by 50 the uniform distribution on Sd~l = {$ eRd : \0\ = 1}, we put R(t) = J X{tB)d6, Hx = c.l.s.{/?(r), t > 0, fi < / < r 2 } , H^ = the orthogonal complement of Hi in H. Then we have (2.1) X(x) - R(\x\) £ //f1 for any x € A. In fact, it is easy to see that, for each fixed t > 0, E{(X(x) - R(\x\))R(t)} depends only on |x| and hence it must vanish, which implies (2.1). We are going to prove that X(tiO) 0 H0 for 0 £ Sd~l. The relation (2.1) implies that X{t\Q) = R{t{) + X' with X' £ H^ and that H0 C Hw e H^ where // ( 0 = c.\.s.{R(t) - R(s),t,s £ [t{,t2]}. Therefore, for the proof of X{t\B) 0 HQ it is enough to show that R(t\) £ H\Q. We now make use of the canonical representation of the Gaussian process {R(t), t > 0} due to McKean [5], which means that R(t)= I f(t,r)dB(r), Jo t>0, 419 EXTREMA OF BROWNIAN MOTION 373 where {B(r), r > 0} is a one-dimensional standard Brownian motion and (2.2) f(t,r) = k(d) J (\-u2)id~3)/2du, 0 Jr/t k(d) being a suitable constant depending only on d. For any s and t with t\ < s < t < t2 we have R(t)-R(s) m)= = [ fts(r)dB(r)+ Jo [ Jit gtx(r)dB(r), f f{r)dB{r), Jo where fls{r) = f(t, r) - f(s, r), f(r) = f{t\, r) and gts(r) is a suitable function. Therefore, if we put //o = c.l.s. J J ' fts{r)dB{r\ t, s G [tut2] H+ = c.l.s. {B(u) - B(r), r, u E [/,, t2]} , then Ho ± H+, Hi0 C Ho © H+ and R(ti) _L W+. From these observations we see that for the proof of R(t\) £ # 1 0 , it is enough to show (2.3) / ' f(r)dB(r) £ HQ. Jo Let LQ be the subspace of L2[0, ?]] spanned by the functions /«(•). t,s £ [t\, ti\. Then the Hilbert space //o is isomorphic to L\ and (2.3) is equivalent to / 0 L\. Now the assumption that d is an odd integer > 3 implies that fts(r), f, s, G |>i,/2], are polynomials of degree d ~ 2 vanishing at r = 0 (use (2.2)). Therefore all the functions in LQ are also polynomials of degree at most d - 2 vanishing at r = 0. On the other hand it is easy to see that / is a polynomial of degree d — 2 with f(0) > 0. Therefore / £ LQ, which finally implies X(?i#) ^ //Q. This completes the proof in the case where d is odd and d > 3. (ii) The proof in the case where d is even can be obtained by the method of descent in which a Brownian motion with ^-dimensional time is viewed as the restriction of a Brownian motion with (d + l)-dimensional time to Rd x {0} c Rd+l and also by using Remark 3. The proof in the case d = 1 is easy. The proof of Lemma 2 is finished. • We are now able to prove Theorem 1. From the assumption on Ak, 1 < k < n, there exist tx and t2 with 0 < t\ < t2 such that A = {x € Rd : t\ < \x\ < t2} includes all Ak. Then, by Lemma 2 the condition (B) is satisfied for XA = {X(x), x G A} 420 374 H. TANAKA U2 5a 5b and by Remark 1 the condition (1.1) is satisfied for the function f(w) = c\w(Ai) + C2w(A\) + • • • + cnW(An), w G C(A), with c = c\ + • • • + cn. Therefore by Lemma 1 the probability distribution of the random variable f(XA) = c\X{A\) + C2X{Ai) + • • • + c„X(An) has a form 7 * /A. This completes the proof of Theorem 1. 3. Proof of Theorem 2 Under the assumption on Aj and Bk in Theorem 2 we can take disjoint closed balls K and L with the following properties: QA)KD\Jj=lAj, LD\Jnk=lBk. (3.2) K is separated from L by the hyperplane n . (3.3) The center a of K and the center b of L are on the straight line that passes through the origin 0 and is perpendicular to n . We consider open balls U\ and Ui with a common radius e and with centers 6a and 5b, respectively, where 6 > 0 is chosen so that 5a g K and 5b £ L (see the figure). We now make use of the Chentsov representation of X(x) ([1]), which asserts that (3.4) X(x) = W(DX), where Dx is the open ball with center x/2 and radius \x\/2, and {W(d£)} is a suitable white noise in R^ associated with the measure Cd|£|-rf+1d£ (cj is a suitable constant), namely, a Gaussian random measure in R^ such that E{W{dQ] = 0 and E{W(d£)2} = c d\£\~d+xd£,- By taking e > 0 small enough, we can assume (3.5) [/, C H Dx\ f| { U Dy\ , .x£K v&L U2c{f)Dy\f){{jDx yeL .xeK If we write X(x) = W{DX) = W(U{) + Xx and X(y) = W(Dy) = W(U2) + Xy, then (3.5) implies that the 2-dimensional random vector (W(Ui), WHJ2)) is independent of the Gaussian family {(Xx, Xy) : x e K,y 6 L}. Therefore we have MX) = cW{U\) + Ju MX) = c'W{U2)+~f2 421 EXTREMA OF BROWNIAN MOTION 375 with c = YTj=\ ch c' = J2*i c'k a n d (W(t/i), W(U2)) is independent of Qufi). Since W(U\) and W{U2) are independent and each of them is a nondegenerate Gaussian random variable with mean 0, the joint distribution of /i(X) and fi(X) has a form ( 7 i <S> 72) * v. 4. Proof of Theorem 3 and Theorem 4 By using the fact that {X(x)~-X(XQ), X G Rd} is identical in law to {X(x — XQ), X G R } for each XQ £ Rd and also by using the assumption Y^j=\cj = St=i c*» w e s e e that the probability distribution of fi(X) — fi{X) is invariant under any simultaneous shift of Aj and B*. Therefore, in proving Theorem 3 we may assume that Aj. and fit satisfy the same assumption as in Theorem 2. Then the joint distribution of f$X) and f2(X) has a form (71 0 72) * v by Theorem 2 and this implies the conclusion of Theorem 3. Before going to the proof of Theorem 4 we introduce some notation. Denote by K the set of all pairs (K\, K2) of disjoint closed balls K\ and K2 with rational centers and rational radii. We put f(K\, K2\o\, &%) ~ X(K\\<j$ — X{K2;<Ji) where each 5. Remarks on a diffusion process in a rf-dimensional Brownian environment This section is to supply an example for the applicability of Theorem 4. We change the notation for a Brownian motion with a ^-dimensional time since we want to use X(t) for a diffusion process. Let W be the space of continuous functions on Rd vanishing at 0. In this section an element W of W is called an environment. We consider the probability measure P on W such that {W(x), x G Rd, P} is a Levy's Brownian motion with a d-dimensional time. Let Q be the space of continuous functions on [0, 00) taking values in Rd. The value of OJ(G £2) at time t is denoted by X(t) = X{t, u) = to(t). For each fixed environment W we consider the probability measure Pw on £2 such that {X(t), t > 0, Pw} is a diffusion process in Rd with generator and starting from 0. Let V be the probability measure on W x Q defined by VidWduj) - P(dW)Pw{duj). Then {X(t),t > 0,V} can be regarded as a process defined on the probability space (W x Q, V), which we call a diffusion process in 422 376 H. TANAKA a (/-dimensional Brownian environment. When d = 1, this model is a diffusion analogue of well-known Sinai's random walk in a random environment 1982) and much is known about the long-term behavior of X(t) such as localization. When d > 2, a similar diffusion model appeared in [3]. Now our interest is the long-term behavior of {X(t), t > 0, V} in the case d > 2. Tanaka [6](see also [7]) proved that, for any dimension d, {X(t), t > 0, Pw} is recurrent for almost all Brownian sample environments W. Mathieu[4] proved that localization takes place for {X(t),t > 0, "P}, in the sense that lim Urn -p(A" 2 max{|X(r)| : 0 < t < ex} > N) = 0. JV—*ooA—too However, in the case d > 2, it seems that the existence of the limiting distribution of {\~2X(ex),V} as A —> co is still an open problem. We give a remark on this problem. We notice the scaling relation {X(t), r > 0 , PXWx} = {\-2X(\4t), r>0, Pw}, where A > 0 and W e W are fixed, Wx denotes an element of W defined by W\(x) = A-'WCA2*), x e Rd, and = means the equality in distribution. This scaling relation combined with W\ = W imply the following: If we can prove that {X(erX), P\w} has the limiting distribution as A —• oo under the condition r = r(X) —> 1, then so does {\~2X(ex), V}. From now on we are interested in {X(t), Pxw}- For W e W we define the sub-level domain D as the connected component of the open set {x e Kd : W(x) < 1} containing 0. Then it is easy to see that D is bounded, P-a.s. By making use of Theorem 4 we see that for W not belonging to some P-negligible subset of W, there exists a point b of local (strict) minimum of W with depth > 1 inside D. Such a point b is characterized by (i) W(b) < W(x) forxGU{b} and (ii) U C D, where U denotes the connected component of the open set {x e Rd : W(x) — W(b) < 1} containing b. It is obvious that the totality of such points b is a finite set, which is denoted by {bk(W), 1 < k < l(W)}. Now suppose l(W) = 1 and put b = bi(W). Then from the argument of [4] we see that (5.1) X(erX) —> b (in probability with respect to P\w) as A —» oo provided r = r(A)(non-random) tends to 1. If l(W) > 2, we do not know whether the limiting distribution of X(erX) exists. Hoping for the best, we think it might be possible to define /?, in one way or another, as a single point among bk(W), 1 < k < l(W), and to prove (5.1) even in the case l(W) > 2, for almost all w. 423 EXTREMA OF BROWNIAN MOTION 377 References [1] [2] [3] [4] [5] [6] [7] N.N. Chentsov: Levy Brownian motion for several parameters and generalized white noise, Theory Probab. Appl. 2 (1957), 265-266 (English translation). P. Levy: Processus Stochastiques et Mouvement Brownien, Gauthier-Viliars, Paris, 1948. E. Marinari, G. Parisi, D. Ruelle and P. Windy: On the interpretation of 1 / / noise, Comm. Math. 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[2] Some properties of one-dimensional diffusion processes (with G. Maruyama), Mem. Fac. Sci., Kyushu Univ., Ser. A, 11 (1957), 117-141. [3] Certain limit theorems concerning one-dimensional diffusion processes, Mem. Fac. Sci., Kyushu Univ., Ser. A, 12 (1958), 1-11. [4] Ergodic property of AT-dimensional recurrent Markov processes (with G. Maruyama), Mem. Fac. Sci., Kyushu Univ., Ser. A, 13 (1959), 157-172. [5]* Additive functionals of the Brownian path (with H.P. McKean), Mem. Coll. Sci., Kyoto Univ., Ser. A, 3 3 (1961), 479-506. [6] Local times on the boundary for multi-dimensional diffusion (with K. Sato), Proc. Japan Acad., 38 (1962), 699-702. [7]* Note on continuous additive functionals of the 1-dimensional Brownian path, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 1 (1963), 251-257. [8]* Existence of diffusions with continuous coefficients, Mem. Fac. Sci., Kyushu Univ., Ser. A, 18 (1964), 89-103. [9] Note on additive functionals of the class (U) of Meyer of Markov processes, Mem. Fac. Sci., Kyushu Univ., Ser. A, 18 (1964), 154-165. [10] Local solutions of stochastic differential equations associated with certain quasilinear parabolic equations, J. Fac. Sci., Univ. Tokyo, Sec. 1,14 (1967), 313-326. [II] On the pre-closedness of the potential operator (with K. Yosida and T. Watanabe), J. Math. Soc. Japan, 20 (1968), 419-421. [12] Propagation of chaos for certain Markov processes of jump type with nonlinear generators I, II, Proc. Japan Acad., 45 (1969), 449-452; 598-600. [13] Purely discontinuous Markov processes with nonlinear generators and their propagation of chaos, Teor. Verojatnost. Primenen., 15 (1970), 599-621. [14]* Propagation of chaos for certain purely discontinuous Markov processes with interactions, J. Fac. Sci., Univ. Tokyo, Sec. I, 17 (1970), 259-272. [15] * An inequality for a functional of probability distributions and its application to Kac's one-dimensional model of a Maxwellian gas, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 27 (1973), 47-52. [16]* On Markov process corresponding to Boltzmann's equation of Maxwellian gas, Proc. Second Japan-USSR Symp. on Probab. Th., 478-489, Lecture Notes in Math., 330, Springer-Verlag, 1973. [17] [18] [19]* [20] [21] [22]* [23] [24]* [25] [26] [27]* [28] [29]* [30] [31] [32]* [33]* An inequality for certain functional of multidimensional probability distributions (with H. Murata), Hiroshima Math. J., 4 (1974), 75-81. Perturbation of drift-type for Levy processes (with M. Tsuchiya and S. Watanabe), J. Math. Kyoto Univ., 14 (1974), 73-92. On the uniqueness of Markov process associated with the Boltzmann equation of Maxwellian molecules, Proc. Intern. Symp. on Stochastic Differential Equations, 1976, 409-425, Kinokuniya, 1978. On the growing up problem for semilinear heat equations (with K. Kobayashi and T. Sirao), J. Math. Soc. Japan, 29 (1977), 407-424. Homogenization of certain one-dimensional discontinuous Markov processes (with M. Horie and T. Inuzuka), Hiroshima Math. J., 7 (1977), 629-641. Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 46 (1978), 67-105. Stochastic differential equation associated with the Boltzmann equation of Maxwellian molecules in several dimensions, Stochastic Analysis, Proc. Intern. Conf. held at Northwestern Univ., 1978, 301-314, Academic Press, 1978. Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J., 9 (1979), 163-177. Central limit theorem for a simple diffusion model of interacting particles (with M. Hitsuda), Hiroshima Math. J., 11 (1981), 415-423. Fluctuation theory for Kac's one-dimensional model of Maxwellian molecules, Sankhya, Ser. A, Part I, 44 (1982), 23-46. Some probabilistic problems in the spatially homogeneous Boltzmann equation, Theory and Application of Random Fields, 258-267, Lecture Notes in Control and Information Sciences, 49, Springer-Verlag, 1983. Limit distribution of the minimum distance between independent and identically distributed d-dimensional random variables (with T. Onoyama and M. Sibuya), Statistical Extremes and Applications, 549-562, Reidel, 1984. Limit theorem for certain diffusion processes with interaction, Stochastic Analysis : Proceedings of the Taniguchi Intern. Symp. on Stochastic Analysis, Katata and Kyoto, 1982, 469-488, Kinokuniya, 1984. Homogenization of diffusion processes with boundary conditions, Advances in Probab., 411-437, Marcel Dekker, 1984. A diffusion process in a singular mean-drift-field (with M. Nagasawa), Z. Wahrscheinlichkeitstheorie verw. Gebiete, 68 (1985), 247-269. Central limit theorem for a system of Markovian particles with mean field interaction (with T. Shiga), Z. Wahrscheinlichkeitstheorie verw. Gebiete, 69 (1985), 439-459. Propagation of chaos for diffusing particles of two types with singular mean field interaction (with M. Nagasawa), Probab. Th. Rel. Fields, 71 (1986), 69-83. 427 [34]* Stochastic differential equations for mutually reflecting Brownian balls (with Y. Saisho), Osaka J. Math., 23 (1986), 725-740. [35]* Limit distribution for 1-dimensional diffusion in a reflected Brownian medium, Seminaire de Probabilites XXI, 246-261, Lecture Notes in Math., 1247, Springer-Verlag, 1987. [36]* Limit distributions for one-dimensional diffusion processes in self-similar random environments, Hydrodynamic Behavior and Interacting Particle Systems, 189-210, IMA Volumes, 9, Springer-Verlag, 1987. [37] On the symmetry of a reflecting Brownian motion defined by Skorohod's equation for multi-dimensional domain (with Y. Saisho), Tokyo J. Math., 10 (1987), 419-435. [38]* Stochastic differential equation corresponding to the spatially homogeneous Boltzmann equation of Maxwellian and non-cutoff type, J. Fac. Sci., Univ. Tokyo, Sec. IA, 34 (1987), 351-369. [39] Diffusion with interaction and collisions between coloured particles and the propagation of chaos (with M. Nagasawa), Probab. Th. Rel. Fields, 74 (1987), 161-198. [40] One-dimensional diffusions and random walks in random environments (with K. Kawazu and Y. Tamura), Probability Theory and Mathematical Statistics, Fifth Japan-USSR Symposium Proceedings, 1986, 170-184, Lecture Notes in Math., 1299, Springer-Verlag, 1988. [41]* Limit theorem for one-dimensional diffusion process in Brownian environment, Stochastic Analysis, 156-172, Lecture Notes in Math., 1322, Springer-Verlag, 1988. [42] On the propagation of chaos for diffusion processes with drift coefficients not of average form (with M. Nagasawa), Tokyo J. Math., 10 (1989), 403418. [43] Time reversal of random walks in one-dimension, Tokyo J. Math., 12 (1989), 159-174. [44] A theorem of Pitman type for simple random walks on Zd (with H. Miyazaki), Tokyo J. Math., 12 (1989), 235-240. [45] Limit theorems for one-dimensional diffusions and random walks in random environments (with K. Kawazu and Y. Tamura), Probab. Th. Rel. Fields, 80 (1989), 501-541. [46] Time reversal of random walks in Rd Tokyo J. Math., 13 (1990), 375-389. [47] Localization of diffusion processes in one-dimensional random environment (with K. Kawazu and Y. Tamura), J. Math. Soc. Japan, 44 (1992), 515-550. [48] Limit theorems of the minimum interpoint distance between any pair of i.i.d. random points in Rd (with S. Kanagawa and Y. Mochizuki), Ann. Inst. Statist. Math., 44 (1992), 121-131. 428 [49] * On the maximum of a diffusion process in a drifted Brownian environment (with K. Kawazu), Seminaire de Probabilites XXVII, 78-85, Lecture Notes in Math., 1557, Springer-Verlag, 1993. [50] Green operators of absorbing Levy processes on the half line, Stochastic Processes, A Festschrift in honour of G. Kallianpur, 313-319, SpringerVerlag, 1993. [51]* Recurrence of a diffusion process in a multidimensional Brownian environment, Proc. Japan Acad., 69, Ser. A (1993), 377-381. [52]* Localization of a diffusion process in a one-dimensional Brownian environment, Comm. Pure Appl. Math,, 47 (1994), 755-766. [53] Localization of one-dimensional random walks in random environment (with K. Kawazu and Y. Tamura), Proc. Intern. Conf. on "Stochastic Processes, Physics and Geometry II", 1991, 444-452, World Scientific, 1995. [54]* Diffusion processes in random environments, Proc. Intern. Congress of Mathematicians, Zurich, Switzerland 1994, 1047-1054, Birkhauser, 1995. [55]* Environment-wise central limit theorem for a diffusion in a Brownian environment with large drift, Ito's Stochastic Calculus and Probability Theory, 373-384, Springer-Verlag, 1996. [56]* A diffusion process in a Brownian environment with drift (with K. Kawazu), J. Math. Soc. Japan, 49 (1997), 189-211. [57]* Limit theorems for a Brownian motion with drift in a white noise environment, Chaos, Solitons & Fractals, 8 (1997), 1807-1816. [58]* Invariance principle for a Brownian motion with large drift in a white noise environment (with K. Kawazu), Hiroshima Math. J., 28 (1998), 129-137. [59] Stochastic differential equations of pure-jumps in relativistic quantum theory (with M. Nagasawa), Chaos, Solitons & Fractals, 10 (1999), 1265-1280. [60] Time dependent subordination and Markov processes with jumps (with M. Nagasawa), Seminaire de Probabilites XXXIV, 257-288, Lecture Notes in Math., 1729, Springer-Verlag, 2000. [61] The principle of variation for relativistic quantum particles (with M. Nagasawa), Seminaire de Probabilites XXXV, 1-27, Lecture Notes in Math., 1755, Springer-Verlag, 2001. [62] A diffusion process with a one-sided Brownian potential (with K. Kawazu, and Y. Suzuki), Tokyo J. Math., 24 (2001), 211-229. [63]* Some theorems concerning extrema of Brownian motion with d-dimensional time, Osaka J. Math., 38 (2001), 369-377. 429 Permissions The editors and World Scientific Publishing Co. Pte. Ltd. would like to thank the original publishers of Tanaka's papers for granting permissions to reprint specific papers in this volume. The following list contains the credit lines for those articles. [5] [7] [8] [14] [15] [16] [19] [22] [24] [27] [29] [32] [33] [34] [35] [36] [38] Reprint from Mem. Coll. Sci, Kyoto Univ., Ser. A, 12, © 1961 by Kyoto University. Reprint from Z. Wahrscheilichkeitstheorie verw. Gebiete, 1, © 1963 by Springer-Verlag. Reprint from Mem. Fac. Sci., Kyushu Univ., Ser. A, 18, © 1964 by Kyushu University. Reprint from J. Fac. Sci., Univ. Tokyo, Sec. I, 17, © 1970 by the University of Tokyo. Reprint from Z. Wahrscheilichkeitstheorie verw. Gebiete, 27, © 1973 by Springer-Verlag. Reprint from Lecture Notes in Mathematics, 330, © 1973 by SpringerVerlag. Reprint from Proc. Intern. Symp. on Stochastic Differential Equations, © 1978 by Kinokuniya Co. Ltd. Reprint from Z. Wahrscheilichkeitstheorie verw. Gebiete, 46, © 1978 by Springer-Verlag. Reprint from Hiroshima Math. J., 9, © 1979 by Hiroshima University. Reprint from Lecture Note in Control and Information Sciences, 49, © 1983 by Springer-Verlag. Reprint from Stochastic Analysis : Proceedings of the Taniguchi Intern. Symp. on Stochastic Analysis, Katata and Kyoto, 1982, © 1984 by Kinokuniya Co. Ltd. Reprint from Z. Wahrscheilichkeitstheorie verw. Gebiete, 69, © 1985 by Springer-Verlag. Reprint from Probab. Th. Rel. Fields, 71, © 1986 by Springer-Verlag. Reprint from Osaka J. Math., 23, © 1986 by Osaka University. Reprint from Lecture Notes in Mathematics, 1247, © 1987 by SpringerVerlag. Reprint from Hydrodynamic Behavior and Interacting Particle Systems, (ed. G. Papanicolaou), IMA Volumes in Mathematics and Its Applications, 9, © 1987 by Springer-Verlag. Reprint from J. Fac. Sci., Univ. Tokyo, Sec. IA, 34, © 1987 by the University of Tokyo. 430 [41] Reprint from Lecture Notes in Mathematics, 1322, © 1988 by SpringerVerlag. [49] Reprint from Lecture Notes in Mathematics, 1557, © 1993 by SpringerVerlag. [51] Reprint from Proc. Japan Academy, 69, © 1993 by the Japan Academy. [52] Reprint from Comm. Pure Appl. Math., 47, © 1994 by John Wiley & Sons, Inc. [54] Reprint from Proc. Intern. Congress of Mathematicians, Zurich, Switzerland 1994, © 1995 by Birkhauser Verlag AG. [55] Reprint from Ito's Stochastic Calculus and Probability Theory, (eds. N. Ikeda et al.) © 1996 by Springer-Verlag. [56] Reprint from J. Math. Soc. Japan, 49, © 1997 by Mathematical Society of Japan. [57] Reprint from Chaos, Solitons & FVactals, 8, © 1997 by Elsevier Science. [58] Reprint from Hiroshima Math. J., 28, © 1998 by Hiroshima University. [63] Reprint from Osaka J. Math., 38, © 2001 by Osaka University. STOCHASTIC PROCESSES Selected Papers of Hiroshi Tanaka Hiroshi Tanaka is noted for his discovery of rhe "Tanaka formula", which is a generalization of the Ito formula in stochastic analysis. This important book is a selection of his brilliant works on srochasric processes and related topics. It contains Tanaka's papers on (i) Drownian morion and stochastic differential equations (additive functionals of Drownian parhs and stochastic differenrial equarions with reflecring boundaries), (ii) rhe probabilistic treatment of nonlinear equarions (Dolrzmann equation, propagorion of chaos and McKean-Vlasov limir), and (iii) srochasric processes in random environment (especially limir theorems on rhe srochasric processes in one-dimensional random environments and rheir refinemenrs). The book also includes essays by Henry McKean, Marc Yor, Shinzo Waranabe and Hiroshi Tanakn on Tanaka's works. ISBN 981-02-4591-2 www. woriascientiiic. com 4673 he

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