8T0CHA8HC elected Papers of Hiroshi Tanaka
Edited by
Makoto Maejima & Tokuzo Shiga
World Scientific
STOCHASTIC PROC...

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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 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): ss,

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)<jbys2 dimensions, but f can still be expressed as a (formal) Hellinger integral 17 1 "

f/n _ [ measure (s: x(s) €db, s

Edited by

Makoto Maejima & Tokuzo Shiga

World Scientific

STOCHASTIC PROCESSES f Selected Papers of Hiroshi Tanaka

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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.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4591-2

Printed in Singapore by Uto-Print

-#*SSF*.--.

-*.

V-

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 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): ss,

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)<jbys2 dimensions, but f can still be expressed as a (formal) Hellinger integral 17 1 "

f/n _ [ measure (s: x(s) €db, s

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