Table Of ContentGrundlehren der
mathematischen Wissenschaften 260
A Series of Comprehensive Studies in Mathematics
Editors
M. Artin S. S. Chern A. Grothendieck E. Heinz
F. Hirzebruch L. Hormander S. Mac Lane W. Magnus
C. C. Moore J. K. Moser M. Nagata W. Schmidt
D. S. Scott J. Tits B. L. van der Waerden
Managing Editors
M. Berger B. Eckmann S. R. S. Varadhan
Grundlehren der mathematischen Wissenschaften
A Series of Comprehensive Studies in Mathematics
A Selection
180. Landkof: Foundations of Modem Potential Theory
181. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications I
182. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications II
183. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications III
184. Rosenblatt: Markov Processes, Structure and Asymptotic Behavior
185. Rubinowicz: Sommerfeldsche Polynommethode
186. Handbook for Automatic Computation. Vol. 2. Wilkinson/Reinsch: Linear Algebra
187. SiegeVMoser: Lectures on Celestial Mechanics
188. Warner: Harmonic Analysis on Semi-Simple Lie Groups I
189. Warner: Harmonic Analysis on Semi-Simple Lie Groups II
190. Faith: Algebra: Rings, Modules, and Categories I
191. Faith: Algebra II, Ring Theory
192. Mallcev: Algebraic Systems
193. P6Iya/Szego: Problems and Theorems in Analysis I
194. Igusa: Theta Functions
195. Berberian: Baer*-Rings
196. Athreya/Ney: Branching Processes
197. Benz: Vorlesungen iiber Geometric der Algebren
198. Gaal: Linear Analysis and Representation Theory
199. Nitsche: Vorlesungen iiber Minimalfliichen
200. Dold: Lectures on Algebraic Topology
201. Beck: Continuous Flows in the Plane
202. Schmetterer: Introduction to Mathematical Statistics
203. Schoeneberg: Elliptic Modular Functions
204. Popov: Hyperstability of Control Systems
205. Nikollskii: Approximation of Functions of Several Variables and Imbedding Theorems
206. Andre: Homologie des Algebres Commutatives
207. Donoghue: Monotone Matrix Functions and Analytic Continuation
208. Lacey: The Isometric Theory of Classical Banach Spaces
209. Ringel: Map Color Theorem
210. Gihman/Skorohod: The Theory of Stochastic Processes I
211. Comfort!Negrepontis: The Theory of Ultrafilters
212. Switzer: Algebraic Topology-Homotopy and Homology
213. Shafarevich: Basic Algebraic Geometry
214. van der Waerden: Group Theory and Quantum Mechanics
215. Schaefer: Banach Lattices and Positive Operators
216. P6Iya/Szego: Problems and Theorems in Analysis II
217. Stenstrom: Rings of Quotients
218. Gihman/Skorohod: The Theory of Stochastic Process II
219. Duvant/Lions: Inequalities in Mechanics and Physics
220. Kirillov: Elements of the Theory of Representations
221. Mumford: Algebraic Geometry I: Complex Projective Varieties
222. Lang: Introduction to Modular Forms
223. Bergh/LOfstrom: Interpolation Spaces. An Introduction
224. Gilbarg!Trudinger: Elliptic Partial Differential Equations of Second Order
Continued after Index
M. I. Freidlin
A. D. Wentzell
Random Perturbations of
Dynamical Systems
Translated by Joseph Sziics
With 20 Illustrations
Springer -Verlag
New York Berlin Heidelberg Tokyo
M. I. Freidlin Joseph Sziics (Translator)
A. D. Wentzell Texas A & M University
c/o VAAP at Galveston
Copyright Agency of the U.S.S.R. P.O. Box 1675
B. Bronnaja 6a Galveston, TX 77553
103104 Moscow U.S.A.
U.S.S.R.
AMS Classifications: 60HXX, 58G32
Library of Congress Cataloging in Publication Data
Freldlin, M. I. (Mark Iosifovich)
Random perturbations of dynamical systems
(Grundlehren der mathematischen Wissenschaften;
260)
Based on Russian ed. of: FluktuaiSii v dinamicheskikh
sistemakh pod delstviem malykh sluchainykh vozmushchenii
A. D. Venttsel'. 1979.
Bibliography: p.
Includes index.
1. Stochastic processes. 2. Perturbation (Mathematics)
I. Venttsel', A. D. II. VentiSel', A. D. FluktuaiSii v
dinamicheskikh sistemakh pod deistviem malykh
sluchainykh vozmushcheniL III. Title. IV. Series.
QA274.F73 1983 519.2 83-4712
Original Russian edition: Fluktuatsii v Dinamicheskikh Sistemakh Pod Deistl'iem
Malykh Sluchainykh Vozmushchenii, Nauka: Moscow, 1979.
© 1984 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1984
All rights reserved. No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New
York 10010, U.S.A.
Typeset by Composition House Ltd., Salisbury, England.
9 8 7 6 5 4 3 2 I
ISBN-13: 978-1-4684-0178-3 e-ISBN-13: 978-1-4684-0176-9
DOl: 10.1007/978-1-4684-0176-9
Foreword
Asymptotical problems have always played an important role in probability
theory. In classical probability theory dealing mainly with sequences of
independent variables, theorems of the type of laws of large numbers,
theorems of the type of the central limit theorem, and theorems on large
deviations constitute a major part of all investigations. In recent years, when
random processes have become the main subject of study, asymptotic
investigations have continued to playa major role. We can say that in the
theory of random processes such investigations play an even greater role
than in classical probability theory, because it is apparently impossible to
obtain simple exact formulas in problems connected with large classes of
random processes.
Asymptotical investigations in the theory of random processes include
results of the types of both the laws of large numbers and the central limit
theorem and, in the past decade, theorems on large deviations. Of course,
all these problems have acquired new aspects and new interpretations in
the theory of random processes.
One of the important schemes leading to the study of various limit
theorems for random processes is dynamical systems subject to the effect
of random perturbations. Several theoretical and applied problems lead to
this scheme. It is often natural to assume that, in one sense or another, the
random perturbations are small compared to the deterministic constitu
ents of the motion. The problem of studying small random perturbations
of dynamical systems has been posed in the paper by Pontrjagin, Andronov,
and Vitt [1]. The results obtained in this article relate to one-dimensional
and partly two-dimensional dynamical systems and perturbations leading
to diffusion processes. Other types of random perturbations may also be
considered; in particular, those arising in connection with the averaging
principle. Here the smallness of the effect of perturbations is ensured by the
fact that they oscillate quickly.
The contents of the book consists of various asymptotic problems arising
as the parameter characterizing the smallness of random perturbations
converges to zero. Of course, the authors could not consider all conceivable
schemes of small random perturbations of dynamical systems. In particu
lar, the book does not consider at all dynamical systems generated by random
vi Foreword
vector fields. Much attention is given to the study of the effect of pertur
bations on large time intervals. On such intervals small perturbations essen
tially influence the behavior of the system in general. In order to take ac
count of this influence, we have to be able to estimate the probabilities of
rare events, i.e., we need theorems on the asymptotics of probabilities of
large deviations for random processes. The book studies these asymptotics
and their applications to problems of the behavior of a random process on
large time intervals, such as the problem of the limit behavior of the in
variant measure, the problem of exit of a random process from a domain,
and the problem of stability under random perturbations. Some of these
problems have been formulated for a long time and others are compara
tively new.
The problems being studied can be considered as problems of the asymp
totic study of integrals in a function space, and the fundamental method
used can be considered as an infinite-dimensional generalization of the well
known method of Laplace. These constructions are linked to contemporary
research in asymptotic methods. In the cases where, as a result of the
effect of perturbations, diffusion processes are obtained, we arrive at prob
lems closely connected with elliptic and parabolic differential equations
with a small parameter. Our investigations imply some new results con
cerning such equations. We are interested in these connections and as a
rule include the corresponding formulations in terms of differential equa
tions.
We would like to note that this book is being written when the theory
of large deviations for random processes is just being created. There have
been a series of achievements but there is still much to be done. Therefore,
the book treats some topics that have not yet taken their final form (part
of the material is presented in a survey form). At the same time, some new
research is not reflected at all in the book. The authors attempted to mini
mize the deficiencies connected with this.
The book is written for mathematicians but can also be used by special
ists of adjacent fields. The fact is that although the proofs use quite intricate
mathematical constructions, the results admit a simple formulation as
a rule.
Contents
Introduction
CHAPTER 1
Random Perturbations IS
§l. Probabilities and Random Variables IS
§2. Random Processes. General Properties 17
§3. Wiener Process. Stochastic Integral 24
§4. Markov Processes and Semigroups 29
§S. Diffusion Processes and Differential Equations 34
CHAPTER 2
Small Random Perturbations on a Finite Time Interval 44
§l. Zeroth Approximation 44
§2. Expansion in Powers of a Small Parameter SI
§3. Elliptic and Parabolic Differential Equations with a Small Parameter at the
Derivatives of Highest Order S9
CHAPTER 3
Action Functional 70
§l. Laplace's Method in a Function Space 70
§2. Exponential Estimates 74
§3. Action Functional. General Properties 79
§4. Action Functional for Gaussian Random Processes and Fields 92
CHAPTER 4
Gaussian Perturbations of Dynamical Systems. Neighborhood of an
Equilibrium Point 103
§l. Action Functional 103
§2. The Problem of Exit from a Domain 108
§3. Properties of the Quasipotential. Examples 118
§4. Asymptotics of the Mean Exit Time and Invariant Measure for the
Neighborhood of an Equilibrium Position 123
§S. Gaussian Perturbations of General Form 132
CHAPTER 5
Perturbations Leading to Markov Processes 136
§l. Legendre Transformation 136
§2. Locally Infinitely Divisible Processes 143
viii Contents
§3. Special Cases. Generalizations 153
§4. Consequences. Generalization of Results of Chapter 4 157
CHAPTER 6
Markov Perturbations on Large Time Intervals 161
§l. Auxiliary Results. Equivalence Relation 161
~2. Markov Chains Connected with the Process (X,o, p.~) 168
§3. Lemmas on Markov Chains 176
§4. The Problem of the Invariant Measure 185
§5. The Problem of Exit from a Domain 192
§6. Decomposition into Cycles. Sublimit Distributions 198
§7. Eigenvalue Problems 203
CHAPTER 7
The Averaging Principle. Fluctuations in Dynamical Systems with
Averaging 212
§l. The Averaging Principle in the Theory of Ordinary Differential
Equations 212
§2. The Averaging Principle when the Fast Motion is a Random Process 216
§3. Normal Deviations from an Averaged System 219
§4. Large Deviations from an Averaged System 229
§5. Large Deviations Continued 237
§6. The Behavior of the System on Large Time Intervals 245
§7. Not Very Large Deviations 249
§8. Examples 253
§9. The Averaging Principle for Stochastic Differential Equations 263
CHAPTER 8
Stability Under Random Perturbations 278
§l. Formulation of the Problem 278
§2. The Problem of Optimal Stabilization 284
§3. Examples 290
CHAPTER 9
Sharpenings and Generalizations 294
§l. Local Theorems and Sharp Asymptotics 294
§2. Large Deviations for Random Measures 303
§3. Processes with Small Diffusion with Reflection at the Boundary 309
References 315
Index 325
Introduction
Let b(x) be a continuous vector field in R'. First we discuss nonrandom per
turbations of a dynamical system
Xt = b(xt)· (1)
We may consider the perturbed system
Xt = b(X" "'t), (2)
where b(x, y) is a function jointly continuous in its two arguments and turning
o.
into b(x) for y = We shall speak of small perturbations if the function '"
giving the perturbing effect is small in one sense or another.
We may speak of problems of the following kind: the convergence of the
solution Xt of the perturbed system to the solution Xt of the unperturbed
system as the effect of the perturbation decreases, approximate expressions of
various accuracies for the deviations Xt - Xt caused by the perturbations,
and the same problems for various functionals of a solution (for example, the
first exit time from a given domain D).
To solve the kind of problems related to a finite time interval we require
less of the function b(x, y) than in problems connected with an infinite
interval (or a finite interval growing unboundedly as the perturbing effect
decreases). The simplest result related to a finite interval is the following: if
the solution of the system (1) with initial condition Xo at t = 0 is unique, then
the solution Xt of system (2) with initial condition X 0 converges to Xt
uniformly in t E [0, T] as Xo -+ Xo and 11"'lloT = sUPOStST I"'tl-+ o. If the
function b(x, y) is differentiable with respect to the pair of its arguments, then
we can linearize it near the point x = Xt, Y = 0 and obtain a linear approxi
mation Ot of Xt - Xt as the solution of the linear system
(3)
under sufficiently weak conditions, the norm sUPostsTIXt - Xt - otl of the
remainder will be o(IXo - xol + ""'''OT). If b(x,y) is still smoother,
2 Introduction
then we have the decomposition
(4)
in which 'Y, depends quadratically on perturbations of the initial conditions
and the right side (the function 'Y, can be determined from a system of linear
differential equations with a quadratic function of "'" (), on the right side),
etc.
We may consider a scheme
(5)
depending on a small parameter e, where "', is a given function. In this case
X: Xo
for the solution with initial condition = Xo we can obtain a decom
position
(6)
in powers of e with the remainder infinitely small compared with en, uniformly
on any finite interval [0, T].
Under more stringent restrictions on the function b(x, y), results of this
kind can be obtained for perturbations "', which are not small in the norm
of uniform convergence but rather, for example, in some LP-norm or another.
As far as results connected with an infinite time interval are concerned,
stability properties of the unperturbed system (1) as t -+ 00 are essential.
Let x* be an equilibrium position of system (1), i.e., let b(x*) = 0. Let this
equilibrium position be asymptotically stable, i.e., for any neighborhood
U3 x* let there exist a small neighborhood V of x* suc°h that for any Xo E V
the trajectory x, starting at Xo does not leave U for t ~ and converges to x*
as t -+ 00. Denote by G* the set of initial points Xo from which there start
solutions converging to x* as °t - + 00. For° an y neighborhood U of x* and any
point Xo E G* there exist () > and T > such that for
the solution X, of system (2) with initial condition Xo does not go out of
U for t ~ T. This holds uniformly in Xo within any compact subset of G* (i.e.,
() and T can be chosen the same for all points Xo of this compactum). This
also implies the uniform convergence of X, to x, on the infinite interval
[0, (0) provided that X 0 -+ Xo, sUPO:5' < 00 I" ', I -+ 0.
On the other hand, if the equilibrium position x* does not have the
indicated stability properties, then by means of arbitrarily small perturba
tions, the solution X, of the perturbed system can be "carried away" from x*
for sufficiently large t even if the initial point X 0 equals x*. In particular,
there are cases where the solution x, of the unperturbed system cannot leave
