Applied Probability Control Economics Information and Communication Modeling and Identification Numerical Techniques Optimization Edited by Advisory Board Applications of Mathematics 11 A V Balakrishnan E Dynkin G Kallianpur K Krickeberg G I Marchuk R Radner T Hida Brownian Motion Translated by the Author and T P Speed With 13 Illustrations Springer-Verlag New York Heidelberg Berlin T Hida T P Speed Department of Mathematics Faculty of Science Nagoya University Chikasu-Ku, Nagoya 464 Japan Department of Mathematics University of Western Australia Nedlands, W.A 6009 Australia Editor A V Balakrishnan Systems Science Department University of California Los Angeles, California 90024 USA AMS Subject Classification (1980): 60j65 Library of Congress Cataloging in Publication Data Hida, Takeyuki, 1927Brownian motion (Applications of Mathematics; Vol 11) Bibliography: p Includes index Brownian motion processes I Title QA274.75.H5213 519.2'82 79-16742 Originally published in Japanese by Iwanami Shoten, Publishers, Tokyo, 1975 All rights reserved No part of this book may be translated or reproduced in any form without written permission from the copyright holder © 1980 by Takeyuki Hida Softcover reprint of the hardcover 1st edition 1980 543 ISBN-13: 978-1-4612-6032-5 e-ISBN-13: 978-1-4612-6030-1 DOl: 10.1007/978-1-4612-6030-1 Preface to the English Edition Following the publication of the Japanese edition of this book, several interesting developments took place in the area The author wanted to describe some of these, as well as to offer suggestions concerning future problems which he hoped would stimulate readers working in this field For these reasons, Chapter was added Apart from the additional chapter and a few minor changes made by the author, this translation closely follows the text of the original Japanese edition We would like to thank Professor J L Doob for his helpful comments on the English edition T Hida T P Speed v Preface The physical phenomenon described by Robert Brown was the complex and erratic motion of grains of pollen suspended in a liquid In the many years which have passed since this description, Brownian motion has become an object of study in pure as well as applied mathematics Even now many of its important properties are being discovered, and doubtless new and useful aspects remain to be discovered We are getting a more and more intimate understanding of Brownian motion The mathematical investigation of Brownian motion involves: a probabilistic aspect, viewing it as the most basic stochastic process; a discussion of the analysis on a function space on which a most interesting measure, Wiener measure, is introduced using Brownian motion; the development of tools to describe random events arising in the natural environment, for example, the function of biological organs; and a presentation ofthe background to a wide range of applications in which Brownian motion is involved in mathematical models of random phenomena It is hoped that this exposition can also serve as an introduction to these topics As far as (1) is concerned, there are many outstanding books which discuss Brownian motion, either as a Gaussian process or as a Markov process, so that there is no need for us to go into much detail concerning these viewpoints Thus we only discuss them briefly Topics related to (2) are the most important for this book, and comprise the major part of it Our aim is to discuss the analysis arising from Brownian motion, rather than Brownian motion itself regarded as a stochastic process Having established this analysis, we turn to several applications in which non-linear functionals of vii viii Preface Brownian motion (often called Brownian functionals) are involved We can hardly wait for a systematic approach to (3) and (4) to be established, aware as we are of recent rapid and successful developments In anticipation of their fruitful future, we present several topics from these fields, explaining the ideas underlying our approach as the occasion demands It seems appropriate to begin with a brief history of the theory Our plan is not to write a comprehensive history of the various developments, but rather to sketch a history of the study of Brownian motion from our specific viewpoint We locate the origin of the theory, and examine how Brownian motion passed into Mathematics The story began in the 1820's In the months of June, July and August 1827 Robert Brown F.R.S made microscopic observations on the minute particles contained in the pollen of plants, using a simple microscope with one lens of focal length about mm He observed the highly irregular motion of these particles which we now call "Brownian motion ", and he reported all this in R Brown (1828) After making further observations involving different materials, he believed that he had discovered active molecules in organic and inorganic bodies Following this, many scientists attempted to interpret this strange phenomenon It was established that finer particles move more rapidly, that the motion is stimulated by heat, and that the movement becomes more active with a decrease in viscosity of the liquid medium It was not until late in the last century that the true cause of the movement became known Indeed such irregular motion comes from the extremely large number of collisions of the suspended pollen grains with molecules of the liquid Following these observations and experiments, but apparently independent of them, a theoretical and quantitative approach to Brownian motion '\:- 't I L 'r1 '\ - \ \ V J T\/ v ~ \' V ,/ r r V ~ f"-, IA '" \ / 1\ I V \ "'- II ~~ /' l I """ ~ 0) we thus obtain u(x, t + r) dx = dx (' u(x - y, t)cp(r, y) dy, (0.1 ) -00 where the functions u and cp can be assumed smooth Further, the function cp can be supposed symmetric in space about the origin, with variance proportional to r: foo y2cp(r, y) dy = Dr, D constant -00 The Taylor expansion of (0.1) for small r gives u(x, t) + TUt(X, t) + o(r) l y) dy, L 1Iu(x, t) - YUx(X' t) + 21 y uxAx, t) - 'ICP(r, 00 00 which, under the assumptions above, leads to the heat equation (0.2) If the initial state of a grain is at some point y say, so that u(x, 0) = b(x - y), then from (0.2) we have u(x, t) = (2nDtt 1/2 exp f - (x 2-;~)2 J (0.3) The u(x, t) thus obtained turns out to be the transition probability function of Brownian motion viewed as a Markov process (see §2.4) Let us point out that formulae (0.2) and (0.3) were obtained in a purely theoretical manner Similarly the constant D is proved to be RT D= Nf' (0.4) where R is a universal constant depending on the suspending material, T the absolute temperature, N the Avogadro number and fthe coefficient offriction It is worth noting that in 1926 Jean Perrin was able to use the formula (0.4) in conjunction with a series of experiments to obtain a reasonably accurate determination of the Avogadro number In this we find a beautiful interplay between theory and experiment x Preface Although we will not give any details, we should not forget that around the year 1900 L Bachelier tried to establish the framework for a mathematical theory of Brownian motion Next we turn to the celebrated work of P Levy As soon as one hears the term Brownian motion in a mathematical context, Levy's 1948 book (second edition in 1965) comes to mind However our aim is to start with Levy's much earlier work in functional analysis, referring to the book P Levy (1951) in which he has organised his work along these lines dating back to 1910 Around that time he started analysing functionals on the Hilbert space E([O, 1]), and the need to compute a mean value or integral of a functional n > n , n! dxn ' ,- Generating function L tnHn(x; ( ) = e-a2t2/2+tx, 00 tEe, n=O (A.32) EXAMPLES 313 A.5 Formulae for Hermite Polynomials n L Hn-k(x; (12)Hk(Y; r2) = Hn(x k=O + Y; (12 + r2) [So Kakutani (1950)] (A.36) is a complete orthonormal system in r,2( R, (iii) Complex Hermite Polynomials [K } (1 e-x2/2a2 dx ) (A.39) Ito (1953a)] Definition Z E C, p, q ~ O Generating function (A.40) EXAMPLES H 0, o(z, z) == 1, H p, o(z, z) = zp, HI l(Z, z) = zz - 1, H 2,l(Z, z) = Z2 Z - 2z, H 2, 2(Z, z) = Z2 Z2 - 4zz + 2, H 3, (z, z) = Z3 Z - 3z , H 3, 2(Z, z) H , 3(Z, z) = Z3 Z3 - 9Z Z2 + 18zz - = Z3 Z2 z + 6z, 6z 314 Appendix 02 oz az H p, q(z, z) - z oz H p, q(z, z) 02 oz ozHp,q(z, z) - + qH p, q(z, z) = (A.41) Z ozHp,q(z, z) + pHp,q(z, z) = o OZ H p, q(z, z) = pH p- 1, q(Z, Z) (A.42) o ozHp,q(z, Z) = qHp,q_1(Z, Z) j H p+ 1, q(Z, Z) - zH p, q(Z, Z) + qH p, q_ (Z, Z) = \ H p, q+ (Z, Z) - zH p, q(Z, Z) + pH p_ 1, q(Z, Z) = \ ,Hp,q(z, z); p ~ 0, q ~ o} {J p.q (A.43) is a complete orthonormal system in the Hilbert space E (C, 2in e - Z Z dz /\ dz ) (A.44) (A.4S) x real Remark Complex Hermite polynomials with parameter Definitioo (J > 0, p, q Generating function co ~ L p, q=o tPtqH p,q" (z z-· (J2) = e-172tl+IZ+iz, tEe ~ Bibliography The following is a list of papers and books that are referred to in our discussion or that deal with related topics Those originally written in Russian are listed here in English if English translations are available Aczel, J (1966) Functional Equations and Their Applications Academic, New York Araki, H (1971), On representations of the canonical commutatioQ relations Commun Math Phys 20, 9-25 Bachelier, L (1941), Probabilites des oscillations maxima C.R Academie Sci Paris 212, 836-838 (Erratum: 213, 220) Balakrishnan, A V (1974), Stochastic optimization theory in Hilbert space Appl Math Optimization 1, 97-120 Bochner, S (1932), Vorlesungen tiber Fouriersche Integrale Leipzig, Akademische Verlagsgesellschaft - (1955), Harmonic Analysis and the Theory of Probability Univ of Calif Press, Berkeley, CA Brown, R (1828), A brief account of microscopical observations made in the months of June, July, and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies Phi/os Mag Ann of Phi/os New ser 4, 161-178 Cameron, R and Martin, W T (1944), Transformations of Wiener integrals under translations Ann Math 45, 386-396 - (1945), Fourier-Wiener transforms of analytic functionals Duke Math J 12,489-507 - (1 947a), Fourier-Wiener transforms offunctionals belonging to L2 over the space C Duke Math J 14,99-107 315 316 - Bibliography (1947b), The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals Ann Math 48, 385-392 Chung, K L., Erdos, P., and Sirao, T (1959), On the Lipschitz's condition for Brownian motion J Math Soc Japan 11, 263-274 Doob, J L (1953), Stochastic Processes Wiley, New York Dynkin, E B and Yushkevich, A A (1969), Markov Processes Theorems and Problems Plenum, New York (Russian original: IhuaTeflbcTBo HaYKa, 1967) Feller, W (1950), An Introduction to Probability Theory and Its Applications, Vol I Wiley, New York (third edition, 1968) - (1966), ibid., vol II (second edition, 1971) Freedman, D (1971), Brownian Motion and Diffusion Holden-Day, San Francisco FUrth, R (1956), Albert Einstein Investigations on the Theory of the Brownian Movement Dover, New York (Translated by Cowper, A D.) Gel'fand, I M and Yaglom, A M (1960), Integration in functional spaces and its applications in quantum physics J Math Phys 1, 48-69 Gel'fand, I M and Vilenkin, N Ya (1964), Generalized Functions, Vol 4, Applications of Harmonic Analysis Academic, New York (Russian original: rocy.uapcTBeHHoe Ih.uaTeflbcTBo,