Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Second Edition

Applied Mathematical Sciences Volume 97 Editors J.E. Marsden L. Sirovich Advisors M. Ghil J.K. Hale T. Kambe J. Keller...

Author: Andrzej Lasota | Michael C. Mackey

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Applied Mathematical Sciences Volume 97 Editors J.E. Marsden L. Sirovich

Advisors M. Ghil J.K. Hale T. Kambe J. Keller K. Kirchgassner B.J. Matkowsky C.S. Peskin J.T. Stuart

Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo

Applied Mathematical Sciences I. John: Partial Differential Equations, 4th ed 2. Sirovich: Techniques of Asymptotic Analysis. 3. Hale: Theory of Functional Differential Equations, 2nd ed. 4. Percus: Combinatorial Methods. 5. von Mises!Friedrichs: fluid Dynamics. 6. Freiberger!Grenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 8. Giacaglia: Perturbation Methods in Non-linear Systems. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space. 10. Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations. II. Wolovich: Linear Multi variable Systems. 12. Berkovitz: Optimal Control Theory. 13. Bluman/Cole: Similarity Methods for Differential Equations. 14. Yoshizawa: Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions. 15. Braun: Differential Equations and Their Applications, 3rd ed. 16. Lefschetz: Applications of Algebraic Topology. 17. Collatz/Wetterling: Optimization Problems. 18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol. I. 19. Mars den/McCracken: Hopf Bifurcation and Its Applications. 20. Driver: Ordinary and Delay Differential Equations. 21. Courant/Friedrichs: Supersonic flow and Shock Waves. 22. Rouche!Habets/Laloy: Stability Theory by Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II. 25. Davies: Integral Transforms and Their Applications, 2nd ed. 26. Kushner!Clark: Stochastic Approximation Methods for Constrained and Unconstrained Systems. 27. de Boor: A Practical Guide to Splines. 28. Keilson: Markov Chain Models-Rarity and Exponentiality. 29. de Veubeke: A Course in Elasticity. 30. • niatycki: Geometric Quantization and Quantum Mechanics. 31. Reid: Sturmian Theory for Ordinary Differential Equations. 32. Meis/Markowitz: Numerical Solution of Partial Differential Equations. 33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III.

34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory. 36. Bengtsson/GhiVK/iiUn: Dynamic Meteorology: Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Uchtenberg!Lieberman: Regular and Chaotic Dynamics, 2nd ed. 39. Piccini/Stampacchia!Vidossich: Ordinary Differential Equations in R". 40. Naylor/Sell.· Linear Operator Theory in Engineering and Science. 41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. 42. Guckenheimer!Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. 43. Ockendonll'aylor: Inviscid fluid flows. 44. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. 45. Glashoff!Gustafson: Linear Operations and Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of SemiInfinite Programs. 46. Wilcox: Scattering Theory for Diffraction Gratings. 47. Hale et al: An Introduction to Infinite Dimensional Dynamical Systems-Geometric Theory. 48. Murray: Asymptotic Analysis. 49. Ladyzhenskaya: The Boundary-Value Problems of Mathematical Physics. 50. Wilcox: Sound Propagation in Stratified fluids. 51. Golubitsky!Schaeffer: Bifurcation and Groups in Bifurcation Theory, Vol. I. 52. Chipot: Variational Inequalities and flow in Porous Media. 53. Majda: Compressible fluid flow and System of Conservation Laws in Several Space Variables. 54. Was ow: Linear Turning Point Theory. 55. Yosida: Operational Calculus: A Theory of Hyperfunctions. 56. Chang/Howes: Nonlinear Singular Perturbation Phenomena: Theory and Applications. 57. Reinhardt: Analysis of Approximation Methods for Differential and Integral Equations. 58. Dwoyer/HussainWoigt (eds): Theoretical Approaches to Turbulence. 59. Sanders/Verhulst: Averaging Methods in Nonlinear Dynamical Systems. 60. GhiVChildress: Topics in Geophysical Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics.

(continued following index)

Andrzej Lasota

Michael C. Mackey

Chaos, Fractals, and Noise Stochastic Aspects of Dynamics Second Edition

With 48 Illustrations

'Springer

Andrzej Lasota Institute of Mathematics Silesian University ul. Bankowa 14 Katowice 40-058, Poland

Michael C. Mackey Center of Nonlinear Dynamics McGill University Montreal, Quebec H3G 1Y6 Canada

Editors

J .E. Marsden Control and Dynamical Systems, 107-81 California Institute of Technology Pasadena, CA 91125 USA

L. Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA

Mathematics Subject Classifications (1991): 60Gxx, 60Bxx, 58F30 Library of Congress Cataloging-in-Publication Data Lasota, Andrzej, 1932Chaos, fractals, and noise ; stochastic aspects of dynamics I Andrzej Lasota, Michael C. Mackey. p. em. - (Applied mathematical sciences ; v. 97) Rev. ed. of: Probabilistic properties of deterministic systems. 1985. Includes bibliographical references and index. ISBN 0-387-94049-9 1. System analysis. 2. Probabilities. 3. Chaotic behavior in systems. I. Mackey, Michael C., 1942. II. Lasota, Andrzej, 1932 . Probabilistic properties of deterministic systems. III. Title. IV. Series:Applied mathematical sciences (Springer-Verlag New York Inc.); v.97. QA1.A647 vol. 97 [QA402] 510s-dc20 [003'.75] 93-10432 Printed on acid-free paper. © 1994 Springer-Verlag New York, Inc. First edition published by Cambridge University Press as Probabilistic Properties of Deterministic Systems. 1985. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Hal Henglein; manufacturing supervised by Vincent R. Scelta. Photocomposed copy prepared from a TeX file. Printed and bound by Berryville Graphics, Berryville, VA. Printed in the United States of America.

9 8 7 6 5 4 3 ISBN 0-387-94049-9 ISBN 3-540-94049-9

SPIN 10851267

Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+ Business Media GmbH

To the memory of

Maria Waiewska- Czyiewska

Preface to the Second Edition

The first edition of this book was originally published in 1985 under the title "Probabilistic Properties of Deterministic Systems." In the intervening years, interest in so-called "chaotic" systems has continued unabated but with a more thoughtful and sober eye toward applications, as befits a maturing field. This interest in the serious usage of the concepts and techniques of nonlinear dynamics by applied scientists has probably been spurred more by the availability of inexpensive computers than by any other factor. Thus, computer experiments have been prominent, suggesting the wealth of phenomena that may be resident in nonlinear systems. In particular, they allow one to observe the interdependence between the deterministic and probabilistic properties of these systems such as the existence of invariant measures and densities, statistical stability and periodicity, the influence of stochastic perturbations, the formation of attractors, and many others. The aim of the book, and especially of this second edition, is to present recent theoretical methods which allow one to study these effects. We have taken the opportunity in this second edition to not only correct the errors of the first edition, but also to add substantially new material in five sections and a new chapter. Thus, we have included the additional dynamic property of sweeping (Chapter 5) and included results useful in the study of semigroups generated by partial differential equations {Chapters 7 and 11) as well as adding a completely new Chapter 12 on the evolution of distributions. The material of this last chapter is closely related to the subject of iterated function systems and their attractors-fractals. In addi-

viii


tion, we have added a set of exercises to increase the utility of the work for graduate courses and self-study. In addition to those who helped with the first edition, we would like to thank K. Alligood (George Mason), P. Kamthan, J. Losson, I. Nechayeva, N. Provatas (McGill), and A. Longtin (Ottawa) for their comments.

A.L. M.C.M.

Preface to the First Edition

This book is about densities. In the history of science, the concept of densities emerged only recently as attempts were made to provide unifying descriptions of phenomena that appeared to be statistical in nature. Thus, for example, the introduction of the Maxwellian velocity distribution rapidly led to a unification of dilute gas theory; quantum mechanics developed from attempts to justify Planck's ad hoc derivation of the equation for the density of blackbody radiation; and the field of human demography grew rapidly after the introduction of the Gompertzian age distribution. From these and many other examples, as well as the formal development of probability and statistics, we have come to associate the appearance of densities with the description of large systems containing inherent elements of uncertainty. Viewed from this perspective one might find it surprising to pose the questions: "What is the smallest number of elements that a system must have, and how much uncertainty must exist, before a description in terms of densities becomes useful and/or necessary?" The answer is surprising, and runs counter to the intuition of many. A one-dimensional system containing only one object whose dynamics are completely deterministic (no uncertainty) can generate a density of states! This fact has only become apparent in the past half-century due to the pioneering work of E. Borel (1909], A. Renyi (1957], and S. Ulam and J. von Neumann. These results, however, are not generally known outside that small group of mathematicians working in ergodic theory. The past few years have witnessed an explosive growth in interest in physical, biological, and economic systems that could be profitably studied using densities. Due to the general inaccessibility of the mathematical lit-

x


erature to the nonmathematician, there has been little diffusion of the concepts and techniques from ergodic theory into the study of these "chaotic" systems. This book attempts to bridge that gap. Here we give a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integr~partial differential equations. We have drawn examples from a variety of the sciences to illustrate the utility of the techniques we present. Although the range of these examples is not encyclopedic, we feel that the ideas presented here may prove useful in a number of the applied sciences. This book was organized and written to be accessible to scientists with a knowledge of advanced calculus and differential equations. In various places, basic concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and stochastic integrals and differential equations are introduced. This material is presented only as needed, rather than as a discrete unit at the beginning of the book where we felt it would form an almost insurmountable hurdle to all but the most persistent. However, in spite of our presentation of all the necessary concepts, we have not attempted to offer a compendium of the existing mathematical literature. The one mathematical technique that touches every area dealt with is the use of the lower-bound function (first introduced in Chapter 5} for proving the existence and uniqueness of densities evolving under the action of a variety of systems. This, we feel, offers some partial unification of results from different parts of applied ergodic theory. The first time an important concept is presented, its name is given in bold type. The end of the proof of a theorem, corollary, or proposition is marked with a •; the end of a remark or example is denoted by a 0. A number of organizations and individuals have materially contributed to the completion of this book. In particular the National Academy of Sciences (U.S.A.), the Polish Academy of Sciences, the Natural Sciences and Engineering Research Council (Canada}, and our home institutions, the Silesian University and McGill University, respectively, were especially helpful. For their comments, suggestions, and friendly criticism at various stages of our writing, we thank J. Belair (Montreal}, U. an der Heiden (Bremen}, and R. Rudnicki (Katowice). We are especially indebted toP. Bugiel (Krakow) who read the entire final manuscript, offering extensive mathematical and stylistic suggestions and improvements. S. James (McGill} has cheerfully, accurately, and tirelessly reduced several rough drafts to a final typescript.

Contents


vii


ix

1

Introduction 1.1 A Simple System Generating a Density of States 1.2 The Evolution of Densities: An Intuitive Point of View 1.3 Trajectories Versus Densities Exercises

1 1 5 9 13

2

The 2.1 2.2 2.3

17

3

4

Toolbox Measures and Measure Spaces Lebesgue Integration Convergence of Sequences of FUnctions Exercises

17 19 31 35

Markov and Frobenius-Perron Operators 3.1 Markov Operators 3.2 The Frobenius-Perron Operator 3.3 The Koopman Operator Exercises

37

Studying Chaos with Densities 4.1 Invariant Measures and Measure-Preserving Transformations

51

37 41

47 49

51

xii

Contents

4.2 4.3 4.4 4.5 5

Ergodic Transformations Mixing and Exactness Using the Frobenius-Perron Koopman Operators for Classifying Transformations Kolmogorov Automorphisms Exercises

The Asymptotic Properties of Densities Weak and Strong Precompactness Properties of the Averages An! Asymptotic Periodicity of { pn!} The Existence of Stationary Densities Ergodicity, Mixing, and Exactness Asymptotic Stability of {pn} Markov Operators Defined by a Stochastic Kernel Conditions for the Existence of Lower-Bound Functions Sweeping The Foguel Alterative and Sweeping Exercises

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

59 65 71

79 83 85 86

88 95 100 102 105 112 123 125 129 136

6

The Behavior of Transformations on Intervals and Manifolds 139 6.1 Functions of Bounded Variation 139 6.2 Piecewise Monotonic Mappings 144 6.3 Piecewise Convex Transformations with a Strong Repellor 153 6.4 Asymptotically Periodic Transformations 156 6.5 Change of Variables 165 6.6 Transformations on the Real Line 172 6.7 Manifolds 175 6.8 Expanding Mappings on Manifolds 183 Exercises 187

7

Continuous Time Systems: An Introduction 7.1 Two Examples of Continuous Time Systems 7.2 Dynamical and Semidynamical Systems 7.3 Invariance, Ergodicity, Mixing, and Exactness in Semidynamical Systems 7.4 Semigroups of the Frobenius-Perron and Koopman Operators 7.5 Infinitesimal Operators 7.6 Infinitesimal Operators for Semigroups Generated by Systems of Ordinary Differential Equations 7.7 Applications of the Semigroups of the Frobenius-Perron and Koopman Operators 7.8 The Hill~Yosida Theorem and Its Consequences

189 190 191 195 199 205 210 215 226

Contents

7.9 7.10 7.11 7.12 8

9

FUrther Applications of the Hille-Yosida Theorem The Relation Between the Frobenius-Perron and Koopman Operators Sweeping for Stochastic Semigroups Foguel Alternative for Continuous Time Systems Exercises

Discrete Time Processes Embedded in Continuous Time Systems 8.1 The Relation Between Discrete and Continuous Time Processes 8.2 Probability Theory and Poisson Processes 8.3 Discrete Time Systems Governed by Poisson Processes 8.4 The Linear Boltzmann Equation: An Intuitive Point of View 8.5 Elementary Properties of the Solutions of the Linear Boltzmann Equation 8.6 FUrther Properties of the Linear Boltzmann Equation 8.7 Effect of the Properties of the Markov Operator on Solutions of the Linear Boltzmann Equation 8.8 Linear Boltzmann Equation with a Stochastic Kernel 8.9 The Linear Tjon-Wu Equation Exercises Entropy Basic Definitions 9.1 9.2 Entropy of pn I When P is a Markov Operator 9.3 Entropy H(pn f) When P is a Frobenius-Perron

9.4

Operator Behavior of pn I from H (Pn f) Exercises

10 Stochastic Perturbation of Discrete Time Systems 10.1 Independent Random Variables 10.2 Mathematical Expectation and Variance 10.3 Stochastic Convergence 10.4 Discrete Time Systems with Randomly Applied Stochastic Perturbations Discrete Time Systems with Constantly Applied 10.5 Stochastic Perturbations 10.6 Small Continuous Stochastic Perturbations of Discrete Time Systems 10.7 Discrete Time Systems with Multiplicative Perturbations Exercises

xiii

232 241 244 246 247 261

251 252 258 261 264 268 270 273 277 280 283 283 289

292 395 300 303 304 306 311

315 320 327 330 333

xiv

Contents

11 Stochastic Perturbation of Continuous Time Systems 11.1 One-Dimensional Wiener Processes (Brownian Motion) 11.2 d-Dimensional Wiener Processes (Brownian Motion) 11.3 The Stochastic Ito Integral: Development 11.4 The Stochastic Ito Integral: Special Cases 11.5 Stochastic Differential Equations 11.6 The Fokker-Planck (Kolmogorov Forward) Equation 11.7 Properties ofthe Solutions of the Fokker-Planck Equation 11.8 Semigroups of Markov Operators Generated by Parabolic Equations 11.9 Asymptotic Stability of Solutions of the Fokker-Planck Equation 11.10 An Extension of the Liapunov Function Method 11.11 Sweeping for Solutions of the Fokker-Planck Equation 11.12 Foguel Alternative for the Fokker-Planck Equation Exercises

335 :i35 344 346 351 355 359

12 Markov and Foias Operators 12.1 The Riesz Representation Theorem 12.2 Weak and Strong Convergence of Measures 12.3 Markov Operators 12.4 Foias Operators 12.5 Stationary Measures: Krylov-Bogolubov Theorem for Stochastic Dynamical Systems 12.6 Weak Asymptotic Stability 12.7 Strong Asymptotic Stability 12.8 Iterated Function Systems and Fractals Exercises

393 393 397 405 411

References

449

Notation and Symbols

457

Index

461

364 368 371 378 386 388 391

417 420 425 432 447

1 Introduction

We begin by showing how densities may arise from the operation of a onedimensional discrete time system and how the study of such systems can be facilitated by the use of densities. If a given system operates on a density as an initial condition, rather than on a single point, then successive densities are given by a linear integral operator, known as the Frobenius-Perron operator. Our main objective in this chapter is to offer an intuitive interpretation of the Frobenius-Perron operator. We make no attempt to be mathematically precise in either our language or our arguments. The precise definition of the Frobenius-Perron operator is left to Chapter 3, while the measure-theoretic background necessary for this definition is presented in Chapter 2.

1.1 A Simple System Generating a Density of States One of the most studied systems capable of generating a density of states is that defined by the quadratic map S(x) = ax(1- x)

for 0 ::; x ::; 1.

(1.1.1)

We assume that a= 4 so S maps the closed unit interval [0, 1] onto itself. This is also expressed by the saying that the state (or phase) space of the system is [0, 1]. The graph of this transformation is shown in Fig. 1.1.1a.

2

1. Introduction

s (a)

(b)

Xj

0 ~~~--~----~~~~-----U

(c)

1

~~~~~~~~~~~r-r-~

Xj

0

200

100

i

FIGURE 1.1.1. The quadratic transformation (1.1.1) with a = 4 is shown in (a). In (b) we show the trajectory (1.1.2) determined by (1.1.1) with :~P = 1rjl0. Panel (c) illustrates the sensitive dependence of trajectories on initial conditions by using x 0 = (7r/10) +0.001. In (b) and (c), successive points on the trajectories have been connected by lines for clarity of presentation.

Having defined S we may pick a.n initial point x 0 e [0, 1] so that the successive states of our system at times 1, 2, ... are given by the trajectory (1.1.2) A typical trajectory corresponding to a given initial state is shown in Figure 1.1.1b. It is visibly erratic or chaotic, as is the case for almost all x 0 • What is even worse is that the trajectory is significantly altered by a slight change

1.1. A Simple System Generating a Density of States

0

3

20

10

FIGURE 1.1.2. The histogram constructed according to equation (1.1.3) with n = 20, N = 5000, and z 0 = 1r /10.

in the initial state, as shown in Figure 1.1.1c for an initial state differing by w- 3 from that used to generate Figure 1.1.1b. Thus we are seemingly faced with a real problem in characterizing systems with behaviors like that of (1.1.1). By taking a clue from other areas, we might construct a histogram to display the frequency with which states along a trajectory fall into given regions of the state space. This is done in the following way. Imagine that we divide the state space [0, 1] into n discrete nonintersecting intervals so the ith interval is (we neglect the end point 1)

[(i -1)/n,i/n)

i = 1, ... ,n.

Next we pick an initial system state x 0 and calculate a long trajectory x 0 , S(x0 ), S2 (x 0 ), ••• , sN (x0 )

of length N where N > > n. Then it is straightforward to determine the fraction, call it /i, of theN system states that is in the ith interval form

fi =;{number of Si(x0 ) E [(i-1)/n,i/n), j = 1, ... ,N}.

(1.1.3)

We have carried out this procedure for the initial state used to generate the trajectory of Figure 1.1.1b by taking n = 20 and using a trajectory of length N = 5000. The result is shown in Figure 1.1.2. There is a surprising symmetry in the result, for the states are clearly most concentrated near 0 and 1 with a minimum at Repeating this process for other initial states leads, in general, to the same result. Thus, in spite of the sensitivity of trajectories to initial states, this is not usually reflected in the distribution of states within long trajectories. However, for certain select initial states, different behaviors may occur. For some initial conditions the trajectory might arrive at one of the fixed points of equation (1.1.1), that is, a point x,. satisfying

!.

x,. = S(x,.).

4

1.. Introduction (a)

x·I

0

(b)

J

I

x·I

0

50 '

100

I

FIGURE 1.1.3. Exceptional initial conditions may confound the study of transformations via trajectories. In (a) we show how an initial condition on the quadratic transformation (1.1.1) with a= 4 can lead to a fixed point z. of S. In (b) we see that another initial condition leads to a period 2 trajectory, although all other characteristics of S are the same.

(For the quadratic map with a = 4 there are two fixed points, x. = 0 and x. = If this happens the trajectory will then have the constant value x. forever after, as illustrated in Figure 1.1.3a. Alternately, for some other initial states the trajectory might become periodic (see Figure 1.1.3b) and also fail to exhibit the irregular behavior of Figures 1.1.1 b and c. The worst part about these exceptional behaviors is that we have no a priori way of predicting which initial states will lead to them. In the next section we illustrate an alternative approach to avoid these problems.

i·)

Remark 1.1.1. Map {1.1.1) has attracted the attention of many mathematicians, Ulam and von Neumann [1947) examined the case when a= 4, whereas Ruelle [1977), Jakobson [1978), Pianigiani [1979), Collet and Eckmann [1980) and Misiurewicz [1981] have studied its properties for values of a< 4. May [1974], Smale and Williams [1976], and Lasota and Mackey [1980), among others, have examined the applicability of {1.1.1) and similar maps to biological population growth problems. Interesting properties re-

1.2. The Evolution of Densities: An Intuitive Point of View

5

lated to the existence of periodic orbits in the transformation {1.1.1) follow from the classical results of Sarkovski'l (1964]. 0

1.2 The Evolution of Densities: An Intuitive Point of View The problems that we pointed out in the previous section can be partially circumvented by abandoning the study of individual trajectories in favor of an examination of the flow of densities. In this section we give a heuristic introduction to this concept. Again we assume that we have a transformation S : [0, 1] -+ [0, 1] (a shorthand way of sayingS maps (0, 1] onto itself} and pick a large number N of initial states To each of these states we apply the map S, thereby obtaining N new states denoted by x~ = S(x~}, x~ = S(xg}, ... , x}.r = S(xO,. ). To define what we mean by the densities of the initial and final states, it is helpful to introduce the concept of the characteristic (or indicator)

function for a set fl. This is simply defined by 1a(x)

1 ifxE!l if X¢ fl .

={ 0

Loosely speaking, we say that a function fo(x) is the density function for the initial states x~, ... , xO,. if, for every (not too small) interval floc (0, 1], we have

f fo(u}du ~ ~ lao

Likewise, the density function tl c [0, 1],

1a0 (xJ).

(1.2.1}

j=l

It {x} for the states xl, ... , x}v satisfies, for

1 N ft(u}du ~ N :~:::)a(x}).

1 a

t

(1.2.2}

i=l

We want to find a relationship between It and fo. To do this it is necessary to introduce the notion of the counterimage of an interval tl c [0, 1] under the operation of the mapS. This is the set of all points that will be in tl after one application of S, or

s- 1 (/l) = {x: S(x) Ell}. As illustrated in Figure 1.2.1, for the quadratic map considered in Section 1.1, the counterimage of an interval will be the union of two intervals.

6

1. Introduction

s

FIGURE 1.2.1. The counterimage of the set [0, zJ under the quadratic transformation consists of the union of the two sets denoted by the heavy lines on the z-axis.

Now note that for any X~J E

a

ac

(0, 1]

if and only if

x1 E s- (a). 1

We thus have the very useful relation 1A(S(x))

= 1s-l(A)(x).

(1.2.3)

With (1.2.3) we may rewrite equation (1.2.2) as (1.2.4) Because Ao and A have been arbitrary up to this point, we simply pick ao = s- 1 (a). With this choice the right-hand sides of (1.2.1) and (1.2.4) are equal and therefore

f

JA

h(u)du =

f

Js-l(A)

fo(u)du.

(1.2.5)

This is the relationship that we sought between fo and h, and it tells us how a density of initial states fo will be transformed by a given map S into a new density h. If a is an interval, say a= [a, x], then we can obtain an explicit representation for f1. In this case, equation (1.2.5) becomes

1"' a

!l(u)du =

f

Js- ([a,z)) 1

fo(u)du,

1.2. The Evolution of Densities: An Intuitive Point of View

7

and differentiating with respect to x gives

ft(x)

=! f

Js-l((a,:l!]}

lo(u}du.

(1.2.6}

It is clear that It will depend on lo· This is usually indicated by writing It = Plo, so that (1.2.6} becomes

Pl(x)

=! f

Js-l([a,:J:))

l(u}du

(1.2.7}

(we have dropped the subscript on lo as it is arbitrary). Equation (1.2.7} explicitly defines the Frobenius-Perron operator P corresponding to the transformation S; it is very useful for studying the evolution of densities. To illustrate the utility of (1.2. 7} and, incidentally, the Frobenius-Perron operator concept, we return to the quadratic map S(x) = 4x(1- x) of the preceding section. To apply (1.2. 7) it is obvious that we need an analytic formula for the counterimage of the interval [0, x]. Reference to Figure 1.2.1 shows that the end points of the two intervals constituting s- 1 ([0,x]} are very simply calculated by solving a quadratic equation. Thus

s- 1 ([o,x]} = (o,!- !v'l-XJ u [! + !v'1- x, 1]. With this, equation (1.2. 7} becomes d 11/2-1/2.,ff=i

Pl(x)=-d X

d 11

l(u}du+dx

0

1/2+1/2,fl-i

l(u}du,

or, after carrying out the indicated differentiation,

Pl(x) =

)=x

4 1-x {I(!- !v'f=X) +I(!+ !v'f=X)}.

(1.2.8}

This equation is an explicit formula for the Frobenius-Perron operator corresponding to the quadratic transformation and will tell us how S transforms a given density I into a new density Pl. Clearly the relationship can be used in an iterative fashion. To see how this equation works, pick an initial density l(x) = 1 for x E [0, 1]. Then, since both terms inside the braces in (1.2.8} are constant, a simple calculation gives

Pl(x)

1

= 2vr::x· 1-x

(1.2.9}

Now substitute this expression for PI in place of I on the right-hand side of (1.2.8} to give

8

1. Introduction

p2f

Pf

2 I I I I I I

I I I

I

I

I

I

I I

I

I

I

I

I I

I

I

I

I

I

I

I \

' ' .......

I

\

__ ., "

\

X

0

X

X

0

0

FIGURE 1.2.2. The evolution of the constant density f(x) = 1, x E [0, 1], by the Frobenius-Perron operator corresponding to the quadratic transformation. Compare the rapid and regular approach of pn J to the density given in equation (1.2.11) (shown as a dashed line) with the sustained irregularity shown by the trajectories in Figure 1.1.1.

P(Pf(x))

= P 2 f(x) = =

4~ {2,;1- ~:~vr-x + 2,;1-l~~~} sb{ v'1+~+ v'1-~}·

2 10 (1. . )

=

In Figure 1.2.2 we have plotted f(x) 1, Pf(x) given by (1.2.9}, and P 2 f(x) given by (1.2.10} to show how rapidly they seem to approach a limiting density. Actually, this limiting density is given by 1

/.(x)

= 1ry'x(1- x)

(1.2.11}

If /. is really the ultimate limit of pn f as n --+ oo, then we should find that P /. when we substitute into equation (1.2.8} for the FrobeniusPerron operator. A few elementary calculations confirm this. Note also the close similarity between the graph of/. in Figure 1.2.2 and the histogram of Figure 1.1.2. Later we will show that for the quadratic map the density of states along a trajectory approaches the same unique limiting density /. as the iterates of densities approach.

=/.

Example 1.2.1. Consider the transformationS: [0, 1]--+ [0, 1] given by

S(x)

= rx

(mod 1},

{1.2.12}

1.3. Trajectories Versus Densities

9

s

FIGURE 1.2.3. The dyadic transformation is a special case of the r-adic transformation. The heavy lines along the x-axis mark the two components of the counterimage of the interval [0, x).

where r is an integer. The notation rx (mod 1) means rx-n, where n is the largest integer such that rx - n ~ 0. This expression is customarily called the r-adic transformation and is illustrated in Figure 1.2.3 for r = 2 {the dyadic transformation) .. Pick an interval [0, x] C [0, 1] so that the counterimage of [0, x] under S is given by

s- 1{[0, x]) =

U[~. ~ + ~]

r r and the Frobenius-Perron operator is thus i=O

d

Pl(x)= dx

r-1

~

i/r+z/r

1

i/r

1 l(u)du="i-

r

r-1

( .

~~ ~+;

)

.

{1.2.13)

This formula for the Frobenius-Perron operator corresponding to the radic transformation {1.2.12) shows again that densities I will be rapidly smoothed by P, as can be seen in Figure 1.2.4a for an initial density l(x) = 2x, x E [0, 1]. It is clear that the density pnl(x) rapidly approaches the constant distribution l.(x) 1, x E [0, 1]. Indeed, it is trivial to show that P1 1. This behavior should be contrasted with that of a typical trajectory (Figure 1.2.4b). D

=

=

1.3 Trajectories Versus Densities In closing this chapter we offer a qualitative examination of the behavior of two transformations from both the flow of trajectories and densities viewpoints.

10

1. Introduction

.,

0

200

tOO

FIGURE 1.2.4. Dynamics of the dyadic transformation. (a) With an initial density J(x) = 2x, x E [0, 1], successive applications of the Frobenius-Perron operator corresponding to the dyadic transformation result in densities approaching j. = 1, x E [0, 1]. (b) A trajectory calculated from the dyadic transformation with x 0 ~ 0.0005. Compare the irregularity of this trajectory with the smooth approach of the densities in (a) to a limit.

Let R denote the entire real line, that is, R = {x: -oo < x < oo}, and consider the transformation S: R - R defined by

S(x) =ax,

a>O.

(1.3.1)

Our study of transformations confined to the unit interval of Section 1.2 does not affect expression (1.2.7) for the Frobenius-Perron operator. Thus (1.3.1) has the associated Frobenius-Perron operator

Pf(x) = (1/a)f(x/a). We first examine the behavior of S for a> 1. Since sn(x) that, for a > 1, X :f: 0,

= anx, we see

and thus the iterates 5n (X) escape from any bounded interval. This behavior is in total agreement with the behavior deduced from the flow of densities. To see this note that

1.3. Trajectories Versus Densities

11

s

0

FIGURE 1.3.1. The transformation S(z), defined by equation (1.3.2), has a single weak repelling point at x 0.

=

By the qualitative definition of the Frobenius-Perron operator of the previous section, we have, for any bounded interval [-A, AJ c R,

!

A

pn f(x) dx

-A

= ~A/a"

f(x) dx.

-A/a"

Since a:> 1, lim

!A

n-+oo -A

pnf(x)dx = 0

and so, under the operation of S, densities are reduced to zero on every finite interval when a: > 1. Conversely, for a: < 1, lim ISR(x)l = 0 n-+oo

for every x E R, and therefore all trajectories converge to zero. Furthermore, for every neighborhood ( -e, e) of zero, we have lim n-+oo

1

~

-~

pn f(x) dx

=

lim

~~/a"

n-+oo

-~/a"

f(x) dx

= ~oo

_ 00

f(x) dx

= 1,

so in this case all densities are concentrated in an arbitrarily small neighborhood of zero. Thus, again, the behaviors of trajectories and densities seem to be in accord. However, it is not always the case that the behavior of trajectories and densities seem to be in agreement. This may be simply illustrated by what we call the paradox of the weak repellor. In Remark 6.2.1 we consider the transformationS: [0, 1]-+ [0, 1] defined by

S(x) = { x /(1 - x) !or x E ([0; i ]] 2X - 1

10r X

E

2• 1 ,

(La. 2)

12

1. Introduction

x,

0

100

200

400

300

500

FIGURE 1.3.2. Dynamics of the weak repellor defined by {1.3.2). (a) The evolution pn f of an initial distribution J(x) = 1, x E (0, 1]. {b) The trajectory originating from an initial point x 0 ~ 0.25.

(see Figure 1.3.1). There we prove that, for every e > 0,

1

pnf(x)dx

= 0.

r

pn f(x) dx

= 1,

1

lim

n-+oo e

Thus, since pn f is a density, lim

n-+oo}0

and all densities are concentrated in an arbitrarily small neighborhood of zero. This behavior is graphically illustrated in Figure 1.3.2a. If one picks an initial point x0 > 0 very close to zero (see Figure 1.3.2b), then, as long as sn(xo) E (0, !] , we have

where a= 1/{1- xo) > 1. Thus initially, for small xo, this transformation behaves much like transformation {1.3.1), and the behavior of the trajectory

Exercises

13

near zero apparently contradicts that expected from the behavior of the densities. This paradox is more apparent than real and may be easily understood. First, note that even though all trajectories are repelled from zero (zero is a repellor), once a trajectory is ejected from (0, ~1 it is quickly reinjected into (0, ~ 1 from ( ~, 11 . Thus zero is a "weak repellor." The second essential point to note is that the speed with which any trajectory leaves a small neighborhood of zero is small; it is given by 2

Xo S n( Xo ) - sn-1( xo ) -_ (1- nxo)[1(n- 1)xo]"

Thus, starting with many initial points, as n increases we will see the progressive accumulation of more and more points near zero. This is precisely the behavior predicted by examining the flow of densities. Although our comments in this chapter lack mathematical rigor, they offer some insight into the power of looking at the evolution of densities under the operation of deterministic transformations. The next two chapters are devoted to introducing the mathematical concepts required for a precise treatment of this problem.

Exercises Simple numerical experiments can greatly clarify the material of this and subsequent chapters. Consequently, the first five exercises of this chapter involve the writing of simple utility programs to study the quadratic map (1.1.1) from several perspectives. Exercises in subsequent chapters will make use of these programs to study other maps. H you have access to a personal computer (preferably with a math coprocessor), a workstation, or a microcomputer with graphics capabilities, we strongly urge you to do these exercises. 1.1. Write a program to numerically generate a sequence of iterates {xn} from Xn+l = S(xn), where S is the quadratic map (1.1.1). Write your program in such a way that the map Sis called from a subroutine (so it may be changed easily) and include graphics to display Xn versus n. When displaying the sequence {xn} graphically, you will find it helpful to connect successive values by a straight line so you can keep track of them. Save this program under the name TRAJ so you can use it for further problems.

1.2. Using TRAJ study the behavior of (1.1.1) for various values of a satisfying 3:::; a:::; 4, and for various initial conditions xo. (You can include an option to generate xo using the random number generator if you wish, but be careful to use a different seed number for each run.) At a given value of a what can you say about the temporal behavior of the sequence

14

1. Introduction

{xn} for different x 0 ? What can you say concerning the qualitative and quantitative differences in the trajectories {xn} for different values of o:? 1.3. To increase your understanding of the results in Exercise 1.2, write a second program called BIFUR. This program will plot a large number of iterates of the map {1.1.1) as o: is varied between 3 and 4, and the result will approximate the bifurcation diagram of (1.1.1). Procedurally, for each value of o:, use the random number generator (don't forget about the seed) to select an initial x0 , discard the first 100 or so values of Xn to eliminate transients, and then plot a large number (on the order of 1000 to 5000) of the Xn vertically above the value of o:. Then increment o: and repeat the process successively until you have reached the maximal value of o:. A good incremental value of o: is ~o: = 0.01 to 0.05, and obviously the smaller ~o: the better the resolution of the details of the bifurcation diagram at the expense of increased computation time. Use the resulting bifurcation diagram you have produced, in conjunction with your results of Exercise 1.2, to more fully discuss the dynamics of {1.1.1). You may find it helpful to make your graphics display flexible enough to "window" various parts of the bifurcation diagram so you can examine fine detail. 1.4. Write a program called DENTRAJ (Density from a 'Irajectory) to display the histogram of the location of the iterates {xn} of {1.1.1) for various values of o: satisfying 3 :::; o: :::; 4 as was done in Figure 1.1.2 for o: = 4. [Constructing histograms from "data" like this is always a bit tricky because there is a tradeoff between the number of points and the number of bins in the histogram. However, a ratio of 20(}-300 of point number to bin number should provide a satisfactory result, so, depending on the speed of your computer (and thus the number of iterations that can be carried out in a given time), you can obtain varying degrees of resolution.) Compare your results with those from Exercise 1.3. Note that at ·a given value of o:, the bands you observed in the bifurcation diagram correspond to the histogram supports (the places where the histogram is not zero).

1.5. Redo Exercise 1.4 by writing a program called DENITER (Density Iteration) that takes a large number N of initial points {x~H~ 1 distributed with some density /o(x), e.g., f 0 (x) could be uniform on [0, 1] for (1.1.1), or /o(x) = 2x, etc., and iterates them sequentially to give {xl}[!. 1 = {S(x~)}: 1 , {xHf: 1 = {S(xf)}:, 1 , etc. Construct your program to display the histogram of the {z{}[!.1 for the initial (j = 0) and successive iterations. Do the histograms appear to converge to an invariant histogram? How does the choice of the initial histogram affect the result after many iterations? Discuss the rate of convergence of the sequence of histograms. 1.6. Prove that /. given by {1.2.11) is a solution of the equation Pf = /, where P, given by {1.2.8), is the Frobenius-Perron operator corresponding to the quadratic map {1.1.1) with o: = 4.

Exercises

15

1.7. This exercise illustrates that there can sometimes be a danger in draw.ing conclusions about the behavior of even simple systems based on numer-

ical experiments. Consider the Frobenius-Perron operator {1.2.13) corresponding to the r-adic transformation {1.2.12) when r is an integer. (a) For every integer r show that /.{x) = 1[o, 1J{x) is a solution of Pf = f. Can you prove that it is the unique solution? {b) For r = 2 and r = 3 use TRAJ, DENTRAJ, and DENITER to study {1.2.12). What differences do you see in the behaviors for r = 2 and r = 3? Why do these differences exist? Discuss your numerical results in light of your computations in (a). 1.8. Consider the example of the weak repellor {1.3.2). (a) Derive the Frobenius-Perron operator corresponding to the weak repellor without looking in Chapter 6. Calculate a few terms of the sequence {pn!} for f(x) = 1[o, 1J(x). {b) Use TRAJ, DENTRAJ and DENITER to study the weak repellor (1.3.2). Discuss your results. Based on your observations, what conjectures can you formulate about the behavior of the weak repellor? In what way do these differ from the properties of the quadratic map (1.1.1) that you saw in Exercises 1.1-1.5?

2 The Toolbox

In this and the following chapter, we introduce basic concepts necessary for understanding the flow of densities. These concepts may be studied in detail before continuing on to the core of our subject matter, which starts in Chapter 4, or, they may be skimmed on first reading to fix the location of important concepts for later reference. We briefly outline here some essential concepts from measure theory, the theory of Lebesgue integration, and from the theory of the convergence of sequences of functions. This material is in no sense exhaustive; those desiring more detailed treatments should refer to Halm08 [1974] and Royden [1968].

2.1

Measures and Measure Spaces

We start with the definition of a a-algebra. Definition 2.1.1. A collection A of subsets of a set X is a a-algebra if: (a) When A E A then X\ A E A; (b) Given a finite or infinite sequence {Ak} of subsets of X, Ak E A, then the union Uk Ak E A; and

(c) X EA. From this definition it follows immediately, by properties (a) and (c), that the empty set 0 belongs to A, since 0 = X \ X. Further, given a

18

2. The Toolbox

sequence {Ak }, Ak that

e A, then the intersection nk Ak e A. To see this, note nAk

=X\U (X\Ak)

k

k

and then apply properties (a) and (b). Finally, the difference A \ B of two sets A and B that belong to A also belongs to A because

A\ B =An (X\ B). Definition 2.1.2. A real-valued function p. defined on au-algebra A is a measure if:

(a) p.(0) = 0; (b) p.(A) 2:: 0 for all A

e A; and

(c) p.(UkAk) = Ekp.(Ak) if {Ak} is a finite or infinite sequence of pairwise disjoint sets from A, that is, A 1 n Aj 0 for i =F j.

=

We do not exclude the possibility that p.(A)

= oo for some A EA.

Remark 2.1.1. This definition of a measure and the properties of aualgebra A as detailed in Definition 2.1.1 ensure that (1) if we know the measure of a set X and a subset A of X we can determine the measure of X \A; and (2) if we know the measure of each disjoint subset Ak of A we can calculate the measure of their union. 0 Definition 2.1.3. U A is a u-algebra of subsets of X and if p. is a measure on A, then the triple (X, A, p.) is called a measure space. The sets belonging to A are called measurable sets because, for them, the measure is defined.

Remark 2.1.2. A simple example of a measure space is the finite set X = {x1, ... , x N}, in which the u-algebra is all possible subsets of X and the measure is defined by ascribing to each element x 1 e X a nonnegative number, say p1• From this it follows that the measure of a subset { X011 ••• , Xa,.} of X is just p 01 + · · · + Pa,. U Pi = 1, then the measure is called a counting measure because it counts the number of elements in the set. 0

Remark 2.1.3. If X = [0, 1] or R, the real line, then the most natural ualgebra is the u-algebra B of Borel sets (the Borel a--algebra), which, by definition, is the smallest u-algebra containing intervals. (The word smallest means that any other u-algebra that contains intervals also contains any set contained in B.) It can be proved that on the Borel u-algebra there

2.2. Lebesgue Integration

19

exists a unique measure p, called the Borel measure, such that p([a, b]) = b - a. Whenever considering spaces X = R or X = Rd or subsets of these (intervals, squares, etc.) we always assume the Borel measure and will not repeat this assumption again. 0 As presented, Definition 2.1.3 is extremely general. In almost all applications a more specific measure space is adequate, as follows: Definition 2.1.4. A measure space (X, A, p) is called D'-finite if there is a sequence {A~e}, A~e e A, satisfying 00

X

= U A~e

and p(A~e) < oo

for all k.

le=l

Remark 2.1.4. If X= R, the real line, and pis the Borel measure, then the A1e may be chosen as intervals of the form [-k, k]. In the d-dimensional space ~, the A~e may be chosen as balls of radius k. 0 Definition 2.1.5. A measure space (X, A, p) is called finite if p(X) < oo. In particular, if p(X) = 1, then the measure space is said to be normalized or probabilistic. Remark 2.1.5. We have defined a hierarchy of measure spaces from the most general (Definition 2.1.3) down to the most specific (Definition 2.1.5). Throughout this book, unless it is specifically stated to the contrary, a measure space will always be understood to be u-finite. 0 Remark 2.1.6. If a certain property involving the points of a measure space is true except for a subset of that space having measure zero, then

we say that property is true almost everywhere (abbreviated as a. e.). 0

2.2 Lebesgue Integration In the material we deal with it is often necessary to use a type of integration more general than the customary Riemann integration. In this section we introduce the Lebesgue integral, which is defined for abstract measure spaces in which no other structures except a u-algebra A and a measure p must be introduced.

Definition 2.2.1. Let (X, A, p) be a measure space. A real-valued function f:X-+ R is measurable if f- 1 (~) e A for every interval~ cR. In developing the concept of the Lebesgue integral, we need the notation

J+(x) = max(O, /(x)) and /-(x)

= max(O, -/(x))

20

2. The Toolbox

X

FIGURE 2.2.1. Illustration ofthe notation j+(z) and r(z).

(see Figure 2.2.1). Observe that

/(z)

= /+(z)- 1-(x)

and

lf(x)l

= f+(x) + 1-(x).

Before presenting the formal definitions for the Lebesgue integral of a function, consider the following. Let f: X -+ R be a bounded, nonnegative measurable function, 0 :5 f(x) < M < oo. Take the partition of the interval [O,M], 0 = ao < a1 h(x)}

and A2

= {x:ft(x) :5 h(x)}.

Then 0=

f

jA1

[ft(x)- h(x)] J'(dx)-

=f

lft(x)- h(x)IJ'(dx)

jA1UA2

=

f

jA2

L

lft(x)- h(x)IJ'(dx).

[ft(x)- h(x)] J'(dx)


25

Hence, from property (12) of Lebesgue integrals, we have lft(x)-h(x)l = 0 a..e., so that h(x) and h(x) differ only on a. set of measure zero. Observe that our argument is quite general, and we have in fact proved the following. Proposition 2.1.1. If h and h are integrable functions such that

L

h(x)IJ(dx) =

then

h =h

L

h(x)IJ(dx)

forAeA

a..e.

Also from property (12) of the Lebesgue integral it is clear that two measurable functions, h and h, differing from each other only on a. set of measure zero, cannot be distinguished by calculating integrals. Thus we sa.y that in the space of measurable functions, every two functions !1, /2, differing only on a. set of measure zero, represent the same element of that space. However, to simplify our notation, we will often write ''measurable function" instead of "an element of the space of measurable functions." Because of property (12) this should not lead to any confusion. With these remarks in mind, we now introduce the concept of an lJ' space. Definition 2.2.6. Let (X, A, IJ) be a. measure space and p a. real number, 1 ~ p < oo. The family of all possible real-valued measurable functions f: X -+ R satisfying

[

lf(x)l" IJ(dx) < oo

(2.2.2)

is the IJ'(X, A, IJ) space. Here we use the term "measurable function" to mean "an element of the space of measurable functions." We shall sometimes write lJ' instead of IJ'(X, A, IJ) if the measure space is understood, or IJ'(X) if A and IJ are understood. Note that if p = 1 then the L 1 space consists of all possible integrable functions. The integral appearing in (2.2.2) is very important for an element f E IJ'. Thus it is assigned the special notation 1

nJnL,. = [[ IJ(x)l" 1J(dx)f "

(2.2.3)

and is called the lJ' norm of f. When property (12) of the Lebesgue integral is applied to Ill", it immediately follows that the condition 11/IIL,. = 0 is equivalent to f(x) = 0 a..e. Or, more precisely, 11/IIL,. = 0 if and only if f is a zero element in lJ' (which is an element represented by a.ll functions equal to zero almost everywhere). Two other important properties of the norm a.re for f E IJ', a E R

(2.2.4)

26

2. The Toolbox

FIGURE 2.2.2. A geometric interpretation of the triangle inequality (2.2.5).

and

II/+ giiLP

~

11/IILP + llgiiLP

for

J,g E V.

(2.2.5}

The first condition, (2.2.4), simply says that the norm is homogeneous. The second is called the triangle inequality. As shown in Figure 2.2.2, if we think of J, g, and J +gas vectors, we can consider a triangle with sides J, g, and J +g. Then, by equation, (2.2.5), the length of the side (J +g) is shorter than the sum of the lengths of the other two sides. From (2.2.4) it follows that for every J E V and real a, the product af belongs to V. Further, from (2.2.5) it follows that for every J,g E V the sum J + g is also an element of V. This is denoted by saying that V is a vector space. Because the value of 11/IILP is interpreted as the length of j, we say that 1

II/- giiLP =

[ [ lf(x)- g(x)IP J.L(dx)]

/P

is the V distance between J and g. It is important to note that the product f g of two functions /, g E V is not necessarily in V, for example, J(x) = x- 1/ 2 is integrable on [0, 1] but [/(x)) 2 = x- 1 is not. This leads us to define the space adjoint to V. Definition 2.2.7. Let (X,A,J.L) be a measure space. The space adjoint to V(X,A,J.L) is v' (X,A,p), where (1/p) + (1/p')

= 1.

Remark 2.2.9. If p = 1, Definition 2.2. 7 of adjoint space fails. The adjoint space, in the case p = 1, by definition, consists of all bounded almost everywhere measurable functions and is denoted by L 00 • Functions that differ only on a set of measure zero are considered to represent the same element. 0


27

It is well known that if I E v and g E v'' then I g is integrable, and hence we define the scalar product of two functions by

(/,g)=

L

f(x)g(x) tt(dx).

An important relation we will often use is the Cauchy-Holder inequality. Thus, if IE LP-and g E v'' then

For this inequality to make sense when f E Lt, g E L 00 , we take the L 00 norm of g to be the smallest constant c such that

lg(x)l :5 c for almost all x E X. This constant is denoted by ess sup lg( x) I, called the essential supremum of g.

Remark 2.2.10. As we almost always work in Lt, we will not indicate the space in which the norm is taken unless it is not Lt. Thus we will write 11/11 instead of 11/IILl· Observe that in Lt the norm has the exceptional property that the triangle inequality is sometimes an equality. To see this, note from property (L3) that

II/+ gll = 11/11 + llgll

for

f '?. 0, g '?. 0; J,g E £ 1.

Thus geometrical intuition in some abstract spaces may be misleading. 0 The concept of the Lt space simplifies the Radon-Nikodym theorem as shown by the following corollary. Corollary 2.2.1. If(X,A,tt) is a measure space and vis a finite measure on A such that v(A) = 0 whenever tt(A) = 0, then there exists a unique element fELt such that

v(A)

=

L

f(x) tt(dx)

forA EA.

One of the most important notions in analysis, measure theory, and topology, as well as other areas of mathematics, is that of the Cartesian product. To introduce this concept we start with a definition. Definition 2.2.8. Given two arbitrary sets At and A2, the Cartesian product of At and A2 (note that the order is important) is the set of all pairs (xt,x2) such that Xt EAt and x2 E A2. This is customarily written as

28

2. The Toolbox

In a natural way this concept may be extended to more than two sets. Thus the Cartesian product of the sets A1 , ••• , Ad is the set of all sequences (xl! ... , xd) such that Xi E Ai, i = 1, ... , d, or

A1 x ··· x Ad= {(xl, ... ,xd):xi

E

Ai fori= 1, ... ,d}.

An important consequence following from the concept of the Cartesian product is that if a structure is defined on each of the factors Ai, for example, a measure, then it is possible to extend that property to the Cartesian product. Thus, given d measure spaces (Xi, A, l'i), i = 1, ... , d, we define (2.2.6)

A to be the smallest u-algebra of subsets of X containing all sets of the form (2.2.7) with Ai E A, i = 1, ... , d, and

(2.2.8) Unfortunately, by themselves they do not define a measure space (X, A, JL). There is no problem with either X or A, but I' is defined only on special sets, namely A = A 1 x · · · x Ad, that do not form a u-algebra. To show that JL, as defined by (2.2.8), can be extended to the entire u-algebra A requires the following theorem. Theorem 2.2.2. If measure spaces (Xi, A, l'i), i = 1, ... , d are given and X, A, and I' are defined by equations (2.2.6), (2.2.7), and (2.2.8), respectively, then there exists a unique extension of I' to a measure defined on A. The measure space (X, A, I') whose existence is guaranteed by Theorem 2.2.2, is called the product of the measure spaces (X11 A 1,JL1), ... , (Xd, ~.I'd), or more briefly a product space. The measure I' is called the product measure. Observe that from equation (2.2.8) it follows that JL(Xl

X ••• X

Xd) = JL(Xl) .. ·JL(Xd)·

Thus, if all the measure spaces (Xi, A, l'i) are finite or probabilistic, then (X, A, JL) will also be finite or probabilistic. Theorem 2.2.2 allows us to define integration on the product space (X, A, JL) since it is also a measure space. A function f: X --+ R may be written as a function of d variables because every point x E X is a sequence x = (x1, ... ,xd), Xi E Xi. Thus it is customary to write integrals on X either as f(x) JL(dx),

L


where it is implicitly understood that x xd, or in the more explicit form

29

= (xl! ... , Xd) and X = X 1 x · · · x

Integrals on the product of measure spaces are related to integrals on the individual factors by a theorem associated with the name of Fubini. For simplicity, we first formulate it for product spaces containing only two factors. Theorem 2.2.3 (Fubini's theorem). Let (X, A, J.L) be the product space formed by (Xl!Al!J.Lt) and (X2,A2,J.L2), and let a J.L integrable function f: X -+ R be given. Then, for almost every Xt, the function f(xl! x2) is J.L2 integrable with respect to x2. Furthermore the function

of the variable Xt is J.Lt integrable and

[

f(xt, x2)J.L2(dx2)} J.L1 (dxt)

{[ 1

=

j j f(xl! x2) J.L(dx1dx2). (2.2.9)

3

X

Theorem 2.2.3 extends, in a natural way, to product spaces with an arbitrary number of factors. If (X, A, J.L) is the product of the measure spaces (X,, At, J.Li), i = 1, ... , d, and f: X-+ R is J.L integrable, then

I . ·I

f(xl! ... , xd) J.L(dxt .. · dxd)

(2.2.10)

X

=

f { ... f [f f(xl! ... ,xd)J.Ld(dxd)]J.Ld-l(dxd_t)···}J.Lt(dxt)· lx1 lxd-1 lxd

Remark 2.2.11. As we noted in Remark 2.1.3, the "natural" Borel measure on the real line R is defined on the smallest u-algebra 8 that contains all intervals. For every interval [a, b] this measure satisfies J.L([a, b]) = b- a. Having the structure (R, 8, J.L), we define by Theorem 2.2.2 the product space (~,8d,J.Ld), where d

d

R =Rx ··· xR,

30

2. The Toolbox

Bd is the smallest u-algebra containing all sets of the form with~

EB,

and {2.2.11) - The measure I'd is again called the Borel measure. It is easily verified that Bd may be alternately defined as either the smallest u-algebra containing all the rectangles [at. b1] x · · · x [ad, bd], or as the smallest u-algebra containing all the open subsets of Jl!l. From {2.2.11) it follows that

J.'d([al, b1] x · · · x [ad, bd])

= {b1 -

a1) · · · (bd- ad),

which is the classical formula for the volume of an n-dimensional box. The same construction may be repeated by starting, not from the whole real lineR, but from the unit interval [0, 1] or from any other finite interval. Thus, from Theorem 2.2.2, we will obtain the Borel measure on the unit square [0, 1] x [0, 1] or on the d-dimensional cube [0,1]d

= [0,1]x .~.

x[0,1].

In all cases (Rd, [0, 1]d, etc.) we will omit the superscript don Bd and I'd and write {Jrl, B, J.£) instead of (Rd, Bd, I'd). Furthermore, in all cases when the space is R, Rd, or any subset ofthese ([0, 1], [0, 1]d, R+ = (0, oo), etc.) and the measure and u-algebra are not specified, we will assume that the measure space is taken with the Borel u-algebra and Borel measure. Finally, all the integrals on R or Rd taken with respect to the Borel measure will be written with dx instead of J.£( dx). D Remark 2.2.12. From the additivity property of a measure (Definition 2.1.2c) it follows that every measure is monotonic, that is, if A and Bare measurable sets and A c B then J.£(A) ~ J.£(B). This follows directly from

= J.£(A U (B \A)) = I'( A)+ J.£(B \A). Thus, if J.£(B) = 0 and A c B, then J.£(A) = 0. However, it could happen J.£(B)

that A C B and B is a measurable set while A is not. In this case, if J.£(B) = 0, then it does not follow that J.£(A) = 0, because J.£(A) is not defined, which is a peculiar situation. It is rather natural, therefore, to require that a "good" measure have the property that subsets of measurable sets of measure zero should also be measurable with, of course, measure zero. If a measure has this property it is called complete. Indeed, it can be proved that, if {X, A, J.£) is a measure

2.3. Convergence of Sequences of Functions

31

space, then there exists a smallest u-algebra A1 :J A and a measure l'l on A1 identical with J1. on A such that (Xl.Al,Jl.l) is complete. Every Borel measure on R (or Rd, [0, 1), [0, 1]d, etc.) can be completed. This complete measure is called the Lebesgue measure. However, when working in R (or ~, etc.), we will use the Borel measure and not the Lebesgue measure, because, with the Lebesgue measure, we encounter problems with the measurability of the composition of measurable functions that are avoided with the Borel measure. 0

2.3 Convergence of Sequences of Functions Having defined V' spaces and introduced the notions of norms and scalar products, we now consider three different types of convergence for a sequence of functions. Definition 2.3.1. A sequence of functions {/n}, (weakly) Cesaro convergent to f E V' if 1 n lim - ""(b" g) n--+oo n L...J

= (/,g)

In

E

V', 1

~ p

for all g E V'.

< oo,

is

(2.3.1)

k=l

Definition 2.3.2. A sequence of functions {/n}, weakly convergent to f E V' if lim (/n,g) =(/,g) n--+oo

In

E

V', 1

~

p

for all g E V'.

Definition 2.3.3. A sequence of functions {/n}, strongly convergent to f E V' if

In

E

lim 11/n - /IILP = 0. n--+oo

V', 1

< oo, is (2.3.2)

~

p

~

oo, is

(2.3.3)

From the Cauchy-Holder inequality, we have

1(/n- /,g)l ~ 11/n- fiiLP ·llgiiLP and, thus, if 11/n - /IILP converges to zero, so must (/n - J,g). Hence 11

strong convergence implies weak convergence, and the condition for strong convergence is relatively straightforward to check. However, the condition for weak convergence requires a demonstration that it holds for all g E v', which seems difficult to do at first glance. In some special and important spaces, it is sufficient to check weak convergence for a restricted class of functions, defined as follows. Definition 2.3.4. A subset K c V' is called linearly dense if for each E V' and e > 0 there are g1, ... ,gn E K and constants .>.l,···•.>.n, such

f

32

2. The Toolbox

that

II/- giiLP < e, where i=1

By using the notion of linearly dense sets, it is possible to simplify the proof of weak convergence. If the sequence {/n} is bounded in norm, that is, 11/niiLP ~ c < oo, and if K is linearly dense in v', then it is sufficient to check weak convergence in Definition 2.3.2 for any g E K. It is well known that in the space V([O, 1]) (1 ~ p < oo) the following sets are linearly dense: K 1 ={the set of characteristic functions 1a(x) of the Borel sets tl. c [0, 1]}, K2 ={the set of continuous. functions on [0, 1]} , Ks = {sin(mrx); n = 1, 2, ... }.

In K 1 it is enough to take a family of sets tl. that are generators of Borel sets on [0, 1], for example, {fl.} could be the family of subintervals of [0, 1]. Observe that the linear density of Ks follows from the Fourier expansion theorem. In higher dimensions, for instance on a square in the plane, we may take analogous sets K 1 and K 2 but replace K 3 with K~

= {sin(m1rx) sin(n1ry):n, m = 1, 2, ... }.

Example 2.3.1. Consider the sequence of functions fn(x) = sin(nx) on L 2([0, 1]). We are going to show that {/n} converges weakly to f 0. First observe that

=

ll/nll£2 = (11 sin2nxdx) 1/2 =I~- si::nr/2

~ 1,

and hence the sequence {11/nll£2} is bounded. Now take an arbitrary function g(x) = sin(m1rx) from Ks. We have 1 (/n,g) = 1 sin(nx) sin(m1rx) dx

=

sin(n- m1r) 2(n- m1r)

sin(n + m1r) 2(n + m1r)

so that lim (/n, g) = (0, g) = 0,

for g E Ks

n-+oo

and {/n} thus converges weakly to f = 0.

0

2.3. Convergence of Sequences of Functions

33

We have seen that, in a given lJ' space, strong convergence implies weak convergence. It also turns out that we may compare convergence in different lJ' spaces using the following proposition.

Proposition 2.3.1. If (X, A, f.J) is a finite measure space and 1 :::; p 1 < P2 :::; oo, then

for every

f

E IJ'2

(2.3.4)

where c depends on JL(X). Thus every element of lJ'2 belongs to IJ'1 , and strong convergence in lJ'2 implies strong convergence in IJ'1 •

Proof. Let

f

E lJ'2 and let P2

< oo. By setting g = IJIP1 , we obtain

Setting p' = P2/P1 and denoting by p the number adjoint top', that is, (1/p) + (1/p') = 1, we have

and, consequently,

11/111~1 :5 IL(X) 11PIIJII1~2,

which proves equation (2.3.4). Hence, if II/IILP2 is finite, then 11/IILPl is also finite, proving that IJ'2 is contained in IJ'1 • FUrthermore, the inequality

implies that strong convergence in IJ'2 is stronger than strong convergence in IJ'1 • If P2 = oo, the inequality (2.3.4) is obvious, and thus the proof is complete. • Observe that the strong convergence of fn to f in £ 1 (with arbitrary measure) as well as the strong convergence of f n to f in lJ' (p > 1) with finite measure both imply lim { fnf.J(dx)

n-+oo}x To see this simply note that

= Jx { /JL(dx).

34

2. The Toolbox

It is often necessary to define a function as a limit of a convergent sequence and/or as a sum of a convergent series. Thus the question arises how to show that a sequence {/n} is convergent if the limit is unknown. The famous Cauchy condition for convergence provides such a tool. To understand this condition, first assume that {/n}, In E V, is strongly convergent to f. Take e > 0. Then there is an integer no such that forn~no

and, in particular, for n ~ no and k ~ 0. From this and the triangle inequality, we obtain

Thus we have proved that, if {/n} is strongly convergent in/.)' to/, then lim

n-+oc

11/n+Tc -/niiLP =

0

uniformly for all k

~

0.

(2.3.5)

This is the Cauchy condition for convergence. It can be proved that all /.)' spaces (1 5 p 5 oo) have the property that condition (2.3.5) is also sufficient for convergence. This is stated more precisely in the following theorem.

Theorem 2.3.1. Let (X, A, JJ) be a measure space and let {/n}, In E V(X, A, JJ) be a sequence such that equation (2.3.5) holds. Then there exists an element f E /.)' (X, A, JJ) such that {In} converges strongly to f, that is, condition (2.3.3) holds. The fact that Theorem 2.3.1 holds for])' spaces is referred to by saying that /.)' spaces are complete. Theorem 2.3.1 enables us to prove the convergence of series by the use of a comparison series. Suppose we have a sequence {gn} c V and we know the series of norms lluniiLP is convergent, that is, 00

L lluniiLP < oo.

(2.3.6)

n=O

Then, using Theorem 2.3.1, it is easy to verify that the series (2.3.7) is also strongly convergent and that its sum is an element of ])'.

Exercises

35

To see this note that the convergence of (2.3. 7) simply means that the sequence of partial sums n

Bn=

~9m m=O

is convergent. To verify that {sn} is convergent, set n t1n

= ~ IIYmiiLP• m=O

From equation (2.3.6) the sequence of real numbers {an} is convergent and, therefore, the Cauchy condition holds for this sequence. Thus

lim lan+A: -ani

n-+oo

=0

uniformly

fork~

0.

Further n+A:

:$;

LP

~ ll9miiLP

= lun+A:- Unl

m=n+l

so finally uniformly

fork~

0,

which is the Cauchy condition for {sn}·

Exercises 2.1. Using Definition 2.1.2 prove the following "continuity properties" of the measure: (a) If {An} is a sequence of sets belonging to A and A1 C A2 C ... , then

(b) If {An} is a sequence of sets belonging to A and A1:) A2:) ... , then

2.2. Let X define

= {1, 2, ... } be the set of positive integers.

For each A

k(n, A)= the number of elements of the set An {1, ... , n}.

c

X

36

2. The Toolbox

Let A be the family of all A of A in X" given by

c X for which there exists ''the average density

J.£(A) = lim .!k(n, A). n-+00 n Is J.£ a measure? [More precisely, is (X, A, J.£) a measure space?] 2.3. Let X = [a, b] be a compact interval and J.' the standard Borel measure. Prove that for a continuous f: [a, b] -+ R the values of the Lebesgue and the Riemann integral coincide. 2.4. Let X = R+ and J.' be the standard Borel measure. Prove that a continuous function f: R+-+ R is Lebesgue integral if and only if lim

a-+oo

and that

Jor

lf(x)ldx < oo,

r f(x)J.£(dx) = lim r f(x) dx.

}R+

n-+oo}o

2.5. Consider the space (X, A, J.£) where X = {1, 2, ... } is the set of positive integers, A all subsets of X and J.' the counting measure. Prove that a function f: X -+ R is integrable if and only if 00

L lf(x)l < oo, k=l

and that

[ f(x)J.£(dx)

= ~ f(k).

[Remark. L 1 (X, A, J.£) is therefore identical with the space of all absolutely convergent sequences. It is denoted by l 1 .] 2.6. From Proposition 2.3.1 we have derived the statement: if 1 ~ Pl < P2 ~ oo and J.£(X) < oo, then the strong convergence of Un} to fin IJ'2(fn, f E LP2 ) implies the strong convergence of {fn} to f in LP1 • Construct an example showing that this statement is false when J.£(X) = oo even if fm f E £Pl n£P2. 2. 7. Let (X, A, J.£) be a finite measure space and let Show that the function

f

E

L 00 (X) be fixed.

1~p 0 and p (X) > 0, while the equality holds if PJ+(x) = 0 or Pf-(x) = 0. Thus, by integrating over the space X, we have

if and only if there is no set A E A, JL(A) > 0, such that Pj+(x) > 0 and Pf-(x) > 0 for x E A, that is, Pj+(x) and Pf-(x) have disjoint support. Since f = j+- 1-, the left-hand integral is simply IIPfll· FUrther, the righthand side is IIPJ+II +liP!_ II= IIJ+II + 111-11 = IIJII, so the proposition is proved. • Having developed some of the more important elementary properties of Markov operators, we now introduce the concept of a fixed point of P. Definition 3.1.2. If P is a Markov operator and, for some f E L 1 , P f = f then f is called a fixed point of P. From Proposition 3.1.1 it is easy to show the following. Proposition

3.1.3. If P f = f then P j+ = j+ and P f- = f-.

Proof. Note that from P f

=f

we have

hence

L[Pj+(x)- j+(x)]JL(dx) + L[Pf-(x)- f-(x)]JL(dx)

= L[Pj+(x) + Pf-(x)]JL(dx)- L[j+(x) + f-(x)]JL(dx) =

LPlf(x)IJL(dx)- Llf(x)IJL(dx)

= IIPiflll-111!111·

3.2. The Frobenius-Perron Operator

41

However, by the contractive property of P we know that

II PI/Ill - Ill/Ill ~ 0. Since both the integrands (Pf+- j+) and (Pf-- f-) are nonnegative, this last inequality is possible only if P j+ = j+ and P 1- = 1-. • Definition 3.1.3. Let (X, A, J.L) be a measure space and the set D(X, A, J.L) be defined by D(X,A,J.L) = {! E L 1 (X,A,J.L):f;;:: 0 and II/II= 1}. Any function f E D(X,A,J.L) is called a density. Definition 3.1.4.

Iff E L 1 (X,A,J.L) J.Lt(A) =

and f;;:: 0, then the measure

L

f(x)J.L(dx)

is said to be absolutely continuous with respect to J.L, and f is called the Radon-Nikodym derivative of J.L 1 with respect to J.L. In the special case that f E D(X, A, J.L), then we also say that f is the density of J.Lt and that J.Lt is a normalized measure. From Corollary 2.2.1 it follows that a normalized measure v is absolutely continuous with respect to J.L if v(A) = 0 whenever J.L(A) = 0. This property is often used as the definition of an absolutely continuous measure. Using the notion of densities we may extend the concept of a fixed point of a Markov operator with the following definition. Definition 3.1.5. Let (X, A, J.L) be a measure space and P be a Markov operator. Any f E D that satisfies P f = f is called a stationary density of P. The concept of a stationary density of an operator is extremely important and plays a central role in many of the sections that follow.

3.2 The Frobenius-Perron Operator Having developed the concept of Markov operators and some of their properties, we are in a position to examine a special class of Markov operators, the Frobenius-Perron operator, which we introduced intuitively in Chapter 1.

We start with the following definitions. Definition 3.2.1. Let (X, A, J.L) be a measure space. A transformation - t X is measurable if

S: X

for all A

EA.

42

3. Markov and Frobenius-Perron Operators

Definition 3.2.2. A measurable transformation S: X -+ X on a measure space (X,A,J£) is nonsingular if J£(S- 1 (A)) = 0 for all A E A such that J£(A) = 0. Before stating a precise definition of the Frobenius-Perron operator, consider the following. Assume that a nonsingular transformation S: X -+ X on a measure space is given. We define an operator P: L 1 -+ L 1 in two steps. 1. Let I E L 1 and I ~ 0. Write

f

l(x)J'(dx).

(3.2.1)

ls-l(A)

Because

s-• (

yA.) ys-'(A.), =

it follows from property (L5) of the Lebesgue integral that the integral (3.2.1) defines a finite measure. Thus, by Corollary 2.2.1, there is a unique element in L 1 , which we denote by PI, such that

f

}A 2. Now let Write I = 1+

Pl(x)J£(dx)

=f

l(x)J£(dx)

for A EA.

Js-l(A)

I E L 1 be arbitrary, - 1- and define

that is, not necessarily nonnegative.

From this definition we have

or, more completely,

f

}A

Pl(x)J£(dx} =

f

l(x)J£(dx),

for A EA.

(3.2.2)

ls-l(A)

From Proposition 2.2.1 and the nonsingularity of S, it follows that equation (3.2.2) uniquely defines P. We summarize these comments as follows. Definition 3.2.3. Let (X, A, J') be a measure space. If S: X -+ X is a nonsingular transformation the unique operator P: L 1 -+ L 1 defined by equation (3.2.2) is called the Frobenius-Perron operator corresponding to S. It is straightforward to show from (3.2.2) that P has the following properties:


+ ).2/2) = ).1Pft + ).2Ph for all It, h E £ 1, ).1, ).1 E R, so P is a linear operator;

{FPl) P{).d1

{FP2) Pf?. 0 iff?. 0; {FP3) [

{3.2.3)

(3.2.4)

and

Pf(x)p,(dx) = [

43

f(x)p,(dx)

(3.2.5)

(FP4) If S., = So . ~. o S and P., is the Frobenius-Perron operator corresponding to S.,, then P., = P"", where Pis the Frobenius-Perron operator corresponding to S.

Remark 3.2.1. Although the definition of the Frobenius-Perron operator P by {3.2.2) is given by a quite abstract mathematical theorem of RadonNikodym, it should be realized that it describes the evolution of f by a transformationS. Properties {3.2.4-3.2.5) of the transformed distribution Pf(x) are exactly what one would expect on intuitive grounds. 0 Remark 3.2.2. From the preceding section, the Frobenius-Perron operator is also a Markov operator. As we wish to emphasize the close connection between the behavior of stochastic systems and the chaotic behavior of deterministic systems, we will formulate concepts and results for Markov operators wherever possible. The Frobenius-Perron operator is a particular Markov operator, and thus any property of Markov operators is immediately applicable to the Frobenius-Perron operator. In some special cases equation {3.2.2) allows us to obtain an explicit form for Pf. As we showed in Chapter 1, if X= (a,b] is an interval on the real lineR, and A= [-a,x], then (3.2.2) becomes

1 :z:

Pf(x)ds =

o

f

1S-l((o,:z:))

f(s)ds,

and by differentiating

Pf(x)

= dxd

{

1s-l((o,:z:))

f(s)ds.

{3.2.6)

It is important to note that in the special case where the transformation

S is differentiable and invertible, an explicit form for P f is available. If S is differentiable and invertible, then S must be monotone. Suppose S is an increasing function and

s-1 has a continuous derivative. Then

s- 1 ([a, x])

= (s- 1 (a), s- 1 (x)],

and from {3.2.6)

Pf(x)

d 1s-'(:z:)

= dx

S- 1 (o)

f(s)ds

d

= f{S- 1 (x))dx(s- 1 (x)].

44

3. Markov and Frobenius-Perron Operators

Pf

-I

0 e-1

0

2

e 3

FIGURE 3.2.1. Operation of the Frobenius-Perron operator corresponding to S(x) = ez, x E R. (a) An initial density l(x) = ~~[-l,lJ(x) is transformed into the density Pl(x) = (2x)- 1 1[.,-1,.,1(x) by S as shown in (b). If S is decreasing, then the sign of the right-hand side is reversed. Thus, in the general one-dimensional case, for S differentiable and invertible with continuous dS- 1 /dx,

(3.2.7}

Example 3.2.1. To see how the Frobenius-Perron operator works, pick S(x) = exp(x). Thus from (3.2.7}, we have Pl(x) = (1/x)l(lnx).

Consider what happens to an initial

f

given by

1 f(x) = 21[-1,1J(x}, and shown in Figure 3.2.1a. Under the action of P, the function I is carried into Pf(x) = (1/2x}1[.,-1,eJ(x) as shown in Figure 3.2.1b. 0 Two important points are illustrated by this example. The first is that for an initial I supported on a set [a, b], PI will be supported on [S(a), S(b)]. Second, Pf is small where (dS/dx} is large and vice versa. We generalize the first observation as follows. Proposition 3.2.1. Let S: X -+ X be a nonsingular transformation and P the associated Probenius-Perron operator. Assume that an f ;:::: 0, f E £ 1 , is given. Then (3.2.8} suppf c s- 1(suppPJ)


45

and, more generally, for every set A E A the following equivalence holds: Pf(x) = 0 for X E A if and only if f(x) = 0 for X E s- 1 (A).

Proof. The proof is straightforward. By the definition of the FrobeniusPerron operator, we have

f

}A

Pf(x)J.L(dx) =

f

ls-l(A)

f(x)J.L(dx)

or

[ 1A(x)Pf(x)J.L(dx) = [ 1s-l(A)(x)f(x)J.L(dx). Thus Pf(x) = 0 on A implies, by property (L2) of the Lebesgue integral, that f(x) = 0 for X E s- 1 (A), and vice versa. Now setting A = X \ supp(P/), we have Pf(x) = 0 for x E A and, consequently, f(x) = 0 for X E s- 1 (A), which means that supp I c X \ s- 1 (A). Since s- 1 (A) = X\ s- 1 (supp(P/)), this completes the proof. • Remark 3.2.3. In the case of arbitrary f E £1, then, in Proposition 3.2.1 we only have: If f(x) = 0 for all X E s- 1 (A), then Pf(x) = 0 for all X EA. That the converse is not true can be seen by the following example. Take S(x) = 2x (mod 1) and let

f(x) Then from (1.2.13) Pf(x) X

E [0,1].

={1

-1

= 0 for

~~x 1 is an integer, on the measure space ([0, 1], 8, IS) where 8 is the Borel u-algebra and IS is the Borel measure (cf. Remark 2.1.3). As we have shown in Example 1.2.1, for any interval [0, x] c [0, 1]

s-1([0,x]) =

ur~. ~ i= 0

rr

+ ~] r

4.1 Measure-Preserving Transformations

53

and the Frobenius-Perron operator P corresponding to S is given by equation {1.2.13):

Pf(x)

r-1 ( · ) = -1 L I .!. + ~ .

ri=O

Thus

r

r

r-1

1 P1 =- L1 = 1 r i=O

and by our previous remark the Borel measure is invariant under the r-adic transformation. 0

Remark 4.1.2. It should be noted that, as defined, the r-adic transfor-

!·

mation is not continuous at However, if instead of defining the r-adic transformation on the interval [0,1] we define it on the unit circle (circle with circumference of 1) obtained by identifying 0 with 1 on the interval [0,1], then it is continuous and differentiable throughout. 0 Example 4.1.2. Again consider the measure space {[0,1], B, J.l.) where J.l. is the Borel measure. Let S: [0, 1] -+ [0, 1] be the quadratic map (S(x) = 4x{1- x) of Chapter 1). As was shown there, for [0, x] c [0, 1],

s- 1{[0,x]) = [o,!- !v'I=XJ u [! + !v'f=X, 1] and the Frobenius-Perron operator is given by

Pf(x)

= 4 )=x{1 (!- !v'l-X) +I(!+ !v'l-X)}. 1-x

Clearly, 1 , 2v'1- X so that the Borel measure J.1. is not invariant under S by Remark 4.1.1. To find an invariant measure we must find a solution to the equation P f = f or P1=

f(x) =

)=-x{ I(!- !v'l-X) +I(!+ !v'l-X) }·

4 This problem was first solved by Ulam and von Neumann [1947] who showed that the solution is given by 1 /.(x) = 11'Jx(1- x)'

{4.1.3)

which justifies our assertion in Section 1.2. It is straightforward to show that/. as given by (4.1.3) does, indeed, constitute a solution to Pf =f. Hence the measure

is invariant under the quadratic transformation S(x)

= 4x{1- x).

0

54

4. Studying Chaos with Densities

Remark 4.1.3. The factor of 1r appearing in equation (4.1.3) ensures that /. is a density and thus that the measure 1'!. is normalized. 0 Example 4.1.3. (The baker transformation). Now let X be the unit square in a plane, which we denote by X = [0, 1] x [0, 1] (see Section 2.2). The Borel 0'-algebra 8 is now generated by all possible rectangles of the form [0, a] x [0, b] and the Borel measure p is the unique measure on 8 such that p([O, a] x [0, b]) = ab. (Thus the Borel measure is a generalization of the concept of the area.) We define a transformation S: X - X by

S(

) _ { (2x, b) 0:::; x < !,O:::; y:::; 1 x, Y (2x -1 1 !y + !) !2
into four triangular ares. In Figure 4.1.3b we show how the unit square is transformed after one application of (x, y)-+ (x+y, x+2y), whereas Figure 4.1.3c shows the result of the full Anosov diffeomorphism. It is clear that the effect of this transformation will be to very quickly scramble, or mix, various regions of the unit square. This property of mixing, also shared by the baker transformation, is most important and is dealt with in more detail in Section 4.3. The determinant of the Jacobian of transformation (4.1.6) is given by

J

= det 1~

~ 1 = 1,

58


so that the transformation is measure preserving; we already noted this result on geometric grounds in Figure 4.1.3. The two eigenvectors associated with S have eigenvalues .X1 =

V5 2 2 3

and .X2

3 V5 = 2 + 2'

hence 0 < .X1 < 1 < .X2• Thus, as for the baker transformation, the Anosov diffeomorphism involves a stretching in one direction and a corresponding compression in the orthogonal direction. With some patience it is possible to derive an explicit formula for the Frobenius-Perron operator corresponding to the Anosov diffeomorphism (4.1.6) using a technique analogous to that employed for the baker transformation of the previous example. However, we can obtain this result immediately from Corollary 3.2.1 since the Anosov diffeomorphism is invertible. An easy calculation gives

s- 1 (x, y) =

(2x- y, y- x)

and thus

Pl(x, y)

(mod 1)

= l(2x- y, y- x),

(4.1.7)

where, as in (4.1.6), the terms 2x- y and y- x should be interpreted modulo 1. From (4.1.7) it is clear that P1 = 1, which corresponds to the fact that S preserves the Borel measure. 0

Remark 4.1.6. Observe that, if we replace the unit square [0, 1] x [0, 1] with the torus, that is, if we identify points (x, 1) with (x, 0) and (1, y) with (0, y), then this example of an Anosov diffeomorphism becomes continuous and differentiable just as the r-adic transformation does when the unit interval is replaced by the unit circle. The word diffeomorphism comes from the fact that the Anosov transformation is invertible, and that both the transformation and its inverse are differentiable. 0

4.2 Ergodic Transformations Because a transformation S has an invariant measure or because the Frobenius-Perron operator P associated with S has a stationary density does not imply that S has interesting statistical properties. For example, if Sis the identity on X, that is, S(x) = x for every x EX, then

s- 1 (A) =A

(4.2.1)

for every A C X, and, consequently, PI = I for every I E L1 • This is, of course, not an interesting transformation. However, even if (4.2.1) holds for just one subset A of X, then the transformation S may be studied on

4.2 Ergodic Transformations

59

the sets A and X\ A separately. To see this, assume that A is fixed and condition (4.2.1) holds. Consider a trajectory x0 , 8(x0 ), 8 2 (x0 ), ••••

Equality (4.2.1) implies that 8 maps A into itself and no element of X\ A is mapped into A. Thus, if x 0 e A then sn(x0 ) e A for all n, and if xo rt A then 8n(x0 ) '1. A for all n. Example 4.2.1. A simple example is 8(k) =

k + 2 for k = 1, ... , 2(N- 1) 1 fork= 2N- 1 {2 fork=2N

operating on the space X= {1, ... , 2N} with the counting measure. This transformation can be studied separately on the sets A = {1, 3, ... , 2N -1} and X \ A = {2, 4, ... , 2N} of odd and even interiors. 0 Any set A satisfying (4.2.1) is called invariant. We require this equality to be satisfied modulo zero (see Remark 3.1.3). Then we can make the following definition. Definition 4.2.1. Let (X, .A, JJ.) be a measure space and let a nonsingular transformation 8: X -+ X be given. Then 8 is called ergodic if every invariant set A e .A is such that either JJ.(A) = 0 or JJ.(X \ A) = 0; that is, Sis ergodic if all invariant sets are trivial subsets of X. From this definition it follows that any ergodic transformation 8 must be studied on the entire space X. Determining ergodicity on the basis of Definition 4.2.1 is, in general, difficult except for simple examples on finite spaces. Thus, for example, the transformation in Example 4.2.1 is not ergodic on the space X of integers, but it is ergodic on the sets of even and odd integers. In studying more interesting examples the following theorem may be of

use. Theorem 4.2.1. Let (X, .A, JJ.) be a measure space and 8: X -+ X a nonsingular transformation. 8 is ergodic if and only if, for every measurable function f: X -+ R,

f(8(x))

= f(x)

for almost all

X

eX

(4.2.2)

implies that f is constant almost everywhere. Proof. We first show that ergodicity implies f is constant. Assume that, as in Figure 4.2.1, we have a function f satisfying (4.2.2), which is not

60


FIGURE 4.2.1. Definition of the sets A and B = B1 U B2.

constant almost everywhere, and that 8 is ergodic. Then there is some r such that the sets A={x:f(x)~r}

and

B={x:f(x)>r}

have positive measure. These sets are also invariant because

8- 1 (A) = {x: 8(x) E A}= {x: f(8(x)) ~ r}

= {x:f(x) ~ r} =A and similarly for B. Because sets A and Bare invariant, 8 is not ergodic, which is a contradiction. Thus, every f satisfying (4.2.2) must be constant. To prove the converse, assume that 8 is not ergodic. Then, by Definition 4.2.1, there is a nontrivial set A E A that is invariant. Set f = 1A, and since A is nontrivial, f is not a constant function. Moreover, since A= 8- 1 (A) we have

/(8(x))

= 1A(8(x)) = 1s-l(A)(x) = 1A(x) = f(x)

and ( 4.2.2) is satisfied by a nonconstant function.

a.e.

•

Remark 4.2.1. It is clear from the proof that it is sufficient to verify only (4.2.2) for bounded measurable functions since in the last part of the proof we used characteristic functions that are bounded. 0 An immediate consequence of Theorem 4.2.1 in combination with the definition of the Koopman operator is the following corollary. Corollary 4.2.1. Let (X, A, JJ) be a measure space, 8: X--+ X a nonsingu-

lar transformation, and U the Koopman operator with respect to 8. Then 8 is ergodic if and only if all the fixed points of U are constant functions. In addition to Theorem 4.2.1 and the preceding corollary, another result of use in checking the ergodicity of 8 using the Frobenius-Perron operator is contained in the following theorem.

4.2 Ergodic Transformations

61

Theorem 4.2.2. Let (X,A,IJ) be a measure space, S:X-+ X a nonsingular transformation, and P the Probenius-Perron operator associated with S. If S is ergodic, then there is at most one stationary density f. of P. Further, if there is a unique stationary density f. of P and f.(x) > 0 a. e., then S is ergodic. Proof. To prove the first part of the theorem 8BSume that S is ergodic and that It and h are different stationary densities of P. Set g = It - h, so that Pg = g by the linearity of P. Thus, by Proposition 3.1.3, g+ and gare both stationary densities of P: (4.2.3)

Since, by 8BSumption, It and h are not only different but are also densities we have Set

A= suppg+ = {x:g+(x) > 0}. and

= suppg- = {x:g-(x) > 0}.

B

It is evident that A and Bare disjoint sets and both have positive (nonzero) measure. By equality (4.2.3) and Proposition 3.2.1, we have

s- 1 (B). Since A and Bare disjoint sets, s- 1 (A) and s- 1 (B) are also disjoint. By Ac

s- 1 (A)

and B

c

induction we, therefore, have

A

c

s- 1 (A) c s- 2 (A) · · · c s-n(A)

B

c

s- 1 (B) c s- 2 (B) ... c s-n(B),

and

where s-n(A) and s-n(B) are also disjoint for all n. Now define two sets by 00

A=

U s-n(A)

00

and

iJ =

U s-n(B). n=O

n=O

These two sets A and iJ are also disjoint and, furthermore they are invariant because 00 00

s- 1 (A) = U s-n(A) = U s-n(A) =A n=l

n=O

62


and

s-l(fJ)

=

us-n(B) = us-n(B) = B. 00

00

n=l

n=O

Neither A nor fJ are of measure zero since A and B are not of measure zero. Thus, A and fJ are nontrivial invariant sets, which contradicts the ergodicity of S. Thus, the first portion of the theorem is proved. To prove the second portion of the theorem, assume that /. > 0 is the unique density satisfying Pf. = /. but that Sis not ergodic. If Sis not ergodic, then there exists a nontrivial set A such that

s- 1 (A) =A and with B = X \A

s- 1 (B) =B.

With these two sets A and B, we may write /.

= 1Af• + 1s/., so that (4.2.4)

s-

1 (A). Thus, The function 1s/. is equal to zero in the set X\ B =A= by Proposition 3.2.1, P{1s/.) is equal to zero in A= X\ Band, likewise, P{1A/.) is equal to zero in B =X\ A. Thus, equality (4.2.4) implies that

1Af•

= P(1Af•)

and

1s/. = P(1s/.).

Since/. is positive on A and B, we may replace 1Af• by !A= 1A/./II1A/.II, and 1s/. by Is= 1s/./ll1s/.ll in the last pair of equalities to obtain

!A= PIA and Is= Pfs. This implies that there exist two stationary densities of P, which is in contradiction to our assumption. Thus, if there is a unique positive stationary density/. of P, then Sis ergodic. •

Example 4.2.2. Consider a circle of radius 1, and let S be a rotation through an angle t/J. This transformation is equivalent to the mapS: [0, 211") -+ [0, 211") defined by (mod 211"). S(x) =x+t/J If tjJ is commensurate with 211" (that is, t/J/211" is rational), then Sis evidently nonergodic. For example, if t/J = 1r /3, then the sets A and B of Figure 4.2.2 are invariant. For any t/J = 27r(k/n), where k and n are integers, we will still find two invariant sets A and B, each containing n parts. As n becomes large the intermingling of the two sets A and B becomes more complicated and suggests that the rotational transformation S may be ergodic for (t/J/211") irrational. This does in fact hold, but it will be proved later when we have more techniques at our disposal.

4.2 Ergodic 'fransformations

63

B

FIGURE 4.2.2. The two disjoint sets A (containing all the arcs denoted by thin lines) and B (containing arcs marked by heavy lines) are invariant under the rotational transformation when t/J/21r is rational. In this example the behavior of the trajectories is moderately regular and insensitive to changes in the initial value. Thus, independent of whether or not (4J/21r) is rational, if the value of 4J is known precisely but the initial condition is located between a and /3, xo E {a, /3), then

{mod 21r) and all of the following points of the trajectory are known with the same accuracy, {/3 - a). 0 Before closing this section we state, without proof, the Birkhoff individual ergodic theorem [Birkhoff, 1931a,b]. Theorem 4.2.3. Let (X, A, p.) be a measure space, 8: X -+ X a measumble transformation, and f: X -+ R an integrable function. If the measure p. is invariant, then there exists an integrable function such that

r

n-1

f*(x) = lim .!_ "f(Sk(x)) n-+oo n L....i

for almost all x E X.

(4.2.5)

k=O

r

Without additional assumptions the limit (X) is generally difficult to determine. However, it can be shown that f*(x) satisfies f*(x)

= f*(S(x))

for almost all x E X,

{4.2.6)

and when p.{X) < oo

L

f*(x)p.(dx) =

L

f(x)p.(dx).

{4.2.7)

Equation (4.2.6) follows directly from (4.2.5) if xis replaced by S(x). The second property, {4.2.7), follows from the invariance of p. and equation

64


(4.2.5). Thus, by Theorem 3.2.1,

[

f(x)JJ(dx) = [

f(S(x))JJ(dx)

= ·· ·

so that integrating equation (4.2.5) over X and passing to the limit yields (4.2. 7) by the Lebesque-dominated convergence theorem when f is bounded. When f is not bounded the argument is more difficult.

Remark 4.2.2. Theorem 4.2.3 is known as the individual ergodic theorem because it may be used to give information concerning the asymptotic behavior of trajectories starting from a given point x E X. AB our emphasis is on densities and not on individual trajectories, we will seldom use this theorem. 0 With the notion of ergodicity we may derive an important and often quoted extension of the Birkhoff individual ergodic theorem. Theorem 4.2.4. Let (X, A, JJ) be a finite measure space and S: X-+ X be measure presenJing and ergodic. Then, for any integrable f, the average of f along the trajectory of S is equal almost everywhere to the average off over the space X; that is 1 n-1

lim n-+oo n

L

f(Sk(x))

k=O

=

JJ

1 (X)

1

f(x)JJ(dx)

a.e.

(4.2.8)

X

Proof. From (4.2.6) and Theorem 4.2.1 it follows that/* is constant almost everywhere. Hence, from (4.2.7), we have

[

f*(x)JJ(dx)

= f* [

so that

/*(x)

JJ(dx)

= JJ(~) [

= J*JJ(X) = [ f(x)JJ(dx)

f(x)JJ(dx),

a.e.

Thus equation (4.2.5) of the Birkhoff theorem and the preceding formula imply (4.2.8), and the theorem is proved. • One of the most quoted consequences of this theorem is the following.

Corollary 4.2.2. Let (X, A, JJ) be a finite measure space and S: X -+ X be measure presenJing and ergodic. Then for any set A E A, JJ(A) > 0, and almost all x E X, the fraction of the points {Sk(x)} in A as k -+ oo is given by JJ(A)/JJ(X). Proof. Using the characteristic function 1A of A, the fraction of points from {Sk(x)} in A is n-1 1 lim lA(Sk(x)). n-+oo n k=O

L

However, from (4.2.8) this is simply JJ(A)/JJ(X).

•

4.3. Mixing and Exactness

65

Remark 4.2.3. Corollary 4.2.2 says that every set of nonzero measure is visited infinitely often by the iterates of almost every x E X. This result is a special case of the Poincare recurrence theorem. 0

4.3 Mixing and Exactness Mixing Transformations The examples of the previous section show that ergodic behavior per se need not be very complicated and suggests the necessity of introducing another concept, that of mixing.

Definition 4.3.1. Let (X, A, I') be a normalized measure space, and S: X-+ X a measure-preserving transformation. Sis called mixing if lim I'(A n s-n(B))

n-+oo

= IJ(A)IJ(B)

for all A,B EA.

(4.3.1)

Condition (4.3.1) for mixing has a very simple interpretation. Consider points X belonging to the set An s-n(B). These are the points such that x E A and sn(x) E B. Thus, from (4.3.1), as n-+ oo the measure of the set of such points is just I'(A)IJ(B). This can be interpreted as meaning that the fraction of points starting in A that ended up in B after n iterations (n must be a large number) is just given by the product of the measures of A and B and is independent of the position of A and B in X. It is easy to see that any mixing transformation must be ergodic. Assume that B E A is an invariant set, so that B = s- 1 (B) and, even further, B = s-n(B) by induction. Take A = X \ B so that I'(A n B) = I'(A n s-n(B)) = 0. However, from (4.3.1), we must have lim I'( An s-n(B))

n-+oo

= I'(A)IJ(B) = (1 -~J(B))I'(B),

and thus I'(B) is either 0 or 1, which proves ergodicity. Many of the transformations considered in our examples to this point are mixing, for example, the baker, quadratic, Anasov, and r-adic. (The rotation transformation is not mixing according to our foregoing discussion.) To illustrate the mixing property we consider the baker and r-adic transformations in more detail.

Example 4.3.1. (See also Example 4.1.3.) In considering the baker transformation, it is relatively easy to check the mixing condition (4.3.1) for generators of the u-algebra B, namely, for rectangles. Although the transformation is simple, writing the algebraic expressions for the counterimages is tedious, and the property of mixing is easier to see pictorially. Consider

66


Figure 4.3.1a, where two sets A and Bare represented with J.'(B) =!·We take repeated counterimages of the set B by the baker transformation and find that after n such steps, s-"(B) consists of 2"- 1 vertical rectangles of equal area. Eventually the measure of AnS-"(B} approaches J.'(A}/2, and condition (4.3.1} is evidently satisfied. The behavior of any pair of sets A and B is similar. It is interesting that the baker transformation behaves in a similar fashion if, instead of examining s-"(B}, we look at S"(B) as shown in Figure 4.3.1b. Now we have 2" horizontal rectangles after n steps and all of our previous comments apply. So, for the baker transformation the behavior of images and counterimages is very similar and illustrates the property of mixing. This is not true for our next example, the dyadic transformation.

0 In general, proving that a given transformation is mixing via Definition 4.3.1 is difficult. In the next section, Theorem 4.4.1 and Proposition 4.4.1, we introduce easier and more powerful techniques for this purpose.

Example 4.3.2. (Cf. Examples 1.2.1 and 4.1.1.} To examine the mixing property (4.3.1) for the dyadic transformation, consider Figure 4.3.2a. Now we take the set B = (0, b] and find that the nth counterimage of B consists of intervals on (0,1] each of the same length. Eventually, as before J.'(A n s-"(B}} - J.£(A)J.£(B). As for the baker transformation let us consider the behavior of images of a set B under the dyadic transformation (cf. Figure 4.3.2b}. In this case, if B = (0, b], then S(B) = [0, 2b] and after a finite number of iterations S"(B) = (0, 1). The same procedure with any arbitrary set B c (0, 1] of positive measure will show that J.'(S"(B)) - 1 and thus the behavior of images of the dyadic transformation is different from the baker transformation. 0

Exact Transformations The behavior illustrated by images of the dyadic transformation is called exactness, and is made precise by the following definition due to Rochlin

[1964]. Definition 4.3.2. Let (X, A, J.') be a normalized measure space and S: X -+ X a measure-preserving transformation such that S(A) E A for each A EA. If lim J.£(S"(A))

n-+oo

=1

for every A E A, J.£(A)

> 0,

(4.3.2}

then Sis called exact. It can be proved, although it is not easy to do so from the definition, that exactness of S implies that S is mixing. As we have seen from the


0

I 5" 1 (B)

0

0

0

0

0

0

67

0 (a)

(b)

FIGURE 4.3.1. Mixing illustrated by the behavior of counterimages and images of a set B by the baker transformation. (a) The nth counterimage of the set B consists of 2"- 1 vertical rectangles, each of equal area. (b) Successive iterates of the same set B results in 2" horizontal rectangles after n iterations.

68

4. Studying Chaos with Densities B

B

0

0

0

0

0 0

(a)

(b)

FIGURE 4.3.2. The behavior of counterimages and images of a set B by the dyadic transformation. (a) Successive counterimages of a set B that result after n such counterimages, in 2" disjoint sets on [0,1]. (b) The behavior of images of a set B generated by the dyadic transformation, which is quite different than that for the baker transformation. (See the text for further details.)

baker transformation the converse is not true. We defer the proof until the next section when we have other tools at our disposal. Condition (4.3.2) has a very simple interpretation. If we start with a set A of initial conditions of nonzero measure, then after a large number of iterations of an exact transformation S the points will have spread and completely filled the space X.

Remark 4.3.1. It cannot be emphasized too strongly that invertible transformations cannot be exact. In fact, for any invertible measure-preserving transformationS, we have p.(S{A)) = p.(S- 1 (S(A))) = p.(A) and by induction p.(S.. {A)) = p.(A), which violates {4.3.2). 0 In this and the previous section we have defined and examined a hierarchy of "chaotic" behaviors. However, by themselves the definitions are a bit sterile and may not convey the full distinction between the behaviors of ergodic, mixing, and exact transformations. To remedy this we present the first six successive iterates of a random distribution of 1000 points in the set X= [0, 1] x [0, 1] by the ergodic transformation {mod 1)

{4.3.3)

in Figure 4.3.3; by the mixing transformation S(x,y)

= (x+y,x+2y)

(mod 1)

(4.3.4)


69

I (d)

I (a)

• 0

0

I (b)

I (e)

I

• 0

0 I (c)

0

• •

0

FIGURE 4.3.3. Successive iterates of a random distribution of 1000 points in (0,0.1) x (0,0.1) by the ergodic transformation (4.3.3). Note how the distribution moves about in the space (0, 1) x [0, 1).

in Figure 4.3.4; and by the exact transformation

S(x,y) = (3x+y,x+3y)

(mod 1)

(4.3.5)

in Figure 4.3.5. Techniques to prove these assertions will be developed in the next two chapters.

70


I (a l

0

0

I (b)

0 I (c)

0

0 (f)

I • •·• _.. _..

• , ....

0

FIGURE 4.3.4. The effect of the mixing transformation (4.3.4} on the same initial distribution of points used in Figure 4.3.3.

4.4. Classifying Transformations

(d)

I (a)

0

71

0

I (b)

0

0

0

0

FIGURE 4.3.5. Successive applications of the exact transformation [equation (4.3.5)]. Note the rapid spread of the initial distribution of points throughout the phase space.

4.4 Using the Frobenius-Perron and Koopman Operators for Classifying Transformations The concepts developed in the previous two sections for classifying various degrees of irregular behaviors (ergodicity, mixing, and exactness) were stated in terms of the behavior of sequences of sets. The proof of ergodicity, mixing, or exactness using these definitions is difficult. Indeed, in all the examples we gave to illustrate these concepts, no rigorous proofs were ever given, although it is possible to do so. In this section we reformulate the concepts of ergodicity, mixing, and

72


exactness in terms of the behavior of sequences of iterates of FrobeniusPerron and Koopman operators and show how they can be used to determine whether a given transformation S with an invariant measure is ergodic, mixing, or exact. The techniques of this chapter rely heavily on the notions of Cesaro, weak and strong convergences, which were developed in Section 2.3. We will first state and prove the main theorem of this section and then show its utility.

Theorem 4.4.1. Let (X, .A, JL) be a normalized measure space, 8: X-+ X a measure preseroing tmnsformation, and P the Frobenius-Perron opemtor corresponding to S. Then (a) Sis ergodic if and only if the sequence {Pn!} is CesAro convergent to 1 for all f E D; (b) S is mixing if and only if { pn!} is weakly convergent to 1 for all /ED; (c) S is exact if and only if { pn!} is strongly convergent to 1 for all /ED. Before giving the proof of Theorem 4.4.1, we note that, since Pis linear, convergence of { pn!} to 1 for f E D is equivalent to the convergence of {Pn!} to (!, 1} for every f E £ 1 • This observation is, of course, valid for all types of convergence: Cesaro, weak, and strong. Thus we may restate Theorem 4.4.1 in the equivalent form.

Corollary 4.4.1. Under the assumptions of Theorem 4.4.1, the following equivalences hold: (a) S is ergodic if and only if n-1

1 lim - '"'(Pk /, g}

n-+oo

n L...J

= (!, 1} (1, g)

k=O

(b) S is mixing if and only if lim (Pn /,g)

n-+oo

= (!, 1} (1, g)

(c) S is exact if and only if lim IIPn f

n-+oo

- (/, 1) II

=0

Proof of Theorem 4.4.1. The proof of part (a) follows easily from Corollary 5.2.3.


73

Next consider the mixing portion of the theorem. AssumeS is mixing, which, by definition, means lim

n-+oo

~(An s-n(B)) =~(A)~( B)

for all A,B EA.

The mixing condition can be rewritten in integral form as lim

f 1A(X)1 8 (sn(x))~(dx) = f 1A(x)~(dx) f 1 8 (x)~(dx). lx lx

n-+oolx

By applying the definitions of the Koopman operator and the scalar product to this equation, we obtain (4.4.1)

Since the Koopman operator is adjoint to the Frobenius-Perron operator, equation (4.4.1) may be rewritten as

or lim (Pnf,g} = (/,1}(1,g}

n-+oo

for f = 1A and g = 1s. Since this relation holds for characteristic functions it must also hold for the simple functions

f

=L

AilA,

and g

=L

i

u,1s1 •

i

Further, every function g E L 00 is the uniform limit of simple functions 9k E L 00 , and every function f E L 1 is the strong (in L 1 norm) limit of a sequence of simple functions be E £ 1 • Obviously, I(Pnj,g}- (/,1}(1,g}j $j(Pnj,g}- {Pn/1c,91c}l

+ I(Pn fk, 9k}- (f~c, 1}{1, 9k}l + l(f~c, 1}(1,g~c}- (!, 1}(1,g}j. (4.4.2)

H llf~c - /II $ f and ll9k - YIILoo $ right-hand side of (4.4.2) satisfy

f,

then the first and last terms on the

I(Pn J,g}- {Pn fk,9k}i $ j(Pnj,g}- (Pnfk,g}j + I(Pnfk,g}- (Pnfk,9k}i $ fii9IILoo + fll/~cll $ f(IIYIILoo +

11!11 +f)

and analogously

1(/k, 1}(1,g~c}- (/, 1}(1,g}l $

f(IIYIILoo +

11/11 +f).

74


Thus these terms are arbitrarily small for small middle term of (4.4.2),

E.

Finally, for fixed k the

converges to zero as n -+ oo, which shows that the right-hand side of inequality (4.4.2) can be as small as we wish it to be for large n. This completes the proof that mixing implies the convergence of (Pn /,g) to (!, 1}(1,g} for all f e £ 1 and g e £ 00 • Conversely, this convergence implies the mixing condition (4.4.1) if we set f = 1A and g = 1s. Lastly, we show that the strong convergence of {pn!} to (!, 1} implies exactness. Assume J.t(A) > 0 and define

/A(x) = (1/J.t(A))1A(x). Clearly, !A is a density. If the sequence {rn} is defined by

then it is also clear that the sequence is convergent to zero. By the definition of rn, we have

J.t(Sn(A)) =

f

J.t(dx)

1sn(A)

={

pn fA(X)J.t(dx)- {

1sn(A)

(Pn fA(x)- 1)J.t(dx)

1sn(A)

pnfA(X)J.t(dx)- Tn•

?; {

(4.4.3)

1sn(A)

From the definition of the Frobenius-Perron operator, we have

{

1sn(A)

pn fA(X)J.t(dx)

={

1s-n(Sn(A))

/A(X)J.t(dx)

and, since s-n(Sn(A)) contains A, the last integral is equal to 1. Thus inequality (4.4.3) gives J.t(Sn(A)) ?: 1 - Tn 1 which completes the proof that the strong convergence of {pn!} to (!, 1} implies exactness. • We omit the proof of the converse (that exactness implies the strong convergence of {pnf} to (/,1}) since we willnever use this fact and its proof is based on quite different techniques (see Lin [1971]). Because the Koopman and Frobenius-Perron operators are adjoint, it is possible to rewrite conditions (a) and (b) of Corollary 4.4.1 in terms of the Koopman operator. The advantage of such a reformulation lies in the fact


75

that the Koopman operator is much easier to calculate than the FrobeniusPerron operator. Unfortunately, this reformulation cannot be extended to condition (c) for exactness of Corollary 4.4.1 since it is not expressed in terms of a scalar product. Thus, from Corollary 4.4.1, the following proposition can easily be stated. Proposition 4.4.1. Let (X, A, J.£) be a normalized measure space, S: X X a measure-preseroing transformation, and U the Koopman operator corresponding to S. Then

(a) S is ergodic if and only if n-1

1 lim - "(!, uk g) n-+oo n L..J

= (!, 1){1, g)

k=O

(b) S is mixing if and only if

lim(!, ung) = (/, l)(l,g)

n-+oo

Proof. The proof of this proposition is trivial since, according to equation (3.3.4), we have

(f,Ung)

= (Pn/,g}

for IE L\ g E L 00 , n

= 1, 2, ... ,

which shows that conditions (a) and (b) of Corollary 4.4.1 and Proposition 4.4.1 are identical. • Remark 4.4.1. We stated Theorem 4.4.1 and Corollary 4.4.1 in terms of L1 and L 00 spaces to underline the role of the Frobenius-Perron operator as a transformation of densities. The same results can be proved using adjoint spaces £P and v' instead of £ 1 and L 00 , respectively. Moreover, when verifying conditions (a) through (c) of Theorem 4.4.1 and Corollary 4.4.1, or conditions (a) and (b) of Proposition 4.4.1, it is not necessary to check for their validity for all I E £P and g E v'. Due to special properties of the operators P and U, which are linear contractions, it is sufficient to check these conditions for f and g belonging to linearly dense subsets of V' and v', respectively (see Section 2.3). 0 Example 4.4.1. In Example 4.2.2 we showed that the rotational transformation S(x) = x + cp (mod 211") is not ergodic when c/J/27r is rational. Here we prove that it is ergodic when r/J/27r is irrational.

It is straightforward to show that S preserves the Borel measure J.£ and the normalized measure J.£/27r. We take as our linearly dense set in v' ([0, 21r])

76


that consisting of the functions {sin kx, cos lx: k, l that, for each function g belonging to this set,

= 0, 1, ... }. We will show

n-1

lim

n-+oo

.!:. "'Ukg(x) = (1,g) n L.J

(4.4.4)

k=O

uniformly for all x, thus implying that condition (a) of Proposition 4.4.1 is satisfied for all f. To simplify the calculations, note that

where i = A. Consequently, it is sufficient to verify (4.4.4) only for g(x) = exp(ikx) with k an arbitrary (not necessarily positive) integer. We have, for k '# 0,

U'g(x)

= g(S'(x)) = eik(z+lt/1)'

so that

n-1

Un(x)

= .!:. L

U1g(x)

n l=O

obeys n-1

Un(x)

1 =n L

eik(z+lt/1)

l=O

1 .

einkt/1 -1

= -e'kz--:-:-:--n eikt/>-1' and llun(x)IIL2

~ nle'kt/1 .1 -

1

{12"' le'kz [einkt/1 -1] 12dx211' }1/2 0

2

~ nleikt/1- 11' Thus un(x) converges in L 2 to zero. Also, however, with our choice of g(x), (1,g)

=

2

1 0

"'

dx eikz_ 21r

1 = -:-[e2wilc _ zk

1] = 0

and condition (a) of Proposition 4.4.1 for ergodicity is satisfied with k '# 0. When k = 0 the calculation is even simpler, since g(x) 1 and thus

=

Un(x)

= 1.


Noting also that

211'

(1, g) =

we have again that

Un (x)

1 0

dx 211"

77

=1,

converges to (1, g).

0

Example 4.4.2. In this example we demonstrate the exactness of the radic transformation S(x) = rx (mod 1}.

From Corollary 4.4.1 it is sufficient to demonstrate that {Pn!} converges strongly to (!, 1) for I in a linearly dense set in .LP([O, 1]}. We take that linearly dense set to be the set of continuous functions. From equation (1.2.13} we have 1 r-1

( ·

ri=O

r

Pl(x) =- L l

~+

= , r )

and thus by induction n-1

pn l(x)

=..!._~I rn ~ (i. rn + -=-) rn · k=O

However, in the limit as n -+ oo, the right-hand side of this equation approaches the Riemann integral of I over [0,1], that is, lim pn l(x)

n-+oo

=

1 1

l(s)ds,

uniformly in x,

0

which, by definition, is just filled. 0

(1, 1). Thus the condition for exactness is ful-

Example 4.4.3. Here we show that the Anosov diffeomorphism S(x,y)

= (x+y,x+2y)

(mod 1}

is mixing. For this, from Proposition 4.4.1, it is sufficient to show that

=

ung(x, Y} g(Sn(x, y)) converges weakly to (1, g) for g in a linearly dense set in V ([0, 1] x [0, 1]). Observe that for g(x,y) periodic in x andy with period 1, g(S(x,y)) = g(x+y,x+2y),g(S 2 (x,y)) = g(2x+3y,3x+5y), and so on. By induction we easily find that

where the an are the Fibonacci numbers given by ao = a1 = 1, an+l an + an_ 1 • Thus, if we take g(x, y) = exp[211"i(kx + ly)J and l(x, y)

= =

78


exp[-211'i(px + qy)), then we have 1 1 (!, ung) = 1 1 exp{211'i[ka2n-2 + la2n-1 - p)x +(ka2n-1 + l~n - q)y]}dx dy, and it is straightforward to show that

(!, U"g)

= {1

if (ka2'!-2 + la2n-1 - p) 0 otherwlSe. Now we show that for large n either

= (ka2n-1 + la2n- q) = 0

ka2n-2 + la2n-1- p or ka2n-1 + la2n- q is different from zero if at least one of k, l, p, q is different from zero. If k = l = 0 but p '::/: 0 or q '::/: 0 this is obvious. We may suppose that either k or l is not zero. Assume k '::/: 0 and that ka2n-2 + la2n-1 - p = 0 for infinitely many n. Thus, ka2n-2 +l- _P_ = O. a2n-1 a2n-1 It is well known [Hardy and Wright, 1959) that . m a2n-2- - = 2- - and l1 n-+oo a2n-1 1 + y'5 hence

[k

lim an= oo, n-+oo

l- _P_]

l

lim (a2n-2) + =~ + n-+oo a2n-1 a2n-1 1 + v'5 . However, this limit can never be zero because k and l are integers. Thus, ka2n-2 + la2n-1- p '::/: 0 for large n. Therefore, for large n,

(!, un ) = { 1 if k = ~ = p = q = 0 g 0 otherwlSe. But (1, g)

=1

1 1 1 exp[211'i(kx + ly))dx dy

={

0 k '::/: 0 or l '::/: 0,

1 k=l=O

so that

(/,1)(1,g)

=1

1 1\1,g)exp[-211'i(px+qy)]dxdy

= { (1, g) 0

={1

if p = q = 0 ifp'::/:0 or q:f:O

ifk=l=p=q=O

0 otherwise.

,~

.

4.5. Kolmogorov Automorphisms

Thus

(1, ung)

79

= {/, 1){1, g)

for large nand, as a consequence, {Ung} converges weakly to {1,g). Therefore, mixing of the Anosov diffeomorphism is demonstrated. 0 In this chapter we have shown how the study of ergodicity, mixing, and exactness for transformations S can be greatly facilitated by the use of the Frobenius-Perron operator P corresponding to S (cf. Theorem 4.4.1 and Corollary 4.4.1). Since the Frobenius-Perron operator is a special type of Markov operator, there is a certain logic to extending the notions of ergodicity, mixing, and exactness for transformations to Markov operators in general. Thus, we close this section with the following definition.

Definition 4.4.1. Let (X,A,JL) be a normalized measure space and P: L1 (X, A,JL)-+ L 1 (X,A,JL) be a Markov operator with stationary density 1, that is, P1 = l.Then we say: (a) The operator P is ergodic if { pn /} is CesAro convergent to 1 for all /ED; {b) The operator P is mixing if { pn!} is weakly convergent to 1 for all fED; and (c) The operator P is exact if { pn!} is strongly convergent to 1 for all /ED.

4.5

Kolmogorov Automorphisms

Until now we have considered three types of transformations exhibiting gradually stronger chaotic properties: ergodicity, mixing, and exactness. This is not a complete list of possible behaviors. These three types are probably the most important, but it is possible to find some intermediate types and some new unexpected connections between them. For example, between ergodic and mixing transformations, there is a class of weakly mixing transformations that, by definition, are measure preserving (on a normalized measure space {X, A, JL)] and satisfy the condition

1

n-1

lim - ~ IJL(A n s-k(B)) - JL(A)JL(B)I n-+oo n L..J

=o

for A,B EA.

lc=O

It is not easy to construct an example of a weakly mixing transformation that is not mixing. Interesting comments on this problem can be found in Brown (1976]. However, Kolmogorov automorphisms, which are invertible and therefore cannot be exact, are stronger than mixing. As we will see later, to

80


some extent they are parallel to exact transformations. Schematically this situation ca be visualized as follows: K -automorphisms

exact

./

'\. mixing

! weakly mixing

! ergodic where K -automorphism is the usual abbreviation for a Kolmogorov automorphism and the arrows indicate that the property above implies the one below. Before giving the precise definition of K -automorphisms, we introduce two simple notations. If S: X-+ X is a given transformation and A is a collection of subsets of X (e.g., au-algebra), then S(A) denotes the collection of sets of the form S(A) for A e A, and s- 1 (A) the collection of s- 1 (A) for A e A. More generally,

n

= 0,±1,±2, ....

Definition 4.5.1. Let (X, A, p.) be a normalized measure space and let S: X-+ X be an invertible transformation such that Sand s- 1 are measurable and measure preserving. The transformation S is called a K -automorphism if there exists au-algebra .Ao c A such that the following three conditions are satisfied:

(i)

s- 1 (.Ao) c .Ao;

(ii) the u-algebra (4.5.1) n=O

is trivial, that is, it contains only sets of measure 1 or 0; and

(iii) the smallest u-algebra containing (4.5.2) is identical to A. The word automorphism comes from algebra and in this case it means that the transformationS is invertible and measure preserving (analogously

4.5 Kolmogorov Automorphisms

81

the word endomorphism is used for measure preserving but not necessarily invertible transformations). Examples 4.5.1. The baker transformation is a. K-a.utomorphism. For .Ao we can take all the sets of the form

.Ao ={Ax [0, 1]: A C [0, 1], A is a. Borel set}. It is easy to verify condition (i) of Definition 4.5.1. Thus, if B =Ax [0, 1], then Bl = s- 1 (B) has the form Bl = Al X [0, 1], where

A1 =~Au(~+ ~A)

(4.5.3)

and thus condition (i) is satisfied. From this follows a. hint of how to prove condition (ii). Namely, from (4.5.3) it follows that the basis A1 of the set B 1 = s- 1 (B) is the union of two sets of equal measure that are contained in the intervals (0, ~] and (~, 1], respectively. Furthermore, set B 2 = 2 (B) has the form A1 x [0, 1] and its basis A2 is the union of four sets of equal measure contained in the intervals [0, l] , ... , [!, 1] . Finally, every set Boo belonging to the u-algebra. (4.5.1) is of the form Aoo x [0, 1] and A 00 has the property that for each integer n the measure of the intersection of Aoo with [k/2n, (k + 1)/2n] does not depend on k. From this it is easy to show that the measure of the intersection of A 00 with [0, x] is a. linear function of x or

s-

1z

1A (y)dy 00

=ex,

where c is a. constant. Differentiation gives for0$x$1. Since 1Aoo is a. characteristic function, either c = 1 or c = 0. In the first case, A 00 as well as Boo have measure 1, and if c = 0, then A 00 and Boo have measure 0. Thus condition (ii) is verified. To verify (iii), observe that .AouS(.Ao) contains not only sets of the form Ax [0, 1] but also the sets of the form A x [0, !] and Ax [!, 1]. Further, .Ao U S(.Ao) U S 2 (.Ao) also contains the sets Ax [0, l], ... , Ax [!, 1] and so on. Thus, by using the sets from the family (4.5.2), we can approximate every rectangle contained in [0, 1] x [0, 1]. Consequently, the smallest ualgebra. containing (4.5.2) is the u-algebra. of Borel sets. 0 Example 4.5.2. The baker transformation considered in the previous example has an important geometrical property. At every point it is contracting in one direction and expanding in the orthogonal one. The transformation S(x,y) = (x+y,x+2y) (mod 1)

82


considered in Example 4.1.4 has the same property. AJJ we have observed the Jacobian of S has two eigenvalues .\11 .\2 such that 0 < .\1 < 1 < .\2. To these eigenvalues correspond the eigenvectors

e1 = (1,! -lv'S) , e2 = (1,! + !v'S) .

e1

e2.

Thus, S contracts in the direction and expands in the direction With this fact it can be verified that S is also a K -automorphism. The and that is, .Ao may be construction of .Ao is related with vectors defined as a u-algebra generated by a class of rectangles with sides parallel to vectors 6 and e2. The precise definition of .Ao requires some technical details, which can be found in an article by Arnold and Avez [1968]. 0 AJJ we observed in Remark 4.1.4, the first coordinate in the baker transformation is transformed independently of the second, which is the dyadic transformation. The baker transformation is a K -automorphism and the dyadic is exact. This fact is not a coincidence. It may be shown that every exact transformation is, in some sense, a restriction of a K -automorphism. To make this statement precise we need the following definition.

el

e2i

;,

Definition 4.5.2. Let (X,A,J.£) and (Y,B,v) be two normalized measure spaces and let S: X -+ X and T: Y -+ Y be two measure-preserving transformations. If there exists a transformation F: Y -+ X that is also measure preserving, namely,

v(F- 1 (A))

= J.£(A)

for A E A

and such that S o F = F o T, then S is called a factor of T. The situation described by Definition 4.5.2 can be visualized by the diagram

y

...!...

y (4.5.4}

X

__!..X

and the condition SoF = FoT may be expressed by saying that the diagram (4.5.4} commutes. We have the following theorem due to Rochlin [1961]. Theorem 4.5.1. Every exact transformation is a factor of a K -automorphism. The relationship between K -automorphisms and mixing transformations is much simpler; it is given by the following theorem. Theorem 4.5.2. Every K -automorphism is mixing.

Exercises

83

The proofs and more information concerning K -automorphisms can be found in the books by Walters [1982] and by Parry [1981].

Exercises 4.1. Study the rotation on the circle transformation (Examples 4.2.2 and 4.4.1) numerically. Is the behavior a consequence of the properties of the transformation or of the computer? Why? 4.2. Write a series of programs, analogous to those you wrote in the exercises of Chapter 1, to study the behavior of tw 0 and K. < 1 such that for every IE D there is an integer no(/) for which

L

pn l(x)Jt(dx) $

K.

for n;:;:: no(/)

and

~t(E) $

6.

(5.3.1)

Note that for every density I the integral in inequality (5.3.1) is bounded above by one. Thus condition (5.3.1) for constrictiveness means that eventually [n;:;:: no(/)] this integral cannot be close to one for sufficiently small sets E. This clearly indicates that constrictiveness rules out the possibility that pn I is eventually concentrated on a set of very small or vanishing measure. H the space X is not finite, we wish to have a definition of constrictiveness that also prevents pn I from being dispersed throughout the entire space. To accomplish this we extend Definition 5.3.1.

Definition 5.3.2. Let (X,A,Jt) be a (u-finite) measure space. A Markov operator Pis called constrictive if there exists 6 > 0, and K. < 1, and a measurable set B of finite measure, such that for every density I there is an integer no (f) for which

f

pn l(x)~t(dx) $

K.

for n;:;:: no(/)

and

~t(E) $

6.

(5.3.2)

j(X\B)UE

Clearly this definition reduces to that of Definition 5.3.1 when X is finite and we take X = B.

Remark 5.3.1. Observe that in inequality (5.3.2) we may always assume that E c B. To see this, take F = BnE. Then (X\B)UE = (X\B)UF and, as a consequence,

r

j(X\B)UE

pn l(x)~t(dx) =

r

j(X\B)UF

pn l(x)~t(dx),

96

5. The Asymptotic Properties of Densities

~~~~~~~~~~~~~~--x

FIGURE 5.3.1. Graph showing convergence of a sequence of functions {/n} to a set :F, where the hatched region contains all possible functions drawn from :F. (See Example 5.3.1 for details.)

and J.&(F) 5 J.&(E). From the definition, one might think that verifying constrictiveness is difficult since it is required to find two constants 6 and K. as well as a set B with rather specific properties. However, it is often rather easy to verify constrictiveness using one of the two following propositions.

Proposition 5.3.1. Let (X,A,J.£) be a finite measure space andP: L1 (X)-+ L 1 (X) be a Markov operator. Assume there is a p > 1 and K > 0 such that for every density f E D we have pn f E V' for sufficiently large n, and lim sup IIPn /IILJ• 5 K.

(5.3.3)

n-+oo

Then P is constrictive. Proof. From (5.3.3) there is an integer n 0 (/) such that IIPn /IILP ~ K

+1

for n 2: no(/).

Thus, by criteria 2 of Remark 5.1.3 the family {Pn /},for n 2: no(/), fED, is weakly precompact. Finally, for a fixed f E {0, 1), criteria 3 of the same remark implies there is a 6 > 0 such that

l

pn f(x)J.&(dx)
.i (/) '# 0 if i is contained in the invariant subset I, and >.i (f) = 0 otherwise. Thus the limit of An! as n -+ oo is not a constant function with respect to x, so that P cannot be ergodic. This is a contradiction; hence, if P is ergodic, {a(i)} must be a cyclical permutation. • Mixing and Exactness

Theorem 5.5.2. Let (X, A, p.) be a normalized measure space and P: L 1 -+ L 1 a constrictive Markov operator. Ifr = 1 in representation {5.3.8) for P, then P is exact.

Proof. The proof is simple. Assume r

= 1, so by {5.4.2) we have

pn+l f(x) = A(/)1x{x) + Qnf(x)

and, thus, lim pn+l f = A(/)1x.

n-+oo

In particular, when f E D then A(/) = 1 since P preserves the norm. Hence, for all f E D, {pn f} converges strongly to 1, and P is therefore exact (and, of course, also mixing). • The converse is surprising, for we can prove that P mixing implies that r = 1.

Theorem 5.5.3. Again let {X, A, p.) be a normalized measure space and P: L 1 -+ £1 a constrictive Markov operator. If P is mixing, then r = 1 in representation (5.9.8). Proof. To see this, assume P is mixing but that r > 1 and take an initial fED given by where c1 = 1/p.(A1). Therefore pn f(x) = C11A(n)(x),

where A(n) = Aa"( 1)· Since P was assumed to be mixing, {pn f} converges weakly to 1. However, note that

5.6. Asymptotic Stability of {P"}

105

Hence { pn!} will converge weakly to 1 only if an(1) = 1 for all sufficiently large n. Since a is a cyclical permutation, r cannot be greater than 1, thus demonstrating that r = 1. • Remark 5.5.1. It is somewhat surprising that in this case P mixing implies P exact. D Remark 5.5.2. Observe that, except for the remainder Qn/, pn+l f behaves like permutations for which the notions of ergodicity, mixing, and exactness are quite simple. D

5.6 Asymptotic Stability of {Pn} Our considerations of ergodicity, mixing and exactness for Markov operators in the previous section were based on the assumption that we were working with a normalized measure space (X, A, f.£). We now turn to a more general situation and take (X,A,f..') to be an arbitrary measure space. We show how Theorem 5.3.1 allows us to obtain a most interesting result concerning the asymptotic stability of {pn!}. We first present a generalization for Markov operators of the concept of exactness for Frobenius-Perron operators associated with a transformation. Definition 5.6.1. Let (X,A,f..') be a measure space and P:L 1 -+ L 1 a Markov operator. Then {pn} is said to be asymptotically stable if there exists a unique /. E D such that P /. = /. and lim

n-+oo

IIPnf- /.II = 0

for every f E D.

(5.6.1)

When Pis a Frobenius-Perron operator, the following definition holds. Definition 5.6.2. Let (X, A, f..') be a measure space and P: L 1 -+ L 1 the Frobenius-Perron operator corresponding to a nonsingular transformation S:X-+ X. If {Pn} is asymptotically stable, then the transformationS is said to be statistically stable. The following theorem is a direct consequence of Theorem 5.3.1. Theorem 5.6.1. Let P be a constrictive Markov operator. Assume there is a set A c X of nonzero measure, f..'( A) > 0, with the property that for every fED there is an integer no(/) such that pnf~)>O

~A~

for almost all x E A and all n > no(/). Then {pn} is asymptotically stable. Proof. Since, by assumption, P is constrictive, representation (5.3.8) is valid. We will first show that r = 1.

106


Assume r > 1, and choose an integer io such that A is not contained in the support of 9io. Take a density I E D of the form I (x) = 9io (x) and let T be the period of the permutation a. Then we have pn-r l(x) = 9i0 (x).

Clearly, pn-r l(x) is not positive on the set A since A is not contained in the support of 9io· This result contradicts (5.6.2) of the theorem and, thus, we must have r = 1. Since r = 1, equation (5.3.10) reduces to pn+l l(x)

= >.(f)g(x) + Qnl(x)

so lim pn I

n ..... oo

= >.(f)g.

If I E D, then limn..... oo pn I E D also; therefore, by integrating over X we have 1 = >.(!). Thus, limn..... oo pn I = g for all I E D and {pn} is asymptotically stable; this finishes the proof. • The disadvantage with this theorem is that it requires checking for two different criteria: (i) that Pis constrictive and {ii) the existence of the set A. It is interesting that, by a slight modification of the assumption that pn I is positive on a set A, we can completely eliminate the necessity of assuming P to be constrictive. To do this, we first introduce the notion of a lower-bound function.

Definition 5.6.3. A function h E £ 1 is a lower-bound function for a Markov operator P: £ 1 --+ £ 1 if for every IE D.

(5.6.3)

Condition {5.6.3) may be rewritten as

where IIEnll

--+

0 as n--+ oo or, even more explicitly, as

Thus, figuratively speaking, a lower.,.bound function h is one such that, for every density I, successive iterates of that density by P are eventually almost everywhere above h. It is, of course, clear that any nonpositive function is a lower-bound function, but, since I E D and thus pn I E D and all densities are positive,

5.6. Asymptotic Stability of {P"}

107

a negative lower bound function is of no interest. Thus we give a second definition.

Definition 5.6.4. A lower-bound function his called nontrivial if h;::: 0 and llhll > 0. Having introduced the concept of nontrivial lower-bound functions, we can now state the following theorem. Theorem 5.6.2. Let P: L 1 -+ L 1 be a Markov operator. { pn} is asymptotically stable if and only if there is a nontrivial lower bound function for

P. Proof. The "only if" part is obvious since (5.6.1) implies (5.6.3) with h = / •. The proof of the "ir' part is not so direct, and will be done in two steps. We first show that lim

n-+oo

IIP"(h - 12)11 = 0

(5.6.4)

for every h, 12 E D and then proceed to construct the function / •. Step I. For every pair of densities fi, 12 E D, the IIP,.(h - 12)11 is a decreasing function of n. To see this, note that, since every Markov operator is contractive, IlP/II ~II/II and, as a consequence,

Now set g = h

- 12 and note that, since /I, 12 ED, c = llg+ll

= IIY-11 = ~11911·

Assume c > 0. We have g = g+ - g- and

IIP,.gll = cii(P,.(g+ /c)- h)- (P,.(g- /c)- h) II.

(5.6.5)

Since g+ fc and g- fc belong to D, by equation (5.6.3), there must exist an integer n1 such that for all n ;::: n1

and

II(P"(g- /c)- h)_ II~ ~llhll. Now we wish to establish upper bounds for IIP,.(g+ /c) - hll and llpn(g- /c)- hll. To do this, first note that, for any pair of nonnegative real numbers a and b, la-bl =a-b+2(a-b)-.

108


Next write

IIPn(g+ /c)- hll

= =

L

IPn(g+ fc)(x)- h(x)IJt(dx)

L

pn(g+ fc)(x)p(dx) -

+2

fx

L

h(x)p(dx)

(Pn(g+ fc)(x)- h(x))- p(dx}

= IIPn(g+ /c}ll-llhll + 2II(Pn(g+ /c)- h}-11 :5 1-llhll + 2 ·lllhll

= 1- !llhll

for n? n1.

Analogously, IIPn(g- /c)- hll :51- !llhll

for n? n1.

Thus equation (5.6.5) gives IIPngll :5 ciiPn(g+ /c)- hll :5 c(2 -llhll}

+ ciiPn(g- /c)- hll

= IIYII (1- !llhll)

for n? n1.

(5.6.6}

From (5.6.6), for any 11, he D, we can find an integer n 1 such that

By applying the same argument to the pair pn 1 11, pn 1 h we may find a second integer n2 such that 11Pn 1 +n 2 (11- h)ll :511Pn1 (11- h)ll (1- !llhll) 2 :51111- hll (1- !llhll) . After k repetitions of this procedure, we have 11Pn1 + ... +n"(/l- h)ll :511/t- hll (1- !llhll)lc' and since llpn(l1- h)ll is a decreasing function of n, this implies (5.6.4). Step II. To complete the proof, we construct a maximal lower-bound function for P. Thus, let p

= sup{llhll: his a lower-bound function for P}.

Since by assumption there is a nontrivial h, we must have 0 < p :5 1. Observe that for any two lower-bound functions h 1 and h2, the function h = max(h 11 h 2) is also a lower-bound function. To see this, note that

Choose a sequence {h;} of lower-bound functions such that llh;ll-+ p. Replacing, if necessary, h; by max(h 1 •.• , h; ), we can construct an increasing

5.6. Asymptotic Stability of {pn}

109

sequence {hj} of lower functions, which will always have a limit (finite or infinite). This limiting function

h.= .lim hi J-+00

is also a lower-bound function since

and, by the Lebesgue monotone convergence theorem, as j-+ oo. Now the limiting function h. is also the maximal lower function. To see this, note that for any other lower function h, the function max( h, h.) is also a lower function and that

II max(h, h.)ll ~ P = I! h. II, which implies h ~ h •. Observe that, since (PJ)-

~

Pf-, for every m and n(n > m},

which implies that, for every m, the function pm h. is a lower function. Thus, since h. is the maximal lower function, pm h. ~ h. and since pm preserves the integral, pmh. = h•. Thus the function /. = h./llh.ll is a density satisfying P /. = ! •. Finally, by equation (5.6.4), we have for fED, which automatically gives equation (5.6.1}. • In checking for the conditions of Theorem 5.6.2 it is once again sufficient to demonstrate that (5.6.3} holds for densities f drawn from a dense set Doc D(X). Remark 5.6.1. Before continuing, it is interesting to point out the connection between Theorems 5.3.1 and 5.6.2 concerning asymptotic periodicity and asymptotic stability. Namely, from the spectral decomposition Theorem 5.3.1 we can actually shorten the proof of asymptotic stability in Theorem 5.6.2. To show this, assume P satisfies the lower-bound function condition (5.6.3). Pick an f E D and choose a number no(!) such that

for n ;?: no(!).

110


From Ia- bl =a- b+ 2{a- b)- we have II(P"/- h)+ll::; IIPn/- hll::; IIP"/11-IIhll +2II{Pn/- h)-11, and since liP" /II= 1, equation (5.6.7) gives II(P"/- h)+ II::; 1-lllhll

for n ~ no(/).

Thus, by Proposition 5.3.2 we know that the operator P is constrictive. Since Pis constrictive it satisfies Theorem 5.3.1 and in particular we have the decomposition formula (5.3.8). Using the assumed existence of a lowerbound function, h we will show that r = 1 by necessity. Assume the contrary and take r ~ 2. Consider two basis functions 9 1 and 92 in the decomposition (5.3.8). From P9, = 9a(i) we obviously have pnm9i = 9i for m = rl and an arbitrary n. However, from (5.6.3) it also follows that pnm9i ~ h- E~m' i = 1,2, so 9i ~ h - E~m for i = 1, 2. This then implies that 9192 > 0, which contradicts the orthogonality of the 9i required by Theorem 5.3.1. We are thus led to a contradiction and therefore must haver = 1. Thus (5.3.8) implies the asymptotic stability of {Pn} with/.= 91 • Hence, by the expedient of using Theorem 5.3.1 we have been able to considerably shorten the proof of Theorem 5.6.2. The results of Theorem 5.6.2 with respect to the uniqueness of stationary densities for asymptotically stable Markov operators may be generalized by the following observation.

Proposition 5.6.1. Let (X, A, J.l.) be a measure space and P: L 1 -+ L1 a Markov operator. If {Pn} is asymptotically stable and/. is the unique stationary density of P, then for every normalized f E L 1 (11/ll = 1) the condition (5.6.7) Pf=f implies that either f

= /.

or f

= -/•.

Proof. From Proposition 3.1.3, equation (5.6.7) implies ~that both j+ and are fixed points of P. Assume 11/+11 > 0, so that f = j+ /11/+11 is a density and Pj =f. Uniqueness of/. implies j =/.,hence

1-

which must also hold for II j+ II = 0. In an analogous fashion,

so that Since 11/11 =

I= J+- 1- = 11/.11, we have lal =

(11/+11-11/-11)/.

=a/•.

1, and the proof is complete.

•

5.6. Asymptotic Stability of {Pn}

111

Before closing this section we state and prove a result that draws the connection between statistical stability and exactness when P is a FrobeniusPerron operator.

Proposition 5.6.2. Let (X, A,~) be a measure space, 8: X -+X a nonsingular transformation such that 8(A) E A for A E A, and P the FrobeniusPemJn operator corresponding to 8. If 8 is statistically stable and/. is the density of the unique invariant measure, then the transformation 8 with the measure forAEA is exact.

Proof. From Theorem 4.1.1 it follows immediately that

~f.

is invariant.

Thus, it only remains to prove the exactness. Assume ~ J, (A) > 0 and define

!A(x)

= [1/~J.(A))/.(x)1A(x)

for

X

EX.

Clearly, fA E D(X,A,~) and lim rn = lim IIPnfA

n-+oo

From the definition

~J.(SR(A)) = f

of~~.,

j Sn(A)

n-+oo

-/.11 = 0.

we have

/.(x)~(dx) ~ f

j Sn(A)

pn!A(x)~(dx)- rn.

(5.6.8)

By Proposition 3.2.1, we know that pn fA is supported on 8n(A), so that

[

Jsn(A)

pn /A(x)~(dx)

= [

Jx

pn /A(x)~(dx)

= 1.

Substituting this result into (5.6.8) and taking the limit as n-+ oo gives

hence 8: X

-+

X is exact by definition.

•

Remark 5.6.1. In the most general case, Proposition 5.6.2 is not invertible; that is, statistical stability of 8 implies the existence of a unique invariant measure and exactness, but not vice versa. Lin [1971) has shown that the inverse implication is true when the initial measure ~ is invariant. 0

112


5. 7 Markov Operators Defined by a Stochastic Kernel As a sequel to Section 5.6, we wish to develop some important consequences of Theorems 5.6.1 and 5.6.2. Let (X, A, JJ) be a measure space and K: X x X -+ R be a measurable function that satisfies

0$ K(x,y)

L

and

K(x,y)dx

(5.7.1)

[dx = JJ(dx)].

= 1,

(5.7.2)

Any function K satisfying (5.7.1) and (5.7.2) is called a stochastic kernel. Further, we define an integral operator P by

Pf(x)

=

L

K(x,y)f(y)dy

(5.7.3)

The operator P is clearly linear and nonnegative. Since we also have

L

LL =L L

Pf(x)dx =

dx

K(x,y)f(y)dy

f(y) dy

K(x, y) dx

=

L

f(y) dy,

Pis therefore a Markov operator. In the special case that X is a finite set and JJ is a counting measure, we have a Markov chain and P is a stochastic matrix. Now consider two Markov operators Pa and .P, and their corresponding stochastic kernels, Ka and Kb. Clearly, PaP, is also a Markov operator, and we wish to know how its kernel is related to Ka and Kb. Thus, write

(Pa.P,)f(x)

= Pa(.P,f)(x) = =

L

Ka(x, z)(.P,f(z)) dz

L z}{ L L{L L Ka(x,

Kb(z, y)f(y)dy} dz

Ka(x, z)Kb(z, y)dz} /(y) dy.

Then PaP, is also an integral operator with the kernel

K(x,y) =

Ka(x,z)Kb(z,y) dz.

(5.7.4}

We denote this composed kernel K by

K=Ka*Kb and note that the composition has the properties:

(5.7.5}

5. 7. Markov Operators Defined by a Stochastic Kernel (i) Ko. * (Kb * Kc)

113

= (Ko. * Kb) * Kc (associative law); and

(ii) Any kernel formed by the composition of stochastic kernels is stochas-

tic. However, in general kernels KA and Kb do not commute, that is, Ko.*Kb =I: Kb* K 0 • Note that the foregoing operation of composition definition is just a generalization of matrix multiplication. Now we are in a position to show that Theorem 5.6.2 can be applied to operators P defined by stochastic kernels and, in fact, gives a simple sufficient condition for the asymptotic stability of {pn}.

Corollary 5.7.1. Let (X,A,JL) be a measure space, K:X x X - R a stochastic kernel, that is, K satisfies (5. 7.1) and (5. 7.2}, and P the corresponding Markov operator defined by (5. 7.9}. Denote by Kn the kernel co1T'esponding to pn. If, for some m,

lxf inf Km(x, y) dx > 0,

(5.7.6)

11

then { pn} is asymptotically stable.

Proof. By the definition of Kn, for every f E D(X) we have pnf(x)

=[

Kn(x,y)f(y)dy.

Furthermore, from the associative property of the composition of kernels, Kn+m(x,y) = [

Km(x,z)Kn(z,y)dz,

so that

pn+m f(x)

=[

Kn+m(X, y)f(y) dy

=[ {[

Km(x,z)Kn(z,y)dz }f(y)dy.

If we set h(x) = inf Km(x, y), 11

then pn+m f(x) '2:. h(x) [

{[

Kn(z, y)dz} /(y) dy

= h(x) [!(y)dy

114


since Kn is a stochastic kernel. Furthermore, since

[

IE D(X),

l(y)dy = 1,

and, therefore, pn+m l(x) ~ h(x)

I

for n ~ 1,

E

D(X).

Thus ~

forn

m+ 1,

which implies that (5.6.3) holds, and we have finished the proof. • In the case that X is a finite set and K is a stochastic matrix, this result is equivalent to one originally obtained by Markov. Although condition (5.7.6) on the kernel is quite simple, it is seldom satisfied when K(x, y) is defined on an unbounded space. For example, in Section 8.9 we discuss the evolution of densities under the operation of a Markov operator defined by the kernel [cf. equation (8.9.6)]

K(x y) '

= { -e11Ei(-y) ' -e11Ei(-x),

where

0<x 0,

and the same holds for all of its iterates Km(x, y). A similar problem occurs with the kernel K(x, y) = g(ax +by), where b # 0 and g is an integrable function defined on R or even on n+ (cf. Example 5.7.2). In these and other cases where condition (5.7.6) is not satisfied, an alternative approach, reminiscent of the stability methods developed by Liapunov, offers a way to examine the asymptotic properties of iterates of densities by Markov operators. Let G be an unbounded measurable subset of a d-dimensional Euclidian space Rd, G c ~, and K: G x G -+ R a measurable stochastic kernel. We will call any measurable nonnegative function V: G -+ R satisfying lim V(x)

lzl->oo

a Liapunov function.

= oo

(5.7.8)

5.7. Markov Operators Defined by a Stochastic Kernel

115

Next, we introduce the Chebyshev inequality through the following proposition.

Proposition 5.7.1. Let (X,A,JL) be a measure space, V:X-+ Ran arbitrary nonnegative measurable function, and for all f E D set E(VIf) = [

V(x)f(x)JL(dx).

If

Ga.= {x: V(x) 0

(5.7.10)

R such that

fooo V(X)f(x)dx+{J

0 :5

Q

< 1, {J 2:: 0 (5.7.11)

for f E D, then { pn} is asymptotically stable.

Remark 5. 7.1. Before giving the proof, we note that sometimes instead of verifying inequality (5.7.11) it is sufficient to check the simpler condition

fa

K(x, y)V(x) dx :5 aV(y) + {J,

(5.7.11a)

116


since (5.7.11a) implies (5.7.11). To see this, note that from (5.7.1la)

L

V(x)Pf(x)dx=

~

LL

V(x)K(x,y)f(y)dxdy

L

[aV(y) + PJ/(y) dy =a

L

V(y)f(y) dy + {3.

Proof. First define the function

En(VIf)

=LV(x)Pnf(x}dx

(5.7.12)

that can be thought of as the expected value of V(x) with respect to the density pn f(x). From (5.7.11) we have directly

En(VIf) ~ aEn-t(VIf) + {3.

(5.7.13}

By an induction argument, it is easy to show that from this equation we obtain En(VIf) ~ [{3/(1- a))+ anEo(VIf). Even though Eo(VIf) is clearly dependent on our initial choice of/, it is equally clear that, for every f such that

Eo(VIf) < oo, there is some integer

(5.7.14}

no= n 0 (/) such that

En(VIf) ~ [{3/(1- a))+ 1

for all n ~no.

(5.7.15}

Now let

Ga

= {x E G: V(x) 1 + [{3/(1- a)], then

En(VIf) a

~! a

(1 +

___L) 0

for n

~no.

(5.7.17)


117

Since V(x) __. oo as lxl __. oo there is an r > 0 such that V(x) > a for lxl > r. Thus the set Ga is entirely contained in the ball lxl :5 r, and we may write

pn+lj(x) = { K(x,y)Pnf(y)dy

la

~

inf K(x,y) {

}Ga

uEGa

~

{

laa

K(x,y)Pnf(y)dy

pnf(y)dy

~

inf K(x,y) { pnf(y)dy lul:5r · Jaa ~ e inf K(x,y) lul:5r

(5.7.18)

for all n ~ no. By setting

h(x) =

inf K(x,y) lu1:5r in inequality {5.7.18) we have, by assumption (5.7.10), that f

llhll > 0. Finally, because of the continuity of V, the set Do c D of all f such that (5.7.14) is satisfied is dense in D. Thus all the conditions of Theorem 5.6.2 are satisfied. • Another important property of Markov operators defined by a stochastic kernel is that they may generate an asymptotically periodic sequence { pn} for every f E D. This may happen if condition (5.7.10) on the kernel is replaced by a different one.

Theorem 5. 7.2. Let K: G x G __. R be a stochastic kernel and P be the corresponding Markov opemtor. Assume that there is a nonnegative A< 1 such that for every bounded B C G there is a o = o(B) > 0 for which

L

K(x,y) dx :5 A

for p,(E)

< o,

y E B,

E

c

B.

(5.7.19)

Assume further there exists a Liapunov function V: G __. R such that (5. 7.11) holds. Then P is constrictive. Consequently, for every f e L 1 the sequence {pn} is asymptotically periodic. Proof. Again consider En {VI!) defined by (5.7.12). Using condition (5.7.11) we once more obtain inequality (5.7.15). Thus by the Chebyshev inequality, with Ga defined as in the proof of Theorem 5. 7.1,

{

jG\Ga

pn f(x) dx = 1- {

laa

:5

pn f(x) dx :5 En(VIf) a

!a (1 + -L) 1-a

for n

~ no(!).

118

Set

5. The Asymptotic Properties of Densities E

= 1(1- .X). Choosing a sufficiently large a that satisfies

a~!E

(1+_L), a 1-

we have

f

P"f(x)dx 5:

for n 2: no(/).

E

(5.7.20)

lc\G .. Consequently, from (5.7.19) we have

f

P"f(x)dx5,

j(G\G,.)uE

f

5: E +

+1

pn- 1 f(y) dy

L

K(x, y) dx

pn- 1 f(y)dy1 K(x,y)dx G\G,. G

G,.

pn- 1 f(y)dy

Using (5.7.19) and (5.7.20) applied to B

f

f

)E

K(x, y) dx.

= Ga we finally have pn- 1 f(y) dy

P" f(x) dx 5:

2E

+.X

5,

2E

+ .X = 1 -

j(G\G,.)UE

P"f(x)dx

)E

L

5: E + 1

f

f

P"f(x)dx+

)G\G,.

Jc .. E

for n ~ no(/) + 1.

Thus, inequality (5.3.2) in Definition 5.3.2 of constrictiveness is satisfied. A simple application of Theorem 5.3.1 completes the proof. • Before passing to some examples of the application of Theorem 5. 7.1 and 5.7.2, we give two simple results concerning the eventual behavior of {P"} when P is a Markov operator defined by a stochastic kernel.

Theorem 5. 7 .3. If there exists an integer m and a g E L 1 such that Km(x,y) 5: g(x), where Km(x,y) is the mth iterate of a stochastic kernel, then the sequence {P"} with P defined by (5.7.3) is asymptotically periodic. Proof. Since Km(x,y)

P" f(x) =

fx

5: g(x) we have

Km(x, y)pn-m f(y) dy '5 g(x)

for n 2: m.

Set h = g and take .X = 0 so by Proposition 5.3.2 the sequence {P"} is asymptotically periodic. •


119

A slight restriction on Km(x,y) in Theorem 5.7.3 leads to a different result, as given in the next result. Theorem 5. 7.4. If there exists an integer m and a g E L 1 such that

Km(x,y) ::5 g(x), where Km(x, y) is the mth iterate of a stochastic kernel, and there is a set A c X with I'(A) > 0 such that 0 < Km(x,y)

forx E A, y E X,

then the sequence {pn} is asymptotically stable. Proof. The proof is a trivial consequence of the constrictiveness of P from Theorem 5.7.3, the assumptions, and Theorem 5.6.1. Example 5.7.1. To see the power of Theorem 5.7.1, we first consider the case where the kernel K(x, y) is given by the exponential integrals in equation (5.7.7). It is easy to show that -eY(-Ei(y)) is decreasing and consequently

inf K(x,y)

O:Sy:Sr

~

min{-Ei{-x),-erEi(-r)}

> 0.

Furthermore, taking V(x) = x, we have, after integration,

fooo xK(x, y) dx = !(1 + y). Therefore it is clear that V(x) =xis a Liapunov function for this system Also, observe that with f(x) = exp( -x), we have when a= f3 =

l·

Pf(x)

=

1

00

K(x,y)e- 11 dy

= e-z.

Thus, the limiting density attained by repeated application of the Markov operator Pis f.(x) = exp( -x). 0 Example 5.7.2. As a second example, let g:R positive function satisfying

L:

g(x)dx = 1 and

m1

=

L:

-+

R be a continuous

lxlg(x)dx

< oo.

Further, let a stochastic kernel be defined by

K(x, y)

= lalg(ax +by),

Ia! > lbl, b =F 0

120


and consider the corresponding Markov operator

Pf(x) Let V(x) =

=

1:

K(x,y)f(y)dy.

lxl, so that we have

/_: K(x, y)V(x) dx = lal /_: lxlg(ax +by) dx =

1:

g(s) Is~ by Ids

~ i:g(s)l~lds+ i:g(s)~~~ds= ~j +1~1· Thus, when a= lb/al and {3 = mtflal, it is clear that V(x) satisfies condition (5.7.11) and hence Theorem 5.7.1 is satisfied . .AJJ will become evident in Section 10.5, in this example P f has the following interesting probabilistic interpretation. If { and f7 are two independent random variables with densities f(x) and g(x), respectively, then

Pf(x)

'h a Wlt

= lal /_: g(ax + by)f(y) dy,

= -C21

andb

Ct = --, C2

is the density ofthe random variable (c1{ +c2t7) [cf. equation (10.1.8)].

D

Example 5. 7 .3. As a final example of the applicability of the results of this section, we consider a simple model for the cell cycle [Lasota and Mackey, 1984]. First, it is assumed that there exists an intracellular substance (mitogen), necessary for mitosis and that the rate of change of mitogen is governed by dm

dt

=g(m},

m(O}

=r

with solution m(r,t). The rate g is a C 1 function on [O,oo) and g(x) > 0 for x > 0. Second, it is assumed that the probability of mitosis in the interval [t, t +At] is given by (m(t))At + u(At), where 4> is a nonnegative function such that q(x) = (x)fg(x) is locally integrable (that is, integrable on bounded sets [0, c]) and satisfies lim Q(x)

Z-+00

= oo,

where Q(x)

=

1z

q(y)dy.

(5.7.21)

Finally, it is assumed that at mitosis each daughter cell receives exactly one-half of the mitogen present in the mother cell. Under these assumptions it can be shown that for a distribution fn-t(x) of mitogen in the (n - 1)st generation of a large population of cells, the mitogen distribution in the following generation is given by

1

00

fn(x)

=

K(x,r)fn-t(r)dr,


121

where

o K(x,r) =

x E [o, !r) x E [lr,oo).

tz q(y)dy]

{ 2q(2x)exp [-

(5.7.22)

It is straightforward to show that K(x,r) satisfies (5.7.1) and (5.7.2) and is, thUB, a stochastic kernel. Hence the operator P: L 1 ( R+) -+ L 1 ( R+) defined by

fooo K(x,r)f(r)dr

Pf(x) =

(5.7.23)

is a Markov operator. To show that there is a unique stationary density /.ED to which {pnf} converges strongly, we use Theorem 5.7.1 under the assumption that

lim inf[Q(2x)- Q(x)] > 1.

(5.7.24)

:I: -tOO

First we consider the integral

1

00

I=

u(Q(2x))Pf(x) dx,

(5.7.25)

where u is a continuoUB nonnegative function. Using equations (5.7.21) through (5.7.23) we can rewrite (5.7.25) as follows: 2

00

I= 21 u(Q(2x))q(2x) dx 1 z exp[Q(y)- Q(2x)]f(y) dy 00 = u(Q(z))q(z) dz exp[Q(y)- Q(z)Jf(y) dy 00 = f(y) dy ioo u(Q(z)) exp[Q(y)- Q(z)]q(z) dz.

1

1z

1

Setting Q(z)-Q(y)

1

= x so q(z)dz = dx we finally obtain the useful equality

1

u(Q(2x))Pf(x) dx =

1

00

00

00

f(y) dy

Note in particular from (5.7.26) that for u(z)

1

1

u(x + Q(y))e-zdx.

=1

(5.7.26)

we have

00

00

Pf(x) dx =

f(y) dy,

which also proves that P is a Markov operator. Now take u(x) = e€z with 0 < £ ~ 1, and V(x) (5. 7.26) it therefore follows that

fooo V(x)Pf(x) dx =

1

1

= u(Q(2x)).

From

00

00

f(y)e£Q('II)dy

e-(l-£)zdx

=1- 100 f(y)eEQ('II)dy.

1-£

0

(5.7.27)

122


Now pick a p > 1 and xo

~

0 such that

Q(2y)- Q(y)

~

p

for y

~

xo.

Then we can write {5.7.27) as

oo V(x)PJ(x)dx ~ - 1 1zo J(y)e'-Q(y)dy + - 1 J(y)e'-Q( l-Eo 1-Ezo 1 1 e-ep 1oo V(y)J(y)dy. ~ -eeQ(zo) + e-'-P

00

211 >dy

0

1-E

1-E

0

e-ep

For the function

a(E)=1-E we have a{O) = 1 and a'{O) = 1- p < 0. Thus for some E > 0 we have a(E) < 1. Take such an E set

1 {3 = --eeQ(zo) 1-E With these values of a and {3 we have shown that the operator P defined by (5.7.22)-{5.7.23) satisfies inequality (5.7.11) of Theorem 5.7.1 under the assumption of (5.7.23). It only remains to be shown that K satisfies (5.7.10). Let ro ~ 0 be an arbitrary finite real number. Consider K(x, r) for 0 ~ r ~ ro and x ~ !r. Then a= a( E),

2

K{x,r)=2q{2x)exp[-1 z q(y)dy]

~ 2q{2x)exp [-

L 2

z q(y)dy]

for 0

~ r ~ ro,x ~ !r

and, as a consequence,

inf K(x,r) > h(x)

O$r$ro

-

= {O2q{2x) exp [- Jof 2z q(y)dy]

for x

< !ro

for x ~ !ro.

Further,

h

1~~

2

00

00 /

h(x) dx =

2q(2x) exp [- [

=exp [-foro q(y)dy]

h

:1:

q(y) dy] dx

> 0;

hence K (x, r) satisfies (5. 7.10). Thus, in this simple model for cell division, we know that there is a globally asymptotically stable distribution of mitogen. Generalizations of this model have appeared in the work of Tyson and Hannsgen [1986], Tyrcha [1988], and Lasota, Mackey, and Tyrcha [1992]. 0

5.8. Existence of Lower-Bound Functions

123

5.8 Conditions for the Existence of Lower-Bound Functions The consequences of the theorems of this chapter for the Frobenius-Perron operator are so far-reaching that an entire theory of invariant measures for a large class of transformations on the interval [0,1], and even on manifolds, may be constructed. This forms the subject of Chapter 6. In this last section, we develop some simple criteria for the existence of lower-bound functions that will be of use in our specific examples of the next chapter. Our first criteria for the existence of a lower bound function will be formulated in the special case when X = (a, b) is an interval on the real line [(a, b) bounded or not) with the usual Borel measure. We will use some standard notions from the theory of differential inequalities [Szarski, 1967]. A function f: (a, b) -+ R is called lower semicontinuous if J~ inf f(x

+ 6) ~

for x E (a, b).

f(x)

It is left lower semicontinuous if J~ inf f(x- 6) ~ f(x)

for x E (a, b).

6>0

For any function/: (a, b) setting d+J(x) X

-+

R, we define its right lower derivative by

= 6-+0 lim inf ~[/(x + 6) -f(x)] u

for x E (a, b).

6>0

It is well known that every left lower semicontinuous function f: (a, b) satisfying

-+

R,

for x E (a, b), is nonincreasing on (a, b). (The same is true for functions defined on a

half-closed interval [a, b).) For every f E Do that is a dense subset of D (Definition 5.6.5) write the trajectory pn I as (5.8.1) for n ~ no(/). Then we have the following proposition. Proposition 5.8.1. Let P:L 1 ((a,b))-+ L 1 ((a,b)) be a Markov operator. Assume that there exists a nonnegative function g E L 1 ((a, b)) and a constant k ~ 0 such that for every f E Do the function In in (5.8.1) are left lower semicontinuous and satisfy the following conditions:

/n(x)

~ g(x)

a. e. in (a, b)

(5.8.2)

124


for all x E (a, b). Then there exists an intenJal ~ a lower function for pn.

c

(a, b) and an e > 0 such that h

(5.8.3)

= ElA

is

Proof. Let xo < x 1 < x 2 be chosen in (a, b) such that

1

Zl

g(x)dx

a

< l and

1b :1:2

g(x)ch

< l·

(5.8.4)

Set

M Since

= l exp[-k(x2- xo)].

llpn /II= 1, condition (5.8.1) implies

1b

(5.8.5)

ln(x) ch = 1.

Now we are going to show that h = El(zo,z 1 ) is a lower function. Suppose it is not. Then there is n';::: no andy E (xo,xt) such that ln•(y) < h(y) =e. By integrating inequality (5.8.3), we obtain for Furthermore, since

X E

[y, X2].

(5.8.6)

In• :::; g, we have

bln•(x)ch5, 1:1:1 g(x)ch+ 1:1:2 g(x)ch+ 1b g(x)ch.

1 a

a

z1

z2

Finally, by applying inequalities (5.8.4) and (5.8.6), we obtain

1b

ln•(x)dx 5:

l + (x2- y)(e/4M) + l:::; !,

which contradicts equation (5.8.5).

•

Remark 5.8.1. In the proof of Proposition 5.8.1, the left lower semicontinuity of In and inequality (5.8.3) were only used to obtain the evaluation for x;::: y. Therefore Proposition 5.8.1 remains true under this condition; for example, it is true if all In are nonincreasing. 0 It is obvious that in Proposition 5.8.1 we can replace (5.8.3) by d-lnfdx;::: -kln and assume In right lower continuous (or assume In nondecreasing;

5.9. Sweeping

125

cf. Remark 5.8.1}. In the case of a bounded interval, we may omit condition (5.8.2} and replace (5.8.3} by a two-sided inequality. This observation is summarized as follows. Proposition 5.8.2. Let (a, b) denote a bounded interoal and let P: L 1 ( (a, b)) -+ L1 ( (a, b)) be a Markov operator. Assume that for each f E Do the functions In in (5.8.1} are differentiable and satisfy the inequality

~df~x) I::; kfn(x},

for all x E (a, b),

(5.8.7)

where k ~ 0 is a constant independent of f. Then there exists an e > 0 such that h = d(a,b) is a lower-bound function. Proof. As in the preceding proof, we have equation (5.8.5}. Set €

= (1/2(b- a)Je-k(b-a).

Now it is easy to show that In ~ h for n ~ no. H not, then /n•(y) < e for some y E (a, b) and n' ~ n 0 • Consequently, by (5.8. 7), fn•(x)::; /n•(y)ekl:z:-yl ::; [1/2(b- a)). This evidently contradicts (5.8.5). The inequality fn proof. •

~

h completes the

5.9 Sweeping Until now we have considered the situation in which the sequence {Pn} either converges to a unique density (asymptotic stability) or approaches a set spanned by a finite number of densities (asymptotic periodicity) for every initial density f. In this section we consider quite a different property in which the densities are dispersed under the action of a Markov operator P. We call this new behavior sweeping, and introduce the concept through two definitions and several examples. Our first definition is as follows. Definition 5.9.1. Let (X, A, J.t) be a measure space and A. c A be a subfamily of the family of measurable sets. Also let P: L1 (X)-+ L 1 (X) be a Markov operator. Then {pn} is said to be sweeping with respect to A. if lim

f

n--+oo}A

pnj(x)J.t(dx)

Since every element two densities

f

=0

for every

f

E

D and A E A •.

(5.9.1)

E L 1 can be written as a linear combination of

for j, ED, for a sweeping operator P, condition (5.9.1) also holds for

f

E L 1.

126


In particular examples, it is sufficient to verify condition (5.9.1) for f E Do, where Do is an arbitrary dense subset of D. That this is so follows immediately from the inequality

L f(x)~S(dx) L fo(x)~S(dx) +II/pn

$;

/oil,

pn

for fED, fo E Do,

(5.9.2) and the fact that both terms on the right-hand side of (5.9.2) can be made arbitrarily small.

Example 5.9.1. Let X= Rand IS be the standard Borel measure. Further, let

Pf(x) = f(x- r),

for jeD

so

pn /(x) = f(x- nr),

(5.9.3)

for fED.

With r > 0 the sequence { pn} is sweeping with respect to the family of intervals .Ao = {( -oo, c): c E R}. To prove this, note that for every

l

c

-oo

pnj(x)dx =

~c -oo

f

E D with compact support we have

f(x- nr)dx =

~c-nr -oo

J(y)dy.

Thus the integral on the right-hand side will eventually become zero since ( -oo, c - nr] n supp f

=0

for sufficiently large n. In an analogous fashion we can also prove that for r < 0 the sequence {Pn}, where P is given by (5.9.3), is sweeping with respect to the family of intervals A1

= {[c, oo): c E R}.

0

Example 5.9.2. Again take X= Rand IS to be the Borel measure. Further, let P be an integral operator with Gaussian kernel

Pj(x)

= ~2 v27ro·

1

00

-oo

exp [- (x;

u

:>

2 ]

f(y)dy.

It is easy to show (see also Example 7.4.1 and Remark 7.9.1) that 2

pnj(x) =

~ /_: exp [ 1 Z--+00

was sufficient to guarantee the asymptotic stability of {pn}. In this example we will show that for the same system (5.7.22)-(5.7.23) the condition limsup[Q(2x)- Q(x)]

. Furthermore u(x)::::; e-Ez for all x, so

1

00

W(y)::::;

exp{-x- E[x + Q(y)- Q(2y)]}dx.

Since, by assumption, Q(2y)- Q(y)::::; p this can also rewritten as

W(y) ::::;

roo e-E(a:-p)-a:dx =

lo

eEP

= o:2(E).

1+E

It is clear that o:2(0) = 1 and that o:~(O) = p- 1 < 0. Thus, there must be an E > 0 such that o: 2(E) < 1. Chose an E such that o:(E) < 1 and define a= min(o:l(E),o: 2(E)). Then W(y)::::; a< 1 for ally 2:: 0 and from (5.9.9) we have

1

1

00

00

V(x)PI(x) dx::::; a

V(x)l(x) dx

for allIED.

Thus by proposition 5.9.1 we have shown that the cell cycle model defined by equations (5.7.21)-(5.7.23) is characterized by a sweeping Markov operator when (5.9.8) holds.

5.10 The Foguel Alternative and Sweeping From Example 5.9.3 it is clear that the demonstration of sweeping is neither necessarily straightforward nor trivial and may, in fact, require a rather

130


strong effort. In this section we present a sufficient condition for sweeping that is sometimes especially helpful in the study of integral Markov operators with stochastic kernels. Let (X, .A, IJ.) be a measure space and P: L 1 (x) - t L 1 (X) be the operator Pf(x)

=

fx

(5.10.1)

K(x,y)f(Y)IJ.(dy)

where K is a stochastic kernel and thus satisfies conditions (5.7.1) and (5.7.2). We have already shown in Section 5.7 that Pis a Markov operator and hence defined for all f e L 1 . However, the right-hand side of (5.10.1) is well defined for every measurable f;?: 0 even though it may, of course, be infinite for some x. With this observation we make the following definitions.

Definition 5.10.1. Let P: L 1 - L 1 be the integral Markov operator (5.10.1) and let f: X - R be a measurable and nonnegative function. We say that f is subinvariant if Pf(x) 5: f(x)

for x eX a.e.

Definition 5.10.2. Let a subfamily .A. c .A be fixed. We say that .A. is regular if there is a sequence of sets An e .A., n = 0, 1 ... , such that 00

UAn=X.

(5.10.2)

n=O

Definition 5.10.3. A nonnegative measurable function cally integrable if

Lf(x)~J.(dx)

< oo

f: X

-t

R is lo-

for A E .A•.

With these definitions we state the following result which will be referred to as the Foguel alternative.

Theorem 5.10.1. Let (X, .A, IJ.) be a measure space and .A. E .A a regular family. Assume that P: L 1 - t L 1 is an integral operator with a stochastic kernel. If P has a locally integrable and positive (f > 0 a. e.) subinvariant function f, then either P has an invariant density or {P"} is sweeping. In the statement of this theorem, there are two implications:

(1) if {P"} is not sweeping, then P has an invariant density; and (2) if {P"} is sweeping, then P has no invariant density.

5.10. The Foguel Alternative and Sweeping

131

Only the first part is hard to prove, and the second part can be demonstrated using condition (5.10.2). To prove the second implication, suppose that {pn} is sweeping and that /. = PJ. is an invariant density. Further define

Then, according to (5.10.2), lim

f

/.(x)p(dx) =

lc-+oo}B,.

f /.(x)p(dx) = Jx

1,

and in particular for some fixed k

(5.10.3) On the other hand, since/.= Pf.,

1

1c

/.(x)p(dx) $

B,.

L i=l

1

1c

/.(x)p(dx)

Ac

=L i=l

1

pn j.(x)p(dx).

Ac

Since {pn} is sweeping by assumption, the right-hand side of this relation converges to zero. This, however, contradicts (5.10.3) and we thus conclude that {pn} is not sweeping. Remark 5.10.1. This theorem was proved by Komorowski and Tyrcha (1989] and the assumptions concerning the regular family A. simplified by Malczak [1992]. Similar theorems when A. is the family of all measurable subsets have been proved by several authors; see Foguel [1966] and Lin [1971].

Example 5.10.1. Let X = R+, and consider the integral operator for x 2:: 0, where '1/J: [0, 1]

--+

(5.10.4)

R is a given integrable function such that

1 1

t/J(z) 2:: 0 and

t/J(z)dz = 1.

(5.10.5)

The operator (5.10.4) appears on the right-hand side of the ChandrasekharMiinch equation describing the fluctuations in the brightness of the Milky Way. This equation will be discussed in Examples 7.9.2 and 11.10.2. Here we are going to study the properties of the operator (5.10.4) alone.

132


Let V: R+ have

-+

R be a nonnegative measurable function. For

f

E

D we

oo 1oo 1oo (X) dy 1 V(x)Pf(x) dx = V(x) dx 1/J y f(y)y z

0

0

1oo f(y)dy 111 1/J (~) V(x)~.

=

Substituting xfy = z this becomes

1

1 1 1

00

00

V(x)Pf(x) dx =

f(y)dy

'1/J(z)V(zy) dz.

(5.10.6)

=1 gives

This equality with V(x)

1

1ooo Pf(x)dx = 1o

00

f(y)dy { '1/J(z)dz

·

lo

=

foo f(y)dy

lo

which, together with the nonnegativity of 1/J, implies that (5.10.4) defines a Markov operator. Now set ffj(x) = x-fj in (5.10.4). Then

(X)

00

=

P/[J(x)

For

1

1/J -

Y

z

dy fJ+ Y 1

= {j1

11

,P(z)zfJ- 1 dz.

(5.10.7)

0

X

/3 ~ 1 we have 1/J(z)zP- 1 ~ 1/J(z) and, as a consequence, for x

~

0.

Thus, by Theorem 5.10.1, the operator P defined by (5.10.4) is either sweeping to zero or has an invariant density. It is easy to exclude the possibility that P has an invariant density. Suppose that there is an invariant density / •. Then the equality (5.10.6) gives

1 1

00

or

1

=

1 1

00

V(y)f.(y) dy

/.(y) dy

1

'1/J(z)V(zy) dz,

1

00

/.(y) dy

,P(z)[V(y)- V(zy)] dz

= 0.

(5.10.8)

Now take V: [O,oo) -+ R to be positive, bounded, and strictly increasing [e.g., V(z) = z/(1 + z)].Then V(y)- V(zy)

and the integral I(y)

=

>0

1

for y

> 0, 0 ~ z

1

,P(z)[V(y)- V(zy)] dz

~

1,


133

is strictly positive for y > 0. Consequently, the product /.(y)I(y) is a nonnegative and nonvanishing function. This shows that the equality {5.10.8} is not satisfied, and thus there is no invariant density for P. Thus, for every '1/J satisfying (5.10.5) the operator P given by equation {5.10.4} is sweeping. This is both interesting and surprising since we will show in Section 11.10 that the stochastic semigroup generated by the Chandrasekhar-Miinch equation is asymptotically stable! D The alternative formulated in Theorem 5.10.1 does not specify the behavior of the sequence {pn} in the case when an invariant density exists. We now formulate a stronger form of the Foguel alternative, first introducing the notion of an expanding operator.

Definition 5.10.4. Let (X,A,p) be a measure space and P:L 1 a Markov operator. We say that P is expanding if

=0

lim p(A - suppPn f)

n-+oo

for

f

E D and p(A) < oo.

-

L 1 be {5.10.9}

The simplest example of an expanding operator is an integral operator with a strictly positive stochastic kernel. In fact, from equation (5.7.3) with K(x,y) > 0 it follows that pnf(x) > 0 for all x EX and n ~ 1. In this case, supp pn f = X and condition (5.10.9) is automatically satisfied. A more sophisticated example of an expanding operator is given by {>.(a:)

Pf(x)

= la

K(x,y)f(y)dy,

(5.10.10}

where K(x, y) is a measurable kernel satisfying

K(x,y) > 0

for a< y < A(x), a< x,

(5.10.11}

and A: [a, oo) - [a, oo) is a continuous strictly increasing function such that

A(x) > x

fora< x.

(5.10.12}

A straightforward calculation shows that P is a Markov operator on L 1{[a, oo)) when

K(x, y) dx = 1

foo

for y

>a.

(5.10.13}

1>.-l(y)

We also have the following.

Proposition 5.10.1. If K and A satisfy conditions (5.10.11)-(5.10.13}, then the Markov operator P: L 1 {[a, oo)) - t L 1 {[a, oo)) defined by (5.10.10) is expanding. Proof. Let

f

E D be given and let

xo

= essinf{x:f(x) > 0}.

134


This means that x 0 is the largest possible real number satisfying l'{supp f

n [0, x 0]) =

0.

Further, let x1 = .x-1(xo). It is evident from the defining equation {5.10.10) that Pf(x) > 0 for .X{{x) > xo or x > x1. Define Xn = .,x-n(xo). By an induction argument it is easy to verify that pn f(x) > 0 for x > Xn· Thus, for an arbitrary measurable set A c [a, oo) we have

I'( A - supp pn /) $

Xn -

a.

(5.10.14)

The sequence {xn} is bounded from below (xn ~a). It is also decreasing since Xn .x- 1{Xn-1) $ Xn-1· Thus {xn} is convergent to a number x. ~ a. Since .X{xn) = Xn-b in the limit as n- oo we have .X{x.) = x •. From inequality {5.10.12) it follows that x. = a, which according to (5.10.14) shows that P is expanding. • For expanding operators, the Foguel alternative can be formulated as follows.

=

Theorem 5.10.2. Let (X,A,J£) be a measure space and A. C A be a regular family of measurable sets. Assume that P: L 1 (X) - L 1 (X) is an expanding integral operator with a stochastic kernel. If P has a locally integrable posititJe (! > 0 a. e.) subintJariant function, then either {pn} is asymptotically stable or it is sweeping. The proof can be found in Malczak {1992). Theorem 5.10.2 can be derived from a new criterion for asymptotic stability given by Baron and Lasota {1993). See Exercise 5.8.

Example 5.10.2. We return to the modeling of the cell cycle (see Example 5. 7.3) by considering the following model proposed by Tyson and Hannsgen

(1986). They assume that the probability of cell division depends on cell size m, so cell size plays the role of the mitogen considered in Example 5.7.3. It is further assumed that during the lifetime of the cell, growth proceeds exponentially, that is, dm -=km. dt When the size is smaller than a given value, which for simplicity is denoted by 1, the cell cannot divide. When the size is larger than 1, the cell must traverse two phases A and B. The end of phase B coincides with cell division. The duration of phase B is constant and is denoted by Ts. The length TA of phase A is a random variable with the exponential distribution prob(TA ~ t)

= e-pt.

At cell division the two daughter cells have sizes exactly one-half that of the mother cell.


135

Using these assumptions it can be shown that the process of the replication of size may be described by the equation

fn+l(x)

= Pfn(x) = [ tT

/tT

K(x,r)fn(r)dr,

(5.10.15)

where fn is the density function of the distribution of the initial size in the nth generation of cells, and the kernel K is given by

_ K(x,r)-

; (kp)u(x)-1-{1'/lc) ( p) (x)-1-(p/lc) '-/A:) {- r\P ku

u

foru$r o.

It can be proved (Baron and Lasota, 1993) that {Pn} is asymptotically stable for every integral operator with stochastic kernel which overlaps supports and has an invariant positive (a.e.) density. Using this result and Theorem 5.10.1, prove Theorem 5.10.2.

6 The Behavior of Transformations on Intervals and Manifolds

This chapter is devoted to a series of examples of transformations on intervals and manifolds whose asymptotic behavior can be explored through the use of the material developed in Chapter 5. Although results are often stated in terms of the asymptotic stability of {Pn}, where Pis a FrobeniusPerron operator corresponding to a transformationS, remember that, according to Proposition 5.6.2, S is exact when {pn} is asymptotically stable and S is measure preserving. In applying the results of Chapter 5, in several examples we will have occasion to calculate the variation of a function. Thus the first section presents an exposition of the properties of' functions of bounded variation.

6.1

Functions of Bounded Variation

There are a number of descriptors of the "average" behavior of a function f: [a, b] -+ R. Two of the most common are the mean value of/, m(f)

= b ~a

1b

f{x) dx,

and its variance, D 2 (/) = m((f -m(/)) 2 ). However, these are not always satisfactory. Consider, for example, the sequence of functions {/n} with fn(x) = sin2mrx, n = 1,2, .... They have the same mean value on [0,1], namely, m(/n) = 0 and the same variance D 2 (/n) = ~; but they behave quite differently for n :» 1 than they do for n = 1. To describe these

140

6. The Behavior of Transformations on Intervals and Manifolds

kinds of differences in the behavior of functions, it is useful to introduce the variation of a function (sometimes called the total variation). Let I be a real-valued function defined on an interval A c R and let [a, b] be a subinterval of A. Consider a partition of [a, b] given by a

=

Xo

and write

< X1 < · · · < Xn = b

(6.1.1)

n

Bn{f) =

L: ll(xi) - l(xi-1)1.

{6.1.2)

i=1

If all possible sums sn{!), corresponding to all subdivisions of [a, b], are bounded by a number that does not depend on the subdivision, I is said to be of bounded variation on [a, b]. Further, the smallest number c such that Bn :$ c for all Bn is called the variation of I on [a, b] and is denoted by I. Notationally this is written as

v!

b

VI= SUPBn{!),

(6.1.3)

a

where the supremum is taken over all possible partitions of the form (6.1.1). Consider a simple example. Assume that I is a monotonic function, either decreasing or increasing. Then

ll(xi) - l(xi-1)1 where 8

_ {1 -1

-

= 8[1(xi) -

l(xi-1)],

for I increasing for I decreasing

and, consequently, n

Bn{f) = 8 L[l(xi) - I(Xi-d] i=1 = 8[1(xn)- l(xo)] = ll(b)- l(a)l. Thus, any function that is defined and monotonic on a closed interval is of bounded variation. It is interesting (the proof is not difficult) that any function I of bounded variation can be written in the form I = h + h, where It is increasing and h is decreasing. Variation of the Sum Let I and g be of bounded variation on [a, b]. Then

6.1. Functions of Bounded Variation

141

and, consequently, b Bn(/

b

+g) $ Bn(/) + Bn(g) :5 VI+ V g. 0

0

Thus (! + g) is of bounded variation and b

b

b

VU+g):SVI+Vg. a

0

0

If It, ... , In are of bounded variation on [a, b], then by an induction argument b

(Vl)

b

b

VU1 +···+In) :5 V It+···+ V In 0

0

(6.1.4)

0

follows immediately. Variation on the Union of Intervals Assume that a < b < c and that the function I is of bounded variation on [a, b]as well as on [b, c). Consider a partition of the intervals [a, b] and [b, c), a

= Xo < X1 < · · · < Xn = b = Yo < Y1 < · · · < Ym = c

(6.1.5)

and the corresponding sums n

Bn

(!) =

[o,b)

L ll(xi)- l(xi-1)1, i=1 m

Bm

(f)=

(b,c)

L

i=1

II(Yi)- I(Yi-1)1.

It is evident that the partitions (6.1.5) jointly give a partition of [a,c]. Therefore, Bn

(f) +

(o,b)

Bm

(f)

= Bn+m (f)

(b,c)

(6.1.6)

[o,c)

where the right-hand side of equation (6.1.6) denotes the sum corresponding to the variation of I over [a, c]. Observe that (6.1.6) holds only for partitions of [a, c] that contain the point b. However, any additional point in the sum Bn can only increase Bm but, since we are interested in the supremum, this is irrelevant. From equation (6.1.6) it follows that b

c

c

VI+VI=VJ. 0

b

0

142

6. The Behavior of 'Iransformatioos on Intervals and Manifolds

Again by an induction argument the last formula may be generalized to 41

where ao < a 1 i = l, ... ,n.

On

Gn

VI+ .. ·+ V I= V/, ao

(V2)

< · · · < an and f

(6.1.7)

is of bounded variation on [ai-l,ai],

Variation of the Composition of Functions Now let g: [a, P] - [a, b] be monotonically increasing or decreasing on the interval [a, P] and let f: [a, b] - R be given. Then the composition fog is well defined and, for any partition of [a, P],

a

= uo < u1 < · · · < Un = P;

(6.1.8)

the corresponding sum is n

Bn(/ o g) = ~ 1/(g(ui))- /(g(ui-1))1. i=l

Observe that, due to the monotonicity of g, the points g(ui) define a partition of [a, b]. Thus, Bn (f o g) is a particular sum for the variation of f and, therefore, b

Bn(/og) $

Vf a

for any partition (6.1.8). Consequently, {3

b

a

a

VJog$V f.

(V3)

(6.1.9)

Variation of the Product Let f be of bounded variation on [a, b] and let g be 0 1 on [a, b]. To evaluate the variation of the product f(x)g(x), x E [a, b], start from the well-known Abel equality, n

n

~ laibi- ai-l bi-ll=~ lbi(ai- ai-l)+ ~-l(bi- bi-1)1. ~1

~1

Applying this equality to the sum [substituting~ = /(xi) and bi n

Bn(/g) = ~ 1/(xi)g(xi) - /(Xi-l)g(Xi-1)1, i=l

= g(xi)]

6.1. Functions of Bounded Variation

143

the immediate result is n

Bn(/g) $;

L {lg(xi)lll(xi) -I(Xi-1)1 + II(Xi-1)llg(xi)- g(Xi-1)1}. i=1

Now, by applying the mean value theorem, we have n

Bn(/9) $; (sup lgl)sn(/) +

L

ll(xi-1)g'(xi)l(xi- Xi-1)

i=1

b

$;(sup lgl) VI+ a

n

L ll(xi-1)g'(xi)l(xi- Xi-1)· i=l

with Xi E (xi-1! Xi)· Observe that the last term is simply an approximating sum for the Riemann integral of the product ll(x)g'(x)l. Thus the function l(x)g(x) is of bounded variation and b

(V4)

b

b

V 19 $; (sup lgl) VI+ 111(x)g'(x)l dx. a

Taking in particular

a

{6.1.10)

a

I= 1,

y b

(V4')

b

g $;

Llg'(x)l dx.

{6.1.11)

However, in this case, the left- and right-hand sides are strictly equal since Bn (g) is a Riemann sum for the integral of g'. Yorke Inequality

Now let I be defined on [0, 1) and be of bounded variation on [a, b] c [0, 1]. We want to evaluate the variation of the product of I and the characteristic function 1[a,b)· Without any loss of generality, assume that the partitions of the interval [0, 1] will always contain the points a and b. Then Bn

(0,1)

(/1[a,bJ) $; Bn (/) [a,b)

+ ll(a)l + ll(b)l.

Let c be an arbitrary point in [a, b]. Then, from the preceding inequality,

Bn

(0,1)

(/1[a,bJ) $; Bn (/) (a,b)

+ ll(b) -I{c) I+ II(c) -l(a)l + 2ll{c)l

b

$; 2 VI+ 2ll(c)j. a

144


It is always possible to choose the point c such that ll(c)l5 b so that Sn

~a 1b ll(x)l dx

b

2 (/1[a,bJ) 52 VI+~

(0,1)

1b lf(x)l dx,

a a

a

which gives ·(V5)

b 2 1b ll(x)l dx. V l1(a,b) 5 2 VI+ -_b 0 a 1

a

6.2

(6.1.12)

a

Piecewise Monotonic Mappings

Two of the most important results responsible for stimulating interest in transformations on intervals of the real line were obtained by R.enyi (1957) and by Rochlin (1964). Both were considering two classes of mappings, namely, {6.2.1) S(x) = r(x) (mod 1),0 5 x 51, where r: [0, 1] --+ [0, oo) is a C 2 function such that infz r' and r(1) is an integer; and the R.enyi transformation S(x)

= rx

(mod 1),0 5 x 51,

> 1, r(O) = 0, (6.2.2)

where r > 1, is a real constant. (The r-adic transformation considered earlier is clearly a special case of the R.enyi transformation.) Using a numbertheoretic argument, Renyi was able to prove the existence of a unique invariant measure for such transformations. Rochlin was able to prove that the Renyi transformations on a measure space with the Renyi measure were, in fact, exact. In this section we unify and generalize the results of Renyi and Rochlin through the use of Theorem 5.6.2. Consider a mapping S: [0, 1] --+ [0, 1] that satisfies the following four properties: (2i) There is a partition 0 = ao < a 1 < · · · < ar = 1 of [0, 1] such that for each integer i = 1, ... , r the restriction of S to the interval [ai-l, ai) is a C 2 function; {2ii) S(ai-l) = 0 fori= 1, ... , r; (2iii) There is a A> 1 such that S'(x) ~A for 0 5 x denote the right derivatives]; and

< 1 [S'(ai) and S"(ai)

6.2. Piecewise Monotonic Mappings

145

s

FIGURE 6.2.1. FUnction S(x) = 3x + ~ sin(7z/4) (mod 1) as an example of a transformation on [0, 1] satisfying the conditions (2i)-(2iv). In this case r = 3, and the counterimage of the set [0, x] consists of the union of the three intervals indicated as heavy lines along the x-axis.

(2iv) There is a real finite constant c such that - S"(z)/[S'(x)] 2 ~ c,

O~x c)..j (>.. - 1). Then (/n)~ $

kfn

{6.2.6)

for n sufficiently large, say n ~ no{!), and thus condition {5.8.3) of Proposition 5.8.1 is satisfied. We now show that the fn are bounded and hence satisfy condition {5.8.2) of Proposition 5.8.1. First note that from equation (6.2.4) we may write r

fn+l(x)

= L,u;(x)fn(Ui(x)). i=l

Thus, since g~ $ 1/>.. and S(ai-1) = 0 fori= 1, ... , r,

1

1

fn+l {0) $ :\ fn{O) + :\

r

L fn(ai_l).

{6.2.7)

i=2

From {6.2.6) it follows that

fn(ai) $ fn(x)elc, so that 1 ~loa' fn(x) dx

~ e-lc fn(ai)ai,

for i = 1, ... , r.

Thus fn(ai) $ elcjai, and from {6.2.7) we have

fn+t{O) $ {1/>..)fn(O) + Lj>.., where

for n

~

no{/),

r

L = :~:::::>lc /ai-l· i=2

Again, using a simple induction argument, it follows that for n

~

no{!),

so

fn(O) $ 1 + (Lj(>..- 1)] for sufficiently large n, say n ~ n1 {!). By using this relation in conjunction with the differential inequality {6.2.6), we therefore obtain

fn(x) $ {1 + [Lj(>.. -1)J}elc,

for 0 $ x

< 1, n

~ n1.

{6.2.8)

Thus, by inequalities (6.2.6) and {6.2.8), all the conditions of Proposition 5.8.1 are satisfied and {pn} is asymptotically stable by Theorem 5.6.2. •

148


s

FIGURE 6.2.2. A piecewise monotonic transformation satisfying the conditions of Theorem 6.2.2.

Theorem 6.2.1 is valid only for mappings that are monotonically increasing on each subinterval [ai_ 1,ai) of the partition of (0, 1]. However, by modification of some of the foregoing properties {2i)-{2iv), we may also prove another theorem valid for transformations that are either monotonically increasing or decreasing on the subintervals of the partition. The disadvantage is that the mapping must be onto for every (ai-l, ai)· We now consider a mapping S: [0, 1] -+ (0, 1] that satisfies a condition slightly different from property {2i): {2i)' There is a partition 0 = a0 < a 1 < · · · < ar = 1 of (0, 1] such that for each integer i = 1, ... , r the restriction of S to the interval (ai-l! ai) is a C 2 function; as well as (2ii)' S((ai-l,ai))

= (0, 1), that is, Sis onto;

(2iii)' There is a A> 1 such that IS'(x)l ~A, for x =f ai, i

= 0, ... ,r; and

(2iv)' There is a real finite constant c such that

IS"(x)I/[S'(x)] 2 5 c,

for x =f ai,i

= 0, ... ,r.

(6.2.9)

(See Figure 6.2.2 for an example.) Then we have the following theorem.

Theorem 6.2.2. If S: (0, 1] -+ (0, 1] satisfies the preceding conditions (2i)'(2iv)' and Pis the Frobenius-Pemm operator associated with S, then {Pn} is asymptotically stable. Proof. The proof proceeds much as for Theorem 6.2.1. Using the same


149

notation as before, it is easy to show that for x E [0, 1],

8- 1 ((0,x))

= U c>.j(>.- 1) such that l/~1 ~ kfn

(6.2.12)

150


Xj+J

0

FIGURE 6.2.3. Successive maxima in the variable z(t) from the Lorenz equations are labeled :z:,, and one maximum is plotted against the previous (z,+l vs. :z:,) after rescaling so that all :z;, E (0, 1).

for sufficiently large n [sa.y n fied. •

~

n 0 (f)], a.nd thus Proposition 5.8.2 is satis-

Example 6.2.1. When u = 10, b = 8/3, a.nd r = 28, then a.ll three variables x, y, a.nd z in the Lorenz [1963] equations,

dx -dt =yz-bx,

-dy = -xz+rz-y' dt

dz dt

= u(y- z),

show very complicated dynamics. If we label successive maxima. in x( t) a.s Xi (i = 0, 1, ... ), plot each maximum against the previous maximum (i.e., Xi+l vs. Xi), a.nd scale the results so that the Xi a.re contained in the interval [0, 1], then the numerical computations show that the points (x;, Xi+l) a.re located approximately on the graph of a. one-dimensional mapping, a.s shown in Figure 6.2.3. AB an approximation to this mapping of one maximum to the next, we ca.n consider the transformation

S(x)

=

{

(~==~X

for x E [0, !] (6.2.13)

(2- a)(1- x) 1-a(1-x)

for x E

(!, 1],

where a= 1-e, shown in Figure 6.2.4 fore= 0.01. Clearly, S(O) = 8(1) = 0, S{!) = 1, a.nd, since S'(x) = (2- a)/(1- ax) 2 , we will always have IS'(x)l > 1 for x E [0, !) if e > 0. Fina.lly, since S"(x) = 2a(2-a)/(1-ax?, IS"(x)l is always bounded above. For x E (!, 1] the calculations a.re similar. Thus the transformation (6.2.13) satisfies a.ll the requirements of Theorem 6.2.2: {pn} is a.symptotica.lly stable a.nd S is exact. 0


151

s

0

FIGURE 6.2.4. The transformation S(z) given by equation (6.2.13) with£= 0.01 as an approximation to the data of Figure 6.2.3.

Remark 6.2.1. The condition that IS'(z)l > 1 in Theorem 6.2.2 is essential for S to be exact. We could easily demonstrate this by using (6.2.13) with e = 0, thus making IS'(O)I = 18'(1)1 = 1. However, even if IS'(z)l = 1 for only one point z E [0, 1), it is sufficient to destroy the exactness, as can be demonstrated by the transformation S(z)

= { z/(1- z) 2x -1

for z E [~, ~] for x E ( 2 , 1] ,

(6.2.14)

which we originally considered in Section 1.3 (paradox of the weak repellor). Now, the condition IS'(x)l > 1 is violated only at the single point x = 0, and, for any IE £ 1 , the sequence {P"!} converges to zero on (0, 1]. Thus, the only solution to the equation PI = I is the trivial solution I 0, and therefore there is no measure invariant under S. This is quite difficult to prove. First write the Frobenius-Perron operator corresponding to S as

=

(6.2.15) Set qn(x) = xln(x), where In = pn lo, and pick the initial density to be lo = 1. Thus q0 (x) = x, and from (6.2.15) we have the recursive formula,

1 ( +X

qn+l(x) = 1 + x qn

1

X

)

(1

X %) + 1 +X qn 2 + 2 '

(6.2.16)

Proceeding inductively, it is easy to prove that q~(x) ~ 0 for all n, so that the functions qn(x) are all positive and increasing. From equation (6.2.16)

152


we have which shows that exists. Write Zo

= 1 and Zlc+t = z~c/{1 + z~c). Then from (6.2.16) we have

Qn+t(z~c) =

1:

Zlc Qn(Zic+t)

+

1

:lcZic Qn

(~ + ~).

Take k to be fixed and assume that limn,_, 00 Qn(x) = Co for Zlc ::::; x ::::; 1 {which is certainly true fork= 0). Since Zlc ::::; + !z~c, taking the limit as n -+ oo, we have

!

Zlc Co= - -1 - 1"1m Qn (Zlc+t ) + - --eo, 1 + Zlc n-+oo 1 + Zlc

so limn..... oo Qn(Zic+l) =Co· Since the functions Qn(x) are increasing, we know that limn-+oo Qn(x) = Co for all x E [z1c+1, 1]. By induction it follows that limn.....oo Qn(x) = Co in any interval [z~c, 1] and, since lim~c.... 00 Zlc = 0, we have limn,_, 00 Qn(x) =Co for all x E {0, 1]. Thus lim fn(x) = eo/x. n-+oo Actually, the limit Co is zero; to show this, assume Co must exist some e > 0 such that

1 1

lim n-+oo

~

¥-

0. Then there

1 1

fn(x) dx

=

~

(eo/x) dx > 1.

However, this is impossible since 11/n II = 1 for every n. By induction, each of the functions f n (x) is decreasing, so the convergence of f n ( x) to zero is uniform on any interval [e, 1] where e > 0. Now, let f be an arbitrary function, and write f = J+- 1-. Given 6 > 0, consider a constant h such that

1 1

Thus, since

1 1

1 1

u-- h)+dx +

u+- h)+dx::::; 6.

IP"' /I : : ; P"'l/1 = P"' J+ + pn f-, we have

1 = 21 1 1

IP"' fldx ::::;

1 +1 1

P"' j+ dx + 1

1

P"'1dx+6

1 1

1

P"'hdx

:::;2h

P"' f- dx P"'(J+- h)dx +

P"'(r- h)dx

6.3. Piecewise Convex Transformations with a Strong Repellor

153

and, since {pn1} converges uniformly to zero on [e, 1], we have fore> 0. Hence the sequence {pn /} converges to zero in L 1 {[e, 1]) norm for every e > 0 and equation PI = I cannot have a solution I E L 1 except I 0. 0

=

6.3 Piecewise Convex Transformations with a Strong Repellor Although the theorems of the preceding section were moderately easy to prove using the techniques of Chapter 5, the conditions that transformation 8 must satisfy are highly restrictive. Thus, in specific cases of interest, it may often not be the case that S' {x) > 1 or IS' {x) I > 1, or that condition {6.2.3) or (6.2.9) is obeyed. However, for a class of convex transformations, it is known that {pn} is asymptotically stable. Consider 8: [0, 1] __. [0, 1] having the following properties: {3i) There is a partition 0 = ao < a1 < · · · < ar = 1 of [0, 1] such that for each integer i = 1, ... , r the restriction of 8 to [ai-l! ai) is a C 2 function; {3ii) S'(x) > 0 and S"{x) right derivatives]; (3iii) For each integer i

~

0 for all x E (0, 1), (S'(ai) and S"(ai) are

= 1, ... , r, S(ai-1) = 0; and

(3iv) S'(O) > 1. An example of a mapping satisfying these criteria is shown in Figure 6.3.1. Remark 6.3.1. Property (3iv) implies that point x = 0 is a strong repellor (see also Section 1.3 and Remark 6.2.1), that is, trajectory {S{x0 ), S 2 (x0 ), ••• }, starting from a point xo E {0, a1), will eventually leave [0, a1). To see this, note that as long as sn(xo) E (0, a1) there is a eE (0, al) such that sn(x0 ) = S(sn- 1 (xo))- 8{0) = S'(e)sn- 1 (xo) ~ .>.sn- 1 (xo),

where ). = 8'(0). By an induction argument, sn(xo) ~ ).Rxo and, since >. > 1, sn(x0 ) must eventually exceed a 1 • After leaving the interval [O,al)

154


s

0

FIGURE 6.3.1. An example of a piecewise convex transformation satisfying the conditions of Theorem 6.3.1.

the trajectory will, in general, exhibit very complicated behavior. If at some point it returns to [0, a1), then it will, again, eventually leave [0, a1). 0 With these comments in mind, we can state the following theorem.

Theorem 6.3.1. Let S: [0, 1] -+ [0, 1] be a transformation satisfying the foregoing conditions {3i}-{3iv), and let P be the Frobenius-Perron operator associated with S. Then { pn} is asymptotically stable. Proof. The complete proof of this theorem, which may be found in Lasota and Yorke [1982], is long and requires some technical details we have not introduced. Rather than give the full proof, here we show only that { pn /} is bounded above, thus implying that there is a measure invariant under S. We first derive the Frobenius-Perron operator. For any x E [0, 1] we have

s- 1 ([0,x]) = where

gi(x)

1

= { s~) (x) a,

r

Ulai-l,g,(x)], i=l

for X E S([a,_l,ai)) for x E [0, 1] \ S([a,-1, a,))

and, as before, S(i) denotes the restriction of S to the interval [ai-l, a,). Thus, as in Section 6.2, we obtain r

Pf(x)

= L:gHx)f(g,(x)).

(6.3.1)

i=l Even though equations (6.2.4) and (6.3.1) appear to be identical, the functions g, have different properties. For instance, by using the inverse function

6.3. Piecewise Convex Transformations with a Strong Repellor

155

theorem, we have g~

= 1/8' > 0

= -8" /[8'] 2 :50.

and g~'

Thus, since g~ > 0 we know that 9i is an increasing function of x, whereas a decreasing function of x since g~' :5 0. Let I E D([O, 1]) be a decreasing density, that is, x :5 y implies l(x) ~ l(y). Then, by our previous observations, l(gi(x)) is a decreasing function of x as is gHx)l(gi(x)). Since PI, as given by (6.3.1), is the sum of decreasing functions, PI is a decreasing function of x and, by induction, so is pnl· Observe further that, for any decreasing density IE D([O, 1]), we have

g~ is

1~

1:~: l(u) du ~ 1:~: l(x) du = xl(x),

so that, for any decreasing density,

l(x) :5 1/x, Hence, for i

~

E (0, 1].

X

2, we must have

gHx)l(gi(x)) :5 gHO)I(g,(O)) :5 gHO) 9i(O)

= gHO),

i

ai-l

= 2, ... ,r.

This formula is not applicable when i = 1 since ao have ~(x)f{g1(x)) :5 g~(O)I(O).

= 0. However, we do

Combining these two results with equation (6.3.1) for P, we can write r

Pl(x) :5 g~(O)I(O) + LYHO)/ai-1· i=2

Set

8'(0) and

= 1/g~(O) = ~ > 1

r

Li;(O)/ai-1

=M

i=2

so

Pl(x) :5

{1/~)1(0)

+ M.

Proceeding inductively, we therefore have

pn l(x) :5

{1/~n)I{O)

+ ~M/(~- 1)

:5 1(0) + ~M/(~- 1).

156


Thus, for decreasing f e D((O, 1]), since /(0) < oo the sequence {P" /}is bounded above by a constant. From Corollary 5.2.1 we therefore know that there is a density, /. e D such that Pf. =/.,and by Theorem 4.1.1 the measure J.I.J. is invariant. •

Example 6.3.1. In the experimental study of ftuid ftow it is commonly observed that for Reynolds numbers R less than a certain value, RL, strictly laminar ftow occurs; for Reynolds numbers greater than another value, RT, continuously turbulent ftow occurs. For Reynolds numbers satisfying RL < R < RT, a transitional type behavior (intermittency) is found. Intermittency is characterized by alternating periods of laminar and turbulent ftow, each of a variable and apparently unpredictable length. Intermittency is also observed in mathematical models of ftuid ftow, for example, the Lorenz equations (Manneville and Pomeau, 1979]. Manneville [1980] argues that, in the parameter ranges where intermittency occurs in the Lorenz equations, the model behavior can be approximated by the transformationS: (0, 1]- [0, 1] given by S(x) = (1 + e)x + (1- e)x2

(mod 1)

(6.3.2)

with e > 0, where x corresponds to a normalized ftuid velocity. This transformation clearly satisfies all of the properties of Theorem 6.2.1 for 0 < e < 2 and is thus exact. The utility of equation (6.3.2) in the study of intermittency stems from the fact that x = 0 is a strong repellor. From Remark 6.3.1 is it clear that any transformationS satisfying conditions (3i)-(3iv) will serve equally well in this approach to the intermittency problem. Exactly this point of view has been adopted by Procaccia and Schuster [1983] in their heuristic treatment of noise spectra in dynamical systems. 0

6.4 Asymptotically Periodic Transformations In order to prove the asymptotic stability of {pn} in the two preceding sections, we were forced to consider transformations S with very special properties. Thus, for every subinterval of the partition of [0, 1], we used either S((ai-l!ai)) = (0, 1) or S(ai-t) = 0. Eliminating either or both of these requirements may well lead to the loss of asymptotic stability of {pn}, as is illustrated in the following example. LetS: (0, 1]- (0, 1] be defined by for x E [0, !) 2x 2x- ! for x E i) { 2x- 1 for x E [4, 1], as shown in Figure 6.4.1. Examination of the figure shows that the Borel measure is invariant since s- 1 ([0, x]) always consists of two intervals whose S(x) =

[j,

6.4. Asymptotically Periodic Transformations

157

s

X

0

FIGURE 6.4.1. An example showing that piecewise monotonic transformation that is not onto might not even be ergodic. (See the text for details.)

union has measure x. However, 8 is obviously not exact and, indeed, is not even ergodic since 8- 1 {[0, !D = [o, !l and 8- 1 {[!, 1]) = l!. 1]. 8 that is restricted to either (0, !J or [!, 1] behaves like the dyadic transformation. The loss of asymptotic stability by {pn} may, under certain circumstances, be replaced by the asymptotic periodicity of {pn}. To see this, consider a mapping 8: [0, 1] -+ [0, 1] satisfying the following three conditions: {4i) There is a partition 0 = ao < a1 < · · · < ar = 1 of [0, 1] such that for each integer i = 1, ... , r the restriction of 8 to (ai-l• ~) is a C 2 function; (4ii) l8'(x)l;::: -X> 1,

x =I= ai, i

= 1, ... , r;

(6.4.1)

(4iii) There is a real constant c such that

l8"(x)l

(8'(x)]2 ~ c < oo,

x 'I ai, i

= 0, ... , r.

(6.4.2)

An example of a transformation satisfying these conditions is shown in Figure 6.4.2. We now state the following theorem.

Theorem 6.4.1. Let 8: [0, 1] -+ [0, 1] satisfy conditions {4i)-(4iii) and let P be the Frobenius-Penvn operator associated with 8. Then, for all fED, { pn I} is asymptotically periodic.

Proof. We first construct the Frobenius-Perron operator corresponding to

158

6. The Behavior of 'Iransformations on Intervals and Manifolds

s

X

----- I

----~'--,--------t--

i

:/

I

I

I

: I

:

:

:

I

I

I

I

I

I

I

I

I

: I

I

I

0~~--~--~~~~~-.-~~~x

FIGURE 6.4.2. An example of a transformation on [0, lJ satisfying the conditions of Theorem 6.4.1.

8. For any x E (0, 1], we have

8- 1((0,x))

r

= UAi(x) i=l

~(x)

and, as before, 9i Therefore,

=

(ai-1•9i(x)) ·x E Ii,g~ > 0 (gi(x), ~) X E !i, g. < 0 { 0 or (ai-l• ai) x ¢ Ii,

= 8~) and 8(i) denotes the restriction of 8 to (ai-t.ai)· Pf(x)=

~~I

f(u)du

""" Js-l([o,z))

=

r d I L dx J~ i=l

f(u)du,

(6.4.3)

a.(z)

where

d

1

dx J~

a.(z)

f(u)du

=

x E Ii,g~ > 0 -gHx)f(gi(x)), x E hg~ < 0 0 x¢h

{ gHx)f(gi(x)),

(6.4.4)

The right-hand side of equation (6.4.3) is not defined on the set of end points of the intervals Ii, 8(ai_ 1 ), and 8(~). However, this set is finite and


159

thus of measure zero. Since a function representing P f that is an element of L 1 is defined up to a set of measure zero we neglect these end points. Equation (6.4.4) may be rewritten as

where ui(x) = lgHx)l and 11, (x) is the characteristic function of the interval h Thus (6.4.3) may be written as r

Pf(x) = Lui(x)f(gi(x))1I,(x).

(6.4.5)

i=1

Equation (6.4.5) for the Frobenius-Perron operator is made more complicated than those in Sections 6.2 and 6.3 by the presence of the characteristic functions 11,(x). The effect of these is such that even when a completely smooth initial function f E L 1 is chosen, P f and all subsequent iterates of f may be discontinuous. As a consequence we do not have simple criteria, such as decreasing functions, to examine the behavior of pn f. Thus we must examine the variation of pn f. We start by examining the variation of P f as given by equation (6.4.5). Let a function f E D be of bounded variation on [0, 1]. From property (V1) of Section 6.1, the Yorke inequality (V5), and equation (6.4.5), 1

1

r

V Pf(x) :5 L:Vfui(x)f(gi(x))1I,(x)] 0

i=1 0

r

:52 L:Vfui(x)/(gi(x))] (6.4.6) Further, by property (V4),

Because, from the inverse function theorem, we have cui, the preceding inequality becomes

Vfui(x)f(gi(x))] :5 ~

and, thus, (6.4.6) becomes

1

~ V /(gi(x)) + c ~

O"i

:5 1/A and

ui(x)f(gi(x)) dx,

~

lu~l

:5

160

6. The Behavior of Transformations on Intervals and Manifolds 1

2

r

VPJ(x) ~X 'LV J(gi(x)) 0

h

i=1

Ill] i. ui(x)f(gi(x))dx.

+2t [c+

{6.4.7)

Define a new variable y = 9i(x) for the integral in {6.4.7) and use property {V3) for the first term to give

yPj(x) ~ Xt1r a~1 J + 2 t1r [ + !Iii1 ] 1o'o,-1 J(y) dy. 0

2

1

o

C

Set L = maJCi 2{c + 1/llil) and use property (V2) to rewrite this last inequality as

1 2 11 2 1 V Pj(x) ~ XV f + L f(y) dy =XV f + L 1

0

0

0

(6.4.8)

0

since J E D([O, 1]). By using an induction argument with inequality {6.4.8), we have

VPnJ~(~)\jJ+L~(~Y· 0

Thus, if ..\

> 2, then

0

y

pnj

and, therefore, for every

(6.4.9)

J=O

~ (~)ny J+ /:2

J E D of bounded variation, 1

lim sup

n-+oo

Vpnj < K,

(6.4.10)

0

where K > ..\Lf(..\- 2) is independent of f. Now let the set :F be defined by

:F = {g E D:

y~ K} . g

From (6.4.10) it follows that pn J E :F for a large enough n and, thus, {pn!} converges to :F in the sense that limn .....oo infpn /E:F llpn J - gil = 0. We want to show that :F is weakly precompact. From the definition of the variation, it is clear that, for any positive function g defined on [0, 1], 1

g(x) - g(y) ~

Vg 0


161

for all x, y E [0, 1]. Since g E D, there is some y E [0, 1] such that g(y) :5 1 and, thus, g(x) :5 K + 1. Hence, by criterion 1 of Section 5.1, :F is weakly precompact. (Actually, it is strongly precompact, but we will not use this fact.) Since :F is weakly precompact, then Pis constrictive by Proposition 5.3.1. Finally, by Theorem 5.3.1, { pn!} is asymptotically periodic and the theorem is proved when A> 2. To see that the theorem is also true for A > 1, consider another transformation S: [0, 1] -+ [0, 1] defined by -

q

S(x) =So··· oS(x)

= Sq(x).

(6.4.11)

Let q be the smallest integer such that Aq > 2 and set X= Aq. It is easy to see that S satisfies conditions (4i)-(4ii). By the chain rule,

IS'(x)l ~ (inf IS'(x)l)q ~ Aq

=X> 2.

Thus, by the preceding part of the proof, {fin} satisfies 1

lim supVPn/

n-+oo

mo, so that 1

VpmI :5 k,

m

> mo.

0

Now, using inequality (6.4.9), we have

:5 k

~up (~); + L I:(~);,

o~ 3 ~q-1

;=o

n

~

(mo + 1)q.

Thus, for n sufficiently large, the variation of pn f is bounded by a constant independent of f and the proof proceeds as before. •

Remark 6.4.1. From the results of Kosjakin and Sandler [1972] or Li and Yorke [1978a], it follows that transformations S satisfying the assumptions of Theorem 6.4.1 are ergodic if r = 2. 0 Example 6.4.1. In this example we consider one of the simplest heuristic models for the effects of periodic modulation of an autonomous oscillator [Glass and Mackey, 1979].

162


/

/ /

/ /

/

/

/ / / /

FIGURE 6.4.3. The periodic threshold 9(t) is shown as a solid curved line, and the activity z(t) as dashed lines. (See Example 6.4.1 for further details.)

Consider a system (see Figure 6.4.3) whose activity x(t) increases linearly from a starting time ti until it reaches a periodic threshold 9(t) at time ti:

(6.4.12) We take x(t)

= ,\(t- ti)

and 9(t)

= 1 + (t),

where is a continuous periodic function with period 1 whose amplitude satisfies 1 ~ sup<J>(t) = -inf(t) = K ~ 0. When the activity reaches threshold it instantaneously resets to zero, and the process begins anew at the starting time, ti+l = ti + -y- 1 x(ti)·

(6.4.13)

In (6.4.13), ti is an implicit function of ti given by (6.4.12) or by

(6.4.14) Equation (6.4.14) has exactly one smallest solution ti ~ ti for every tiER. We wish to examine the behavior of the starting times ti. Set F(ti) = ti(ti) + -y- 1 x(ti(ti)) so that the transformation S(t) = F(t)

(mod 1)

gives the connection between successive starting times. Many authors have considered the specific cases of (t) -y- 1 = 0, so ti = ti+l and, thus, ti+l is given implicitly by

(6.4.15)

= K sin 2?Tt,

6.4. Asymptotically Periodic 'Iransformations

163

Here, to illustrate the application of the material of this and previous sections, we restrict ourselves to the simpler situation in which t/>(t) is a piecewise linear function of t and 8 given by

!J

_ { 4Kt + 1 - K t E (0, (}(t)- 4K(1-t)+I-K tE (!,I). The calculation of F(t) depends on the sign of .X- 4K. For example, if .X > 4K, a simple computation shows that

tf + ~) f:f

~!pt+(l+~)

F(t) = {

~:;:pt + ( l

1

tE [-a,!(1-,8)-a]

t E {!(1 - ,8) -a, 1 -a],

6 16 ( .4' )

where a= 4Kf'y, ,8 = 4Kf..X, and a= (I- K)f..X. Since 0 $; ,8 < 1, it is clear that F'(t) > 1 for all t E [-a, !(1 - ,8)- a]. However, if (1 - a)/(1 + ,8) < -1, then l8'(t)l > 1 for all t and {P"} is asymptotically periodic by Theorem 6.4.1. Should it happen in this case that 8 is onto for every subinterval of the partition, then {pn} is asymptotically stable by Theorem 6.2.2. Despite the obvious simplifications in such models they have enjoyed great popularity in neurobiology: the "integrate and fire" model (Knight, I972a,b]; in respiratory physiology, the "inspiratory off switch" model [Petrillo and Glass, 1984]; in cardiac electrophysiology, the "circle model" [Guevara and Glass, 1982]; and in cell biology, the "mitogen" model [Kauffman, 1974; Tyson and Sachsenmaier, 1978]. 0 Example 6.4.2. An interesting problem arises in the rotary drilling of rocks. Usually the drilling tool is in the form of a toothed cone (mass M and radius R) that rotates on the surface of the rock with tangential velocity u. At rest the tool exerts a pressure Q on the rock. In practice it is found that, for sufficiently large tool velocities, after each impact of a tooth with the rock the tool rebounds before the next blow. The energy of each impact, and thus the efficiency of the cutting process, is a function of the angle at which the impact occurs. Let x be the normalized impact angle that is in the interval (0, I]. Lasota and Rusek [1974] have shown that the next impact angle is given by the transformation 8: [0, 1]-+ [0, 1] defined by

8(x) = x + aq(x)- v'faq(x)] 2 + 2axq(x)- aq(x)[1 + q(x)] (mod 1), (6.4.I7) where q(x) =I+ int[(1- 2x)f(a- 1)]; int(y) denotes the integer part of y, namely, the largest integer smaller than or equal to y, and a= F/(F-1),

164


where

F=Mu2 /QR is Freude's number, the ratio of the kinetic and potential energies. The Freude number F contains all of the important parameters charact!lrizing this process. It is moderately straightforward to show that with S =So S, IS'(x)l > 1 ifF> 2. However, the transformation (6.4.17) is not generally onto, so that by Theorem 6.4.1 the most that we can say is that for F > 2, if Pis the Frobenius-Perron operator corresponding to S

then {pn} is asymptotically periodic. However, it seems natural to expect that {pn} is in fact asymptotically stable. This prediction is supported experimentally, because, once u > (2QR/M) 112 , there is a transition from smooth cutting to extremely irregular behavior (chattering) of the tool. 0

Example 6.4.3. Kitano, Yabuzaki, and Ogawa [1983] experimentally examined the dynamics of a simple, nonlinear, acoustic feedback system with a time delay. A voltage x, the output of an operational amplifier with response time -y- 1 , is fed to a speaker. The resulting acoustic signal is picked up by a microphone after a delay T (due to the finite propogation velocity of sound waves), passed through a full-wave rectifier, and then fed back to the input of the operational amplifier. Kitano and co-workers have shown that the dynamics of this system are described by the delay-differential equation -y- 1:i:(t)

= -x(t) + J.tF(x(t- r)),

(6.4.18)

-lx+ !I+!

(6.4.19)

where

F(x) =

is the output of the full-wave rectifier with an input x, and J.l. is the circuit loop gain. In a series of experiments, Kitano et al. found that increasing the loop gain J.l. above 1 resulted in very complicated dynamics in x, whose exact nature depends on the value of -yr. To understand these behaviors they considered the one-dimensional difference equation, Xn+l

= J.tF(xn),

derived from expressions (6.4.18) and (6.4.19) as -y- 1 -+ 0. In our notation this is equivalent to the map T: [-J.t/(J.t- 1),J.t/2] -+ [-J.t/(J.t- 1),J.t/2], defined by J.t(1 +x) for x E (6.4.20) T(x) = { ( ] -J.tX for X E -!, j .

[-f--r, -!]

6.5. Change of Variables

165

for 1 < J.t ::5; 2. Make the change of variables J.t ,J,t J.t+ 1 x----+x--J,t-1 2J,t-1

so that (6.4.20) is equivalent to the transformation 8: [0, 1]- [0, 1], defined by 1 S(x') = { J.tX for x' E [0, 1/J.t] (6.4.21) 2- J.tX 1 for x' E (1/J.t, 1]. For 1 < J.t ::5; 2, the transformation S defined by (6.4.21) satisfies all the conditions of Theorem 6.4.1, and 8 is thus asymptotically periodic. If J.t = 2, then, by Theorem 6.2.2, 8 is statistically stable. Furthermore, from Remark 6.4.1 it follows that 8 is ergodic for 1 < J.t < 2 and will, therefore, exhibit disordered dynamical behavior. This is in agreement with the experimental results. 0

Remark 6.4.2. As we have observed in the example of Figure 6.4.1, piecewise monotonic transformations satisfying properties (4i)-(4iii) may not have a unique invariant measure. If the transformation is ergodic, and the invariant measure is thus unique by Theorem 4.2.2, then the invariant measure has many interesting properties. For example, in this case Kowalski [1976] has shown that the invariant measure is continuously dependent on the transformation. 0

6.5 Change of Variables In the three preceding sections, we have examined transformations 8: [0, 1] [0, 1] with very restrictive conditions on the derivatives S'(x) and S"(x). However, most transformations do not satisfy these conditions. A good example is the quadratic transformation,

= 4x(1- x), for x E [0, 1]. For this transformation, S'(x) = 4- 8x, and IS'(x)l < 1 for x E (i, i). Furthermore, IS"(x)j[S'(x)J21 = !(1- 2x)- 2 , which is clearly not bounded S(x)

at x = !· However, iteration of any initial density on [0, 1] indicates that the iterates rapidly approach the same density (Figure 1.2.2), leading one to suspect that, for the quadratic transformation, { pn} is asymptotically stable. In this section we show how, by a change of variables, we can sometimes utilize the results of the previous sections to prove asymptotic stability. The idea is originally due to Ruelle [1977] and Pianigiani [1983]. Theorem 6.5.1. Let 8: [0, 1] - [0, 1] be a transformation satisfying properties (2i)' and (2ii)' of Section 6.2, and Ps be the Frobenius-Perron operator

166


corresponding to S. If there exists an a. e. positive 0 1 function 4> e L 1 ([0, 1]) such that, for some real ~ and c,

= IS'(x)lr/>(S(x)) >

p (X ) -

and

r/>(x)

_

~ A

> 11

(6.5.1)

0<xN} is asymptotically stable. Proof. Set g(x)

1 r = jj4;jj Jo

for x

rJ>(u) du,

and consider a new transformation T defined by

T(x)

= g(S(g- 1 (x)))

(6.5.4)

with associated Frobenius-Perron operator ~. From (6.5.4), T(g(x)) g(S(x)), so dT dg dg dS dg dx = dS dx or T '( g(x )) = g'(S(x))S'(x) g'(x) .

=

Using (6.5.3) this may be rewritten as

T '( g(x ))

= S'(x)r/>(S(x)) r/>(x)

.

Hence, by (6.5.1), we have IT'(g)l ~ ~ > 1. Further, by comparing this equation with (6.5.1), we see that p(x) = IT'(g(x))!. It follows that

1 d ( 1 ) r/>(x) dx p(x)

1

d (

= r/>(x) dx

so that, from inequality (6.5.2),

I

1 ) IT'(g)l

T"(g)

= ±(T'(g)]2 114>11'

I

T"(g) [T'(g)] 2 ~ cllr/>11 < oo.

Thus the new transformation T satisfies all the conditions of Theorem 6.2.2, and {~} is asymptotically stable as is {J>N} by (6.5.14). • Example 6.5.1. Consider the quadratic transformation S(x) with x e [0, 1] and set 1 r/>(x) = -7rv''x=7(1::=-=x=.=)

= 4x(1- x) (6.5.5)

6.5. Change of Variables

167

Using equations (6.5.3) and (6.5.5), it is easy to verify that all the conditions of Theorem 6.5.1 are satisfied in this case and, thus, for the quadratic transformation, {pn} is asymptotically stable. Note that with l/J as given by (6.5.5), the associated function g, as defined by (6.5.3), is given by du 1 1. 1 (1-2z), g(z) =-11:r: =---sin11" o y'u(1 - u) 2 11"

(6.5.6)

and thus (6.5.7) Hence, when S(z) by

= 4z(1-z), the transformation T: [0, 1]-+ [0, 1), defined T(z) =go So g- 1 (z),

(6.5.8)

is easily shown to be

2z

T(z)

= { 2(1- z)

for z E for z E

[0, ~) [~, 1].

(6.5.9)

[The transformation defined by (6.5.9) is often referred to as the tent map or hat map.] The Frobenius-Perron operator, Fr, corresponding to T is given by PTf(z) ~~ (~z) + ~~ (1- ~z),

=

and, by Theorem 6.2.2, {P~} is asymptotically stable. Furthermore, it is clear that /. 1 is the unique stationary density of PT, soT is, in fact, exact by Theorem 4.4.1. Reversing the foregoing procedure by constructing a transformation S = g- 1 o Tog from T given by (6.5.9) and from g, g- 1 given by equations (6.5.6) and (6.5.7) yields the transformation S(z) = 4z(1-z). From this {Pff} is asymptotically stable, and l/J, given by (6.5.5), is the stationary density of Ps. 0 These comments illustrate the construction of a statistically stable transformation S with a given stationary density from an exact transformation T. Clearly, the use of a different exact transformation T1 will yield a different statistically stable transformation 8 1 , but one that has the same stationary density asS. Thus we are led to the next theorem.

=

Theorem 6.5.2. LetT: (0, 1)-+ (0, 1) be a measumble, nonsingular tronaformation and let l/J E D((a,b)), with a and b finite or not, be a given positive density, that is, lfJ > 0 a. e. Let a second tmnsformation 8: (a, b) -+ (a, b) be given by S = g- 1 o Tog, where g(z) =

1:r: lfJ(y)dy,

a< z 1 for S: R-+ R is not sufficient for the asymptotic stability of {P"}. This is amply illustrated by the specific example S(x) = 2x, which was considered in Section 1.3. There are, however, transformations on the real line for which the asymptotic stability of {Pn} can be demonstrated; one example is S(x) = .8 tan("Yx + 6), I.B"YI > 1. This section will treat a class of such transformations. Assume the transformation S: R -+ R satisfies the following conditions: (6i} There is a partition · · · a_ 2 < a_ 1 < ao < a 1 < a2 · · · of the real line such that, for every integer i 0, ±1, ±2, ... , the restriction S(i) of S to the interval (ai-l! ai) is a 0 2 function;

=

(6ii) S((ai-11 a,))

= R;

(6iii) There is a constant A > 1 such that IS' (x) I ~ A for x ::/: a,, i = 0, ±1, ±2, ... i (6iv) There is a constant L ~ 0 and a function q E L 1 (R) such that (6.6.1} where 9i

= s~)l' for i =

0, ±1, ... i and

(6v) There is a real constant c such that IS"(x)l

[S'(x)]2 ~ c,

for x ::/: ai, i

= 0, ±1, ....

(6.6.2)

Then the following theorem summarizes results of Kemperman [1975], Schweiger [1978), Jablonski and Lasota [1981), and Bugiel [1982). Theorem 6.6.1. If S: R-+ R satisfies conditions (6i)-(6v) and P is the associated Frobenius-PeTTOn operator, then {Pn} is asymptotically stable. Proof. We first calculate the Frobenius-Perron operator. To do this note

6.6. 'Iransformations of the Real Line

that

173

U> + U 1, is asymptotically stable. We have S'(x)P"Y - cos2 ('"Yx + 6) hence IS' (x) I ~ /3-y. Further

S"(x) [S'(x)] 2 =

so that

1 .

--p sm[2('"Yx + 6)]

I

S"(x) 1 · [S'(x)] 2 :5 1131"

I

6.7

0

Manifolds

The last goal of this chapter is to show how the techniques described in Chapter 5 may be used to study the behavior of transformations in higherdimensional spaces. The simplest, and probably most striking, use of the Frobenius-Perron operator in d-dimensional spaces is for expanding mappings on manifolds. To illustrate this, the results of Krzyzewski and Szlenk (1969), which may be considered as a generalization of the results of Renyi presented in Section 6.2, are developed in detail in Section 6.8. However, in this section we preface these results by presenting some basic concepts from the theory of manifolds, which will be helpful for understanding the

176


geometrical ideas related to the Krzyiewski-Szlenk results. This elementary description of manifolds is by no means an exhaustive treatment of differential geometry. First consider the paraboloid z = x 2 + y2 • This paraboloid is embedded in three-dimensional space, even though it is a two-dimensional object. If the paraboloid is the state space of a system, then, to study this system, each point on the paraboloid must be described by precisely two numbers. Thus, any point m on the paraboloid with coordinates (x, y, x 2 + y2 ) is simply described by its x, y-coordinates. This two-dimensional system of coordinates may be described in a more abstract way as follows. Denote by M the graph of the paraboloid, that is, M = {(x,y,z): z = x 2 + y 2 },

and, as a consequence, there is a one-to-one transformation 4>: M --+ R2 described by 4>(x,y,z) = (x,y) for (x,y,z) eM. Of course, other coordinate systems on M are possible, that is, another one-to-one mapping, 4>*: M --+ R 2 , but 4> is probably the simplest one. Now let M be the unit sphere, M

= {(x,y,z):x2 +y2 + z2 = 1}.

In this example it is impossible to find a single smooth invertible function 4>: M --+ R 2 • However, six functions ¢i: M --+ R 2 may be defined as follows:

4>t(x,y,z) = (x,y), 4>2(x, y, z) = (x, y), 4>s(x, y, z) = (x, z), 4>4(x, y, z) = (x, z), 4>s(x, y, z) = (y, z), 4>6(x,y,z) = (y,z),

for z > 0; for z < 0; for y > 0; for y < 0; forx>O; for x < 0.

Each of these functions 4>i maps a hemisphere of M onto an open unit disk. This coordinate system has the property that for any m E M there is an open hemisphere that contains m and on each of these hemispheres one 4>i is defined. In the same spirit, we give a general definition of a smooth manifold.

Definition 6. 7 .1. A smooth d-dimensional manifold consists of a topological Hausdorff space M and a system {4>i} of local coordinates satisfying the following properties: (a) Each function 4>i is defined and continuous on an open subset Wi C M and maps it onto an open subset U;. = 4>i(W;.) of Rd. The inverse functions 4>i 1 exist and are continuous (i.e., 4>i is a homeomorphism of Wi onto Ui)i (b) For each mE M there is a Wi such that mE Wi, that is, M = Ui W;.;

6.7. Manifolds

177

(c) If the intersection Win W; is nonempty, then the mapping 4Ji o 4Jj 1 , which is defined on 4J; (Wi n W;) c ~ and having values in ~, is a 0 00 mapping. (Note that a topological space is called a Hausdorff space if every two distinct points have nonintersecting neighborhoods.) Any map 4Ji gives a coordinate system of a part of M, namely, Wi. A local coordinate of a point mE Wi is 4Ji(m). Having a coordinate system, we may now define what we mean by a Ole function on M. We say that 1: M -+ R is of class Ole if for each 4Ji: Wi -+ Ui the composed mapping I o 4Ji 1 is of class c~e on ui. Next consider the gradient of a function defined on the manifold. For I: Rd -+ R the gradient of I at a point x E Rd is simply the vector (sequence of real numbers),

81(x)) _ (81(x) grad I( X ) 8 , ... , 8 • Xd

Xt

For I: M -+ R of class 0 1 , the gradient of calculated in local coordinates as follows:

I at a point

grad l(m) = (Dz 1 (m}I, ... ,Dz,(m}f),

m E M can be

(6.7.la}

where (6.7.lb} Thus the gradient is again a sequence of real numbers that depends on the choice of the local coordinates. The most important notion from the theory of manifolds is that of tangent vectors and tangent spaces. A continuous mapping 'Y= [a, b] -+ M represents an arc on M with the end points 'Y(a} and 'Y(b}. We say that 'Y starts from m = 'Y(a). The arc 'Y is Ole if, for any coordinate system (jJ, the composed function 4Jo'Y is of class Ole. The tangent vector to 'Y at a point m = 'Y(a) in a coordinate system (jJ is defined by d [

dt 4J('Y(t)}

]

t=a =

1

(~ ' .. • '~ }, d

(6.7.2}

e' ...

where, again, the numbers '~d depend on the choice of the coordinate system (jJ. Of course, 'Y must be at least of class 0 1. Two arcs 'Yt and 'Y2 starting from m are called equivalent if they produce the same coordinates, that is, (6.7.3} where 'Yt(at) = 'Y2(a2) = m. Observe that, if (6.7.3} holds in a given system of coordinates (jJ, then it holds in any other coordinate system. The class of

178

6. The Behavior of 'Ii'ansformations on Intervals and Manifolds

all equivalent arcs produces the same sequence (6. 7.3) for any given system of coordinates. Such a class represents the tangent vector. Tangent vectors are denoted by the Greek letters { and 'I· Assume that a tangent vector in a coordinate system ,p has components e1, ... , {d. What are the components in another coordinate system'¢? Now,

e

d

d

dt [1/J('y(t))] = dt [H(,P('y(t)))], where H = 1/J o,p- 1 and, therefore, setting

d('f/Jo-y)/dt =('I\ ... ,,d),

,i = L: aniej. 8x; d

(6.7.4)

;= 1

Equation (6.7.4) shows the transformations of the tangent vector coordinates under the change of coordinate system. Thus from an abstract (tensor analysis) point of view the tangent vector at a point m is nothing but a sequence of numbers in each coordinate system given in such a way that these numbers satisfy condition (6.7.4) when we pass from one coordinate system to another. From this description it is clear that the tangent vectors at m form a linear space, the tangent space, which we denote by Tm· Now consider a transformation F from a ~dimensional manifold Minto a ~dimensional manifold N, F: M-+ N. The transformation F is said to be of class Ck if, for any two coordinate systems ,P on M and 1/J on N, the composed function 1/J oF o ,p- 1 is of class Ck, or its domain is empty. Let be a tangent vector at m, represented by a 0 1 arc "(: [a, b] -+ M starting from m. Then F o 'Y is an arc starting from F(m), and it is of class 0 1 if F is of class 0 1 • The tangent vector to F o 'Y in a coordinate system 1/J is given by

e

!

Setting

q

[1/J oF o 'Y]t=a

= ('I\ ...

1

"'d).

= 1/J oF o ,p- 1, where ,P is a coordinate system on M, 8 ej ,i = L: 8x; d

(fi

(6.7.5)

i= 1

results. Equation (6.7.5) gives the linear transformation of a tangent vector eat m to a tangent vector 'I at F(m) without explicit reference to the arc 'Y. This transformation is called the differential of F at a point m and is denoted by dF(m), and thus symbolically 'I= dF(m){.

Note that the differential ofF is represented in any two coordinate systems, ,P on M and 1/J on N, by the matrix

i,j = 1, ... ,d.

6.7. Manifolds

179

The same matrix appears in the formula for the gradient of the composed function: If F: M -+ N and f: N -+ R are C 1 functions, then the differentiation of(! oF) o f/J- 1 = (! o 1/J- 1 ) o (1/J oF o f/J- 1 ) gives grad(/ o F)(m) = (D:r: 1 (m)(f oF), ... , D:r:Am)(f oF)), where

d

D:r:,(m)(f 0 F)=

La:. [!(1/J- (x))]:r:=tt>(F(m)) ~:~. 1

j=l

J

I

This last formula may be written more compactly as grad((/ o F)(m))

= (grad f)(dF(m)).

Observe that now dF(m) appears on the right-hand side of the vector. Finally observe the relationship between tangent vectors and gradients. Let f: M-+ R be of class C 1 and let -y: [a, b] -+ M start from m. Consider the composed function f o -y: [a, b] -+ R that is also of class C 1 • Using the local system of coordinates,

I

o 'Y = (! o f/J- 1 ) o (f/J o -y),

and, consequently, (6.7.6) Observe that the numbers D:r:J and {i depend on ¢J even though the lefthand side of (6.7.6) does not. Equation (6.7.6) may be more compactly written as (6.7.7) In order to construct a calculus on manifolds, concepts such as the length of a tangent vector, the norm of a gradient, and the area of Borel subsets of M are necessary. The most effective way of introducing these is via the Riemannian metric. Generally speaking the Riemannian metric is a scalar product on Tm. This means that, for any two vectors {t,{2 E Tm, there corresponds a real number denoted by ({1!{2 ). However, the coordinates

<ef, ... ,et>, <eJ, ... ,eg> depend on the coordinate system f/J. Thus the rule that allows ({11 { 2 ) to be calculated given ({f), ({~)must also depend on f/J. These facts are summarized in the following definition. Definition 6. 7.2. A Riemannian metric on the manifold M is a system of functions i,j = l, ... ,d, gf;(m): M-+ R, such that

180


(a) For any choice of local coordinates : W -+ U. We can find d normalized vectors (6.7.12)

182


orthogonal with respect to this form, that is, d

<e~c,e,) =

:E uf;<m>eLel = 6~c,, i,j=l

where

61c1

= 1 if k = l, and 61c1 = 0, k ::F l, is the Kronecker delta. Write Vq,(m)

=

et et ...

ej

det .. · .. · .. · e~

This same procedure can be carried out algebraically by setting Vq,(m) = ldet(gf;(m))l- 1/ 2 • Function Vq,(m) has a simple heuristic interpretation. Vectors e11· .. , ed, which correspond to the components (6.7.12), are orthogonal and normal~ ized. Thus the volume spanned by them should be equal to 1. The volume spanned by the representation of el, ... , edina local coordinate system is Vq,(m). Thus Vq,(m) gives the volume in local coordinates corresponding to a unit volume in M. By using this interpretation we define the measure J1. of a Borel set B c Was

J.J.(B)

=

1

4>(B)

dx

Vq,(-l(x))

.

(6.7.13}

The idea leading to this definition is obvious, as the elementary volume dx = dx1 · · · dxd in Rd corresponds to the volume Vq,(- 1(x}}dx1 · · · dxd in M. Thus, in order to reproduce the "original" volume in M, we must divide dx by Vq,. It can be shown that J.J.(B) defined by (6.7.13} does not depend on the choice of , which is quite obvious from the heuristic interpretation of Vq,{m). Analogous considerations lead to the definition of the determinant of the differential of a C 1 transformation F from a d-dimensional manifold Minto ad-dimensional manifold N. Take a point mE M and define ldet dF{m)l where matrix

dal Vt~~(F(m)), Vq,(m) = ldx

Ida/dxl denotes the absolute value of the determinant of the d x d {Jqi) (8x· ' i,j = 1, ... ,d. J

It can be shown that this definition does not depend on the choice of coordinate systems and t/J in M and N, respectively. Note also that the determinant per se is not defined, but only its absolute value. This is because our manifolds M, N are not assumed to be oriented.

6.8. Expanding Mappings on Manifolds

183

The following c8.J.culation will justify our definition of ldet dF(m)l. Let B be a small set on M, and F(B) its image on N. What is the ratio IJ(F(B))/IJ(B)? From equation (6.7.13),

f

IJ(F(B)) IJ(B)

Setting u

dy

= }t/J(F(B)) V(t/J- 1(y))

If

dx }"'(B) V"'(- 1 (x)).

= 1/J oF o 4>- 1 and substituting y = u(x),

I I

IJ(F(B)) f du dx IJ(B) =}"'(B) dx Vt/J(F(- 1 (x))

If

dx }"'(B) V"'(- 1 (x))

results. Thus, for small B containing a point m, we have approximately IJ(F(B)) IJ(B)

~

ldul V"'(m) = ldet(dF(m))l. dx Vt/JF(m))

(6.7.14)

6.8 Expanding Mappings on Manifolds With the background material of the preceding section, we now turn to an examination of the asymptotic behavior of expanding mappings on manifolds. We assume that M is a finite-dimensional compact connected smooth (C 00 ) manifold with a Riemannian metric. As we have seen in Section 6.7, this metric induces the natural (Borel) measure 1J and distance p on M. We use 1/'(m)l to denote the length of the gradient off at point mE M. Before starting and proving our main result, we give a sufficient condition for the existence of a lower-bound function in the same spirit as contained in Propositions 5.8.1 and 5.8.2. We use the notation of Section 5.8. Proposition 6.8.1. Let P: L 1 (M) --+ L 1 (M) be a Markov operator and if we assume that there is a set Do, dense in D, so that for every f E Do the trajectory pnf=f,, for n ?: no(/), (6.8.1)

is such that the functions In are C 1 and satisfy 1/~(m)l $ kfn(m),

formE M,

(6.8.2)

where k ?: 0 is a constant independent of I, then there exists e > 0 such that h =elM is a lower-bound function for P. Proof. The proof of this proposition proceeds much as for Proposition 5.8.2. As before, 11/nll = 1. Set

e = [1/2J.t(M)]e-kr,

184


where

r=

sup p(mo,ml)· mo,m1EM Let "Y( t), a :$ t :$ b, be a piecewise smooth arc joining points mo = "Y(a) and m1 =')'(b). Differentiation of/no')' gives (see inequality (6.7.11)]

Id(/n~~(t))]l :$1/~("Y(t))I·II"Y'(t)ll :$ klb'(t)ll/n("Y(t)) so that /n(ml) :$ fn(mo) exp { k

1b

lb'(s)ll ds}.

Since "Y was an arbitrary arc, this gives /n(mt) :$ /n(mo)ekp(mo,ml) :$ fn(mo)ekr. Now suppose that h =elM is not a lower-bound function for P. This means that there must be some n' > no and m 0 E M such that /n• (mo) < e. Therefore, for m1 EM, which contradicts 11/nll = 1 for all n >no(/). Thus we must have /n ~ h = elM for n > no. • Next we turn to a definition of an expanding mapping on a manifold.

Definition 6.8.1. Let M be a finite-dimensional compact connected smooth (C00 ) manifold with Riemannian metric and let 11- be the corresponding Borel measure. A C 1 mapping S: M -+ M is called expanding if there exists a constant A> 1 such that the differential dS(m) satisfies

(6.8.3)

e

at each m E M for each tangent vector E Tm· With this definition, Krzyzewski and Szlenk (1969] and Krzyzewski [1977] demonstrate the existence of a unique absolutely continuous normalized measure invariant under S and establish many of its properties. Most of these results are contained in the next theorem.

Theorem 6.8.1. Let S: M -+ M be an expanding mapping of class C 2 , and P the Frobenius-Perron operator con-esponding to S. Then {pn} is asymptotically stable. Proof. From equation (6.7.5) with F = S, since Sis expanding, 7J =F 0 for any =F 0, and, thus, the matrix (8ui/8x;) must be nonsingular for every mEM.

e

6.8. Expanding Mappings on Manifolds

185

In local coordinates the transformation 8 has the form

x- tj>(S(¢>- 1 (x)))

= u(x)

and consequently is locally invertible. Therefore, for any point m e M the counterimage s- 1 (m) consists of isolated points, and, since M is compact, the number of these points is finite. Denote the counterimages of m by m11 ... , m1c. Because 8 is locally invertible there exists a neighborhood W of m and neighborhoods Wi of mi such that 8 restricted to Wi is a one to one mapping from Wi onto W. Denote the inverse mapping of 8 on Wi by 9i· We have 8 o 9i = lw11 where lw1 is the identity mapping on Wi and, consequently, (dB) o (dgi) is the identity mapping on the tangent vector space. From this, in conjunction with (6.8.3), it immediately follows that ll(dui)ell ~

Now take a set B

(1/-X)IIeu.

(6.8.4)

c W, so 1c

s- 1 (B) = U9i(B), i=l

and, by the definition of the Frobenius-Perron operator,

L:1 1c

f

Pf(m)p.(dm)

=f

jB

/(m)p.(dm)

ls-l(B)

=

i=l

f(m)p.(dm).

91(B)

This may be rewritten as 1 Jl. (B)

1 B

Pf(m)p.(dm)

~ p.(gi(B)) 1 = L.., (B) · ( ·(B)) i=l Jl. p. g,

1 91(B)

f(m)p.(dm).

If B shrinks tom, then 9i(B) shrinks to 9i(m),

p.(~) and (

·~B))

P. g,

1

91(B)

L

P/(m)p.(dm)- Pf(m) a.e.

f(m)p.(dm) - f(gi(m)) a.e.,

i = 1, ... ,k.

Moreover, by (6.7.14),

p.~{~))

-ldet(dgi(m))j.

Thus, by combining all the preceding expressions, we have 1c

Pf(m) =

L ldet(dgi(m))lf(gi(m)), i=l

which is quite similar to the result in equation (6.2.10).

(6.8.5)

186

f


Now let Doc D(M) be the set of all strictly positive C 1 densities. For E Do, differentiation of P/(m) as given by (6.8.5) yields I(P/)'1 = E~=~ I(Ji(f o gi))'l Pf Ei=l J,(f 0 g,)

< E~=l IJ:I(f o 9i) + 2::-1 Jill' o 9illdgil -

E~=l Ji(f o 9i)

E~=l Ji(f o 9i)

< max IJ:I +max I/' 0 9illd9il -

i

Ji

i

(/ 0 9i)

'

where Ji = ldet dgi(m)l. From equation (6.8.4), it follows that ldgil ~ so that sup I(P1)'1 < c + .!. sup

Pf

-

>..

1/>..,

ltJ

I '

where c =sup IJHm>l. i,m Ji(m)

Thus, by induction, for n = 1, 2, ... , we have sup I(Pn/)'1 < ~ + _!:._supJfl. pn / - ).. - 1 )..n / Choose a real K

> >.cj(>.- 1), then sup I(Pn f)' I < K pnf -

(6.8.6)

for n sufficiently large, say n > n 0 (/). A straightforward application of Proposition 6.8.1 and Theorem 5.6.2 finishes the proof. • Example 6.8.1. Let M be the two-dimensional torus, namely, the Cartesian product of two unit circles: M = {(m11m2):m1 = eiz 1 ,m2 = eiz 2 ,X1X2 E R}. M is evidently a Riemannian manifold, and the inverse functions to (6.8.7)

define the local coordinate system. In these local coordinates the Riemannian metric is given by 9ik = 6;k, the Kronecker delta, and defines a Borel measure 11- identical with that obtained from the product of the Borel measures on the circle. We define a mapping S: M -+ M that, in local coordinates, has the form (mod 211").

(6.8.8)

Exercises

Thus S maps each point (m 1 ,m2) given by (6.8.7) to the point where m1 = exp(i(3xl + X2)] and ffl2 = exp(i(xl + 3x2)].

187

(m1 ,m2),

We want to show that S is an expanding mapping. From (6.8.8) we see that dS(m) maps the vector = (e,e 2) into the vector (3e+e,e 1 +3e). Also, sinceg;A: = D;A:, hence (e 1 ,e)= (e 1 ) 2+(e) 2 from (6.7.8). Thus 2 2 lldS(m)ell = (3e + e) + (e + 3e) 2 2 22 2 = 4!<e ) + <e ) l + 6(e + ?: 4llell 2,

e

e>

and we see that inequality (6.8.3) is satisfied with.\= 2, therefore Sis an expanding mapping. Further, if Pis the Frobenius-Perron operator corresponding to S, then, by Theorem 6.8.1, {Pn} is asymptotically stable. It is also possible to show that S is measure preserving, so by Proposition 5.6.2 this transformation is exact. This proves our earlier assertion in Section 4.3. 0

Exercises 6.1. Let (X, A, p.) be a measure space and let S: X-+ X be a nonsingular transformation. Fix an integer k ?: 1. Prove that SA: is statistically stable if and only if S is statistically stable. 6.2. Consider the transformationS: (0, 1]-+ (0, 1] defined by -{2x ~

S(x)-

a:

O$x
Using the result proved in Exercise 6.1 show that Sis statistically stable.

6.3. Consider the transformationS: (0, 1]-+ (0, 1] of the form S(x)

= cx(1- x2 ).

Fix cat the value for which S maps (0, 1] onto [0, 1] and, using the change of variables formulas (Theorem 6.5.1), show that for this value of c the transformations S is statistically stable. 6.4. Consider the transformations S: (0, 1]-+ (0, 1] of the form S(x)

= cx2 (1- x).

Again fix c to be that value at which S maps (0, 1] onto (0, 1] and use a change of variables and the Birkhoff individual ergodic theorem to show that lim sn(x) 0, for x E (0, 1] a.e. n-+oo

=

188


Observe that S has periodic points of period 3 and thus is chaotic in the sense of Sark.ovskil and Li-Yorke [Jama, 1989; Li and Yorke, 1975]. 6.5. Consider the transformation S: [0, l] -+ [0, l] defined by

S(x)

= x [-a+ 1 !x],

where l = (b/a)- 1 > 0. Find b = bo such that S maps [0, l] onto [0, l] and using the change of variables formulas prove that S is statistically stable. 6.6. Consider the ''tent" transformationS: [0, 1]-+ [0, 1] of the form

-{ex O$x
S(x) -

·

where 1 < c $ 2. Let P be the corresponding Frobenius-Perron operator. Determine the values of parameter c for which {pn} is asymptotically periodic but not asymptotically stable (r ~ 2 in formula. (5.3.8)). 6.7. Let M be the two-dimensional torus described in Example 6.8.1. Consider the transformation S: M -+ M which in local coordinates has the form S(xt, x2) = (auxt + a12X2, a21X1 + a22x2). Assume that the coefficients ai; are positive integers and find conditions concerning the matrix (ai;) which imply the exactness of S. 6.8. Using TRAJ, BIFUR, DENTRAJ, and DENITER numerically study the behavior of transformations defined in Exercises 6.5 and 6.6 for a < b $ bo, and 0 < c < 2, respectively.

7 Continuous Time Systems: An Introduction

In previous chapters we concentrated on discrete time systems because

they offer a convenient way of introducing many concepts and techniques of importance in the study of irregular behaviors in model systems. Now we turn to a study of continuous time systems. Continuous and discrete systems differ in several important and interesting ways, which we will touch on throughout the remainder of this book. For example, in a continuous time system, complicated irregular behaviors are possible only if the dimension of the phase space of the system is three or greater. As we have seen, this is in sharp contrast to discrete time processes that can have extremely complicated dynamics in only one dimension. Further, continuous time processes in a finite-dimensional phase space are in general invertible, which immediately implies that exactness is a property that will not occur for these systems (recall that noninvertibility is a necessary condition for exactness). However, systems in an infinite-dimensional phase space, namely, time delay equations and some partial differential equations, are generally not invertible and, thus, may display exactness. This chapter is devoted to an introduction of the concept of continuous time systems, an extension of many properties developed previously for discrete time systems, and the development of tools and techniques specifically designed for studying continuous time systems.

190

7. Continuous Time Systems: An Introduction

7.1

Two Examples of Continuous Time Systems

Here a continuous time process in a phase space X is given by a family of mappings St: X -+ X, t ;::: 0.

AB illustrated in Figure 7.1.1, the value St(:r:0 ) is the position ofthe system at a time t that started from an initial point :r:0 E X at time t = 0. We consider only those processes in which the dynamical law S does not explicitly depend on time so that the property (7.1.1)

holds. This simply means that the dynamics governing the evolution of the system are the same on the intervals (0, t'] and [t, t + t']. Example 7.1.1. A well-known example of a continuous time process is given by an autonomous d-dimensional system of ordinary differential equations d:r: (7.1.2) - =F(:r:)

dt

where :r: = (:r:11 ... ,:r:d) and F: Jld -+ Jld is sufficiently smooth to ensure the existence and uniqueness of solutions, such as F is C 1 and satisfies IF(:r:)l ~ et + .BI:r:l, with et and .B finite. In this case, St(:r:0 ) is the solution of (7.1.2) with the initial condition (7.1.3)

In this example time t need not be restricted to t ;::: 0, and the system can also be studied for t ~ 0. As we will see in Section 7.8, this is a commonly encountered situation for problems in finite dimensional phase spaces. 0

Example 7.1.2. Consider the delay-differential equation

~~t)

= F(x(t),x(t -1))

(7.1.4)

with the initial condition forTE (-1,0).

(7.1.5)

For rather simple restrictions on F (namely, that F is C 1 and IF(:r:, y)l ~ a(y) + ,B(y)l:r:l, where et and .B are arbitrary continuous functions of y), there is a unique solution to (7.1.4} with (7.1.5) [see Hale, 1977]. Let X be the space of all functions (-1,0]-+ R with the usual uniform convergence topology. Given :r:0 E X and the solution :r: of (7.1.4) and (7.1.5}, we may define

St:r:0 (T) = :r:(t + T},

forTE [-1, 0].

(7.1.6}

7.2. Dynamical and Semidynamical Systems

X

191

s1 (x >

FIGURE 7.1.1. The trajectory of a continuous time process in the phase space X. At timet= 0 the system is at x0 , and at timet it is at St(x0 ).

If x(t) is the solution of (7.1.4)-(7.1.5), then, since F is not an explicit function oft, x(t +a), a~ 0, is also a solution to the problem. Using this fact it is easy to verify that transformation (7.1.6) satisfies property (7.1.1), although it is impossible to define Stx(r) fort< 0. 0 A very important difference exists between these two examples with respect to their invertibility. Thus, although the solution to the system of ordinary differential equations in Example 7.1.1 may be studied fort~ 0, in general no solution exists for the differential-delay equation of Example 7.1.2 when t < 0. This lack of invertibility is generally the case for delay-differential equations and, indeed, for many continuous time systems whose phase space X is not finite dimensional (e.g., some partial differential equations).

7.2 Dynamical and Semidynamical Systems It is possible to establish many results for continuous time processes in a phase space X endowed with no other property than a measure ,.,, as was done in earlier chapters for discrete time processes. However it is simpler to consider continuous time processes in a measure space that is also equipped with a topology. Thus, from this point on, let X be a topological Hausdorff space and A the u-algebra of Borel sets, that is, the smallest u-algebra that contains all open, and thus closed, subsets of X. Dynamical Systems

Definition 7.2.1. A dynamical system {StheR on X is a family of transformations St: X-+ X, t E R, satisfying (a) S0 (x)

= x for all x EX;

(b) St(St'(x)) = St+t'(x) for all x EX, with t,t' E R; and (c) The mapping (t,x)-+ St(x) from X x R into X is continuous.

192


Remark 7.2.1. System (7.1.2) of ordinary differential equations, introduced in the preceding section, is clearly an example of a dynamical system. 0 Remark 7.2.2. It is clear from the group property of Definition 7.2.1 that for all t E R. Thus, for all toE R, any transformation St0 of a dynamical system {StheR is invertible. 0 In applied problems the space X is customarily called the phase space of the dynamical system {StheR. whereas, for every fixed x 0 E X, the function St(x0 ), considered as a function of t, is called a trajectory of the system. The trajectories of a dynamical system {StheR in its phase space X are of only three possible types, as shown in Figure 7.2.1a,b,c for X = R 2 • First (Figure 7.2.1a), the trajectory can be a stationary point x0 such that for all t E R. Second, as shown in Figure 7.2.1b, the trajectory can be periodic with period w > 0, that is, for all t E R. Finally, the trajectory can be nonintersecting (see Figure 7.2.1c), by which we mean that

It is straightforward to show that the trajectory of a dynamical system cannot be of the intersecting nonperiodic form shown in Figure 7.2.1d. To demonstrate this, assume the contrary, that, for a given x 0 E X, we have

By applying Bt-t 1 to both sides of this equation, we have Bt-tl (St 1(x0 ))

= Bt-t1(St~ (x0 )).

By the group property (b) of Definition 7.2.1, we also have St-t1 (Sh (x 0 ))

= St(x 0 )

and Bt-tt(St~(x0 ))

Hence, with w = (t 2

-

= St+(t~-tt)(x

0

tl), our assumption leads to St(x0 )

= St+w(x0 ),

).

7.2. Dynamical and Semidynamical Systems

193

X

X

()

• ; • s, (;)

~=Sw(;)

(b)

(a)

X

X

(c)

(d)

FIGURE 7.2.1. Trajectories of a dynamical system in its phase space X. In (a) the trajectory is a stationary point, whereas in (b) the trajectory is a periodic orbit. Trajectory (c) is of the nonintersecting type. The intersecting trajectory shown in (d) is not possible in a dynamical system.

implying that the only possible intersecting trajectories of a dynamical system are periodic. However, it is often the case that the evolution in time of data is observed to be of the intersecting nonperiodic type. For example, the tw~ dimensional projection of the trajectory of a three-dimensional system might easily be of this type. The projection of a trajectory of a dynamical system is called the trace of the system. The following is a more precise definition. Definition 7.2.2. Let X and Y be two topological Hausdorff spaces, (St(y))

for all t E R.

Proof. Let Y be the space of all continuous functions from R into X (note that the elements of space Y are functions, not points). Let a dynamical system {St' h'eR, St': Y --+ Y, operating on Y, be a simple shift so that starting from a given y E Y we have, after the operation of St', a new function y(t + t'). This may be represented by a diagram, y(t)

~ y(t + t'),

or, more formally, St'(y)(t)

Define a projection 4>: Y

--+

= y(t + t').

X by 4>(y) = y(O),

then projection 4> is just the evaluation of y at point t = 0. Let g: R be an arbitrary continuous function so that, by our definitions,

St'(g)(t)

--+

X

= g(t + t')

and 4>(St'(g)) = St•g(O} = g(t'},

showing that g is the trace of a trajectory of the dynamical system {St' h• eR operating in Y, namely, a trajectory starting from the initial point g. Further, Y will be a topological Hausdorff space, and (t', y) --+ St' (y) a continuous mapping if we equip the function space Y with the topology of uniform convergence on compact intervals. This, coupled with the trivial observation that St'+t"(Y) = St•(St"(y)) and So(Y) = y, completes the proof of the theorem. •

Remark 7.2.3. Note that the proof of this theorem rests on the identification of the functions on X as the objects on which the new dynamical system {St' h•eR operates. 0

7.3. Invariance, Ergodicity, Mixing, and Exactness

195

Semidynamical Systems

Definition 7.2.3. A semidynamical system {Sth~o on X is a family of transformations Bt: X -+ X, t E R+, satisfying

= x for all x EX; Bt(St'(x)) = Bt+t•(x) for all x EX, with t,t' E R+;

(a.) So(x) (b)

(c) The mapping (t,x)-+ Bt(x) from X x R+ into X is continuous.

Remark 7.2.4. The only difference between dynamical and semidynamical systems is contained in the group property [compare conditions (b) of Definitions 7.2.1 and 7.2.3]. The consequence of this difference is most important, however, because semidynamical systems, in contrast to dynamical systems, are not invertible. It is this property that makes the study of semidynamical systems so important for applications. Henceforth, we will confine our attention to semidynamical systems. 0 Remark 7.2.5. An examination of the proof of Theorem 7.2.1 shows that it is also true for semidyna.mical systems. 0 Remark 7 .2.6. On occasion a. family of transformations { Bt }t~ 0 satisfying properties (a.) and (b) will be called a. semigroup of transformations. This is because property (b) in Definition 7.2.3 ensures that transformations Bt form an Abelian semigroup in which the group operation is the composition of two functions. Thus a. semidyna.mical system is a. continuous semigroup. 0

Remark 7.2.7. The area. of topological dynamics examines the behavior of semidyna.mical systems from a topological perspective. Here, however, since we are primarily interested in highly irregular behaviors, our main tools will be measures on X. 0

7.3 Invariance, Ergodicity, Mixing, and Exactness in Semidynamical Systems Inva.riance and the Individual Ergodic Theorem From the continuity property (c) of Definition 7.2.3, all our transformations Bt are measurable, that is, for all A E A,

196


where, as usual, Bt 1 (A) denotes the counterimage of A, namely, the set of all points x such that St(x) EA. Thus we can state the following definition.

Definition 7.3.1. A measure p. is called invariant under a family {St} of measurable transformations St: X -+ X if for all A EA.

(7.3.1)

As for discrete time processes, we will say interchangeably either that a measure is invariant under {St} or that transformations {St} are measure preserving when equation (7.3.1) holds. Given a finite invariant measure p., we can formulate a continuous time analog of Theorem 4.2.3, which is also known as the Birkhoff individual ergodic theorem. Theorem 7.3.1. Let p. be a finite invariant measure with respect to the semidynamicalsystem {Sth>o, and let f: X-+ R be an arbitrary integrable function. Then the limit -

J*(x)

11T

= Tlim -T ..... oo

f(St(x)) dt

{7.3.2)

0

exists for all x E X except perhaps for a set of measure zero.

Proof. This theorem may be rather easily demonstrated using the corresponding discrete time result, Theorem 4.2.3, if we assume, in addition, that for almost all x EX the integrand f(St(x)) is a bounded measurable function of t. Set

1 1

g(x)

=

f(St(x)) dt

and assume at first that T is an integer, T = n. Note also that the group property {b) of semidynamical systems implies that

Then the integral on the right-hand side of (7.3.2) may be written as 1

rT f(St(x)) dt = ;;1 Jor f(St(x)) dt

T Jo

1

n-1

=;; L

1

f(St(x))dt

1

f(St-k(SA:(x))) dt

A:+l

k=O k

1

n-1

= ;; L

A:+l

k=O k

7.3. Invariance, Ergodicity, Mixing, and Exactness n-1

197

1

L 1o f(St'(S~c(x))) dt' n

1 = -

k=O

1

n-1

=- l:u(S~c(x)). n k=O

However, S~c

= 81 o S~c-1 = 81

k

o o o 81 =Sf, so that

11n f(St(x)) dt = lim -1n-1 L g(S~(x)),

lim n-+oo

n

n-+oo

0

n

k=O

and the right-hand side exists by Theorem 4.2.3. Call this limit f*(x). If T is not an integer, let n be the largest integer such that n < T. Then we may write

1r f(St(x)) dt = Tn . n1r 1r lo f(St(x)) dt + T ln f(St(x)) dt.

T lo

As T -+ oo, the first term on the right-hand side converges to f*(x), as we have shown previously, whereas the second term converges to zero since f(St(x)) is bounded. • As in the discrete time case, the limit f*(x) satisfies two conditions: (C1) f*(St(x)) and

(C2)

i

= f*(x),

f*(x) dx

=

i

a.e. in x for every t

f(x) dx.

~

0,

(7.3.3)

(7.3.4)

Ergodicity and Mixing We now develop the notions of ergodicity and mixing for semidynamical systems. Exact semidynamical systems are considered in the next section. Under the action of a semidynamical system {Stlt>o, a set A E A is called invariant if fort~ 0. (7.3.5) Again we require that for every t ~ 0 the equality (7.3.5) is satisfied modulo zero (see Remark 3.1.3). By using this notion of invariant sets, we can define ergodicity for semidynamical systems.

Definition 7.3.2. A semidynamical system {Sth>o, consisting of nonsingular transformations St: X -+ X is ergodic if every invariant set A E A

198


is such that either JL(A) = 0 or JL(X\A) = 0. (Recall that a set A for which JL(A) = 0 or JL(X \A)= 0 is called trivial.)

Example 7.3.1. Again we consider the example of rotation on the unit circle, originally introduced in Example 4.2.2. Now X= (0, 211") and (mod 211").

St(x)=x+wt

(7.3.6)

St is measure preserving (with respect to the natural Borel measure on the circle) and, for w i= 0, it is also ergodic. To see this, first pick t = to such that wto/211" is irrational. Then the transformation St0 : X-+ X is ergodic, as was shown in Example 4.4.1. Since St0 is ergodic for at least one to, every A must be trivial by Definition (invariant) set A that satisfies St;; 1 (A) 4.2.1. Thus, any set A that satisfies (7.3.5) must likewise be trivial, and the semidynamical system {Sth>o with St given by (7.3.6) is ergodic. 0

=

Remark 7.3.1. It is interesting to note that, for any to commensurate with 211"/W (e.g., to= 11"/w), the transformation St0 is not ergodic. This curious result illustrates a very general property of semidynamical systems: For a given ergodic semidynamical system {Sth2:o, there might be a specific to for which St 0 is not ergodic. However, if at least one St 0 is ergodic, then the entire semidynamical system {St h>o is ergodic. 0 We now turn our attention to mixingin semidynamical systems, starting with the following definition.

Definition 7.3.3. A semidynamical system {Sth2:o on a measure space (X, A, JL) with a normalized invariant measure JL is mixing if lim JL(A n 8t" 1 (B)) = JL(A)JL(B)

t-+oo

for all A,B EA.

(7.3.7)

Thus, in continuous time systems, the interpretation of mixing is the same as for discrete time systems. For example, consider all points x in the set An St" 1 (B), that is, points x such that x E A and St(x) E B. From (7.3.7), for large t the measure of these points is just JL(A)JL(B), which means that the fraction of points starting in A that eventually are in B is given by the product of the measures of A and B in the phase space X. By Definition 7.3.3 the semidynamical system {Sth2:o, consisting of rotation on the unit circle given by (7.3.6), is evidently not mixing. This is because, given any two nontrivial disjoint sets, A, B E A, the left-hand side of (7.3.7) is always zero for wt = 21rn (nan integer), whereas JL(A)JL(B) i= 0. A continuous time system that is mixing is illustrated in Example 7. 7.2. Remark 7 .3.2. The concepts of ergodicity and mixing are also applicable to dynamical systems. In this case, condition (7.3. 7) can be replaced by lim JL(A n St(B))

t-+oo

= JL(A)JL(B)

(7.3.8)

7.4. Semigroups

199

since

Exactness

Definition 7.3.4. Let (X, A, p.) be a normalized measure space. A measurepreserving semidynamical system {Sth2:o such that St(A) E A for A E A is exact if lim p.(St(A))

t-+oo

=1

for all A E A, p.(A)

> 0.

(7.3.9)

Example 11.1.1 illustrates exactness for a continuous time semidynamical system.

Remark 7.3.3. As in discrete time systems, exactness of {Sth>o implies that {Sth2:o is mixing. 0 Remark 7.3.4. Due to their invertibility, dynamical systems cannot be exact. This is easily seen, since p.(S-t(St(A))) = p.(A) and, thus, the limit in (7.3.9) is p.(A) and not 1, for all A EA. If the system is nontrivial and contains a set A such that 0 < p.(A) < 1, then, of course, condition (7.3.9) is not satisfied. 0

7.4 Semigroups of the Frobenius-Perron and Koopman Operators As we have seen in the discrete time case, many properties of dynamical systems are more easily studied by examining ensembles of trajectories rather than single trajectories. This is primarily because the ensemble approach leads to semigroups of linear operators, and, hence, the techniques of linear functional analysis may be applied to a study of their properties. Since, for any fixed tin a semidynamicalsystem {Sth>o, the transformation St is measurable, we can adopt the discrete time definitions of the Frobenius-Perron and Koopman operators directly for the continuous time case. Frobenius-Perron Operator Assume that a measure p. on X is given and that all transformations St of a semidynamicalsystem {St}t2:0 are nonsingular, that is, for each A E A such that p.(A)

= 0.

200


Then, analogously to (3.2.2), the condition

f

}A

=f

Ptf(x)p.(dx)

ls;- 1 (A)

f(x)p.(dx)

(7.4.1)

forAEA

for each fixed t ~ 0 uniquely defines the Frobenius-Perron operator Pt: L 1 (X)-+ L 1 (X), corresponding to the transformation St. It is easy to show, with the aid of (7.4.1), that Pt has the following properties: (FP1) Pt(Adt

+ >.2!2) = AtPth + >.2Pth,

>.11 >.2 E R; {FP2) (FP3)

Pt! ~ 0,

L

(7.4.2) if I~ 0;

=

Ptf(x)p.(dx)

(7.4.3)

L

f(x)p.(dx),

for all

f

E L1 •

{7.4.4)

Thus, for every fixed t, the operator Pt: L 1 (X) -+ L 1 (X) is a Markov operator. The entire family of Frobenius-Perron operators Pt: L 1 (X) -+ L 1 (X) satisfies some properties similar to (a) and (b) of Definition 7.2.3. To see this, first note that since St+t' = StoSt', then s;_;t, = S;, 1 (8t" 1 ) and, thus,

f

1A

Pt+t'f(x)p.(dx)

=f f(x)p.(dx} = f Js-t+t' (A) 1s-t' = f Pt•f(x)p.(dx) 1

1

(s-t 1 (A))

f(x)p.(dx)

ls; 1 (A)

=

i

Pt(Pt•f(x))p.(dx).

This implies that for all

f E L 1 {X), t, t' ~ 0

(7.4.5)

and, thus, Pt satisfies a group property analogous to (b) of Definition 7.2.3. Further, since S0 (x) = x, we have S0 1 (A) =A and, consequently,

f

}A

Pof(x)p.(dx)

= f

Js0 (A) 1

f(x)p.(dx)

=f

}A

f(x)p.(dx)

implying that

Pol=!

(7.4.6)

Hence Pt satisfies properties (a) and (b) of the definition of a semidynamical system.

7.4. Semigroups

201

The properties of Pt in (7.4.2)-(7.4.6) are important enough to warrant the following definition.

Definition 7.4.1. Let (X, A, JJ) be a measure space. A family of operators L 1 (X), t ~ 0, satisfying properties (7.4.2)-(7.4.6) is called a stochastic semigroup. Further, if, for every f E L 1 and to~ 0,

Pt:

L 1 (X)

-+

lim IIPtf - Ptofll t-+to

= 0,

then this semigroup is called continuous. A very important and useful property of stochastic semigroups is that (7.4.7) and, thus, from the group property (7.4.5), the function t-+ is a nonincreasing function of t. This is simply shown by

IIPt+t'ft- Pt+t•hll

IIPt/1- Pthll

= 11Pt·(Ptf1- Pth)ll:::; IIPtft- Pthll,

which follows from (7.4. 7). By using this property, we may now proceed to prove a continuous time analog of Theorem 5.6.2.

Theorem 7.4.1. Let {Pth>o be a semigroup of Markov operators, not necessarily continuous. Assume that there is an hE L 1 , h(x) ~ 0, llhll > 0 such that lim II(Ptf- h)_ II= 0 for every fED. (7.4.8) t-+oo

Then there is a unique density/. such that Ptf. = /. for all t thermore, for every fED. lim Ptf = /. t-+oo

Proof. Take any to > 0 and define P (7.4.8) lim

n-+oo

= Pt

II(P I- h)_ II= o

0

so that Pnto

lim

Pf = /.

0. Fur(7.4.9)

= pn. Then, from

for each fED.

Thus, by Theorem 5.6.2, there is a unique /. E D such that P /. n-+oo

~

= /. and

for every fED.

Having shown that Ptf. = /. for the set {to, 2t0 , ••• }, we now turn to a demonstration that Ptf. = /. for all t. Pick a particular time t', set ft = Pt' /., and note that /. = pn /. = Pntof•· Therefore,

I!Pt•f•- f•ll

= IIPt'(Pntof*)- /.11 = IIPnta(Pt•!.)- /.II = IIPn(Pt•f.)- /.II = IIPnft -f·ll·

(7.4.10)

202


Thus, since, and the left-hand side of (7.4.10) is independent of n, we must have IIPt' /./.II = 0 so Pt' /. = J•. Since t' is arbitrary, we have Pd. = /. for all t ~ 0. Finally, to show (7.4.9), pick a function f E D so that IIPtf- /.11 liFt!- .Pt/.11 is a nonincreasing function. Pick a subsequence tn = nto. We know from before that limn-oo IIPt,.!- /.II = 0. Thus we have a nonincreasing function that converges to zero on a subsequence and, hence

=

lim

t-oo

IIPt!- /.II = o. •

Remark 7.4.1. The proof of this theorem illustrates a very important property of stochastic semigroups: namely, a stochastic semigroup {Pth;::o is called asymptotically stable if there exists a unique /. E D such that

P/. = /.

and if condition (7.4.9) holds for every IE D.

0

Remark 7.4.2. From the above definition, it immediately follows that the asymptotic stability of a semigroup {Pt h;::o implies the asymptotic stability of the sequence {P{!} for arbitrary to > 0. The proof of Theorem 7.4.1 shows that the converse holds, that is, if for some to > 0 the sequence {Pt~} is asymptotically stable, then the semigroup {Pth;::o is also asymptotically stable. 0 Stochastic semigroups that are not semigroups of Frobenius-Perron operators can arise, as illustrated by the following example. Example 7.4.1. Let X by Ptf(x)

= R, I

E

L1 (X), and define Pt: L 1 (X)

= /_: K(t,x,y)J(y)dy,

Pof(x) = l(x),

where

y)2] .

1 [ (xK(t,x,y) = ~ exp - 2a2t

It may be easily shown that the kernel K(t, x, y) satisfies: (a) K(t,x,y) ~ 0; (b) /_: K(t, x, y) dx

(c) K(t+t',x,y)

= 1; and

= /_: K(t,x,z)K(t',z,y)dz.

--+

L 1 (X)

(7.4.11)

(7.4.12}

7.4. Semigroups

203

From these properties it follows that Pt defined by (7.4.11) forms a continuous stochastic semigroup. The demonstration that {Pth>o defined by (7.4.11) and (7.4.12) is not a semigroup of Frobenius-Perron operators is postponed to Remark 7.10.2. That (7.4.11) and (7.4.12) look familiar should come as no surprise as the function u(t,x) = Ptf(x) is the solution to the heat equation

au

u2

8t = 2 with the initial condition u(O,x)

8 2u 8x2

= l(x)

fort> O,x E R

forxER.

0

(7.4.13)

(7.4.14)

The Koopman Operator Again let {Sth>o be a semigroup of nonsingular transformations St in our topological Hausdorff space X with Borel u-algebra A and measure J.l.· Recall that the St are nonsingular if, and only if, for every A E A such that JJ(A) = 0, JJ(S; 1 (A)) = 0. Further, let IE L 00 (X). Then the function Utf, defined by Utf(x) = I(St(x)), (7.4.15) is again a function in L 00 (X). Equation (7.4.15) defines, for every t ;:::: 0, the Koopman operator associated with the transformation St. The family of operators {Uth~o, defined by (7.4.15), satisfies all the properties of the discrete time Koopman operator introduced in Section 3.3. It is also straightforward to show that {Uth~o is a semigroup. To check this, first note from the defining formula (7.4.15) that

Ut+t•l(x) = I(St+t•(x)) = I(St(St•(x))) = Ut(Ut•l(x)), which implies

=

Ut+t'l Ut(Ut•l) for all IE L 00 • Furthermore, Uol(x) = I(So(x)) = l(x), or Uol =I

for all IE L 00 ,

so that {Uth>o is a semigroup. Finally, the-Koopman operator is adjoint to the Frobenius-Perron operator, or

(Ptl, g) = (!, Utg}

for all

I

E L 1 (X), g E L 00 (X) and t ;:::: 0. (7.4.16)

The family of Koopman operators is, in general, not a stochastic semigroup because Ut does not map L 1 into itself (though it does map L 00 into itself) and satisfies the inequality ess sup IUtfl ::5 ess sup Ill

204


~. FIGURE 7.4.1. Plots of /(z) and Te/(z) = f(x- ct), for c > 0.

instead of preserving the norm. In order to have a common notion for families of operators such as {Pt} and {Ut}, we introduce the following definition.

Definition 7.4.2. Let L = LP, 1 ~ p ~ oo. A family {Tth;::o of operators, 1t: L - L, defined for t ~ 0, is called a semigroup of contracting linear operators (or a semigroup of contractions) if Tt satisfies the following conditions: (a) Tt(l..dt

+ l..2h) = l..tTdt + l..2Tth, for ft, h e L, >..to >..2 e R;

(b) IITtfiiL ~ II/IlL for IE L; (c)

To/=/,

(d) Tt+t'f

for

f

E L; and

= 1t(Tt•/), for IE L.

Moreover, if for

lim !lTd- TtofiiL = 0, t-+to

f

E L, to ~ 0,

then this semigroup is called continuous. Example 7.4.2. Consider the family of operators {Tth;::o defined by (see Figure 7.4.1) Ttf = f(x-ct) for x E R, t ~ 0. (7.4.17) These operators map L = LP(R), 1 ~ p ~ oo, into itself, satisfy properties (a)-(d) of Definition 7.4.2., and form a semigroup of contractions. To see that property (b) holds for Tt. use the "change of variables" formula,

IITt!IFr_P =

i:

lf(x- ct)IP dx

= /_: 1/(y)IP dy = 11/II':,P

when p < oo, and the obvious equality,

1111/IILoo = ess sup 1/(x- ct)l = ess sup 1/(x)l = 11/IILoo Ill

Ill

7.5. Infinitesimal Operators

0

c!

I

205

I+C!

FIGURE 7.4.2. Function llct,l+ctJ(x) -l[o,ce)(x)l versus x.

whenp = oo. The remaining properties (a), (c), and (d) follow immediately from the definition of 7t in equation (7.4.17). Finally, we note that if p = 1 then this semigroup of contractions is continuous. To see this, first use

ll7t/- TtofiiLl

= /_: 1/(x- ct) -

f(x- cto)l dx

= /_: 1/(y)- f(y- c(to- t))l dy and note that the right-hand side converges to zero by Corollary 5.1.1. A slightly more complicated calculation shows that {Tth>o is a continuous semigroup of contractions for every 1 ~ p < oo. However, in £ 00 the semigroup {7t}t~o given by (7.4.17) is not continuous except in the trivial case 1[o,l). We then have when c 0. This may be easily shown by setting f

=

=

Ttf(x)

= 1[o,lJ(X- ct) = 1[ct,ct+1J(x)

and, as a consequence, 117t/- !IlL"" for 0

= ess sup j1(t,l+ctJ(x)- 1[o,ct)(x)j = 1

"' < ct < 1. ThusiiTd- filL""

does not converge to zero as t - 0. This

may be simply interpreted as shown in Figure 7.4.2 where the hatched areas

corresponding to the function l1(1,1ctl - 1[o,ct)l disappear as t - 0 but the heights do not. 0

7.5 Infinitesimal Operators The problems associated with the study of continuous time processes are more difficult than those encountered in discrete time systems. This is partially due to concerns over continuity of processes with respect to time. Also, equivalent formulations of discrete and continuous time properties

206


may appear more complicated in the continuous case because of the use of integrals rather than summations, for example, in the Birkhoff ergodic theorem. However, there is one great advantage in the study of continuous time problems over discrete time dynamics, and this is the existence of a new tool-the infinitesimal operator. In the case of a semidynamical system {St }t~o arising from a system of ordinary differential equations (7.1.2), the infinitesimal operator is simply the function F(x). This connection between the infinitesimal operator and F( x) stems from the formula lim x(t) - x(O) = F(xo),

t

t-+0

where x(t) is the solution of (7.1.2) with the initial condition (7.1.3). This can be rewritten in terms of the transformations St as lim St(xo) - xo

= F(xo).

t

t-+0

This relation offers some insight into how the infinitesimal operator may be defined for semigroups of contractions in general, and for semigroups of the Frobenius-Perron and Koopman operators in particular. Definition 7.5.1. Let L = V', 1 ~ p ~ oo, and {Tth~o be a semigroup of contractions. We define by I>(A) the set of all f E L such that the limit

Af = lim Ttf- f

(7.5.1)

t

t-+0

exists, where the limit is considered in the sense of strong convergence (cf. Definition 2.3.3). Thus (7.5.1) is equivalent to lim ''Af- Ttft

t-+0

'II = L

0.

Operator A: I>( A) -+ Lis called the infinitesimal operator. It is evident that the subspace V(A) is linear or that

>..tft + >..2/2

E I>(A)

for all

/t, /2 E I>(A), and >..17 >..2 E R.

Furthermore, operator A: V(A)-+ Lis linear or A(>..tft + >..2!2) =>.tAft+ >..2Ah

for all

/t, hE V(A) and >..t, >..2 E R.

In general, the domain V(A) of operator A is not the entire space L. Before deriving the infinitesimal operators for the Frobenius-Perron and Koopman semigroups, we consider the following example. Example 7.5.1. Let X = R and L = V'(R), 1 ~ p < oo. Consider a semigroup {Tth~o on L defined, as in Example 7.4.2, by

Ttf(x) = /(x- ct)


207

cf. Figure 7.4.1). By the mean value theorem, iff is 0 1 on R, then

f(xwhere 191 :5 1 and f' continuous on R, then

ct;-

f(x) = -cf'(x _(Jet),

= dJ jdx.

Thus, if

Af =lim Ttlt-+0 t

!'

is bounded and uniformly

I= -cf',

and the limit is uniform on R and consequently strong in VXJ. Further, iff {and thus/') has compact support {zero outside a bounded interval}, then the limit is strong in every LP, 1 :5 p :5 oo. Thus, all such f belong to 'D(A) and for them A is just differentiation with respect to x and multiplication by -c. 0 In studying infinitesimal operators and many other problems of analysis, functions that are equal to zero outside a compact set play an important role. It is customary to call such mappings functions with compact support. This notion does not coincide very well with our definition of support given by equation {3.1.8) even though it is commonly accepted. Thus, we will also use this traditional terminology, hoping that it will not lead to confusion or misunderstanding. Having introduced the notion of infinitesimal operators, and illustrated their calculation in Example 7.5.1, we now wish to state a theorem that makes explicit the relation among semigroups of contractions, infinitesimal operators, and differential equations. First, however, we must define the strong derivative of a function with values in L = LP. Given a function u: l:i - L, where l:i c R, and a point t 0 E l:i, we define the strong derivative u'(to) by '( ) li u(t) - u(to) uto= m , t-+to t- to where the limit is considered in the sense of strong convergence. This definition is equivalent to lim llu(t)- u(to)- u'(to)ll t-+to t- to L

= 0.

(7.5.2)

By using this concept, we can see that the value of the infinitesimal operator for f E V{A), Af, is simply the derivative of the function u(t) = Ttl at t 0. The following theorem gives a more sophisticated relation between the strong derivative and the infinitesimal operator.

=

Theorem 7.5.1. Let {Tt}t~o be a continuous semigroup of contractions acting on L, and A: V(A) - L the corre8p0nding infinitesimal operator. Further, let u(t) =Ttl for fixed I E 'D{A). Then u(t) satisfies the following properties:

208


(1) u(t) E 'D(A) fort~ 0;

(2) u'(t) exists fort~ 0; and (3) u(t) satisfies the differential equation fort~

u'(t) = Au(t)

0

(7.5.3)

and the initial condition

u(O) =f.

(7.5.4)

Proof. Fort= 0, properties (1)-(3) are satisfied by assumption. Thus we may concentrate on t > 0. Let t 0 > 0 be fixed. By the definition of u(t), we have u(t) - u(to) = Ttf- Ttof t-to t-to Noting that Tt = Tt-t 0 Tt 0 for t > to this differential quotient may be rewritten as ~'---"-=

~---"".;....

u(t)- u(to) = Tto (Tt-tof-

t-to Because

f

E V(A),

t-to

1)

fort> to.

(7.5.5)

the limit of

Tt-tof- I t- to exists as t -+ to and gives Af. Thus the limit of (7.5.5) as t -+ to also exists and is equal to T't, 0 Af. In an analogous fashion, if t < to, we have Tto = TtTt 0 -t and, as a consequence, u(t) - u(to)

t-to

= Tt (Tt

0

-d- 1)

to-t

fort to.


209

Since the limit of the differential quotient on the left-hand side exists as t-+ to, the limit on the right-hand side also exists as t-+ to, and we obtain u'(to)

= ATt f, 0

which proves that Tt 0 f E 1>(A) and that u'(to) = Au(to).

•

Remark 7.5.1. The main property of the set V(A) that follows directly from Theorem 7.5.1 is that, for f E V(A), the function u(t) =Ttl is a solution of equations (7.5.3) and (7.5.4). Moreover, the solution can be proved to be unique. Unfortunately, in general1>(A) is not the entire space L, although it can be proved that, for continuous semigroups of contractions, V(A) is dense in L. 0 In Theorem 7.5.1, the notion of a function u: [0, oo) -+ L, where L is again a space of functions, may seem strange. In fact, u actually represents a function of two variables, t and x, since, for each t ~ 0, u(t) E II'. Thus we frequently write u(t)(x) = u(t,x), and equation (7.5.3) is to be interpreted as an equation in two variables. Applying this theorem to the semigroup considered in Examples 7.4.2 and 7.5.1 with L =II', 1 :5 p < oo, it is clear that this semigroup satisfies equation (7.5.3), where u(t,x)

= Ttf(x) = f(x- ct)

Af=

dJ -c-, dx

and

f

E V(A).

These relations can, in turn, be interpreted as meaning that u( t, x) satisfies the first-order partial differentiation equation

au at

au

+ c ax

with the initial condition u(O,x)

= 0

(7.5.7)

= f(x).

Remark 7.5.2. It is important to stress the large difference in the two interpretations of this problem as embodied in equations (7.5.3) and (7.5.7). From the point of view of (7.5.7), u(t, x) is thought of a8 a function of isolated coordinates t and x that evolve independently and whose derivatives 8uf8t and 8ujax are evaluated at specific points in the (t,x)-plane. However, in the semigroup approach that leads to (7.5.3), we are considering the evolution in time of a family of functions, and the derivative du(t)fdt is to be thought of as taken over an entire ensemble of points. This is made somewhat clearer when we take into account that u(t) = Ttl has a time derivative u'(t 0 ) at a point t 0 if (7.5.2) is satisfied, that is, lim t-+to

joo Iu(t)(x)u(to)(x) - u'(to)(x)IP dx = 0. t - to _ 00

210


Moreover, u(t)(x) and u'(t)(x) with fixed tare defined as functions of x up to a set of measure zero. D

7.6 Infinitesimal Operators for Semigroups Generated by Systems of Ordinary Differential Equations We now turn to an explicit calculation of the infinitesimal operators for the semigroups {Pth2:o and {Uth2:o generated by ad-dimensional system of ordinary differential equations dx

-

dt

(7.6.1a)

=F(x)

or (7.6.1b)

i = 1, ... ,d,

where x = (xt, ... ,xd). The semigroup of transformations {Stlt2:0 corresponding to equations (7.6.1) is defined by the formula St(x0 )

= x(t),

(7.6.2)

where x(t) is the solution of (7.6.1) corresponding to the initial condition (7.6.3) We will assume that the F, have continuous derivatives 8F,j8x;, i,j = 1, ... , d, and that for every x 0 e Jld the solution x(t) exists for all t E R. This guarantees that (7.6.2) actually defines a group of transformations. Because of a well-known theorem on the continuous dependence of solutions of differential equations on the initial condition, {St}t;::o is a dynamical system (see Example 7.1.1). As the derivative of the infinitesimal operator AK for the Koopman operator is simpler, we start from there. By definition we have

Therefore Utf(x 0 )

-

f(x 0 )

t

f(St(x 0 )) - f(x 0 ) _ f(x(t)) - f(x 0 ) t t _

so that, iff is continuously differentiable with compact support, then by the mean value theorem Utf(xo)t- /(xo)

=

tJz, i=l

(x(fJt))xWJt)

=

t !~~:, i=l

(x(6t))F,(x(6t)),

7.6. Semigroups and Differential Equations

211

where 0 < 8 < 1. Now by using equation (7.6.2), we obtain

Utf(~rP)t- l(:rP) =

t !:~:,

(SBt(xo))F,(SBt(xo)).

(7.6.4)

i=l

Since the derivatives

I z, have compact support

uniformly for all x 0 • Thus (7.6.4) has a strong limit in L 00 , and the infinitesimal operator AK is given by (7.6.5) Observe that equation (7.6.5) was derived only for functions I with some special properties, namely, continuously differentiable I with compact support. These functions do not form a dense set in L 00 , which is not surprising since it can be proved that the semigroup {Uth>o is not, in general, continuous in L 00 • It does become continuous in a subspace of L 00 consisting of all continuous functions with compact support (see Remark 7.6.2). Hence, if I is continuously differentiable with compact support, then by Theorem 7.5.1 for such I the function

u(t,x) = Utf(x) satisfies the first-order partial differential equation (7.5.3). From (7.6.5) it may be written as (7.6.6)

Remark 7.6.1. It should be noted that the same equation can be immediately derived for u(t, x) = I (St (x)) by differentiating the equality u(t, S_t(x)) = l(x) with respect to t. In this case I may be an arbitrary continuously differentiable function, not necessarily having compact support. However, in this case (7.6.6) is satisfied locally at every point (t,x) and is not an evolution equation in L 00 (cf. Remark 7.5.2}. 0 We now turn to a derivation of the infinitesimal operator for the semigroup ofFrobenius-Perron operators generated by the semigroup of (7.6.la). This is difficult to do if we start from the formal definition of the FrobeniusPerron operator, that is,

[ Ptl(x)p.(dx)

}A

= [

1s; 1 (A)

l(x)p.(dx)

forA EA.

212


However, the derivation is straightforward if we start from the fact that the Frobenius-Perron and Koopman operators are adjoint, that is,

(Ptf,g}

= (/,Utg},

(7.6.7)

Subtract {/,g) from both sides of (7.6.7) to give

{Pd- /,g)=(!, Utg- g} or, after division on both sides by t, {(Ptf- /)/t,g}

= {/, (Utg- g)ft}.

(7.6.8)

Now let f E V(App) and g E V(AK), where App and AK denote, respectively, the infinitesimal operators for the semigroups of Frobenius-Perron and Koopman operators. Take the limit as t-+ 0 in (7.6.8) to obtain (Appf,g}

= {/,AKg}.

(7.6.9)

However, from equation (7.6.5) the right-hand side of (7.6.9) can be written as

provided g is a continuously differentiable function with compact support. If we write out this scalar product explicitly and note that X = Rd and dp. = dx1 · · · dxd = dx, we obtain

for f E V(App ), which is also continuously differentiable. Since g has compact support,

and thus

7.6. Semigroups and Differential Equations

213

which is a d-dimensional version of the "integration by parts" formula. From this and equation (7.6.9), we finally obtain

(AFPI,g}

= (-

t. {}~:i) ,g).

This formula holds for every continuously differentiable I e 1>( AF p) and for every continuously differentiable g with compact support. Such a function g is automatically contained in 'D(AK)· Therefore (7.6.10) for continuously differentiable I E 'D(AFP ). Again, by using Theorem 7.5.1, we conclude that the function

u(t,x)

= Ptf(x)

satisfies the partial differential equation (continuity equation) d

au+ 2: a(uFi) = o. 8t

i=l

(7.6.11)

OXi

Example 7.6.1. As a special case of the system (7.6.1) of ordinary differential equations, let d = 2n and consider a Hamiltonian system whose dynamics are governed by the canonical equations of motion (Hamilton's equations) i

= 1, ... ,n,

(7.6.12)

where H(p, q) is the system Hamiltonian. In systems of this type, q and p are referred to as the generalized position and momenta, respectively, whereas His called the energy. Equation (7.6.11) for Hamiltonian systems takes the form 8u + 8u 8H _ 8u 8H = O,

at

t oq, i=l

{}p,

{}pi

oq,

which is often written as

ou 8t + [u,H] = 0, where [u, H] is the Poisson bracket of u with H. For Hamiltonian systems, the change with time of an arbitrary function g of the variables q1, ... , qn, Ph ... , Pn is given by

214


In particular, if we take g to be a function of the energy H, then dg dt

= dg

dH dHdt

= dg (H H] =0 dH

I

since [H, H] = 0. Thus any function of the generalized energy His a constant of the motion. 0

Remark 7.6.2. The semigroup ofFrobenius-Perron operators {Pth~o corresponding to the system {Sth~o generated by equation (7.6.1) is continuous. To show this note that, since St is invertible (S;- 1 B-t), by Corollary 3.2.1 we have (7.6.13) Ptf(x) = I(B-t(x))Lt(x),

=

where Lt is the Jacobian of the transformation B-t· Thus, for every continuous I with compact support,

uniformly with respect to x. This implies that lim

t-to

IIPtf- Ptolll = t-to lim f IPtl(x)- Ptol(x)l dx = 0 } Rd.

since the integrals are, in actuality, over a bounded set. Because continuous functions with compact support form a dense subset of L 1 , this completes the proof that {Pt}t~o is continuous. Much the same argument holds for the semigroup {Uth~o if we restrict ourselves to continuous functions with compact support. In this case, from the relation

Utf(x) = I(St(x)), it immediately follows that Utf is uniformly convergent to Ut 0 l as t - to. For this class of functions the proof of Theorem 7.5.1 can be repeated, thus showing that equation (7.5.3) is true for I E 'D(AK ). 0 In the whole space L 00 , it may certainly be the case that {Uth~o is not a continuous semigroup. As an example, consider the differential equation

dx dt

= -c

whose corresponding dynamical system is Stx = x- ct. Thus the semigroup {Uth~o is given by Utf(x) = l(x- ct). As we know from Example 7.4.2, when c -::/= 0, this semigroup is certainly not continuous in L 00 • The continuity of {Pt }t~ 0 is very important since it proves that the set 'D(AFP) is dense in L 1 . Using equation (7.6.13) it may also be shown that 'D(AFP) contains all I with compact support, that have continuous firstand second-order derivatives.

7.7. Applications of Semigroups

215

7.7 Applications of the Semigroups of the Frobenius-Perron and Koopman Operators After developing the concept of the semigroups of the Frobenius-Perron operators in Section 7.4 and introducing the general notion of an infinitesimal operator in Section 7.5 and of infinitesimal operators for semigroups generated by a system of ordinary differential equations in Section 7.6, we are now in a position to examine the utility and applications of these semigroups to questions concerning the existence of invariant measures and ergodicity. This material forms the core of this and the following section.

Theorem 7. 7.1. Let (X, .A, JJ) be a measure space, and St: X-+ X a family of nonsingular tmnsformations. Also let Pt: L 1 -+ £ 1 be the FrobeniusPerron opemtor corresponding to {Sth~o· Then the measure JLJ(A) is invariant with respect to

=

L

{Sth~o

f(x)JL(dx)

if and only if Ptl

=f

for all t ~ 0.

Proof. The proof is trivial, since the invariance of JL1 implies for A

e .A,

which, with the definition of Pt, implies Ptf =f. The converse is equally • Now assume that JJ/ is invariant. Since by the preceding theorem we know that Ptf = f, and easy to prove.

A

f=limPtf-f FP

t-+0

t

1

then Appf = 0. Thus the condition Appf = 0 is necessary for 1'1 to be invariant. To demonstrate that Appf = 0 is also sufficient for 1'1 to be invariant is not so easy, since we must pass from the infinitesimal operator to the semigroup. To deal with this very general and difficult problem, we must examine the way in which semigroups are constructed from their infinitesimal operators. This construction is very elegantly demonstrated by the Hill~Yosida theorem, which is described in Section 7.8. Analogously to the way in which the semigroup of the Frobenius-Perron operator is employed in studying invariant measures of a semidynamical system {Sth~o, the semigroup of the Koopman operator can be used to study the ergodicity of {Sth~o· We start by stating the following theorem.

Theorem 7.7.2. A semidynamicalsystem {Sth>o, with nonsingulartmnsformations St: X -+ X, is ergodic if and only if the fixed points of {Uth~o are constant functions.

216


Proof. The proof is quite similar to that of Theorem 4.2.1. First note that if {Sth~o is not ergodic then there is an invariant nontrivial subset CCX, that is, fort~ 0. 8t" 1 (C) = C By setting

f = 1c, we have Ud = 1c o St = 18 ;-l(C} = 1c =f.

Since Cis not a trivial set, f is not a constant function (cf. Theorem 4.2.1). Thus, if {Sth~o is not ergodic, then there is a nonconstant fixed point of {Uth~o· Conversely, assume there exists a nonconstant fixed point f of {Uth~o· Then it is possible to find a number r such that the set

C

= {x:f(x) < r}

is nontrivial (cf. Figure 4.2.1). Since, for each t

8t" 1 (C)

~

0,

= {x: St(x) E C} = {x: f(St(x)) < r} = {x:Utf < r} = {x:f(x) < r} = C,

subset Cis invariant, implying that {St}t~o is not ergodic. • Proceeding further with an examination of the infinitesimal operator generated by the Koopman operator, note that the condition Utf = f, t ~ 0, implies that . Utf-f AK f =hm =0. t-+0

t

Thus, if the only solutions of AK f = 0 are constant, then the semidynamical system {St}t~o must be ergodic.

Example 7. 7.1. In this example we consider the ergodic motion of a point on a d-dimensional torus, which is a generalization of the rotation of the circle treated in Example 7.3.1. We first note that the unit circle 8 1 is a circle of radius 1, or

8 1 = {m:m = eiz,x E R}. Formally, the d-dimensional torus Td is defined as the Cartesian product of d unit circles 8 1 ' that is, Td

.~. x8 1 = {(m1, ... ,md):mk = eiz~,Xk E R,k = 1, ... ,d}

= S1 x

(cf. Example 6.8.1 where we introduced the two-dimensional torus). Td is clearly a d-dimensional Riemannian manifold, and the functions mk = eiz,,

7. 7. Applications of Semigroups

217

k = 1, ... , d, give a one to one correspondence between points on the torus Td and points on the Cartesian product d

(0,21r)x · · · x[0,21r).

(7.7.1)

x,.

The have an important geometrical interpretation since they are lengths on 8 1. The natural Borel measure on 8 1 is generated by these arc lengths and, by Fubini's theorem, these measures, in turn, generate a Borel measure on~- Thus, from a measure theoretic point of view, we identify Td with the Cartesian product (7.7.1), and the measure p, on ~ with the Borel measure on R,d. We have, in fact, used exactly this identification in the intuitively simpler cases d = 1 (r-adic transformation; see Example 4.1.1 and Remark 4.1.2) and d = 2 (Anosov diffeomorphism; see Example 4.1.4 and Remark 4.1.6). The disadvantage of this identification is that curves that are continuous on the torus may not be continuous on the Cartesian product (7.7.1). Thus we consider a dynamical system {StheR that, in the coordinate system {x,.}, is defined by (mod 211"). We call this system rotation on the torus with angular velocities w11 .. ., Since det(dSt(x)fdx) = 1, the transformation St preserves the measure. We will prove that {StheR is ergodic if and only if the angular velocities w1, ... , Wd are linearly independent over the ring of integers. This linear independence means that the only integers k1, ... , kd satisfying

Wd·

(7.7.2) are kt = · · · = kd = 0. To prove this, we will use Theorem 7.7.2. Choose f E L 2 (Td) and assume Utf f for t E R, where Utf f o St is the group of Koopman operators corresponding to St. Write f as a Fourier series

=

=

f(xl! ... , Xd)

=L

Ok1 ···kd

exp(i(k1x1

+ · · · + kdxd)],

where the summation is taken over all possible integers k11 . .. , kd. Substitution of this series into the identity f(x) = f(St(x)) yields

L a1c

1 ••• ,.d

=L

exp(i( kt Xt

+ · · · + kdxd)]

a,., ... ,.d exp(it(w1k1

exp(i(k1x1

+ · ·· + wdkd)]

+ ··· + kdxd)).

As a consequence we must have fortER

(7.7.3)

218


and all sequences k11 ••• , kd. Equation (7.7.3) will be satisfied either when ak 1 ••• k., = 0 or when (7.7.2) holds. If w1 , ••• ,wd are linearly independent, then the only Fourier coefficient that can be different from zero is ao...o. In this case, then, /(x) = ao ...o is constant and the ergodicity of {StheR is proved. Conversely, if the w1 , .•• , Wd are not linearly independent, and condition (7.7.2) is thus satisfied for a nontrivial sequence kt, ... , kd, then {7.7.3) holds for ak 1 ••• k., = 1. In this case the nonconstant function

/(x) satisfies /(x)

= f(St(x))

= exp[i(k1x1 + · · · + kdxd)]

and {StheR is not ergodic.

0

Remark 7. 7.1. The reason why rotation on the torus is so important stems from its frequent occurrence in applied problems. As a simple example, consider a system of d independent and autonomous oscillators k= l, ... ,d,

(7.7.4)

where q1, ... , qd are the positions of the oscillators and Pl, ... , Pd are their corresponding velocities. For this system the total energy of each oscillator is given by Ek = ~p~ + ~w~q~, k = 1, ... ,d, and it is clear that the Ek are constants of the motion. Assuming that E1, ... , Ed are given and positive, equations (7.7.4} may be solved to give

Pk(t) = Akwk cos(wkt + ak),

qk(t) = Ak sin(wkt + ak),

where Ak = V2Ek/wk and the ak are determined, modulo 211", by the initial conditions of the system. Set Pk = Pk/Akwk and iik = qk/Ak so that the vector (p( t), q(t)) describes the position of a point on a d-dimensional torus moving with the angular velocities w1 , ••• ,wd. Thus, for fixed and positive E1, ... , Ed, all possible trajectories of the system (7.7.4) are described by the group {StheR of the rotation on the torus. At first it might appear that the set of oscillators described by (7.7.4) is a very special mechanical system. Such is not the case, as equations (7.7.4) are approximations to a very general situation. We present an argument below that supports this claim. Consider a Hamiltonian system

dqk

8H

dt = 8pk'

dpk dt

8H

= - 8qk

k= l, ... ,d.

Typically the energy H has the form

H(p,q)

=! :~::::a;k(q)P;Pk + V(q), j,k

(7.7.5)


219

where the first term represents the kinetic energy and V is a potential function. Because the first term in H is associated with the kinetic energy, the quadratic form E;,k a;~~:(q) is symmetric and positive definite. Further, if rf is a stable equilibrium point, then

avl

-o

k= 1, ... ,d

8qk q=qOand the quadratic form,

is also positive definite (we neglect some special cases in which it might be semidefinite). Further, we assume that H{O,q0 ) = V(q0 ) = 0 since the potential is only defined up to an additive constant. Thus, developing H in a Taylor series in the neighborhood of {0, q0 ), and neglecting terms of order three and higher, we obtain

H(p, q)

=!L j,k

a;~~:P;Pk +!

L b;~~:(q;- qJ)(q~~:- q2)

{7.7.6)

j,k

where a;~~: = a;~~:(q0 ) and b;~~: = (82 Vj8q;8q~~:)lqo. Both the quadratic forms E;,k a;~~: and E ·,~~: b;k are symmetric and positive definite. With approximation (7.7.6), t~e original Hamiltonian equations (7.7.5) may be rewritten as

(7.7.7) where the variables Pk and q~~: - q2 denote, respectively, the deviation of the system from the equilibrium point {0, q0 ). Since matrices A = (a;~~:) and B = (b;~~:) are symmetric and positive definite, there exists a nonsingular matrix C such that (Gantmacher, 1959)

and

1

CA- CT

=

1~ ... 0)i (

with positive elements Ai on the diagonal. By introducing new variables q- qO = cTij and p = C- 1p into equations (7.7.7), we obtain

{7.7.8) This new system is completely equivalent to our system {7. 7.4) of independent oscillators with angular velocities w~ = .X~~:. Finally we note that, although our approximation shows the correspondence between rotation on the torus and Hamiltonian systems, the terms

220


we neglected in our expansion of H might play a very important role in modifying the eventual asymptotic behavior of a Hamiltonian system. 0

Remark 7.7.2. Note that the statement and proof of Theorem 7.7.2 are virtually identical with the corresponding discrete time result given in Theorem 4.2.1. Indeed, necessary and sufficient conditions for ergodicity, mixing, and exactness using the Frobenius-Perron operator, identical to those in Theorem 4.4.1, can be stated by replacing n by t. Analogously, conditions for ergodicity and mixing in continuous time systems using the Koopman operator can be obtained from Proposition 4.4.1 by setting n = t. Since all of these conditions are completely equivalent we will not rewrite them for continuous time systems. 0 Example 7. 7.2. To illustrate the property of mixing in a continuous time system we consider a model for an ideal gas in R,3 adapted from Cornfeld, Fomin, and Sinai [1982). However, our proof of the mixing property is based on a different technique. At any given moment of time the state of this system is described by the set of pairs

where Xi denotes the position, and vi, the velocity of a particle. We emphasize that y is a set of pairs and not a sequence of pairs, which means that the coordinate pairs (xi, vi) are not taken in any specific order. Physically this means that the particles are not distinguishable. It is further assumed that the gas is sufficiently dilute, both in spatial position and in velocity, so that the only states that must be considered are such that in every bounded set B c It> there is, at most, a finite number of pairs (xi, vi)· The collection of all possible states of this gas will be denoted by Y, and we assume that the motion of each particle at the gas is governed by a group of transformations St: Y --+ Y given by or, more compactly, by St(Y) = {st(Xi,vi)}, where {stheR is the family of transformations in It> such that Bt(x,v)

= (x+vt,v).

Thus particles move with a constant speed and do not interact. The surprising result, proved below, is that this system is mixing. To study the asymptotic properties of {StheR, we must define a aalgebra and a measure on Y. We do this by first introducing a special measure on Jt>, which is the phase space for the motion of a single particle. Let g be a density on R,3. As usual, the measure associated with g is m 9 (A) =

L

g(v)dv


221

for every Borel set A c R.3, and the measure m in Jl6 = R.3 x R.3 is defined as the product of the usual Borel measure and m 9 , that is,

From a physical point of view this definition of the measure simply reflects the fact that the particle positions are uniformly distributed in R3 , whereas the velocities are distributed with a given density g, for instance, the Maxwellian g(v) = c exp( -lvl 2). With these comments we now proceed to define a u-algebra and a measure on Y. Let Bt, ... , Bn be a given sequence of bounded Borel subsets of R6 for an arbitrary n, and k1, ... , kn be a given sequence of integers. We use C(Bt. ... , Bn; kt. ... , kn) to denote the set of all y = {(xi, vi)} such that the number of elements (xi, vi) that belong to B; is equal to k;, that is,

= {y E Y:

C(Bt, ... , Bn; kt. ... , kn)

#(ynBt)

= kt. ... ,

#(ynBn)

= kn},

(7.7.9) where # Z denotes the number of elements of the set Z. Sets of the form (7. 7.9) are called cylinders. If the sets Bt. ... , Bn are disjoint, then the cylinder is said to be proper. For every proper cylinder, we define ~t(C(Bt,

=

... , Bn; kt. ... , kn))

[m{Bt)]k1 • • • [m{Bn)]k,. [- ~ {B·)] k '-··k I exp L...Jm ' . n·

1·

(7.7.10)

i=l

From {7.7.10) it follows immediately that ~t(C(Bt. ... , Bn; kt. ... , kn)

= ~t(C(Bt; kt)) · · ·~t(C(Bn; kn))

(7.7.11)

whenever the sets B1, ... , Bn are mutually disjoint. It is also easy to calculate the measure of C(Bt. B2; kt. k2) when B1 and B 2 are not disjoint by writing C as the union of proper cylinders. Thus, y belongs to C(Bt, B2; kt. k2) if, for some r ::::;; min{kt. k2), the set B~ = B 1 \ B 2 contains kt - r particles, B 0 = Bt n B2 contains r particles, and B~ = B2 \ B1 has k2 - r particles. AB a consequence, k

C(Bt, B2; kt. k2)

= U[C(Br; kt -

r) n C(B 0 ; r) n C(B~; k2- r)J,

r=O

I'

C(B B . k k )) ( t. 2' t. 2

= ~ [m(B?)Jkl-r[m{BoW[m(B~)Jk2-r ~

{kt- r)!r!{k2- r)!

· exp[-m(Br)- m(no)- m(Bg)]

(7.7.12)

222


By employing arguments of this type we can calculate the measure JJ of any cylinder. However, the formulas for arbitrary cylinders are much more complicated as it is necessary to sum these various contributions first with respect to q

= (;)

parameters r 1 , ••• , r 9 , corresponding to all possible

intersections Bin B;, i

#

j, then with respect to (

~)

parameters cor-

responding to all possible intersections Bin B; n B1, i # j # l, and so forth. With respect to the 0'-algebra, we define A to be the smallest 0'-algebra that contains all the cylinders or, equivalently, all proper cylinders. Using standard results from measure theory, it is possible to prove that JJ given by (7.7.10) for proper cylinders can be uniquely extended to a measure on A and that the characteristic functions of proper cylinders

form a linearly dense subset of L 2 (Y, A, I'). We omit the proof of these facts as they are quite technical in nature and, instead, turn to consider the asymptotic properties of system {StheR on the phase spaceY. First we note that the measure I' is normalized. To show this, take an arbitrary bounded Borel set B. Then 00

Y= ua(B;k) k=O

since every y belongs to one of the cylinders C(B; k), namely, the one for which #(y n B) = k. As the cylinders C(B; k), k = 0, 1, ... , are mutually disjoint, we have JJ(Y)

= EJJ(C(B;k)) = lc=O

E[m~)Jk

e-m(B)

= 1.

lc=O

Second, the measure JJ is invariant with respect to {StheR· To show this, note that for every cylinder St(C(Bl, · · · 1 Bni k1, · · · 1 kn)) = C(st(Bl), ... 1 Bt(Bn)i k11 ... 1 kn)· It is clear that (x,v) E Bt(B;) if and only if (x,v) E B;, where x = x- vt, v = v, and, as a consequence,

m(st(B;)) =

II •e(BJ)

g(v)dxdv

=ILJ

g(v)dXdv = m(B;).

7.7. Applications of Semigroups From this equality, m(st(B;)) therefore, have

= m(B;)

p.(St(C(Bt, ... , Bni k1, ... , kn)))

223

and, from equation {7.7.10), we,

= p.(C(B1, ... , Bni k1, ... , kn))

for every proper cylinder. Writing JJ.t(E) = p.(St(E)) forE E A, we define for every fixed t a measure Jl.t, on A that is identical with J1. for proper cylinders. Since J1. is uniquely determined by its values on cylinders, we must have Jl.t(E) = p.(E) for all E E A, and thus the invariance of Jl. with respect to St is proved. With these results in hand, we now prove that the dynamical system {StheR is mixing. Since the characteristic functions of proper cylinders are linearly dense in L 2 (Y,A,p.), by Remark 7.7.2 it is sufficient to verify the condition (7.7.13) for every two proper cylinders C1 and C2. Since Utlcl (y)

and {lei, 1}

= lc (St(y)) = ls_e(Cl)(Y) 1

= p.(C;), condition (7.7.13) is equivalent to lim p.(S-t(Cl) n C2)

t-+OO

,

= p.(Cl)JJ.(C2)·

(7.7.14)

We will verify that (7.7.14) holds only in tlie simplest case when each of the cylinders C; is determined by only one bounded Borel set. Thus we assume (7.7.15) C; = C(B;;kj), j = 1,2. (This is not an essential simplification, since the argument proceeds in exactly the same way for arbitrary proper cylinders. However, in the general case the formulas are so complicated that the simple geometrical ideas behind the calculations are obscured.) When the C; are given by (7.7.15), the right-hand side of equation (7. 7.14) may be easily calculated by (7. 7.10). Thus

To compute the left-hand side of equation (7.7.14), observe that

so

p.(S-t(Cl) n C2)

= p.(C(s_t(Bl)i kl) n C(B2; k2)) = p.(C(s-t(Bl), B2; k1, k2)).

(7.7.17)

224


With (7.7.12) we have k

°

p(S-t(Cl) n 2) = ~

[m(Bf)]kl-r[m(BtW[m(B~)jk2-r

(k1 - r)lr!(k2- r)l

· exp(-m(BD- m(Bt)- m(B~)], where B1 = 8-l(Bl) \ B2, Bt = Lt(Bl) n B2, and B~ From our definition of m, we have

=

= B2 \

(7.7.18)

8-t(Bt).

j j 1s (x+vt,v)g(v)dxdv. 1

B2

Since B 1 and B2 are bounded, 1s 1 (x+vt,v) = 0 for almost every point (x, v) E B 2 if t is sufficiently large (except for some points at which v = 0). Thus, by the Lebesgue dominated convergence theorem, (7.7.19) Furthermore, since B~ = B 2 \ Bt, it follows that (7.7.20) Finally, since B1

= Lt(B1) \

Bt and 8t is measure preserving,

m(BD = m(8-t(Bt))- m(Bt) = m(B1) - m(Bt), and lim m(BD

t-+oo

= m(B1).

(7.7.21)

Passing to the limit in equation (7.7.18) and using (7.7.19) through (7.7.21) gives

which, together with (7.7.16), proves (7.7.14). From this proof, it should be clear that mixing in this model is a consequence of the following two facts. The first is that, for disjoint B 1 and B2 and given kt and k2, the events consisting of B 1 containing kt particles and B2 containing k2 particles are independent [this follows from equation (7.7.11)]. Second, for every two bounded Borel sets Bt and B2, the


225

sets Lt{B1) and B2 are "almost" disjoint for large t. Taken together these produce the surprising result that mixing can appear in a system without particle interaction. 0

Example 7.7.3. The preceding example gave a continuous time, dynamical system that was mixing. The phase space of this system was infinite dimensional. This fact is not essential. There is a large class of finite dimensional, mixing, dynamical systems that play an important role in classical mechanics. In this example we briefly describe these systems. An exhaustive treatment requires highly specialized techniques from differential geometry and cannot be given within the measure-theoretic framework that we have adopted. All necessary information can be found in the books by Arnold and Avez [1968), by Abraham and Marsden [1978), and articles by Anosov [1967) and by Smale [1967). Let M be a compact connected smooth Riemannian manifold. Having M, we define the sphere bundle E as the set of all pairs {m,~), where m is an arbitrary point of M and ~ is a unit tangent vector starting at m. This definition can be written as E

= {{m,{):m EM,{ E Tm, 11{11 = 1}.

It can be proved that E, with an appropriately defined metric, is also a lliemannian manifold. Thus a measure #JE is automatically given on E. In a physical interpretation, M is the configuration space of a system that moves with constant speed and E is its phase space. To describe precisely the dynamical system that corresponds to this interpretation we need only the concept of geodesics. Let "'(: R -+ M be a C 1 curve. This curve is called a geodesic iffor every point mo ="'((to) there is an e > 0 such that for every m 1 = "Y(tl), with lt1- to I $ e, the length of the arc "Y between the points mo and m 1 is equal to the distance between mo and m1. It can be proved that, for every {m, ~) E E, there exists exactly one geodesic satisfying "'f{O)

= m,

'Y'{O)

= ~.

!I"Y'(t)ll

=1

fortE R.

(7.7.22)

We define a dynamical system {StlteR on E by setting St(m,~)

= ("Y(t),"'f'(t))

fortE R,

where the geodesic "Y satisfies (7.7.22). This system is called a geodesic flow. In the case dim M = 2, the geodesic flow has an especially simple interpretation: It describes the motion of a point that moves on the surface M in the absence of external forces and without friction. The motion described by the geodesic flow looks quite specific but, in fact, it represents a rather general situation. If M is the configuration space of a mechanical system with the typical Hamiltonian function (see Remark 7.7.1),

H(q,p) =

! L a;~c(q)P;Pk + V(q), j,lc

226


then it is possible to change the Riemannian metric on M in such a way that trajectories of the system become geodesics. The behavior of the geodesic flow depends on the geometrical properties of the manifold M and most of all on its curvature. In the simplest case, dim M = 2, the curvature K is a scalar function and has a clear geometrical interpretation. In order to define K at a point m E M, we consider, in a neighborhood W of m, a triangle made by three geodesics. We denote the angles of that triangle by a1, a2, as, and its area by u. Then K(m) =lim[(at+ a2 +as -11')/u], where the limit is taken over a sequence of neighborhoods that shrinks to the point m. In the general case, dim M > 2, the curvature must be defined separately for every two-dimensional section of a neighborhood of the point m. (Thus, in this case, the curvature becomes a tensor.) When the curvature of M is negative, the behavior of the geodesic flow is quite specific and highly chaotic. Such flows have been studied since the beginning of the century, starting with Hadamard (1898]. Results were first obtained for manifolds with constant negative curvature and then finally completed by Anosov (1967]. It follows that the geodesic flow on a compact, connected, smooth Riemannian manifold with negative curvature is mixing and even a K-flow (a continuous time analog of K-automorphism). This fact has some profound consequences for the foundations of classical statistical mechanics. A heuristic geometrical argument of Arnold (1963] shows that the Boltzmann-Gibbs model of a dilute gas (ideal balls with elastic collisions) may be considered as a geodesic flow on a manifold with negative curvature. Thus, such a system is not only ergodic but also mixing. A sophisticated proof of the ergodicity and mixing of the Boltzmann-Gibbs model has been given by Sinai (1963, 1970]. D

7.8 The Hille-Yosida Theorem and Its Consequences Theorem 7.8.1 (Hille-Yosida). Let A:V(A)-+ L be a linear operator, where V(A) C Lis a linear subspace of L. In order for A to be an infinitesimal operator for a continuous semigroup of contractions, it is necessary and sufficient that the following three conditions are satisfied: (a) V(A) is dense in L, that is, every point in L is a strong limit of a sequence of points from V(A);

(b) For each f E L there exists a unique solution g E V(A) of the resolvent equation (7.8.1) ).g-Ag=/;

7.8. The Hille-Yosida Theorem and Its Consequences

227

{c) For every g E V(A) and,\> 0, ll-\g- AgiiL ~

AII9IIL·

(7.8.2)

Further, if A satisfies {a)-(c), then the semigroup corresponding to A is unique and is given by fEL,

(7.8.3)

where A.x = -\AR.x and R.xf = g (the resolvent operator) is the unique solution of ,\g-Ag = f. Consult Dynkin [1965) or Dunford and Schwartz [1957) for the proof. Operator A.x -\AR.x can be written in several alternative forms, each of which is useful in different situations. Thus, after substitution of g = R.xf into (7.8.1), we have

=

(7.8.4) By applying the operator R.x to both sides of (7.8.1) and using g = R.xf, we also obtain -\R.xg- R.xAg g for g E V(A). (7.8.5)

=

Equations (7.8.4) and (7.8.5) immediately give

R.xAf = AR.xf

for

f

E

V(A).

(7.8.6)

Equation (7.8.4) also gives

AR.xf = (,\R.x - I)f

for f E L,

(7.8.7)

=

where I is the identity operator (If f for all f). Thus we have three possible representations for A.x: the original definition, (7.8.8) or, from (7.8.7), (7.8.9) and, finally, from (7.8.6), {7.8.10) The representations in (7.8.8) and (7.8.9) hold in the entire space L, whereas (7.8.10) holds in V(A). From conditions (b) and (c) of the Hille-Yosida theorem, using g = R.xf, it follows that (7.8.11)

228


Consequently, using (7.8.9), IIA~!IIL = 11>.2 R~f- >.filL ~ 11>.2 R~!IIL

so that the operator

exp(tA~)

+ 11>./IIL ~ 2>.11/IIL.

can be interpreted as the series

etA,. f =

oo tn

L --;A~/, n.

(7.8.12)

n=O

which is strongly convergent. In addition to demonstrating the existence of a semigroup {Tt}t~o corresponding to a given operator A, the Hille-Yosida theorem also allows us to determine some properties of {Tth~o· One very interesting corollary is the following. Suppose we have an operator A: V(A) - t L (remembering that L = V) that satisfies conditions (a)-(c) of the Hille-Yosida theorem, and such that the solution g = R~f of equation (7.8.1} has the property that R~f 2: 0 for f 2: 0. Then, as we will show next, Ttf 2: 0 for every f 2: 0. To see this, note that from (7.8.9} we have (7.8.13} where

et~ R" f 2

=L -

oo tn An

n=O

n.1

(>.R~tf.

(7.8.14}

Further, for any f 2: 0, R~f 2: 0 and, by induction, ~/ 2: 0. Thus, from (7.8.14}, since >. > 0 and t 2: 0, exp(t>.2 R~)f 2: 0 and so, from (7.8.13}, exp(tA~}/ 2: 0. Finally, from (7.8.3}, we have Ttl 2: 0 since it is the limit of nonnegative functions. Now suppose that L = L 1 and that the operator >.R~ preserves the integral, that is,

>.

LR~f(x)J.L(dx} L L L =

for all f E L 1 , >. > 0.

f(x)J.L(dx}

(7.8.15}

We will show that these properties imply that

Ttf(x)J.L(dx) =

f(x)J.L(dx},

f 2: 0, t 2: 0.

This is straightforward. Since (7.8.14} is strongly convergent, and using equation (7.8.15}, we obtain

L

2

eU R"

f(x)J.L(dx}

= ~ t:~n

L(>.R~)n

=?; 7

Jx f(x)J.L(dx}

oo tn).n

=

et~

L

f(x)J.L(dx)

{

f(x)J.L(dx).

(7.8.16)


229

Now,

by the use of equation (7.8.16), and the claim is demonstrated. These two results may be summarized in the following corollary.

Corollary 7.8.1. Let A: I>(A) - L1 be an operator satisfying conditions

(a)-(c) of the Hille-Yosida theorem. If the solution g = R>./ of (7.8.1) is such that >.R>. is a Markov operator, then {Tth~o generated by A is a continuous semigroup of Markov operators. In fact, in this corollary only conditions (a) and {b) of the Hille-Yosida theorem need be checked, as condition (c) is automatically satisfied for any Markov operator. To see this, set f = ).g - Ag and write inequality {7.8.2) in the form

11/11 ~ 11>-R>./11. This is always satisfied if >.R>. is a Markov operator, as we have shown in Section 3.1 [cf. inequality {3.1.6)]. The Hille-Yosida theorem has several other important applications. The first is that it provides an immediate and simple way to demonstrate that Appf = 0 is a sufficient condition that /.Lf is an invariant measure. Thus, A/= 0 implies, from (7.8.10), that A>./ = 0 and from (7.8.12) etA,.

I= f.

This, combined with (7.8.3), gives

Ttl=!

for all t

~

0.

Thus, in the special case Appf = 0 this implies that Ptf = f and thus 1-'1 is invariant. By combining this result with that of Section 7.7, we obtain the following theorem.

Theorem 7.8.2. Let {Sth~o be a semidynamicalsystem such that the corresponding semigroup of Frobenius-Perron operators is continuous. Under this condition, an absolutely continuous measure /.L1 is invariant if and only if Appf = 0. Consider the special case where App is the infinitesimal operator for a d.dimensional system of ordinary differential equations {cf. equation 7.6.10).

230


Then the necessary and sufficient condition that Jl.J be invariant, that is AFP! = 0, reduces to d

L8(/Fi) =0 i=1 8xi

(7.8.17)

for continuously differentiable f e L 1 • This result was originally obtained by Liouville using quite different techniques and is known as Liouville's theorem.

Remark 7.8.1. Equation (7.8.17) is also a necessary and sufficient condition for the invariance of the measure Jl.J(A) =

L

f(x)p.(dx)

even if f is an arbitrary continuously differentiable function that is not necessarily integrable on R!-. This is related to the fact that operators Ptf as given by (7.6.13) can also be considered for nonintegrable functions. Thus, if one wishes to determine when the Lebesgue measure

p.(A)

=

L

dxt ... dxd

=

L

dx

is invariant, it is necessary to substitute its density J(x) This gives

= 1 into (7.8.17). (7.8.18)

as a necessary and sufficient condition for the invariance of the Lebesgue measure. (In many sources, equation (7.8.18) is called Liouville's equation, even though it is a special case of equation (7.8.17).] 0

Remark 7.8.2. It is quite straightforward to show that Hamiltonian systems (see Example 7.6.1) satisfy (7.8.18) since

t

i= 1

[~ (8H) + ~ (- 8H)] = O 8qi

Bpi

Bpi

8qi

automatically, and thus they preserve the Lebesgue measure. 0 Returning now to the problem of determining the ergodicity of a semidynamical system {Sth>o, recall that Utg = g implies AKg = 0. Using this relation and Theorem 7-:1.2 we are going to prove the following theorem. {Bth~o be a semidynamical system such that the corresponding semigroup {Pt} of Frobenius-Perron operators is continuous.

Theorem 7.8.3. Let


231

Then {Sth~o is ergodic if and only if AKg = 0 has only constant solutions in L 00 • Proof. The "if'' part follows from Theorem 7. 7.2. The proof of the "only ir' part is more difficult since, in general, the semigroup {Ut} is not continuous and we cannot use the Hille-Yosida theorem. Thus, assume that AKg = 0 for some nonconstant g. Choose an arbitrary f E L 1 and define the realvalued function ,P by the formula

,P(t) = (/, Utg) = {Pt/,g). Due to the continuity of {Pt}, function ,P is also continuous. Further, we have

,P(t + ~- ,P(t)

= ( /, Ut+hgh- Utg) = (Ptf, Uhgh-

g)

Since AKg = 0, passing to the limit as h

-+

,P'(t)

for h

> 0, t ~ 0.

0, we obtain

= (Ptf,AKg) = 0.

Function ,P is continuous with the right-hand derivative identically equal to zero, implying that ,P{t) = ,P{O) for all t ~ 0. Consequently,

(!, Utg- g)

= ,P(t) -

,P{O)

=0

fort~

0.

Since f is arbitrary this, in turn, implies that Utg = g for t ~ 0, which, by Theorem 7. 7.2, completes the proof. • In particular, if { Sth~o is a semigroup generated by a system of ordinary differential equations then, from equation (7.6.5), AK f = 0 is equivalent to {7.8.19) for continuously differentiable f with compact support. However, it must be pointed out that {7.8.19) is of negligible usefulness in checking ergodicity, because the property "AK f = 0 implies f constant for all functions in L 00 " must be checked and not just the continuously differentiable functions. This is quite different from the situation where one is using the Liouville theorem (7.8.17) to check for invariant measures. In the latter case, it is necessary to find only a single solution of AFP! = 0.

Example 7.8.1. Theorem 7.8.3 allows us easily to prove that Hamiltonian systems {see Example 7.6.1) are not ergodic. To show this, note that for a

232


·Hamiltonian system defined by equation (7.6.12), equation (7.6.5) becomes

AK/ =

t

[8/ 8H _ 8/ 8H]

i=l

8qi 8pi

8pi 8qi

= [f,H].

Take f E L 00 to be any nonconstant function of the energy H. By Example 7.6.1, we know that AKf 0 since

=

[/(H), H]

= 88/H [H, H] = 0

and therefore Hamiltonian systems are not ergodic on the whole space. However, if we fix the total energy, or the energy for each degree of freedom as in Remark 7.7.1, then the system may become ergodic. 0

7.9 Further Applications of the Hille-Yosida Theorem Thus far we have used the Hille-Yosida theorem to demonstrate some simple properties of semigroups that followed directly from properties of the infinitesimal operator A and the resolvent equation (7.8.1). In these cases the semigroups were given. Now we are going to show a simple application of the theorem to the problem of determining a semigroup corresponding to a given infinitesimal operator A. Let X= Rand L = L 1 (R), and consider the infinitesimal operator

Af = tf/

(7.9.1)

dx2

that can, of course, only be defined for some f E £ 1 • Let 'D(A) be the set of all f E L 1 such that f"(x) exists almost everywhere, is integrable on R, and f'(x) = /'(0) /"(s) ds.

+loa:

In other words, 'D(A) is the set of all f such that/' is absolutely continuous and f" is integrable on R. We will show that there is a unique semigroup corresponding to the infinitesimal operator A. The set 'D(A) is evidently dense in L 1 (even the set of functions is dense in £ 1), therefore we may concentrate on verifying properties (b) and (c) of the Hille-Yosida theorem. The resolvent equation (7.8.1) has the form

coo

tfg ).g- dx2

= /,

(7.9.2)

7.9. FUrther Applications of the Hille-Yosida Theorem

233

which is a second-order ordinary differential equation in the unknown function g. Using standard arguments, the general solution of (7.9.2) may be written as

g(x)

1 = Cte-az + C2eaz +-

1z

2a zo

1z

e-a(z-y) f(y) dy- -1 ea(z-y) f(y) dy 2a zl

where a = v'>., and Ct. C2, xo, and Xt are arbitrary constants. To be specific, pick x 0 = -oo, Xt = +oo, and set

K(x- y) = (1/2a}e-al(z-y)l,

(7.9.3)

Then the solution of (7.9.2} can be written in the more compact form

g(x) =

C1e-az

+ C2eaz + /_: K(x- y)f(y) dy.

(7.9.4}

The last term on the right-hand side of (7.9.4) is an integrable function on R, since

/_: dx /_: K(x- y)f(y) dy = /_: K(x- y) dx /_: f(y) dy

rJC' = ~1 }_oo f(y) dy.

(7.9.5}

Thus, since neither exp( -ax) nor exp( ax) are integrable over R, a necessary and sufficient condition for f to be integrable over R is that 0 1 = 0 2 = 0. In this case we have shown that the resolvent equation (7.9.1} has a unique solution g E L 1 given by

g(x)

= R>.f(x) = /_: K(x- y)f(y) dy,

(7.9.6}

and thus condition (b) of the Hille-Yosida theorem is satisfied. Combining equations (7.9.5} and (7.9.6) it follows immediately that the operator >.R>. preserves the integral. Moreover, >.R>. ~ 0 iff~ 0, so that >.R>. is a Markov operator. Thus condition (c) of the Hille-Yosida theorem is automatically satisfied, and we have shown that the operator ~ / dx 2 is an infinitesimal operator of a continuous semigroup {Tth2:o of Markov operators, where (7.9.7} and R>. is defined by (7.9.3} and (7.9.6}. It is interesting that the limit (7.9.7} can be calculated explicitly. To do this, denote by rjJf the Fourier transformation of f, that is,

rPJ(w)

= /_: e-iwz f(x) dx.

234


The Fourier transformation of K(x) given by equation (7.9.3) is

1/(>. + w2 ), where).= o. 2 • Since, by (7.9.6), R>.l is the convolution of the functions K and I, and it is well known that

(7.9.8) where I* g denotes the convolution of I with g, the Fourier transformation of R';.l is As a consequence, the Fourier transformation of the series in (7.9.7) is 00

L

n=O

(>.

tn ).2n )n .2tj(>. + w2 )]tPJ(w).

Thus the Fourier transformation of Ttl is lim exp( ->.t) exp[>.2 tj(>. + w2 )]tPJ(w) = exp{ -w2 t)tPJ(w).

>.-+oo

Using the fact that exp( -w2 t) is the Fourier transformation of

~ exp( -x2 /4t)

v47rt

and (7.9.8), we then have

1

00

Ttf(x) =

r:;-::; 1

v47rt

exp[-(x- y) 2 /4t]l(y) dy.

-oo

(7.9.9)

·

Hence, using the semigroup method we have shown that u(t, x) = Ttf(x) is the solution of the heat equation

with the initial condition

u(O,x)

= l(x).

Remark 7.9.1. It is a direct consequence of the elementary properties of the differential quotient (see Definition 7.5.1) that if A is the infinitesimal operator corresponding to a semigroup {Tth~o, then cA is the infinitesimal operator corresponding to {Tcth>o· Thus, since we have proved that A= ~ / dx 2 is the infinitesimal operator corresponding to the semigroup {Tt }t~ 0 given by (7.9.9), we know immediately that

7.9. Further Applications of the Hille-Yosida Theorem

235

has a corresponding infinitesimal operator equal to (u2 l2)(d?-ldx 2 ). (This is in perfect agreement with our observations in Example 7.4.1.) For simplicity, we have omitted the coefficient (u 2 12) in the foregoing calculations. 0 The proofthat tP I dx2 is an infinitesimal operator for a stochastic semigroup on R may be extended to Rd. Thus, for example, the operator

L n

Af=ilf=

i=l

£Pf -2

8x,

(7.9.10)

on Rd may be shown to be an infinitesimal operator for a stochastic semigroup, as can n lJ2f {7.9.11) Af = a,; {) .{) . '

L

x, x,

i,j=l

where the a,; are constant, or sufficiently regular functions of x, and Lt,; a,;{te; is positive definite. The procedure for proving these assertions is similar to that for operator tP I dx 2 on R, but requires some special results from the theory of partial differential equations and functional analysis, allowing us to extend the definitions of the differential operators (7.9.10) and (7.9.11). Operators such as tP ldx 2 , (7.9.10), or (7.9.11) may be considered not on the whole space (R or Rd), but also on bounded subspaces. However, in this case other boundary conditions must be specified, for example, Af= d?-f

dz2

with

I

I

df = 0 and df = 0 dxa dxb is an infinitesimal operator for a stochastic semigroup. More details concerning such general elliptic operators may be found in Dynkin [1965]. Finally, we note that all semigroups that are generated by second-order differential operators are not semigroups of Frobenius-Perron operators for a semidynamical system and, thus, cannot arise from deterministic processes. This is quite contrary to the situation for first-order differential operators, as already discussed in Section 7.8. Remark 7.9.2. Equation (7.8.3) of the Hille-Yosida theorem allows the construction of the semigroup {Tth~o if the resolvent operator RA is known. As it turns out, the construction of the resolvent operator when the continuous semigroup of contractions is given is even simpler. Thus it can be shown that (Dynkin, 1965) for

f

E

L, A > 0.

(7.9.12)

236


In (7.9.12} the integral on the half-line [0, oo) is considered as the limit of Riemann integrals on [0, a] as a - oo. This limit exists since

It is an immediate consequence of (7.9.12} that for every stochastic semigroup Tt: £1 -+ L 1 , the operator >.R>. is a Markov operator. To show this note first that, for f ~ 0, equation (7.9.12} implies >.R>. ~ 0. Furthermore, for f ~ 0,

IIR>./11

=[

R>./(x) dx

1

00

=

1

=

00

e->.t { [ Ttf(x)

dx} dt

e->.tll/11 dt =~II/II.

In addition to demonstrating that >.R>. is a Markov operator, (7.9.12} also demonstrates that the semigroup corresponding to a given resolvent R>. is unique. To see this, choose g E L 00 and take the scalar product of both sides of equation (7.9.12} with g. We obtain

(g, R>./)

=

1

00

e->.t(g, Ttf) dt

for>.> 0,

which shows that (g, R>./), as a function of>., is the Laplace transformation of (g, 7t/) with respect tot. Since the Laplace transformation is one to one, this implies that (g, Ttl) is uniquely determined by (g, R>.f). Further, since g E L 00 is arbitrary, {Ttl} is uniquely determined by {R>./}. The same argument also shows that for a bounded continuous function u{t), with values in £1, the equality implies u(t)

= Ttf.

0

Some of the most sophisticated applications of semigroup theory occur in treating integra-differential equations. Thus we may not only prove the existence and uniqueness of solutions to such equations, but also determine the asymptotic properties of the solutions. One of the main tools in this area is the following extension of the H~Yosida theorem, generally known as the Phillips perturbation theorem. Theorem 7.9.1. Let a continuous stochastic semigroup {Tth>o and a Markov operator P be given. Further, let A be the infinitesimal operator of {Tth~o· Then there exists a unique continuous stochastic semigroup {l't h>o for which Ao=A+P-1


237

(I is the identity operator on L 1) is the infinitesimal operator. Furthermore, the semigroup {Pt}t~o is defined by 00

Ptf

= e-t LTn(t)l

IE Ll,

(7.9.13)

n=O

where To(t) =

7t and (7.9.14)

Tn(t)l = 1t To(t- r)PTn-t(r)l dr.

Proof. Denote by R~(A) the resolvent corresponding to operator A, that is, g = R~(A)I is the solution of >-.g-Ag= I

Since {Tth~o is a stochastic semigroup, )..R~(A) is a Markov operator (see Remark 7.9.2). Now we observe that the resolvent equation for operator

Ao, >-.g-Aog

may be rewritten as

=I,

(7.9.15)

(>-. + 1)g- Ag = I+ Pg.

Thus (7.9.15) is equivalent to g = R~+l(A)I + R~+l(A)Pg.

From ineq11ality (7.8.11) we have IIR~+l(A)Pgll ~ is a Markov operator, this becomes

(7.9.16)

IIPull/().. + 1). Since P

Thus, equation (7.9.16) has a unique solution that can be constructed by the method of successive approximations. The result is given by 00

9 = R~(Ao)l = L[R~+l(A)P]nRMt(A)I,

(7.9.17)

n=O

and the existence of a solution g to (7.9.15) is proved. Further, from (7.9.17) it follows that R~(Ao)l;;::: 0 for I;;::: 0 and that

oo (

1

IIR~(Ao)lll = ~ ).. + 1

)n+l IIIII = -xlllll 1

for I;;::: 0.

238


Thus ~R~ (Ao) is a Markov operator and Ao satisfies all of the assumptions of the Hille-Yosida theorem. Hence the infinitesimal operator A0 generates a unique stochastic semigroup and the first part of the theorem is proved. Now we show that this semigroup is given by equations (7.9.13) and (7.9.14). Using (7.9.14) it is easy to show by induction that

IITn(t)/11 5 (t" /nl)ll/11·

(7.9.18)

Thus, the series (7.9.13) is uniformly convergent, with respect to t, on bounded intervals and Ptf is a continuous function oft. Now set

Q~,nf = so

1 e-~tTn(t)f 00

dt,

n=0,1, ... ,

Q~,of = fooo e-~tTtf dt = R~(A)f

and

Q~,nf =

1 e-~t 1 {1 e-~tTo(t00

{lot To(t- r)PTn-t(T)/ dT} dt

00

=

00

r)PTn-t(r)f dt} dr

1 e-~'T 1 e-~tTo(t)PTn-t(r)f =1 e-~tTo(t)P 1 00

=

00

{

00

dt} dr

00

e-MTn-t(r)f dr} dt

{

= R~(A)PQ~,n-d· Hence, by induction, we have Q~,n = (R~(A)P]" R~(A).

Define

Q~f =

1 e-~t 00

Ptf dt

and substitute equation (7.9.13) to give

Q~f = ~

1

00

e-(Ml)tTn(t)f dt

= ~ Q~+t,n/

00

= L(R~+t(A)P]"R~+t(A)f. By comparing this result with (7.9.17), we see that

R~(Ao)f =

fooo e-~tptf dt.

Q~ = R~(Ao)

or

(7.9.19)


239

From (7.9.19) (see also the end of Remark 7.9.2), it follows that {Ptfh>o is the semigroup corresponding to Ao. • -

Example 7.9.1. Consider the integro-differential equation 8u( t, x) &t

u 2 82u(t, x)

+u(t,x) = 2

1Jx2

1

00

+

-oo

K(x,y)u(t,y)dy, t > 0, x E R

(7.9.20)


u(O,x) = <jJ(x)

xER.

(7.9.21)

We assume that the kernel is measurable and stochastic, that is,

K(x,y)

~0

and

L:

K(x,y)dx = 1.

To treat the initial value problem, equations (7.9.20) and (7.9.21), using semigroup theory, we rewrite it in the form

du dt

= (A+P-I)u,

u(O) = (x) = e-t ft{

Jo

1 y'2u 21r(t- r)

1

00

exp(-(x- y) 2 f2u 2 (t- r)]·

-oo

·g.,.(y) dy}dr + e-t ~ roo exp(-(x- y) 2 /2u 2 t]l/>(y) dy. 2u211't 1-oo This shows directly that the function u(t, x) = Ptlf>(x) is continuous and strictly positive fort> 0 and every 4> E L 1 (R), even if 4> and the stochastic kernel K are not continuous! Finally, we will come back to this semigroup approach in Section 11.10 and use it to demonstrate some asymptotic properties of the solution u( t, x). 0

Example 7.9.2. As a second example of the applicability of the Phillips perturbation theorem, we consider the first-order integro-differential equation

8u(t,x) 8u(t,x) 8t + ax +u(t,x)=

1

00

a:

K(x,y)u(t,y)dy, t

with u(t, 0) = 0 and u(O, x)

> 0,

X ~

0

= lf>(x).

(7.9.24) (7.9.25)

Again the kernel K is assumed to be measurable and stochastic, that is,

K(x, y)

~0

1 11

and

K(x, y) dx = 1.

(7.9.26)

Equation (7.9.24) occurs in queuing theory and astrophysics (BharuchaReid, 1960]. In its astrophysical form,

K(x, y) = (1/y)'I/J(xfy),

{7.9.27)

and, with this specific expression for K, equation (7.9.24) is called the Chandrasekhar-Miinch equation. As developed by Chandrasekhar and Miinch (1952], equation (7.9.24) with K as given by (7.9.27) describes fluctuations in the brightness x of the Milky Way as a function of the extent of the system t along the line of sight. The unknown function u( t, x) is the probability density of the fluctuations, and the given function 1/J in {7.9.27) is related to the probability density of light transmission through interstellar gas clouds. This function satisfies 1/l(z)

~0

1 1

and

1/J(z) dz = 1

and, thus, K as given by (7.9.27) automatically satisfies (7.9.26).

(7.9.28)

7.10. Frobenius-Perron and Koopman Operators

241

To rewrite (7.9.24) as a differential equation in £ 1 , recall (see Example 7.5.1) that -d/dx is the infinitesimal operator for the semigroup Ttf(x) = f(x- t) defined on L 1 (R). On L 1 ([0,oo)),

Ttf(x)

= 1[o,oo)(x- t)f(x- t)

(7.9.29)

plays an analogous role. Proceeding much as in Example 7.5.1, a simple calculation shows that for continuously differentiable f with compact support in [0, oo) the infinitesimal operator corresponding to the semigroup in (7.9.29) is given by Af = -df jdx. Further, it is clear that u(t,x) = Ttf(x) satisfies u(t, 0) = 0 for t > 0. Hence we may rewrite equations (7.9.24)(7.9.25) in the form

du dt

= (A+P-I)u,

where

1

u(O}

= t/>,

(7.9.30)

00

Pf(x) =

K(x,y)f(y)dy.

By Theorem 7.9.1 there is a continuous unique semigroup {Pth>o corresponding to the infinitesimal operator A+ P-I. For every t/> E V(A), the function u(t) = Ptt/> is a solution of (7.9.30). 0

7.10 The Relation Between the Frobenius-Perron and Koopman Operators The semigroup ofFrobenius-Perron operators {Pt} and the semigroup {Ut} of Koopman operators, both generated by the same semidynamical system {Sth>o, are closely related because they are adjoint. However, each describes the behavior of the system {Bth~o in a different fashion, and in this section we show the connection between the two. Equation (7.4.16}, (Ptf, g) = (!, Utg}, which says that Pt and Ut are adjoint, may be written explicitly as

[ g(x)Ptf(x)JJ(dx)

=[

f(x)g(St(X))JJ(dx)

For some A C X such that A and Bt(A) are in A, take f(x) = 0 for all x ¢A and g = 1x\St(A) so the preceding formula becomes

[ 1x\St(A)(x)Ptf(x)JJ(dx) = [ f(x)1x\St(A)(St(x))JJ(dx)

=

L

f(x)1x\St(A)(Bt(X))JJ(dx).

242


The right-hand side ofthis equation is obviously equal to zero since St(x) ¢ X\ St(A) for x E A. The left-hand side is, however, just the L 1 norm of the integrand, so that

ll1x\S,(A)Ptlll

= 0.

This, in turn, implies

or for x ¢ St(A).

Ptf(x) =0

(7.10.1)

Thus the operator Pt "carries" the function I, supported on A, forward in time to a function supported on a subset of St(A) (see Example 3.2.1 and Proposition 3.2.1). Figuratively speaking, we may say that the density is transformed by Pt analogously to the way in which initial points x are transformed into St(x). Now consider the definition of the Koopman operator,

Utf(x) Assume

IE L

00

= I(St(x)).

is zero outside a set A, so we have

I(St(x))

=0

if St(x) ¢A.

(7.10.2)

This, in turn, implies that

Utf(x)

=0

(7.10.3)

In contrast to Pt, therefore, Ut may be thought of as transporting the function supported on A, backward in time to a function supported on Bt 1 (A). These observations become even clearer when {St} is a group of transformations, that is, when the group property holds for both positive and negative time, for all t, t' E R,x EX, and all the St are at least nonsingular. In this case, St 1 (x) (7.10.3) becomes Utf(x) =0 for x ¢ B-t(A).

= B-t(x) and

If, in addition, the group { St} preserves the measure p., we have

( Ptf(x)p.(dx)

h

= (

k_,(~

l(x)p.(dx)

= (

which gives

Ptl(x) = 1(8-t(x))

h

1(8-t(x))p.(dx),

7.10. Frobenius-Perron and Koopman Operators or, finally,

Ptf(x)

= U_tf(x).

243

(7.10.4)

Equation (7.10.4} makes totally explicit our earlier comments on the forward and backward transport of densities in time by the Frobenius-Perron and Koopman operators. Furthermore, from (7.10.4) we have directly that

lim[(Ptf- /)/t]

t--+0

and, thus, for

= t--+0 lim[(U-tf- !)ft]

f in a dense subset of £ 1 , (7.10.5)

This relation was previously derived, although not explicitly stated, for dynamical systems generated by a system of ordinary differential equations [cf. equations (7.6.5) and (7.6.10)].

Remark 7.10.1. Equation (7.10.4) may, in addition be interpreted as saying that the operator adjoint to Pt is also its inverse. In the terminology of Hilbert spaces [and thus in L 2 (X)] this means simply that {Pt} is a semigroup of unitary operators. The original discovery that {Ut}, generated by a group {St} of measure-preserving transformations, forms a group of unitary operators is due to Koopman [1931]. It was later used by von Neumann [1932] in his proof of the statistical ergodic theorem. 0 Remark 7.10.2. Equation (7.10.1) can sometimes be used to show that a semigroup of Markov operators cannot arise from a deterministic dynamical system, which means that it is not a semigroup of Frobenius-Perron operators for any semidynamical system {Sth>o· For example, consider the semigroup {Pt} 'iiven by equations (7.4.11} and (7.4.12): Ptf(x) Setting f(y)

1

= J21W2t

joo f(y) exp [- (x _ y)2] dy. -oo

2u2t

(7.10.6)

= 1[o, 1J(y), it is evident that we obtain Ptf(x) > 0

for all x and t > 0.

However, according to (7.10.1), if Ptf(x) was the Frobenius-Perron operator generated by a semidynamicalsystem {Sth~o, then it should be zero outside a bounded interval St([O, 1]). [The interval St([O, 1]) is a bounded interval since a continuous function maps bounded intervals into bounded intervals.] Thus {Pt}, where Ptf(x) is given by (7.10.6), does not correspond to any semidynamical system. 0

244

7.11


Sweeping for Stochastic Semigroups

The notion of sweeping for operators as developed in Section 5.9 is easily extended to semigroups. We start with the following.

Definition 7.11.1. Let (X,A,J.&) be a measure space and A. C A be a given family of measurable sets. A stochastic semigroup ,Pt: L 1 (X) -+ L1 (X) is called sweeping with respect to A. if lim

f

t-+oo}A

Ptf(x)J.&(dx) = 0

for IE D and A EA•.

(7.11.1)

As in the discrete time case, it is easy to verify that condition (7.11.1) for a sweeping semigroup {Pth>o also holds for every IE L 1 (X). Alternately, if Doc Dis dense in D, th;n it is sufficient to verify (7.11.1) for IE Do. In the special case that X c R is an interval (bounded or not) with endpoints a and {3, a< {3, we will use notions analogous to those in Definition 5.9.2. Namely, we will say that a stochastic semigroup Pt: L 1 (X)-+ L 1 (X) is sweeping to a, sweeping to {3, or simply sweeping if it is sweeping with respect to the families Ao, A 11 or A 2 defined in equations (5.9.5)(5.9.7), respectively. Example 7.11.1. Let X = R. We consider the semigroup generated by the infinitesimal operators cd./dx and (u 2 /2)f12/dx 2 discussed in Example 7.5.1 and Remark 7.9.1. The operator cd./dx corresponds to the semigroup Ptf(x) = l(x- ct) which, for c > 0, is sweeping to +oo and for c < 0 to -oo. The verification of these properties is analogous to the procedure in Example 5.9.1. Thus, for c > 0 we have

l-oob

Ptf(x) dx

=

lb-oo

l(x- ct) dx

=

lb-ct -oo

l(y) dy

=0

when I has compact support and t is sufficiently large. For c argument is similar. The operator (u 2 /2)f12 /dx 2 generates the semigroup

Ptf(x) =

k

/_:oo exp [- (x2:2~)2] l(y) dy

which is evidently sweeping since, for

1 b

a

I

E D,

b-a Ptf(x)dx :5 ~-+ 0 21ru 2 t

as t-+ oo.

0

0. This is not a coincidence. It is evident from Definitions 5.9.1 and 7.11.1 that if {.Pt}t~o is sweeping, then {Pt~} is also sweeping for an arbitrary to > 0. The converse is more delicate, but is assured by the following result. Theorem 7.11.1. Let (X, A, Jl.) be a measure space, A. c A be a given family of measurable sets, and Pt: L 1 (X)- L 1 (X) a continuous stochastic semigroup. If for some to > 0 the sequence {JTo} is sweeping, then the semigroup {Pth>o is also sweeping. Proof. Fix an e > 0 and such that

f

E D. Since Pt is continuous there is a 6

liFt!- Ill$ e

>0

for 0$ t $6.

Let 0 = So < St < • •· < Sk be a partition of the interval [0, t 0 ] such that for i Define

h

=

= to

= 1, ... , k.

P.J. Every value to ~ 0 can be written in the form t = nto + si + r,

where nand i are integers (n Therefore, Ptf

= 0,1, ... ; i = i, ... ,k)

and 0 $ r < 6.

= Pt~P. 1 Pd = Pt~fi + ~~P.; (Pd- !).

Since IIPd- /II$ e and Pt~ and P. 1 are contractive, we have IIPt~P.;(Pd-

!)II$ e.

As a consequence, for every A E A.

L

Ptf(x)J.t(dx) $

LPt~fi(x)Jl.(dx) +e.

Evidently, n - oo as t - oo and the integrals on the right-hand side converge to zero, thus completing the proof. • The main advantage of Theorem 7.11.1 is that it allows us to obtain many corollaries concerning sweeping for semigroups from previous results for iterates of a single operator. As an example, from Theorem 7.11.1 and Proposition 5.9.1 we have the following: Proposition 7.11.1. Let (X,A,Jl.) be a measure space, and A. c A be a given family of measurable sets. Furthermore, let Pt: L 1 (X)- L 1 (X) be a

246


continuous stochastic semigroup for which there exists a Bielecki function V: X--+ R, a constant 'Y < 1, and a point to > 0 such that

LV(x)Pt0 /(x)~(dx) ~ LV(x)f(x)~(dx) 'Y

for fED.

Then the semigroup {Pto }t~o is sweeping.

Proof. Since the operator Pt0 satisfies the conditions of Proposition 5.9.1, the sequence{~} is sweeping. Theorem 7.11.1 completes the proof. • More sophisticated applications of Theorem 7.11.1 will be given in the next section.

7.12 Foguel Alternative for Continuous Time Systems We start from a question concerning the relationship between the existence of an invariant density for a stochastic semigroup {Pth>o and for an operator Pto with a fixed t 0 . Clearly, if/. is invariant with respect to Pt so Pt/. = /. for all t ~ 0, then/. is invariant for every operator Pto· The converse is, however, unfortunately false. Rather we have the following result.

Proposition 7.12.1. If Pt: L1 (X) --+ L1 (X) is a continuous stochastic semigroup and if Pto fo = fo for some to > 0 with fo E D, then

/.(x) is a density and satisfies Ptf.

= t~ 1to Ptfo(x) dt = j.

for all t ~ 0.

Proof. From the definition of /. we have

L/.(x)~(dx) L[t~

1to Ptfo(x) dt]

= :0 1to

[L Ptfo(x)~(dx)]

=1. Furthermore, 1 rto

Pt/.

= to Jo

~(dx)

=

Ps+tfo ds

1 lto+t t P.fo ds

= to

dt

Exercises

1 1to 1 1to+t = -too Pefo ds + -t Pefo ds Ot 0

= /.

247

1 1t -t P8 fo ds oo

+ _!_

ft (Pe+tofo - P.Jo) ds. to lo

Since Pt 0 /o = /o we have Pe+tofo- Palo thus completing the proof. •

= 0 and the last integral vanishes,

Now, using Theorems 5.9.1, 5.9.2, and 7.12.1, it is easy to establish the following alternative. Theorem 7.12.1. Let (X, .A, p.) be a measure space, and .A. c .A be a given regular family of measurable sets. FUrthermore, let Pt: L1 (X)-+ L1 (X) be a continuous stochastic semigroup such that for some to > 0 the operator Pt 0 satisfies the following conditions:

( 1) Pt0 is an integral operator given by a stochastic kernel; and

(2) There is a locally integrable function/. such that I'tof• :::; /.

and /. > 0 a.e.

Under these conditions, the semigroup {Pt h>o either has an invariant density, or it is sweeping. If an invariant density exists and, in addition, Pt0 is an expanding operator, then the semigroup is asymptotically stable.

Proof. The proof is quite straightforward. Assume first that { Pt }t~o is not sweeping so by Theorem 7.11.1 the sequence {JTo} is also not sweeping. In this case, by Theorem 5.10.1 the operator Pt 0 has an invariant density. Proposition 7.12.1 then implies that { Pth~o must have an invariant density. In the particular case that Pt0 is also an expanding operator, it follows from Theorem 5.10.2 that {JTo} is asymptotically stable. Finally, Remark 7.4.2 implies that {Pt h>o is also asymptotically stable. In the second case that {Pt }t~o is sweeping, {JTo} is also, and by Th~ orem 5.10.1 the operator Pt0 does not have an invariant density. As a consequence, { Pt h~o also does not have an invariant density. •

Exercises 7.1. Let A: L-+ L be a linear bounded operator, that is, IIAII

= sup{IIA/11: 11/11 :::; 1} < oo.

Using a comparison series prove that

248


(a) etA f = E~=0 (tn fn!)An f is strongly convergent in L for t E R and

/EL,

(b) eo has independent increments if, for any sequence of times to < t 1 < · · · < t"' the random variables etl - eto' et2 - eto . .. 'etn - etn-1 are independent. Further, if for any h and t 2 and Borel set B

c R, (8.2.5)

does not depend on t'' then the continuous time stochastic process {et} has stationary independent increments. Before giving the definition of a Poisson process, we note that a stochastic process {et} is called a counting process if its sample paths are nondecreasing functions of time with integer values. Counting processes will be denoted by {Nt}t~o·

Definition 8.2.5. A Poisson process is a counting process {Nth>o with stationary independent increments satisfying: (a) No= 0;

(8.2.6a)

(b) }~(1/t) prob{Nt ~ 2}

= 0;

(8.2.6b)

= t-+0 lim(1/t)prob{Nt = 1}

(8.2.6c)

(c) The limit ~

exists and is positive; and

8.2. Probability Theory and Poisson Processes

255

(d) prob{ Nt = k} as functions of t are continuous. A classic example of a. Poisson process is illustrated by a. radioactive substance placed in a. chamber equipped with a. device for detecting and counting the total number of atomic disintegrations Nt that have occurred up to a. time t. The amount of the substance must be sufficiently large such that during the time of observation there is no significant decrease in the mass. This ensures that the probability (8.2.5) is independent oft'. It is an experimental observation that the number of disintegrations that occur during any given interval of time is independent of the number occurring during any other disjoint interval, thus giving stationary independent increments. Conditions (a.)-(c) in Definition 8.2.5 have the following interpretations within this example: No = 0 simply means that we start to count disintegrations from time t = 0. Condition (b) states that two or more disintegrations are unlikely in a. short time, whereas (c) simply means that during a. short time t the probability of one disintegration is proportional tot. Also, the classical derivations of the Boltzmann equation implicitly assume that molecular collisions are a. Poisson process. This fact will turn out to be important later. It is interesting that from the properties of the Poisson process we may derive a. complete description of the way the process depends on time. Thus we may derive an explicit formula. for P~c(t) = prob{Nt = k}.

(8.2.7)

This is carried out in two steps. First we derive an ordinary differential equation for p~c(t), and then we solve it. In our construction it will be useful to rewrite equations (8.2.6a.) through (8.2.6c) using the notation of (8.2.7): (8.2.8a.) Po(O) = 1, 1 00 lim - LPi(t) = 0, t-+0 t

(8.2.8b)

~ = lim(1/t)pl(t).

(8.2.8c)

i=2

and t-+0

To obtain the differential equation for P~c(t), we first start with Po(t), noting that Po(t +h) may be written as

Po(t +h) = prob{Nt+h = 0} = prob{Nt+h- Nt + Nt- No = 0}. Since Ntis not decreasing, hence (Nt+h- Nt) + (Nt- No)= 0 if and only if (Nt+h - Nt) = 0 and (Nt -No) = 0. Thus,

Po(t +h) = prob{(Nt+h - Nt) = 0 and (Nt -No) = 0}

256

8. Discrete Time Processes/Continuous Time Systems

= prob{Nt+h -

Nt

= 0} prob{Nt -No = 0}

= prob{Nh- No= O}prob{Nt- No= 0} = Po(h)Po(t),

(8.2.9)

where we have used the property of stationary independent increments. From (8.2.9) we may write

Po(t +h)- Po(t)

= Po( h)- 1Po(t).

h

(8.2.10)

h

Since L::oPi(t) = 1, we have Po(h) -1 h

= _Pt(h) h

_ .!:_ ~p·(h)

h.L...J

I

1

i=2

and, thus, by taking the limit of both sides of (8.2.10) ash-+ 0, we obtain

dPo(t) dt

= -.\Po(t) .

The derivation of the differential equation for fashion. Thus

(8.2.11) P~c(t)

proceeds in a similar

P~c(t +h)= prob{Nt+h = k}

= prob{Nt+h- Nt + Nt- No= k}

= prob{Nt- No= k and Nt+h- Nt = 0} +prob{Nt- No= k -1 and Nt+h- Nt = 1} k

+ Eprob{Nt- No= k- i

and Nt+h- Nt = i}

i=2 k

= Pk(t)Po(h) + Pk-l(t)pt(h) + LPk-i(t)pi(h). i=2

As before, we have

P~c(t + h~- P~c(t)

=

k

k

Po(h~- 1Pk(t) + Plt) Plc-l(t) + LPk-i(t)pi(h), i=2

and, by taking the limit as h -+ 0, we obtain (8.2.12) The initial conditions for Po(t) and p~c(t), k ~ 1, are just Po(O) = 1 (by definition), and this immediately gives p~c(O) = 0 for all k ~ 1. Thus, from (8.2.11), we have (8.2.13)

8.2. Probability Theory and Poisson Processes

2

0

3

4

257

5

FIGURE 8.2.1. Probabilities Po(t), p1(t), P:l(t) versus .Xt for a Poisson process.

Substituting this into (8.2.12) when k = 1 gives dp 1 (t)

dt

= -APl(t) + Ae-.\t

whose solution is

Pl(t)

= Ate->.t.

Repeating this procedure for k = 2, ... we find, by induction, that {8.2.14) The behavior of Pk(t) as a function oft is shown in Figure 8.2.1 fork= 0, 1, and 2. Figure 8.2.2 shows Pk(t) versus k for several values of At.

Remark 8.2.1. Note that in our derivation of equation {8.2.12) we have only used h > 0 and, therefore, the derivative pl, on the left-hand side of {8.2.12) is, in fact, the right-hand derivative of Pk· However, it is known [Szarski, 1967] that, if the right-hand derivative pl, exists and the Pk are continuous [as they are here by assumption {d) of Definition 8.2.5], then there is a unique solution to {8.2.12). Thus the functions {8.2.14) give the unique solution to the problem. 0 Although the way we have introduced Poisson processes and derived the expressions for Pk (t) is the most common, there are other ways in which this may be accomplished. However, all these derivations, as indicated by properties (a)-(c) of Definition 8.2.5, show that a Poisson process results if the events counted by Nt are caused by a large number of independent factors, each of which has a small probability of incrementing Nt.

258


>.1•0.1

0

>.t•I.O

I

L---..L---L.---1.

2

0

3

4

Pk(t)

0.1

>.t •10

0

0

.d

10

II

I

I

20

FIGURE 8.2.2. Plots of p,.(t) versus k for a Poisson process with >.t or 10.

= 0.1, 1.0,

8.3 Discrete Time Systems Governed by Poisson Processes A particular sample path for a Poisson process might look like the one shown in Figure 8.3.1. In this section we develop some ideas and tools that will allow us to study the behavior of a deterministic discrete time process given by a nonsingular transformation 8: X -+ X on a measure space (X, A, I') coupled with a Poisson process {Nt}t~o· The coupling is such that, even though the dynamics are deterministic, the times at which the transformation S operates are determined by the Poisson process. Thus we consider the situation in which each point x E X is transformed into SNt(x). This may be written symbolically as X-+

8Nt(x)

for times in the interval [0, oo). Specifically, we consider the following problem. Given an initial distribution of points x E X, with density f, how does this distribution evolve in time? We denote the time-dependent density by u(t,x) and set u(O,x) = f(x). The solution of this problem starts with a calculation of the probability

8.3. Discrete Time Systems Governed by Poisson Processes

259

FIGURE 8.3.1. A sample path for a Poisson process.

that (8.3.1) for a given set A c .A and time t > 0. This probability depends on two factors: the initial density f and the counting process {Nth>o· To be more precise, we need to calculate the measure of the set (8.3.2) This, in turn, requires some assumptions concerning the product space n X X given by

0 x X= {(w,x):w E O,x EX} that contains all sets of the form (8.3.2). In the space n x X we define (see Theorem 2.2.2) a product measure that, for the sets C x A, C E :F, A E .A, is given by prob(C)#Lt(A), and we denote it by Prob(C x A) or Prob(C x A) where, as usual,

#Lt(A)

=

= prob(C)#Lt(A),

(8.3.3)

L

f(x)#L(dx).

This measure is denoted by "Prob" since it is a probability measure. Equation (8.3.3) intuitively corresponds to the assumption that the initial position x and the stochastic process {Nth>o are independent. Now we may proceed to calculate the 'ineasure of the set (8.3.2). This set may be rewritten as the union of disjoint subsets in the following way: 00

{(w,x):SN,(w>(x)

E

A}= U{Nt(w) = k,Sk(x) E A} k=O 00

= U{Nt(w) = k} x {Sk(x) e A}. k=O

260


Thus the Prob of this set is 00

L Prob{Nt(w) = k, S"(x) E A}

Prob{SN• E A}=

k=O 00

= Lprob{Nt = k}pj(X E s-"(A)) k=O

EP~c(t)

=

k=O

EP~c(t)

=

[

f(x)IJ.(dx)

ls-"(A)

1

pk f(x)IJ.(dx)

(8.3.4)

A

lc=O

so that Prob{SN• E A}=

f EP~c(t)Pic f(x)IJ.(dx)

forAEA,

(8.3.5)

}Ak=-0

where, as before, P denotes the Frobenius-Perron operator associated with 8, and we have assumed that B: X--+ X is nonsingular. The integrand on the right-hand side of (8.3.5) is just the desired density, u(t,x): 00

u(t,x)

= LP~c(t)P" f(x).

(8.3.6)

k=O

[Note that the change in order of integration and summation in arriving at (8.3.5) is correct since IIPic /II = 1 and ~~oP~c(t) = 1. Thus the sequence on the right-hand side of (8.3.6) is strongly convergent in £ 1 .] Differentiating (8.3.6) with respect tot and using (8.2.12), we have

&(t,x) 0t

=

f: dp~c(t)

pk f(x)

dt

k=O 00

00

= -~ LPk(t)Pic f(x) + ~ LPk-1(t)Pk f(x). k=O

k=1

Since the last two series are strongly convergent in £ 1 , the initial differentiation was proper. Thus we have

&~ x) = -~u(t, x) + ~ EP~c(t)Pk+l f(x) k=O 00

= -.Xu(t, x) + ~p LP~c(t)Pk f(x) k=O

= -.Xu(t,x) + ~Pu(t,x).

8.4. Linear Boltzmann Equation

261

Therefore u( t, x) satisfies the differential equation

&u~x) = -~u(t,x) + ~Pu(t,x)

(8.3.7)

with, from (8.3.6), the initial condition

u(O,x)

= f(x).

We may always change the time scale in (8.3.7) to give

&u~x) = -u(t,x) + Pu(t,x).

(8.3.8)

From a formal point of view, equation (8.3.7) is a generalization of the system of differential equations (8.2.11) and (8.2.12) derived for the Poisson process. Consider the special case where X is the set of nonnegative integers {0, 1, ... }, p. is a counting measure, and S(x) = x + 1. For a single point n~ 1, Pf(n) = f(n -1) and when n

= 0, P/(0) = 0. Thus, from (8.3.7), we have 8u(t,x) 8t

=-~u(t,n)+~u(t,n-1)

and

8u(t, 0) 8t

n~1

= -~u(t, O),

which are identical with equations (8.2.12) and (8.2.11), respectively, except that the initial condition is more general than for the Poisson process since u(O, n) = f(n).

8.4 The Linear Boltzmann Equation: An Intuitive Point of View Our derivation in the preceding section of equation (8.3.8) for the density u(t,x) was quite long as we wished to be precise and show the connection with Poisson processes. In this section we present a more intuitive derivation of the same result, using arguments similar to those often employed in statistical mechanics. Assume that we have a hypothetical system consisting of N particles enclosed in a container, where N is a large number. Each particle may change its velocity x = (v 11 v2, v2) from x to S(x) only by colliding with the walls of the container. Our problem is to determine how the velocity

262


distribution of particles evolves with time. Thus we must determine the function u( t, x) such that

N

L

u(t,x)dx

is the number of particles having, at time t, velocities in the set A. The change in the number of particles, whose velocity is in A, between t and t + fl.t is given by

N

L

u(t+tl.t,x)dx-N

L

u(t,x)dx.

(8.4.1)

From our assumption, such a change can only take place through collisons with the walls of the container. Take fl.t to be sufficiently small so that a negligible number of particles make two or more collisions with a wall during fl.t. Thus, the number of particles striking the wall during a time fl.t with velocity in A before the collision [and, therefore, having velocities in S(A) after the collision] is N>.tl.t

L

(8.4.2)

u(t,x)dx,

where >.N is the number of particles striking the walls per unit time. In this idealized, abstract example we neglect the quite important physical fact that the faster particles are striking the walls of the container more frequently than are the slower particles. Conversely, to find the number of particles whose velocity is in A after the collision, we must calculate the number having velocities in the set s- 1 (A) before the collision. Again, assuming fl.t to be sufficiently small to make the number of double collisions by single particles negligible, we have N>.fl.t

f

u(t,x)dx.

(8.4.3)

ls-l(A)

Hence the total change in the number of particles with velocity in the set A over a short time fl.t is given by the difference between (8.4.3) and (8.4.2): N>.fl.t { ls-l(A)

u(t,x)dx- N>.fl.t

f

}A

u(t,x)dx.

(8.4.4)

By combining equation (8.4.1) with equation (8.4.4), we have N

f [u(t+fl.t,x)-u(t,x)]dx = >.N fl.t { f

}A

u(t,x)dx-

ls-l(A)

and, since

f

ls-l(A)

u(t,x)dx =

f

}A

Pu(t,x)dx,

f

}A

u(t,x)dx},

8.4. Linear Boltzmann Equation

263

where Pis the Frobenius-Perron operator associated with S, we have

N L[u(t+J)..t,x) -u(t,x)]dx

= >.N J)..t L[-u(t,x) +Pu(t,x)]dx.

(8.4.5)

Equation (8.4.5) is exact to within an error that is small compared to J)..t. By dividing through in (8.4.5) by /),.t and passing to the limit J)..t - t 0, we obtain

L

Ou~x)dx=>. L[-u(t,x)+Pu(t,x)]dx,

which gives

8u~x) = ->.u(t,x) + >.Pu(t,x). Thus we have again arrived at equation (8.3.7). In this derivation we assumed that the particle, upon striking the wall, changed its velocity from x to S(x), where 8: X - t X is a point-to-point transformation. An alternative physical assumption, which is more general from a mathematical point of view, would be to assume that the change in velocity is not uniquely determined but is a probabilistic event. In other words, we might assume that collision with the walls of the container alters the distribution of particle velocities. Thus, if before the collision the particles have a velocity distribution with density g, then after collision they have a distribution with density Pg, where P: L 1 (X) - t L 1 (X) is a Markov operator. So, assume as before that u( t, x) is the density of the distribution of particles having velocity x at time t, so

Ni u(t,x)dx is the number of particles with velocities in A. Once again,

>.N J)..t L u(t,x)dx is the number of particles with velocity in A that will collide with the walls in a time /),.t, whereas

>.N /),.t

L

Pu(t,x)dx

is the number of particles whose velocities go into A because of collisions over a time /),.t. Thus,

- >.N /),.t

L

u(t, x) dx + >.N /),.t

L

Pu(t, x) dx

264


is the net change, due to collisions over a time !:it in the number of particles whose velocities are in A. Combining this result with (8.4.1), we immediately obtain the balance equation {8.4.5), which leads once again to {8.3.7). The only difference is that P is no longer a Frobenius-Perron operator corresponding to a given one-t~one deterministic transformationS, but it is an arbitrary Markov operator. Since in our intuitive derivations of {8.3.7) presented in this section, we used arguments that are employed to derive a Boltzmann equation, we will call equation {8.3.7) a linear abstract Boltzmann equation corresponding to a collision (Markov) operator P. To avoid confusion with the usual Boltzmann equation, bear in mind that x corresponds to the particle velocity and not to position. Indeed, it is because we assume that the only source of change for particle velocity is collisions with the wall, that drift and external force terms do not appear in (8.3.7). Our next goal will not be to apply equation (8.3. 7) to specific physical systems. Rather, we will demonstrate the interdependence between the properties of discrete time deterministic processes, governed by S: X -+ X or a Markov operator, and the continuous time process, determined by {8.3.7). The next four sections are devoted to an examination of the most important properties of (8.3.7), and then in the last section we demonstrate that the Tjon-Wu representation of the Boltzmann equation is a special case of {8.3.7).

8.5

Elementary Properties of the Solutions of the Linear Boltzmann Equation

To facilitate our study of the linear Boltzmann equation {8.3.7), we will consider the solution u( t, x) as a function from the positive real numbers, R+, into L 1 u:R+-+ L 1 • Thus, by writing {8.3.8) in the form du

dt

= (P- I)u,

(8.5.1)

where Pis a given Markov operator and I is the identity operator, we may apply the Hille-Yosida theorem 7.8.1 to the study of equation {8.3.8). All three assumptions (a)-(c) of the Hille-Yosida theorem are easily shown to be satisfied by the operator (P-I) of equation {8.5.1). First, since A = P-I is defined on the whole space, L 1 , V(A) = L 1 and property (a) is thus trivially satisfied. To check property {b), rewrite the resolvent equation>../- Af = g using

8.5. Linear Boltzmann Equation: Solutions

A

=P -

265

I to give (>-.+1)/-Pf=g.

(8.5.2)

Equation (8.5.2) may be easily solved by the method of successive approximations. Starting from an arbitrary /o, we define fn by

().. + 1)/n- Pfn-1 = g, so, as a consequence, 1

/n=(>-.+ 1)"'

pn~ ~ 1 A:-1 J0+~()..+1)kp g.

(8.5.3)

Since IIPkgll $ llgll, the series in (8.5.3) is, therefore, convergent, and the unique solution f of the resolvent equation (8.5.2) is (8.5.4)

Remark 8.5.1. The method of successive approximations applied to an equation such as (8.5.2) will always result in a solution (8.5.3) that converges to a unique limit, as n --+ oo, when IIPII $ ).. + 1. The limiting solution given by (8.5.4) is called a von Neumann series. 0 To check that the linear Boltzmann equation satisfies property (c) of the Hill~Yosida theorem, integrate (8.5.4) over the entire space X to give

E

I R>.g(x)JJ(dx) = (>-.: 1)k I pk-1g(x)JJ(dx) lx k=l lx 1 I = {; (>-. + 1)k lx g(x)JJ(dx) 00

11

= -)..

X

g(x)JJ(dx)

1

= -, )..

where we used the integral-preserving property of Markov operators in passing from the first to the second line. Thus,

L

AR>.g(x)JJ(dx)

= 1,

and, since >-.R>. is linear, nonnegative, and also preserves the integral, it is a Markov operator. Thus condition (c) is automatically satisfied (see

Corollary 7.8.1). Therefore, by the Hill~Yosida theorem, the linear Boltzmann equation (8.3.8) generates a continuous semigroup of Markov operators, {f-'t}t~O·

266


To determine an explicit formula for

Pt, we first write

so lim

>.-+oo

A>.f = Pf- f.

Thus, by the Hill~Yosida theorem and equation (7.8.3), the unique semigroup corresponding to A = P - I is given by

A! =

et(P-1) j,

(8.5.5)

and the unique solution to equation (8.3.8) with the initial condition u(O, x) = f(x) is u(t,x) = et(P- 1 ) f(x). (8.5.6) Although we have determined the solution of (8.3.8) using the Hill~ Yosida theorem, precisely the same result could have been obtained by applying the method of successive approximations to equation {8.5.1). However, our derivation once again illustrates the techniques involved in using the Hill~Yosida theorem and establishes that (8.3.8) generates a continuous semigroup of Markov operators. Finally, we note that if P in equation (8.3.8) is a Frobenius-Perron operator corresponding to a nonsingular transformation 8, the solution can be obtained by substituting equation (8.2.14) into equation {8.3.6). In addition to the existence and uniqueness of the solution to (8.3.8), other properties of Pt may be demonstrated. Property 1. From inequality (7.4.7) we know that, given ft,/2 E Lt, the norm (8.5.7) IIPtft-Pthll is a nonincreasing function of time t. Property 2. If for some f E £ 1 the limit

/. = t-+00 lim A! exists, then, for the same

(8.5.8)

f, lim Pt(Pf)

t-+oo

=f•.

(8.5.9)

To show this, we prove even more, namely that lim Pt{f- Pf) = 0

t-+oo

(8.5.10)


for all

f

267

E £ 1 • Now,

(8.5.11) and

oo tn

f>t(Pf)

oo

= e-t L

lpn+l f n=O n.

= e-t L

tn-1

( -1)1pnf. n=1 n ·

Taking the norm of I'tf- i't(Pf), we have

III'tf- Pt(P!)II

~ e-t ~ [ ~ - (~=-:)I] pn /II + e-t IIIII 00

ltn

tn-1

I

~ e-t ~ nl - (n- 1)1 11/11 + e-t IIIII· If t

= m, an integer, then 00

I

tn tn-1 e-t ~ nl - (n- 1)1

I=

(

2e-m :1m -

1)

2

since almost all of the terms in the series cancel. However, by Stirling's formula, ml = mme-mv'11rm9m, where 9m -+ 1 as m -+ oo. Thus for integer t, converges to zero as t -+ oo. Since, by property 1 this quantity is a nonincreasing function, then (8.5.10) is demonstrated for all t -+ oo. Finally, inserting (8.5.8) into (8.5.10) gives the desired result, (8.5.9).

Remark 8.5.2. Note that the sum of the coefficients of Pt! given in equation {8.5.11) is identically 1, and thus the solutions of the linear Boltzmann equation u(t, x) = Pt!(x) bear a strong correspondence to the averages An! studied earlier in Chapter 5, with n and t playing analogous roles. 0 Property 9. The operators P and Pt commute, that is, P Pt! = PtPf for all f E £ 1 . This is easily demonstrated by applying P to (8.5.11):

Properly 4. If for some f E L 1 the limit (8.5.8), /. = limt-oo Pt!, exists, then/. is a fixed point of the Markov operator P, that is,

P/. = / •.

268


To show this, note that if

then, by (8.5.9),

which gives the desired result. Further, the same argument shows that, if /. = limn-+oo Pt"f exists for some subsequence {tn}, then P/. = j •. Property 5. HPj.=/. for some/. e £ 1 , then also iH. = j •. This is also easy to show. Write P/. = /. as

(P-1)/.

= 0.

Since (P-I) = A is an infinitesimal operator, and every solution of A/ = 0 is a fixed point of the semigroup (see Section 7.8), we have immediately that Pt!. = j •.

8.6 Further Properties of the Linear Boltzmann Equation AB shown in the preceding section, the solutions of the linear Boltzmann equation are rather regular in their behavior, that is, the distance between any two solutions never increases. Now we will show that, under a rather mild condition, Pt! always converges to a limit. Recall our definition of precompactness (Section 5.1) and observe that every sequence {/n} that is weakly precompact contains a subsequence that is weakly convergent. Analogously, if for a given f, the trajectory {Pd} is weakly precompact, then there exists a sequence {tn} such that {Pt"f} is weakly convergent as tn - oo. To see this, take an arbitrary sequence of numbers t~ - oo and, then, applying the definition of precompactness to {Pt:, I}, choose a weakly convergent subsequence {Pt" I} of {Pt:, I}. Theorem 8.6.1. If the trajectory exists a fixed point of P.

{Pd} is weakly precompact, then there

Proof. If { Pt!} is weakly precompact, then there exists a sequence {tn} such that (8.6.1) weakly exists. This implies the weak convergence of

(8.6.2)


269

However, from (8.5.10), we have lim

n-+oo

Pt .. U- Pf) = 0,

and, thus, from equations (8.6.1) and (8.6.2), we have

P/. =

/.,

which establishes the claim. Note also from property 5 of Pt (Section 8.5) that this implies Pt!. = /.. •

Theorem 8.6.2. For a given f E L 1 , if the trajectory {AI} is weakly precompact, then

A! strongly converges to a limit.

Proof. From Theorem 8.6.1 we know that Pt .. I converges wea.kly to an/. that is a fixed point of P and Pt. Write f E L 1 in the form

I= I-/.+/•. Assume that for every c

> 0 the function f- /. may be written in the form

I - /. = Pg - g + r,

(8.6.3)

where g E L 1 and llrll ~ c. (We will prove in the following that this representation is possible.) By using (8.6.3), we may write

Pt! = AU- /. + /.) = Pt(Pg- g) + Pt!. + Ptr· However

Pt!. = /.

and, thus,

From (8.5.10), the first term on the right-hand side approaches zero as t-+ oo, whereas the second term is not greater than c. Thus

11Pt!-/.ll~2c for t sufficiently large, and, since e is arbitrary, lim

t-+oo

IIA/- 1.11 = o,

which completes the proof if (8.6.3) is true. Suppose (8.6.3) is not true, which implies that

I-/. ¢ closure(P- I)L 1 (X). This, in turn, implies by the Hahn-Banach theorem (see Proposition 5.2.3) that there is a go E L 00 such that

(/-/.,go) :f:. 0

(8.6.4)

270


and

{h,go} for all hE closure(P-

I)L1 (X).

=0

In particular

((P- I)Pnf,go} = 0, since (P- I)Pn f E (P- I)L 1 (X), so

(pn+l /,go}= (Pn/,go} for n

= 0, 1, .... Thus, by induction, we have (pnf,go} =(!,go).

(8.6.5)

Furthermore, since e-t :E~=O tn fn! = 1, we may multiply both sides of (8.6.5) by e-ttn fn! and sum over n to obtain

or

(8.6.6)

(Ptf,go} =(!,go}. Substituting t = tn and taking the limit as t--+ oo in (8.6.6) gives

(/.,go}= {/,go}, and, thus,

(!.-/,go}= 0, which contradicts equation (8.6.4). Thus (8.6.3) is true.

•

8.7 Effect of the Properties of the Markov Operator on Solutions of the Linear Boltzmann Equation From the results of Section 8.6, some striking properties of the solutions of the linear Boltzmann equation emerge. The first of these is stated in the following corollary. Corollary 8.7.1. If for f E £ 1 there exists agE £ 1 such that t ~ 0,

(8.7.1)

then the (strong) limit

(8.7.2)

8. 7. Markov Operator and Boltzmann Equation

271

exists. That is, either i'tf is not bounded by any integrable function or Ptf is strongly converyent.

Proof. Observe that {Ptf} is weakly precompact by our first criterion of precompactness; see Section 5.1. Thus the limit {8.7.2) exists according to Theorem 8.6.2. • With this result available to us, we may go on to state and demonstrate some important corollaries that give information concerning the convergence of solutions Ptf of {8.3.8) when the operator P has various properties.

Corollary 8. 7 .2. If the {Markov) opemtor P has a positive fixed point /., /.(x) > 0 a.e., then the strong limit, limt-+oo Ptf, exists for all/ E L 1 • Proof. First note that when the initial function

f

satisfies

l/1 :5 c/.

{8.7.3)

for some sufficiently large constant c > 0, we have

IPn /I :5 pn(c/.)

= cPn /. = cf•.

Multiply both sides by e-ttn fnl and sum the result over n to give

The left-hand side of this inequality is just

IPtfl, so that

IA/1:::; c/., and, since Ptf is bounded, by Corollary 8.7.1 we know that the strong limit limt-+oo Ptf exists. In the more general case when the initial function f does not satisfy (8.7.3), we proceed as follows. Define a new function by

!. (x) _ { f(x) c

-

0

if 1/(x)l :5 cf.(x) if 1/(x)l > cf.(x).

It follows from the Lebesgue dominated convergence theorem that lim

C-+00

Thus, by writing /

11/c -/II = 0.

= /c + / - /c, we have

At= Ptfc + PtU -!c)·

272

Since


/c satisfies

1/cl $ cj.,

from (8.7.3) we know that {F't/c} converges strongly. Now take e > 0. Since {f'tfc} is strongly convergent, there is a to> 0, which in general depends on c, such that for t ~ to, t' ~ 0.

(8.7.4)

fort~O

(8.7.5)

Further, for a fixed but sufficiently large c. From equations (8.7.4) and (8.7.5) it follows that for t ~ to, t' ~ 0, which is the Cauchy condition for {Ftf}. Thus {Fd} also converges strongly, and the proof is complete. • The existence of the strong limit (8.7.2) is interesting, but from the point of view of applications we would like to know what the limit is. In the following corollary we give a sufficient condition for the existence of a unique limit to (8.7.2), noting, of course, that, since (8.7.2) is linear, uniqueness is determined only up to a multiplicative constant.

Corollary 8. 7 .3. Assume that in the set of all densities f E D the equation Pf = f has a unique solution/. and /.(x) > 0 a.e. Then, for any initial density, f E D (8.7.6) lim Pt! = /., t-+oo

and the convergence is strong.

Proof. The proof is straightforward. From Corollary 8. 7.2 the limt-+oo Pt! exists and is also a nonnegative normalized function. However, by property 4 of Pt (Section 8.5), we know that this limit is a fixed point of the Markov operator P. Since, by our assumption, the fixed point is unique it must be /.,and the proof is complete. • In the special case that P is a Frobenius-Perron operator for a nonsingular transformation S: X --+ X, the condition P /. = /. is equivalent to the fact that the measure

1-'J.(A) =

L

/.(x)J.t(dx)

is invariant with respect to S. Thus, in this case, from Corollary 8.7.2 the existence of an invariant measure 111. with a density /.(x) > 0 is sufficient for the existence ofthe strong limit (8.7.2) for the solutions of (8.3.8). Since,

8.8. Linear Boltzmann Equation with a Stochastic Kernel

273

for ergodic transformations/. is unique (cf. Theorem 4.2.2), these results may be summarized in the following corollary.

Corollary 8.7.4. Suppose S:X-+ X is a nonsingular tmnsfonnation and P is the cotTesponding Frobenius-Perron operator. Then with respect to the trajectories {i'tf} that generate the solutions of the linear Boltzmann equation (8.3.8): 1. If there exists an absolutely continuous invariant measure J.L1. with a positive density j.(x) > 0 a.e., then for every f E L 1 the strong limit, limt-+oo iH exists; and

2. If, in addition, the tmnsfonnation S is ergodic, then lim

t-+oo

AI=/.

(8.7.7)

for allIED. Now consider the more special case where (X, A, J.L) is a finite measure space and S: X -+ X is a measure-preserving transformation. Since S is measure preserving,/. exists and is given by

/.(x) = 1/J.L(X)

for x EX.

Thus limt-+oo Pt! always exists. Furthermore, this limit is unique, that is, lim

t-+oo

Pt/ = /. = 1/J.L(X)

(8.7.8)

if and only if Sis ergodic (cf. Theorem 4.2.2). In closing this section we would like to recall that, from Definition 4.4.1, a Markov operator P: L 1 -+ L 1 is exact if and only if the sequence { pn!} has a strong limit that is a constant for every f E L 1 . Although the term exactness is never used in talking about the behavior of stochastic semigroups, for the situation where (8.7.8) holds, then, the behavior of the trajectory {Ptf} is precisely analogous to our original definition of exactness. Figuratively speaking, then, we could say that S is ergodic if and only if { Pt h>o is exact.

8.8 Linear Boltzmann Equation with a Stochastic Kernel In this section we consider the linear Boltzmann equation 8u(t x)

-~';........:..

Ot

+u(t,x) = Pu

274


where the Markov operator P is given by

Pf(x) and K( x, y): X x X

-+

=

L

(8.8.1)

K(x,y)f(y)dy

R is a stochastic kernel, that is,

K(x,y)

~

(8.8.2)

0

L

and

(8.8.3)

K(x,y) dx = 1.

For this particular formulation of the linear Boltzmann equation, we will show some straightforward applications of the general results presented earlier. The simplest case occurs when we are able to evaluate the stochastic kernel from below. Thus we assume that for some integer m the function inf11 Km(x, y) is not identically zero, so that

lxf infKm(x,y)dx>O

(8.8.4)

11

(Km is them times iterated kernel K). In this case we will show that the strong limit {8.8.5) lim i'tl = /. t-+oo exists for all densities solution of

f

E

f(x)

D, where /. is the unique density that is a

=

L

(8.8.6)

K(x,y)f(y)dy.

The proof of this is quite direct. Set

h(x)

= inf Km(x, y). 11

By using the explicit formula (8.5.11) for the solution

However, for n

~

i'tl, we have

m, we may write

pn f(x)

=

L

Km(x, y)pn-m /{y) dy

~ h{x),

8.8. Linear Boltzmann Equation with a Stochastic Kernel

275

and thus the explicit solution Ptf becomes

Thus we have immediately that Ptf(x)- h(x) ;::: A

(

-e-t

Lm Itn) h(x), n.

n=0

so that (Ptf- h)- $

(

L 1tn) h. m

e-t

n=O

n.

Since, however, e-ttn-+ 0 as t-+ oo, we have lim

t-+oo

IICA/- h)-11 = o,

and, by Theorem 7.4.1, the strong limit/.. of (8.8.5) is unique. Properties 4 and 5 of the solution Ptf, outlined in Section 8.5, tell us that / .. is the unique solution of Pf = /, namely, equation (8.8.6). Thus the proof is complete. Now we assume, as before, that K(x, y) is a stochastic kernel for which there is an integer m and agE L 1 such that Km(x,y) $ g(x)

for x,y EX.

(8.8.7)

Then the strong limit lim Ptf

(8.8.8)

t-+oo

exists for all f E L 1 • AB before, to prove this we use the explicit series representation of Ptf, noting first that, because of (8.8.7), we have, for n;::: m,

IPn/(x)l

= IPm(pn-m/(x))l $ $ g(x) [

[

Km(x,y)IPn-mf(y)idy

lpn-m /(y)l dy $ g(x)llfll.

276


Thus we can evaluate I'tf as

IP.JI ,s.-•~:IP"/1+ (·-•

~ e-t

t t~ IPn/I

n=O

n.

.f._, ~u) 11/11

+ gllfll·

Further, setting m

r

= c "E IPn/1, n=O

tn

c= sup e -t I• n. O 0, (8.9.2}

= (v 2 /2} (note that xis not a spatial coordinate) and u(t,x)

= const

00

1 a:

F(t v) ' dv. Jv-x

Equation (8.9.2}, called the Tjon-Wu equation [Barnsley and Cormille, 1981], is nonlinear because of the presence of u(t, y- z)u(t, z) in the integrand on the right-hand side. Thus the considerations of this chapter are of no help in studying the behavior of u( t, x) as t -+ oo. However, note that exp( -x) is a solution of (8.9.2}, a fact that we can use to study a linear problem. Here we will investigate the situation where a small number of particles with an arbitrary velocity distribution f are introduced into a gas, containing many more particles, at equilibrium, so that u.(x) = e:xp( -x). We want to know what the eventual distribution of velocities of the small number of particles tends to. Thus, on the right-hand side of (8.9.2}, we set u(t, y- z) = u.(y- z) = e:xp[-(y- z}], so the resulting linear Tjon-Wu equation is of the form

8u(t, x) 8t

+ u(t, x) =

1

00

a:

dy {" e-(ll-z>u(t, z) dz, Y

lo

X

> 0.

(8.9.3}

278


Equation (8.9.3) is a special case of the linear Boltzmann equation of this chapter with a Markov operator defined by

1 1" =1 00

Pf(x) =

..1!.. d y

:r:

for

f

e- 0 for some m, then we could determine limt-+oo Ptf. However, inf, K(x, y) = 0, and further composition of the kernel with itself leads to analytically complex results. Second, if we were able to find a g(x) :2:: Km(x,y) for some m, then the results of the preceding section could be applied. However, the maximum of K(x, y) in y occurs at y = x and -exp(x)Ei(-x) is not integrable. As before, compositions of K(x,y) become so complicated that it is difficult to work with them. A third alternative is the following. Note that J(x) = exp(-x) is a fixed point of (8.9.4). If we can show that exp( -x) is the unique fixed point of (8.9.4), then we may apply Corollary 8.7.3 to show that lim u(t,x)

t-+oo

= t-+oo lim Pt!(x) = e-:r:

(8.9.7)

for all densities J E D((O, oo)). From Pf = f and (8.9.4), we have

f(x)

=

1 1" 00

:r:

..1!.. d y

e-o be a Poisson process and S:R-+ R a nonsingular mapping. Consider the following procedure: In a time t > 0 a point x E R is transformed into S(x) + Nt. Given an initial density distribution function f of the initial point x find the density u(t, x) of S(x) + Nt. (As in Section 8.3 assume that the position x of the initial point and the process Nt are independent.) Prove that u(t, x) satisfies the differential equation

&~x) = -.Xu(t,x) + .Xu(t,x -1)

for

t > 0, x

E R,

which does not depend explicitly on S. (.X is defined in (8.2.8c).] Explain this paradox. 8.4. Derive formula (8.5.5) for the solution of the linear Boltzmann equation by the use of the Phillips perturbation theorem. 8.5. Consider the linear Boltzmann equation (8.5.1) and corresponding semigroup {Fth2::0· Assuming that P: L 1 -+ L 1 is a constrictive operator, prove that exists for every

f

E L 1•

8.6. Again consider the linear Boltzmann equation (8.5.1) and assume that P: L 1 (X,A,Jt)-+ L 1 (X,A,Jt) is sweeping with respect to a family A. CA. Prove that the semigroup {Fth2::o is sweeping with respect to the same family. 8.7. The nonlinear Tjon-Wu equation (8.9.2) may be written in the form

du dt

= -u+P(u,u),

Exercises

P(f, g)(x)

=

1 -dy1"

281

00

:1:

y

l(y- z)g(z) dz.

0

Verify that the series 00

u(t)

= e-t L)1 - e-t)nun n=O

with 1

Un

=-

n

n-1

L P(u~c, Un-1-lc),

UO

=IE V(R+)

k=O

is uniformly convergent on compact subintervals of R+ and satisfies the

nonlinear Tjon-Wu equation with the initial condition u(O) 1988).

=I

(Kielek,

9 Entropy

The concept of entropy was first introduced by Clausius and later used in a different form by L. Boltzmann in his pioneering work on the kinetic theory of gases published in 1866. Since then, entropy has played a pivotal role in the development of many areas in physics and chemistry and has had important ramifications in ergodic theory. However, the Boltzmann entropy is different from the Kolmogorov-Sinai-Qrnstein entropy (Walters, 1975; Parry, 1981] that has been so successfully used in solving the problem of isomorphism of dynamical systems, and which is related to the work of Shannon [see Shannon and Weaver, 1949]. In this short chapter we consider the Boltzmann entropy of sequences of densities { pn!} and give conditions under which the entropy may be constant or increase to a maximum. We then consider the inverse problem of determining the behavior of { pn!} from the behavior of the entropy.

9.1

Basic Definitions

If (X, A, p,) is an arbitary measure space and P: £ 1 - £ 1 a Markov operator, then under certain circumstances valuable information concerning the behavior of { pn!} (or, in the continuous time case, { pt!}) can be obtained from the behavior of the sequence

H(Pnf)

=

L

1J(Pn f(x))p,(dx),

where 1J(u) is some function appropriately defined for u ~ 0.

{9.1.1)

284

9. Entropy

1/2

FIGURE 9.1.1. Plot of function fl(u)

= -ulogu.

The classical work of Boltzmann on the statistical properties of dilute gases suggested that the function 1J should be of the form fJ(u)

= -u log u,

TJ(O)

= 0,

(9.1.2)

and gi,ves us our definition of entropy. Definition 9.1.1. If I~ 0 and~(/) E L 1 then the entropy of I is defined by

H(f)

=

L

TJ(/(x))J.'(dx).

(9.1.3)

Remark 9.1.1. If J.'(X) < oo, then the integral (9.1.3) is always well defined for every I ~ 0. In fact, the integral over the positive parts of TJ(/(x)), [TJ(/(x))]+ = max[O,TJ(/(x))]. is always finite. Thus H(/) is either finite or equal to -oo. 0 Since we take TJ(O) = 0, the function TJ(u) is continuous for all u ~ 0. The graph of 1J is shown in Figure 9.1.1. One of the most important properties of 1J is that it is convex. To see this, note that TJ"(u)

= -1/u

so TJ" (u) < 0 for all u > 0. From this it follows immediately that the graph of 1J always lies below the tangent line, or TJ(u) ~ (u- v)TJ'(v) + TJ(v)

(9.1.4)

for every u, v > 0. Combining (9.1.4) with the definition of 1J given in equation (9.1.2) leads to the Gibbs inequality u-ulogu~v-ulogv

for u,v > 0,

which we shall have occasion to use frequently.

(9.1.5)

9.1. Basic Definitions

285

If f and g are two densities such that 71(/(x)) and f(x) logg(x) are integrable, then from (9.1.5) we have the useful integral inequality

-i

f(x) log /(x)JJ(dx) :5 -

i

f(x) log g(x)JJ(dx).

(9.1.6)

and the equality holds only for f =g. Inequality (9.1.6) is often of help in proving some extremal properties of H(/) as shown in the following. Proposition 9.1.1. Let JJ(X) < oo, and consider all the possible densities f defined on X. Then, in the family of all such densities, the maximal entropy occurs for the constant density

/o(x) = 1/JJ(X),

(9.1.7)

and for any other f the entropy is strictly smaller.

f

Proof. Pick an arbitrary

E D

so that the entropy of f is given by

H(f) =- [

f(x) log /(x)JJ(dx)

and, by inequality (9.1.6),

H(f) :5 - [ f(x) log /o(x)JJ(dx) =-log

[JJ(~)] [!(x)JJ(dx)

or

H(f) ::::; -log

[JJ(~)] ,

and the equality is satisfied only for f = fo· However, the entropy of /o is simply H(/o) =-

i JJ(~)

log

[JJ(~)] JJ(dx) =-log [JJ(~)],

so H(/) :5 H(/o) for all f E D. • If JJ(X) = oo, then there are no constant densities and this proposition fails. However, if additional constraints are placed on the density, then we may obtain other results for maximal entropies as illustrated in the following two examples. Example 9.1.1. Let X= [0, oo) and consider all possible densities f such that the first moment of f is given by (9.1.8)

286

9. Entropy

Then the density

(9.1.9) maximizes the entropy. The proof proceeds as in Proposition 9.1.1. From inequality (9.1.6) we have, for arbitrary f E D satisfying (9.1.8),

-1

00

H(J) :5

f(x)

1

log(.Xe--':~:) dx

00

=-log.X

1

00

f(x)dx+

-Xxf(x)dx

=-log.X+l.

Also, however, with H(J0) =

/o given by (9.1.9),

-1

00

.Xe-.\:1: log(.Xe--':~:) dx = -log .X+ 1

and thus H(J) :5 H(Jo) for all

f E D satisfying (9.1.8).

0

Example 9.1.2. For our next example take X= (-oo,oo) and consider all possible densities f E D such that the second moment of f is finite, that is, (9.1.10) /_: x 2J(x) dx = u 2. Then the maximal entropy is achieved for the Gaussian density

2

/o(x)

1 - exp (- x 2 ) = -V27ru 2 2u

(9.1.11)

.

As before, we calculate that, for arbitrary fED satisfying (9.1.10),

H(J) :5- /_: /(x) log [

.;2~u2 exp (- ;;2 )] dx

roo

roo

=-log [.; 1 2] f(x) dx + 2\ x2 J(x) dx 27ru 1-oo q 1-oo 1 =!-log [ -- ] . 2 V27ru2 Further

H(Jo) = -

roo /o(x) log /o(x) dx = ! -log [ v21ru ~] 1-oo 2

so that the entropy is maximized with the Gaussian density (9.1.11).

0

9.1. Basic Definitions

287

These two examples are simply special cases covered by the following simple statement. Proposition 9.1.2. Let {X,A,J.L) be a measure space. Assume that a sequence 91, ... , 9m of measurable functions is given as well as two sequences of real constants 91, ... , 9m and 111, ... , lim that satisfy

_

fx g;(x) exp[-v;g;(x)]J.L{dx) g; = fx n~1 exp[-v;g;(x)]J.L(dx), where all of the integrals are finite. Then the maximum of the entropy H(J) for all f E D, subject to the conditions

g; =

g;(x)f(x)J.L(dx),

[

occurs for

n~1 exp[-v;g;(x)]

;{! ( ) 0 X

i=l, ... ,m

fx n~1 exp[-ll;g;(x)]J.L(dx) •

-

Proof. For simplicity, set

=

Z

1IT X

exp[-v;g;(x)]J.L(dx)

i=1

so

m

fo(x)

= z- 1 II exp[-v;g;(x)J. i=1

From inequality (9.1.6), we have

H(J)

~=:....

[

f(x) log fo(x)J.L(dx)

f

lx

f(x) [-log Z-

"$: v;g;(x)] J.L(dx) •=1

=log Z

+ ~ 11; [

=log Z

+ ~ v;g;.

f(x)g;(x)J.L(dx)

m

i=1 Furthermore, it is easy to show that m

H(Jo) =log Z

+ ~lli9i i=1

and thus H(J) ~ H(Jo).

•

288

9. Entropy

Remark 9.1.2. Note that if m = 1 and g(x) is identified as the energy of a system, then the maximal entropy occurs for lo(x) =

z-le-vg(z)'

which is just the Gibbs canonical distribution function, with the partition function Z given by

Further, the maximal entropy

H(/o)

= log Z + vg

is just the thermodynamic entropy. AB is well known, all of the results of classical thermodynamics can be derived with the partition function Z and the preceding entropy H(/0 ). Indeed, the contents of Proposition 9.1.2 have been extensively used by Jaynes [1957] and Katz [1967] in an alternative formulation and development of classical and quantum statistical mechanics. 0 Thus, the simple Gibbs inequality has far-reaching implications in pure mathematics as well as in more applied fields. Another inequality that we will have occasion to use often is the Jensen inequality: If 17(u), u ~ 0 is a function such that 11" ~ 0 (i.e., the graph of 11 is convex), P: £P-+ £P 1 ~ p ~ oo, is a linear operator such that P1 = 1, and PI ~ 0 for all I~ 0, then for every IE LP, I~ 0, whenever P17(/) exists.

(9.1.12)

The proof of this result is difficult and requires many specialized techniques. However, the following considerations provide some insight into why it is true. Let 17(y) be a convex function defined for y ~ 0. Pick u, v, and z such that 0 ~ u ~ z ~ v. Since z E [u, v] there exist nonnegative constants, a and {j, with a + {3 = 1, such that

z = au+{3v. Further, from the convexity of 11 it is clear that 17(z) ~ r, where

r

= a17(u) + {j1J(v).

Thus 17(z) ~ r gives

17(au + {jv)

~

a17(u) + /317(v).

9.2. Entropy of P" / when P is a Markov Operator

289

Further, it is easy to verify by induction that for every sequence 0 $ u 1

log

L

f(x)

[~~:n}"

log[~~=~] J'(dx) = H(f I g),

0

9.3 Entropy H(Pn f) when P is a Frobenius-Perron Operator Inequalities (9.2.1} and (9.2.3} of Theorems 9.2.1 and 9.2.2 are not strong. In fact, the entropy may not increase at all during successive iterations of f. This is always the case when P is the Frobenius-Perron operator corresponding to an invertible transformation, which leads to the following theorem.

Theorem 9.3.1. Let (X, A, J') be a finite measure space and S: X --t X be an invertible measure-presennng transformation. If P is the FrobeniusPerron operator corresponding to S, then H(pn f)= H(f) for all n. Proof. If Sis invertible and measure preserving, then by equation (3.2.10} we have Pf(x) = f(S- 1 (x)) since J- 1 1. If P 1 is the Frobenius-Perron

=

9.3. Entropy H(Pn f) when P is a Frobenius-Perron Operator

293

operator corresponding to s- 1 , we also have Ptf{x) = f(S(x)). Thus PtPf = PPtf = f, so P1 =p-l. From Theorem 9.2.1 we also have ~

H(PtP!}

H(Pf)

~

H{!},

but, since P1 Pf = p-lpf = /, we conclude that H(Pf} = H{!}, so H(Pn f) = H(f) for all n. • Remark 9.3.1. For any discrete or continuous time system that is invertible and measure preserving the entropy is always constant. In particular, for a continuous time system evolving according to the set of differential equations x = F(x}, the entropy is constant if div F = 0 (see equation (7.8.18}]. Every Hamiltonian system satisfies this condition. D However, for noninvertible (irreversible) systems this is not the case, and we have the following theorem.

Theorem 9.3.2. Let (X, A, p.) be a measure space, p.(X) = 1, 8: X -+ X a measure-preseroing transformation, and P the Frobenius-Perron operator corresponding to S. If S is exact then lim H(Pn!} = 0

n-oo

for all f E D such that H(f) > -oo. Proof. Assume initially that f is bounded, that is, 0 ~ f 0 :5 pn f

~

pnc = cPn1

~

c. Then

= c.

Without any loss of generality, we can assume that c > 1. Further, since 77(u) :50 for u ~ 1, we have (note p.(X} = 1 and Hmax =OJ {9.3.1} where

An= {x: 1 ~ pnf(x)

~

c}.

Now, by the mean value theorem (using 77{1} =OJ, we obtain

lin

TJ(Pn f(x)}p(dx}l

=in ~k :5 k

where

{

ITJ(Pn f(x)}- TJ(1}Ip(dx}

JAn

IPn f(x)- 1lp.(dx}

LIPn

f(x} -1lp.(dx} = IIPnf -111,

k = sup l77'(u)l· l$u$c

294

9. Entropy

Since S is exact, from Theorem 4.4.1, we have for all J E D and thus lim {

n-+oo}A..

liP"J- 111 -+ 0 as n -+ oo

t7(Pn J(x))JJ(dz)

= 0.

From inequality (9.3.1), it follows that H(P" f) converges to zero. Now relax the assumption that J is bounded and write J in the form

!=It +h, where

I 1 (x) -_ and

h

=

{0 J(x)

if J(x) > c if 0 ~ J(x) ~ c

J - It. Fixing E > 0, we may choose c sufficiently large so that llhll

<E

and H(h)

> -E.

Write P" J in the form

.P"J = (1-6)Pn ( 1 ~ ft) +6pn (~h),

6

where 6 = llhll· Now ft/(1- 6) is a bounded density, and so from the first part of our proof we know that for n sufficiently large

Furthermore,

6H

(pn (~h)) = H(Pn

h) -log (

~)

L

pn h(x)JJ(dz)

= H(Pn h) -llhlllog (~) = H(Pnh) + 6log 6. Since H(P" h) ~ H(h)

> -E, this last expression becomes

6H ( pn

(~h)) ~ -E + 6 log 6.

Combining these results and inequality (9.1.14), we have

H(Pnf)

~ (1- 6)H (pn ( 1 ~ 6 ~t)) +6H ~

-t(1 - 6) -

E

(pn (~h))

+ 6 log 6

= -2e + 6e + 6 log 6.

(9.3.2)

9.4. Behavior of pnf from H(Pnf)

295

Since f..'( X) = 1, we have H(pn f) ~ 0. Further since 6 < e and e is arbitrary, the right-hand side of (9.3.2) is also arbitrarily small, and the theorem is proved. •

Example 9.3.1. We wish to compare the entropy of the baker transformation S(

i•

) _ { (2x, b), 0~x< 0~y~1 (2x - 1'21y + 1) x, Y 2, 1 2< - x -< 1 , 0 < - y -< 1,

originally introduced in Example 4.1.3, with that of the dyadic transformation. Observe that the x-coordinate of the baker transformation is transformed by the dyadic transformation 81(x)

= 2x

(mod 1).

From our considerations of Chapter 4, we know that the baker transformation is invertible and measure preserving. Thus by Theorem 9.3.1 it follows that the entropy of the sequence {pn !}, where P is the FrobeniusPerron operator corresponding to the baker transformation, is constant for every density f. Conversely, the dyadic transformation 81 is exact. Hence, from Theorem 9.3.2, the entropy of {Pf !}, where P 1 is the Frobenius-Perron operator corresponding to 8 11 increases to zero for all bounded initial densities f. 0 Remark 9.3.2. Observe that in going from the baker to the dyadic transformation, we are going from an invertible (reversible) to a noninvertible (irreversible) system through the loss of information about the y-coordinate. This loss of information is accompanied by an alteration of the behavior of the entropy. An analogous situation occurs in statistical mechanics where, in going from the Liouville equation to the Boltzmann equation, we also lose coordinate information and go from a situation where entropy is constant (Liouville equation) to one in which the entropy increases to its maximal value (Boltzmann H theorem). 0

9.4 Behavior of pnf from H(Pnf) In this section we wish to see what aspects of the eventual behavior of pn f can be deduced from H(Pn f). This is a somewhat difficult problem, and the major stumbling block arises from the fact that '1 changes its sign. Thus, because of the integration in the definition of the entropy, it is difficult to determine f or its properties from H(f). However, by use of the spectral representation Theorem 5.3.1 for Markov operators, we are able to circumvent this problem.

296

9. Entropy

In our first theorem we wish to show that, if H(pn f) is bounded below, then P is constrictive. This is presented more precisely in the following theorem. Theorem 9.4.1. Let (X, A, JJ) be a measure space, JJ(X) < oo, and P: £ 1 -+ £ 1 a Markov operator such that P1 = 1. If there exists a constant c > 0 such that for every bounded f E D H(Pn f)~ -c

for n sufficiently large,

then P is constrictive.

Proof. Observe that P1 = 1 implies that Pf is bounded for bounded f. Thus, to prove our theorem, it is sufficient to show that the set F of all bounded f E D that satisfy

H(/)

~

-c

is weakly precompact. We will use criterion 3 of Section 5.1 to demonstrate the weak precompactness of F. Since 11/11 = 1 for all f E D, the first part of the criterion is satisfied. To check the second part take e > 0. Pick l = e- 1JJ(X}, N = exp[2(c + l)/e] and 6 = ef2N, and take a set A c X such that JJ(A) < 6. Then

rf(x)JJ(dx) = hlr f(x)JJ(dx) + h2r f(x)JJ(dx),

(9.4.1)

h where

A1

= {x E A:f(x) ~ N}

A2

= {x E A: /(x) > N}.

and

The first integral on the right-hand side of (9.4.1} clearly satisfies

{ f(x)JJ(dx)

}At

~ N6 = ef2.

In evaluating the second integral, note that from H (/) that

{ f(x) log /(x)JJ(dx} jA2

~ C-

~ c+

{

jX\A2

r

~

-c, it follows

f(x) log /(X)JJ(dx)

'lmaxP.( dx) lx\A2 ~ c + (1/e)JJ(X} = c + l.

9.4. Behavior of pn f from H(Pn f)

297

Therefore

f

jA2 or

f(x)log Np(dx)

f

< c+ l

c+l

}A f(x)p(dx) < log N 2

e

= 2·

L

Thus

f(x)p(dx) < e

and :F is weakly precompact. Thus, by Definition 5.3.3, the operator Pis constrictive. • Before stating our next theorem, consider the following. Let (X,A,p) be a finite measure space, 8: X - X a nonsingular transformation, and P the Frobenius-Perron operator corresponding to 8. Assume that for some c > 0 the condition

H(Pnf) 2:: -c holds for every bounded f E D and n sufficiently large. Since P is a Markov operator and is constrictive, we may write P f in the form given by the spectral decomposition Theorem 5.3.1, and, for every initial/, the sequence {pn!} will be asymptotically periodic.

Theorem 9.4.2. Let (X, A, p) be a normalized measure space, 8: X -X a measure-preseroing tmnsfonnation, and P the Frobenius-PeTTOn opemtor corresponding to 8. If lim H(Pn f) = 0 n-+oo

for all bounded f E D, then 8 is exact.

Proof. It follows from Theorem 9.4.1 that Pis constrictive. Furthermore, since 8 is measure preserving, we know that P has a constant stationary density. From Proposition 5.4.2 we, therefore, have r

pn+l f(x)

= L Aa-"(i)(/)IA, (x) + Qnf(x) i=l

If we can demonstrate that r shown 8 to be exact. Pick

f(x)

as an initial

f.

If T

= 1, then from Theorem 5.5.2 we will have = [1/p(A1)]1A

(x) is the asymptotic period of pn f, then we must have 1

However, by assumption,

lim H(Pn f)

n-+oo

= 0,

298

9. Entropy

and, since the sequence {H(pn.,. f)} is a constant sequence, we must have

Note that, by Proposition 9.1.1, H(J} = 0 only if

f(x)

= 1x(x).

So, clearly, we must have

[1/I'(At}]1A 1 (x)

= 1x(x).

This is possible if and only if A1 is the entire space X, and thus r = 1. Hence Sis exact. • This theorem in conjunction with Theorem 9.3.2 tells us that the convergence of H(Pn f) to zero as n --+ oo is both necessary and sufficient for the exactness of measure-preserving transformations. If the transformation is not measure preserving then an analogous result using the conditional entropy may be proved. To see this, suppose we have an arbitrary measure space (X, .A, I') and a nonsingular transformationS: X--+ X. Let P be the Frobenius-Perron operator corresponding to Sand g ED (g > 0} the stationary density of P so Pg =g. Since Sis not measure preserving, our previous results cannot be used directly in examining the exactness of S. However, consider the new measure space (X, .A, Jl}, where

jj(A)

=

i

g(x)l'(dx)

for A E .A.

Since Pg = g, therefore ji is an invariant measure. Thus, in this new space the corresponding Frobenius-Perron operator P is defined by

f

}A

Ph(x}ji(dx}

and satisfies P1

=f

ls-l(A)

h(x}ji(dx}

for A E .A

= 1. This may be rewritten as

f [Ph(x)]g(x)l'(dx} = f

}A

ls-l(A)

h(x)g(x)l'(dx}.

However, we also have

f

ls-l(A)

so that (i'h}g

h(x)g(x)l'(dx}

= f P(h(x)g(x)}~£(dx} }A

= P(hg} or Ph= (1/g)P(hg).

9.4. Behavior of P" f from H(P" /)

299

Furthermore, by induction,

finh

= (1/g)Pn(hg).

In this new space {X, A, jJ.), we may also calculate the entropy il(Pn h) as

H(Ph) =-

i

pnh(x) log[finh(x)]il{dx)

g(~) pn(h(x)g(x)) log [ pn(~~rx))] g(x)p(dx)

=- [

= H(P.. (hg) I g). Observe that hE D(X,A,ji,) is equivalent to

h

~0

and

[

h(x)g(x)p(dx)

= 1,

which is equivalent to hg E D(X, A, p). Set I= hg, so

il(Pnh)

= H(Pn I I g).

We may, therefore, use our previous theorems to examine the exactness of S in the new space (X, A, jJ) or its asymptotic stability in the original space (X, A, p), that is, S is statistically stable in (X, A, p) if and only if {9.4.2) for all

I ED

such that

I/ g is bounded.

Example 9.4.1. Consider the linear Boltzmann equation [equation {8.3.8)]

8u(t,x) 8t +u(t,x)=Pu(t,x), with the initial condition u(O,x) = l(x), which we examined in Chapter 8. There we showed that the solution of this equation was given by

u(t,x)

= et(P-I) l(x) = Al(x),

and et(P-I) is a semigroup of Markov operators. From Theorem 9.2.2 we know immediately that the conditional entropy H(Pt! I 1.) is continually increasing for every 1. that is a stationary density of P. Furthermore, by (9.4.2) and Corollary 8.7.3, if l.(x) > 0 and 1. is the unique stationary density of P, then lim H(Pt!

t-+oo

I 1.) = H(f. I 1.) = 0.

Thus, in the case in which 1. is positive and unique, the conditional entropy for the solutions of the linear Boltzmann equation always achieves its maximal value.

300

9. Entropy

Exercises 9.1. Let X= {(xt, ... ,x~~:) E Rlc:x1 ~ 0, ... ,x~~: ~ 0}. Consider the family Fm 1 ···m• of densities 1: X - R+ such that

1 ·1 00

• •

00

xd(xt, ... ,

x~~:)dxt · · · dx~~: = Tni > 0,

i

= 1, ... ,k.

Find the density in Fm 1 ···m• that maximizes the entropy. 9.2. Let X = {(x, y) e R 2 : y ~ ajxl} where a is a constant. Consider the family Fma of densities 1: X - R+ such that

j

LYI(x,y)dxdy = m

> 0.

Show that for a> 0 there is a density in Fma having the maximal entropy and that for Q ~ 0 the entropy in r mOt is unbounded. 9.3. Consider the space X = { 1, ... , N} with the counting measure. In this space 'D(X) consists of all probabilistic vectors (It = 1(1), ... , IN = f(N)) satisfying

lc=l

Show that j,, = 1/N, k = 1, ... , N maximizes the entropy. For which vector is the entropy minimal? 9.4. Consider the heat equation

au

62

a2 u

for t

&t-2ax2

> 0, x

E

R,

and prove that every positive solution u( t, x) corresponding to the bounded initial u(O, x) = l(x), IE D with compact support, satisfies d dtH(u)

(a )2

= J+oo -oo u ax lnu

dx ~ 0.

9.5. Consider the differential equation

au

62

&t = 2

B2u

fJx2 -

a

ax (b(x)u)

for t

> 0, 0 ~ x

~

1

with the boundary value conditions

Uz(t, 0)

= Uz{t, 1) = 0

for t > 0.

Assume that b is a (J2 function and that b(O) = b(1) = 0. Without looking for the explicit formula for the solutions (which, for arbitrary b, is difficult) prove the following properties:

Exercises

301

(a) For every solution

1 1

u(t,x)dx = const.

(b) For every two positive normalized solutions u1 and u 2

d~H(u1 I u2) =

1

2

1

u1

(!In ::) dx ~

0,

(Risken, 1984; Sec. 6.1.)

9.6. Write a program called CONDENT (conditional entropy) to study the value

1 1

H(f I g)=-

/(x)log

[~~=~] dx

for

f, g E D([O, 1]).

Compare for different pairs of sequences {/n}, {gn} C D([O, 1]) the asymptotic behavior of 11/n- 9niiL1 and H(/n I 9n)· 9.7. Let (X,A,JS) be a measure space. Prove that for every two sequences {/n}, {gn} CD the convergence H(fn I Un) - 0 implies 11/n- YniiLl - 0. Is the converse implication also true? Exercise 9.6 can be helpful in guessing the proper answer (Loskot and Rudnicki, 1991). 9.8. Consider a density f,.: R 3

-

R+ of the form

= o: exp(-Pixl 2 + kx), where lxl 2 = x~ + x~ + x~ and kx = k1x1 + k2x2 + ksxs. f,.(x)

Assume that a

sequence of densities InC D(~) satisfies

r 9i(x)fn(x) dx = }Rar 9i(x)f,.(x) dx,

}Ra

with go(x) = lxl 2 and 9i(x) H(/n)- H(f,.) implies 11/n nicki, 1991).

i

= 0, 1,2,3,

= Xi, i = 1, 2, 3. Prove that the convergence

-/.II- 0

(Elmroth, 1984; Loskot and Rud-

10 Stochastic Perturbation of Discrete Time Systems

We have seen two ways in which uncertainty (and thus probability) may appear in the study of strictly deterministic systems. The first was the consequence of following a. random distribution of initial states, which, in turn, led to a. development of the notion of the Frobenius-Perron operator and an examination of its properties a.s a. means of studying the asymptotic properties of flows of densities. The second resulted from the random application of a. transformation S to a. system and led na.tura.lly to our study of the linear Boltzmann equations. In this chapter we consider yet another source of probabilistic distributions in deterministic systems. Specifica.lly, we examine discrete time situations in which a.t each time the value Xn+l = S(xn) is reached with some error. An extremely interesting situation occurs when this error is small and the system is ''primarily" governed by a. deterministic transformation S. We consider two possible ways in which this error might be sma.ll: Either the error occurs rather rarely and is thus small on the average, or the error occurs constantly but is sma.ll in magnitude. In both cases, we consider the situation in which the error is independent of S(xn) and are, thus, led to first reca.ll the notion of independent random variables in the next section and to explore some of their properties in Sections 10.2 and 10.3.

304

10. Stochastic Perturbation of Discrete Time Systems

10.1

Independent Random Variables

Let (0, F, prob) be a probability space. A finite sequence of random variables (et, ... , is called a k-dimensional random vector. Equivalently, we could say that a random vector = ( is a measurable transformation from n into Rk. Measurability means that for every Borel subset B C Rk the set

ek)

e e, ... ,ek)

belongs to F. Thus, having a k-dimensional random vector (e1, ... ,eA:), we may consider two different kinds of densities: the density of each random component and the joint density function for the random vector (e1, ... , Let the density of be denoted by f,(x), and the joint density of = (et •... ,eA:) be /(xt, ... , X A:)· Then by definition, we have

ei

ei

e

ek).

(10.1.1)

and

j .. ·j

f(xt, ... ,xA:)dxt" ·dxA: = prob{(et, ... ,eA:) E B},

forB c Rk,

B

Jik,

where B, and B are Borel subsets of R and respectively. In this last integral take B= B 1 x Rx ··· x R

._____., k-l times

so that we have prob{(et. ...• eA:)

=

e B} = prob{et e Bt}

k{I . ·I 1

f(x, x2, ... , XA:) dx2 .. ·

dxk} dx.

(10.1.2)

R"-1

By comparing (10.1.1) with (10.1.2), we see immediately that

ft(x)

=

I· . I

J(x,x2, ... ,xA:)dx2 ..

·dxk.

(10.1.3)

R"-1

Thus, having the joint density function f for (e1, ... ,eA:), we can always find the density for from equation (10.1.3). In an entirely analogous fashion,

el

10.1. Independent Random Variables

305

h can be obtained by integrating f(xlt x, ... , Xk) over Xt, xs, ... , Xk· The same procedure will yield each of the densities /;. However, the converse is certainly not true in general since, having the density /; of each random variable (i = 1, ... 'k), it is not usually possible to find the joint density I of the random vector (elt ... 'ek>· The one important special case in which this construction is possible occurs when ek are independent random variables. Thus, we have the following theorem.

ei

e1, ... ,

Theorem 10.1.1. If the random variables 6, ... ,ek are independent and have densities It, ... , fk, respectively, then the joint density function for the random vector (6' ... 'ek) is given by (10.1.4)

where the right-hand side is a product.

Proof. Consider a Borel set B

c

Rk of the form (10.1.5)

where B 1 , ••• , Bk

c R are Borel sets.

and, since the random variables

Then

el' ... 'ek are independent,

With this equation and (10.1.1}, we obtain

=

j .. ·j It (xt) .. ·fk(xk) dx1 .. · dxk.

(10.1.6)

B

Since, by definition, sets of the form {10.1.5) are generators of the Borel subsets in Rk, it is clear that (10.1.6) must hold for arbitrary Borel sets B c Rk. By the definition of the joint density, this implies that It (x 1 ) • • ·fk(xk) is the joint density for the random vector (e1 , .•• ,ek>· • As a simple application of Theorem 10.1.1, we consider two independent random variables and with densities It and f2, respectively. We wish to obtain the density of + Observe that, by Theorem 10.1.1, the random vector 2) has the joint density ft(xt)f2(x2)· Thus, for an

e1 e2 e1 e2. (et.e

306


arbitrary Borel set B

c R, we have

II

prob{~1 +~2 E B} =

11(xl)f2(x:~)dxldx2,

z1+z:aEB

or, setting x = x1 +x2 andy= x2, prob{el +e2

e B}

=II L{/_:

11(x- y)f2(y)dxdy

BxR

=

1:

11(x- y)f2(y)dy} dx.

From the definition of a density, this last equation shows that f(x) =

is the density of

el + e2·

(10.1.7)

11(x- y)f2(y) dy

Remark 10.1.1. From the definition of the density, it follows that, if has a density/, thence has a density (1/lcl)f(x/c). To see this, write prob{ce e A}= prob {~

e

1 f(y) dy = - f I(~) dx. j(l/c)A c 1c 1 )A

e !A}= f c

Thus, from (10.1.7), if e1 and e2 are independent and have densities 11 and /2, respectively, then (c 1 ~1 + C:J~2 ) has the density 1 f(x) = 1C1C2 1

joo dy. -oo 11 (x-C1 y) h(1L) C:J

(10.1.8)

0

10.2 Mathematical Expectation and Variance In previous chapters we have, on numerous occasions, used the concept of mathematical expectation in rather specialized situations without specifically noting that it was, indeed, the mathematical expectation that was involved. We now wish to explicitly introduce this concept in its general sense. Let (O,F,prob) be a probability space and let e:n--+ R be a random variable. Then we have the following definition.

e

Definition 10.2.1. If is integrable with respect to the measure "prob," then the mathematical expectation (or mean value) of is given by

E(~)

e

=In e(w) prob(dw).

10.2. Mathematical Expectation and Variance

307

Remark 10.2.1. By definition, E(e) is the average value of e. A more illuminating interpretation of E(e) is given by the law of large numbers (see equation (10.3.4)]. D In the case when is a constant, = c, then it is trivial to derive E(c). Since prob{O} = 1 for any constant c, we have

e

e

E(c) = c

k

prob(dw)

=c.

(10.2.1)

Now we show how the mathematical expectation may be calculated via the use of a density function. Let h: Ric - R be a Borel measurable function, that is, h- 1 (a) is a Borel subset of Ric for each interval a. Further, let = (6' ... '{~c) be a random vector with the joint density function f(xl! ... , x~c). Then we have the following theorem.

e

Theorem 10.2.1. If hf is integrable, that is,

j ···I h(x1· · ·x~c)f(xl· · ·x~c)dx 1 · · ·dx~c < oo, R~

then the mathematical expectation of the random variable h 0

eexists and

is given by E(h 0 e)

=

I ... I

h(xl, ... 'x~c)f(xb ... 'x~c) dxl ... dx~c.

{10.2.2)

R•

Proof. First assume that h is a simple function, that is, n

h(x)

= L.\i1A (x) 1

i=l where the Ai are mutually disjoint Borel subsets of Ric such that UiAi Then n

h({(w))

=L

n

.\ilA,(e(w))

=L

.\il~-l(A,)(w),

i=l i=l and, by the definition of the Lebesgue integral, E(h o e)=

1

h({(w)) prob{dw)

0

Further' since

=

t

.\i prob{C 1 (Ai)}.

i=l

f is the density for e' we have

prob{C 1 (Ai)}

= prob{e E Ai} = }A, f f(x) dx,

= Ric.

308


As a consequence,

Thus, for the h that are simple functions, equality {10.2.2) is proved. For an arbitrary h, hf integrable, we can find a sequence {hn} of simple functions converging to h and such that lhnl :S lhl. From equality {10.2.2), already proved for simple functions, we thus have E(hn o e)=

f

}R•

hn(x)f(x) dx.

By the Lebesgue dominated convergence theorem, since follows that f h(e(w)) prob(dw) = f h(x)f(x) dx,

Jn

lhn/1

:S

lhl/,

it

JR•

which completes the proof. • In the particular case that k (10.2.2)

= 1 and h(x) = x, we have from equation

E(e) =

1:

xf(x)dx.

(10.2.3)

Thus, if /(x) is taken to be the mass density of a rod of infinite length, then E(e) gives the center of mass of the rod. From Definition 10.2.1, it follows that, for every sequence of random variables e11 ••• 1 e1c and COnstants .X11 ••• 1 .X1c, we have E

t,

(t,.xiei) = .xiE(ei>

(10.2.4)

since the mathematical expectation is simply a Lebesgue integral on the probability space (O,F,prob). Moreover, the mathematical expectation of exists whenever all of the exist. We now turn to a consideration of the variance, starting with a definition.

Ei .xiei

E(ei)

e:

Definition 10.2.2. Let 0 - R be a random variable such that m = E(e) exists. Then the variance of is

e

(10.2.5) if the corresponding integral is finite. Thus the variance of a random variable is just the average value of the square of the deviation of away from m. By the additivity of the mathematical expectation, equation (10.2.5) may also be written as

e

e

(10.2.6)

10.2. Mathematical Expectation and Variance

309

e

If has a density f(x), then by the use of equation (10.2.2), we can also write

whenever the integral on the right-hand side exists. Finally, we note that for any constant >.,

Since in any application there is a certain inconvenience in the fact that D 2 (e) does not have the same dimension as it is sometimes more convenient to use the standard deviation of e' defined by

e,

u(e)

= VD2 (e).

For our purposes here, two of the most important properties of the mathematical expectation and variance of a random variable are contained in the next theorem.

e

Theorem 10.2.2. Let 6, ... , ek be independent random variables such that E(e,), D 2 (e,), i = 1, ... , k exist. Then E(6 · · · ek)

= E(6) · · · E(ek)

(10.2.7)

and

(10.2.8)

Proof. The proof is easy even in the general case. However, to illustrate again the usefulness of (10.2.2), we will prove this theorem in the case when all thee, have densities. Thus, assume that has density j,, i = 1, ... , k, and pick h(xll ... 'Xk) = Xl ••• Xk. Since the ell ... ' ek are independent random variables, by Theorem 10.1.1, the joint density function for the random vector (ell ... ' ek) is

e,

Hence, by equation (10.2.2),

E(el · · ·ek)

=I··· I

x1· · ·xk!I(xl) · · · !k(xk)dxl·. ·dxk

R"

= /_: xd1(x1)dx1 · · · /_: Xk/k(xk) dxk = E(ed ... E(ek), and (10.2. 7) is therefore proved.

310


= m 1, so that 2 D (et + · · · + e~c) = E((et + · · · + e~c- mt -

Now set E(e,)

= E

(.t

,,,=1

(e,- m,)(e;-

2

· · ·- m~c) )

m;)) .

Since the et, ... ,e~c are independent, (et- mt), ... , (e~c- m~c) are also independent. Therefore, by (10.2.4) and (10.2.7), we have 1c

D 2 (et

+ ... +e~c) = EE((e,- m,) 2 ) + EE((e,- m,)(e;- m;)) i~j

i=l

1c

= ED2 (e,) + E<E(e,)- m,)(E(e;)- m;). i=l

Since E(e,) =

i"#j

ms, equation (10.2.8) results immediately.

•

e,

Remark 10.2.2. In Theorem 10.2.2, it is sufficient to assume that the are mutually independent, that is, is independent of t for i ':/: j. 0 To close this section on mathematical expectation and variance, we give two versions of the Chebyshev inequality, originally introduced in a special context in Section 5.7.

e.

e;

Theorem 10.2.3. If e is nonnegative and E(e) exists, then prob{e ~a} 5 E(e)/a

If e is arbitrary but such that m

for every a> 0.

(10.2.9)

= E(e) and D 2 (e) exist, then

prob{le- ml ~ e} 5 D 2 (e)/e2

for every e > 0.

(10.2.10)

Proof. By the definition of mathematical expectation,

E(e)

= f e(w) prob{dw) ~ f Jo

~

aJ

J{w:E(w)~a}

e(w) prob(dw)

prob(dw) = aprob{e

{w:E(~o~)~a}

~a},

which proves (10.2.9). (This is, of course, analogous to equation (5.7.9).] Now replace by {e- m) 2 and a by e 2 in (10.2.9) to give

e

prob{(e- m) 2 ~ e 2 } 5 (1/e2 )E((e- m) 2 )

= (1/e2 )D2 (e),

which is equivalent to (10.2.10) and completes the proof.

•

10.3. Stochastic Convergence

311

10.3 Stochastic Convergence There are several different ways in which the convergence of a sequence {en} of random variables may be defined. For example, if en E V(O,F,prob), then we may define both strong and weak convergences of {en} to in V(O) space, as treated in Section 2.3. In probability theory some of these types of convergence have special names. Thus, strong convergence of {en} in £ 2 (0), defined by the relation

e

(10.3.1) is denoted by l.i.m.en =

e

and called convergence in mean. A second type of convergence useful in the treatment of probabilistic phenomena is given in the following definition. Definition 10.3.1. A sequence Un} of random variables is said to be stochastically convergent to the random variable if, for every e > 0,

e

lim prob{len- el ~ e}

n-+oo

= 0.

{10.3.2)

The stochastic convergence of {en} toe is denoted by st-limen

=e.

(10.3.3)

Note that in terms of V norms, the mathematical expectation and variance of a random variable may be written as

and .o2(e)

=In le-

2

ml prob(dw)

= lie- mii~2(0)•

This observation allows us to derive a connection between stochastic convergence and strong convergence from the Chebyshev inequality, as contained in the following proposition. Proposition 10.3.1. If a sequence {en} of mndom variables, en E V(O), is strongly convergent in V(O) toe, then {en} is stochastically convergent to Thus, convergence in mean implies stochastic convergence.

e.

=

Proof. We only consider p < oo, since for p oo the proposition is trivial. Applying the Chebyshev inequality (10.2.9) to len- eiP, we have prob{len - ei.P ~ eP} ~ (1feP)E(len - ei.P)

312


or, equivalently,

which completes the proof. • A third type of convergence useful for random variables is defined next. Definition 10.3.2. A sequence {en} of random variables is said to converge ahnost surely to (or to converge to with probability 1) if

e

e

lim en(w) = e(w)

n-+oo

for almost all w. Equivalently, this condition may be written as prob{ lim en(w) = e(w)} = 1. n-+oo

Remark 10.3.1. For all of the types of convergence we have defined (strong and weak V convergence, convergence in mean, stochastic convergence, and almost sure convergence), the limiting function is determined up to a set of measure zero. That is, if and are both limits of the sequence {en}, then and ( differ only on a set of measure zero. 0 We now show the connection between almost sure and stochastic convergence with the following proposition.

e

e

e

Proposition 10.3.2. If a sequence of random variables {en} converges almost surely to then it also converges stochastically to

e,

e.

Proof. Set

71n(w) = min(l, len(w)- e(w)l).

e,

Clearly, l77nl 5 1. If {en} converges almost surely to then {77n} converges to zero almost surely, and, by the Lebesgue dominated convergence theorem, lim ll77niiLl(Sl)

n-+oo

= n-+00 lim { 71n(w)prob(dw) = 0. ln

By Proposition 10.3.1 this implies that {71n} converges stochastically to zero. Since in the definition of stochastic convergence it suffices to consider only c < 1, it then follows that prob{len- el ~ c}

= prob{77n ~ c}

forO 0, then {P"} is asymptotically stable. Proof. We start with the observation that for every fixed x the product g(x- S(y))P"- 1 f(y), considered as a function of y, does not vanish everywhere. As a consequence, P" f(x)

={

}Rd

g(x-S(y))P"- 1 f(y) dy > 0

for all x E :gJ,n ~ 1,f ED.

The asymptotic stability of {P"} is thus proved by applying Theorem 5.6.1.

•

It is interesting that we may also prove the uniqueness of a stationary density f. of P defined by {10.5.5) without the rather restrictive conditions required by Corollary 10.5.1.

Theorem 10.5.2. Let the operator P: L 1 -+ £ 1 be defined by equation {10.5.5) and let g E D. If g(y) > 0 for all y E Rd and if a stationary density f. for P exists, then f. is unique. Proof. Assume there are two stationary densities for h· Set f = It - h, so we clearly have

Ff=f.

P,

namely,

It

and

{10.5.9)

We may write f = f+- f- by definition, so that, if It :f:. h, then neither f+ nor f- are zero. Since Pf+ = f+ {by Proposition 3.1.3), from {10.5.5) we have {10.5.10)

324

10. Stochastic Perturbation of Discrete Time Systems (a)

5

J

rl

n lh

(b)

5

(c)

5

(d)

5

J

l

h

0 FIGURE 10.5.1. Asymptotic periodicity illustrated. Here we show the histograms obtained after iterating 104 initial points uniformly distributed on (0, 1] with a= ~ • .X= and(}= ft in equation (10.5.8). In (a) n = 10; (b) n = 11; (c) n = 12; and (d) n = 13. The correspondence of the histograms for n = 10 and n = 13 indicates that, with these parameter values, numerically the sequence of densities has period 3.

¥o,

10.5. Constantly Applied Stochastic Perturbations

325

and similarly for P f-. Since f+ is not identically zero and g is strictly positive, the integral in (10.5.10) is a. nonzero function for every x, and, thus, Pf+(x) > 0 for all x. Clearly, too, Pf-(x) > 0 for all x and, thus, the supports of Pf+ and Pf- are not disjoint. By Proposition 3.1.2, then, we must have IIPfll < llfll, which contradicts equality (10.5.9). Thus, ft and h must be identical almost everywhere if they exist. •

Remark 10.5.3. It certainly may happen that there is no solution to Ff =fin D. As a. simple example, consider S(x) = x for all x E R. Take g to be the Gaussian density g(x)

= ~exp(-x2 /2), v2rr

so the operator P defined in (10.5.5) becomes Ff(x)

=

00

1

-oo

1

rn= exp[-(x- y) 2 /2]f(y) dy. v2rr

Note that Ff(x) is simply the solution u(t, x) of the heat equation (7.4.13) with u 2 = 1 at time t = 1, assuming an initial condition u(O, y) = f(y). Since this solution is given by a. semigroup of operators [cf. equations (7.4.11) and (7.9.9)], it can be shown that

:::;

00

v2rrn

1

exp[-(x- y) 2 /2n]f(y) dy

1

100

f(y)dy=

pn f(x) = . ~ 1 ~

v 2rrn

-oo

-oo

1

~·

v 2rrn

Thus pn f converges uniformly to zero as n -+ oo for all f E D, and there is no solution to Ff =f. 0 If these conditions for the existence and uniqueness of stationary densities of P are strengthened somewhat, we can prove that {pn} is asymptotically stable. In fact from our results of Theorem 5.7.1, we have the following corollary.

Corollary 10.5.2. Let the opemtor P: L 1 -+ L 1 be defined by equation (10.5.5) and let g E D. If there is a Liapunov function V: Rd -+ R such that

f

}Rd.

g(x- S(y))V(x) dx:::; aV(y)

+ {3

(10.5.11)

for some nonnegative constants a, {3, a< 1, and

f inf g(x- S(y)) dx > 0 }Rd.ifll~r for every r

> 0, then {pn} is asymptotically stable.

(10.5.12)

326


Remark 10.5.4. Note that condition (10.5.12) is automatically satisfied if g: Rd. - R is positive and continuous and S: Rd. - Rd. is continuous because

inf g(x- S(y))

lvl~r

for every x E

Ir'-.

= min g(x- S(y)) > 0 lvl~r

0

Example 10.5.2. Consider a point moving through is determined by Xn+l = S(xn) +en,

Ir'- whose trajectory

where S: Rd. - ~ is continuous and satisfies for lxl ~ M,

IS(x)l ~ ~lxl,

(10.5.13)

eo,

where ~ < 1 and M > 0 are given constants. Assume that e11 ... are independent random variables with the same density g, which is continuous and positive, and such that E(en) exists. Then {.Pn} defined by (10.5.5) is asymptotically stable. To show this, it is enough to confirm that condition (10.5.11) is satisfied. Set V(x) = lxl, so

f g(x- S(y))V(x) dx = f g(x- S(y))lxl dx Jnd lnd =f

Jnd g(x)lx + S(y)l dx

~

r g(x)(lxl + IS(y)l) dx Jnd

= IS(y)l + Jnd r g(x)lxl dx. From (10.5.13) we also have IS(y)l < ~1111 -

+ lzi~M max IS(x)l

so that

f g(x- S(y))V(x) dx ~ ~IYI + max IS(x)l + f g(x)lxl dx. kd ~SM ~ Thus, since E(en) exists, equation (10.5.11) is satisfied with a= ~ and {3

r

= Jnd g(x)lxl dx + lzi~M max IS(x)l.

0

10.6. Small Continuous Stochastic Perturbations

327

It is important to note that throughout it has not been necessary to require that 8 be a nonsingular transformation. Indeed, one of the goals of this section was to demonstrate that the addition of random perturbations to a singular transformation may lead to interesting results. However, if 8 is nonsingular, then the Frobenius-Perron operator P corresponding to 8 exists and allows us to rewrite (10.5.5) in an alternate form that will be of use in the following section. By definition,

Ff(x) =

f

jRtJ

g(x- 8(y))f(y) dy.

Assume 8 is nonsingular, therefore the Frobenius-Perron and Koopman operators corresponding to 8 exist. Let ha:(Y) = g(x- y), so we can write PJ as

Ff(x) =

f

jRtJ

ha:(8(y))f(y) dy = (/, Uha:} = (Pj, ha:},

or, more explicitly,

Ff(x) =

f

jRtJ

g(x- y)Pf(y)dy.

(10.5.14)

By a change of variables, {10.5.14) may also be written as

Ff(x) =

f

}Rd

g(y)Pf(x-y)dy.

(10.5.15)

Remark 10.5.5. Observe that ford= 1, equations (10.5.14) and (10.5.15) could also be obtained as an immediate consequence of equation (10.1.7) applied to equation (10.5.1) since {n and 8(xn) are independent. D

10.6 Small Continuous Stochastic Perturbations of Discrete Time Systems This section examines the behavior of the system

e > 0,

(10.6.1)

where 8: Jld --+ Rd is measurable and nonsingular. As in the preceding section, we assume the {n to be independent random variables each having the same density g. Since the variables e{n have the density (1/e)g(x/e), see Remark 10.1.1, equation (10.5.15) takes the form

Pd(x) =

!e }Rd f g (!!.) Pf(x- y) dy e

{10.6.2)

328


and gives the recursive relation (10.6.3) that connects successive densities /n of Xn· The operator Pe can also be written, via a change of variables, as

= }Rd f g(y)Pf(x- ey) dy.

Pd(x)

(10.6.4)

Since

f

}Rd

g(y)Pf(x)dy

= Pf(x),

we should expect that in some sense lime-o Pef(x) = P /( x ). To make this more precise, we state the following theorem. Theorem 10.6.1. For the system defined by equation (10.6.1)

= 0,

lim IIPd- Pfll

e-+0

for all f E L\

where P is the Frobenius-PefTOn operator corresponding to S and Pe is given by (10.6.4). Proof. Since P and Pe are linear we may restrict ourselves to f E D. Write

=f

Pf(x)

}Rd

g(y)Pf(x)dy,

then

Pd(x)- Pf(x)

=f

}Rd

g(y)[Pf(x- ey)- Pf(x)] dy.

Pick an arbitrarily small 6 > 0. Since g and P f are both integrable functions on _Rd., there must exist an r > 0 such that

f

JIIII~r

g(y)dy:::;;

~

and

1lzl~r/2

Pf(x)dx:::;;

~-

To calculate the norm of Pel - P f,

IIPd-Pfll::=;;

f f

}Rd }Rd

g(y)IPf(x-Ey)-Pf(x)ldxdy,

we split the integral into two parts,

IIPd-P/11 :s;It+h where

It=

f f

}Rd A111~r

g(y)IPf(x- ey)- Pf(x)l dxdy,

10.6. Small Continuous Stochastic Perturbations

and

I2 =

f

}R4

1.

329

g(y)IPJ(x- ey)- Pf(x)l dxdy.

l11l?!:r

We consider each in turn. With respect to I 11 note that, since the function Pf is integrable, by Corollary 5.1.1, we may assume

L 4

IPf(x- ey)- Pf(x)l dx :5

~

for e :5 e0 with e0 sufficiently small. Hence

61.

I 1 :5 g(y) dy :5 2 l11l:$r

61

2

R4

g(y) dy =

62·

In examining I 2 , we use the triangle inequality to write

I2:5

f

1.

JR 4 l11l?!:r

g(y)Pf(x-ey)dxdy+

Change the variables in the first integral to v

f

1.

j R 4 l11l?!:r

g(y)Pf(x-ey)dxdy=

=

1.

f

}R4 l11l?!:r

f4

JR

1.

g(y)Pf(x)dxdy.

= y and z = x- ey, then

1.

g(v)Pf(z)dzdv

lvl;?!r

lvl?!:r

g(v) dv :5

~-

Further, we also have

f

}Rd

1.

l11l?!:r

g(y)Pf(x)dxdy :5

~

so that I2 :5 6/2. Thus

IIPd- P/11 :56

for any e :5 min (!,eo),

that is, lim

€-+0

IIPd- pIll

=0

•

As an immediate consequence of Theorem 10.6.1 we have the following corollary.

Corollary 10.6.1. Suppose that S and g are given and that for every small e, 0 < e < e0 , the opemtor P€, defined by (10.6.4), has a stationary density fe. If the limit /.=lim /€ €--+0

330


exists then /. is a stationary density for the Ftobenius-PefTOn operator corresponding to S.

Proof. Write Since Pe is contractive,

Thus fe + Pe(/• - /e) -+ /. as E -+ 0 and, as a consequence, Pd. -+ / •• However, Theorem 10.6.1 also tells us that Pd.-+ P/., soP/.=/.. II

10.7 Discrete Time Systems with Multiplicative Perturbations Up to now in this chapter we have confined our attention to situations in which a discrete time system is perturbed in an additive fashion, for example, (10.4.1), (10.5.1), and {10.6.1). We now turn to a consideration of the influence of perturbations that appear in a multiplicative way. Since in many applied problems this arises because of noise in parameters, it is also known as parametric noise. Specifically, we examine a process {10.7.1) where 8: R+ -+ R+ is continuous and positive a.e. and, as before, the en are independent random variables, each distributed with the same density g. We denote the density of Xn by /n, and our first task in the study of {10.7.1) will be to derive a relation connecting fn+l and In· Using exactly the same approach employed in Section 10.5, let h: R+ -+ R+ be an arbitrary bounded and Borel measurable function. The expectation of h(xn+l) is given by (10.7.2) However, using (10.7.1) we also have E(h(xn+l))

= E(enS(xn)) = h(zS(y))fn(y)g(z) dy dz

11 00

00

1oo 1

00

=

h(x)fn(y)g (s(y))

S~y) dxdz,

(10.7.3)

10.7. Discrete Time Systems with Multiplicative Perturbations

331

where we used a change of variables z = x / S(y) in passing from the second to third lines of (10.7.3). Equating (10.7.2) and (10.7.3), and using the fact that h was arbitrary by assumption, we arrive at (10.7.4) which is the desired relation. From (10.7.4) we may also write fn+l given by

P!(x)

= Pin

= 1oo f(y)g (s~y))

where the operator

sty) dy,

P,

(10.7.5)

is a Markov operator with a stochastic kernel

K(x,y) =g

(s~y)) S~y)"

(10.7.6)

Our first result is related to the generation of asymptotic periodicity by multiplicative noise. Though originally formulated by Horbacz (1989a], the proof we give is different from the original. Theorem 10.7.1. Let the Markov opemtor P:L1 (R+)-+ L 1 (R+) be defined by (10.7.5). Assume that g ED,

0 < S(x) $ax+ {3 and am< 1

with m =

forx

100

~

0,

xg(x) dx,

(10.7.7)

(10.7.8)

where a and {3 are nonnegative constants. Then P is constrictive. As a consequence the sequence {pn} is asymptotically periodic. Proof. Once again we will employ Theorem 5. 7.2 in the proof. We first show that (5.7.11) holds for the kernel (10.7.6) with V(x) = x. We have

1oo xPf(x)dx= 1oo xdx 1oo g (s~y)) sty/(y)dy = 100 f(y)dy 100 g (s~y)) s~y) dx. Using the change of variables z

= x/S(y) and then (10.7.7) we obtain

00 00 00 1 xPf(x)dx = 1 f(y)S(y)dy 1 zg(z)dz 00 00 = m 1 f(y)S(y) dy $am 1 yf(y) dy +{3m.

332


Thus, we have verified inequality (5.7.11). We next show that the kernel K(x, y) given by (10.7.6) satisfies inequality (5.7.19) of Theorem 5.7.2. Fix an arbitrary positive >. < 1 and choose a bounded set B C R+. Since g is uniformly integrable there must be a 61 > 0 such that

Define

6 = 61 inf S(y). veB

Then for p(E)

< 6 we have p(E/S(y)) < 61 and

h K(x,y)dx

= hg

(s(y))

=f

S;y)dx

g(x)dx

}E/S(v)

~ >.

for y E Band p(E)

< 6,

and all of the conditions of Theorem 5.7.2 are satisfied. Thus Pis constrictive and a simple application of the spectral decomposition Theorem 5.3.1 finishes the proof. • We close with a second theorem concerning asymptotic stability induced by multiplicative noise. Theorem 10.7.2. If the Markov operator P: L 1 (R+)- L 1 (R+) defined by (10.7.5) satisfies the conditions of Theorem 10.7.1 and, in addition, g(x) > 0, then {P"} is asymptotically stable. Proof. Note that, for fixed x, the quantity X

g ( S(y)

)

1 - 1 S(y) pn- f(y),

as a function of y, does not vanish everywhere. Consequently,

pn f(x) =

1oo g (s(y)) S;y) pn-1 f(y)dy > 0 for all x E R+,n;:::: 1,/ ED,

and Theorem 5.6.1 finishes the proof of the asymptotic stability of {P"} .

•

Theorems 10.7.1 and 10.7.2 illustrate the behaviors that may be induced by multiplicative noise in discrete time systems. A number of other results concerning asymptotic periodicity and asymptotic stability induced by multiplicative noise may be proved, but rather than giving these we refer the reader to Horbacz [1989a,b].

Exercises

333

Exercises 10.1. Let {n: 0 -+ Jld.. , n = 1, 2, ... , be a sequence of independent random vectors, and let cpn: Rd.. -+ ~ be a sequence of Borel measurable functions. Prove that the random vectors 17n(w) = 'Pn({n(w)) are independent. 10.2. Replace inequality (10.7.7) in Theorem 10.7.1 by

0 :5 8(x) :5 ax,

a
0 and r > 0 such that {Pn}

10.7. Consider the system

. Xn+l = 8(xn) +en with additive noise. Note that with the definitions y = ez, T = e 8 , and 17 = e~ this can be rewritten in the alternative form

Yn+l

= 17nT(lnyn)

334


as if there were multiplicative noise. Using this transformation, discUBB the results for multiplicative noise that can be obtained from the theorems and corollaries of Section 10.5. 10.8. As a counterpoint to the previous examples, note that if

= enS(xn) and we set y = lnx, 'I= 1ne, and T = lnS, then Xn+l

Yn+l

= T(e11") + '7n

results. Examine the results for additive noise that can be obtained using this technique on the theorems of Section 10.7 pertaining to multiplicative noise.

11 Stochastic Perturbation of Continuous Time Systems

In this chapter continuous time systems in the presence of noise are considered. This leads us to examine systems of stochastic differential equations and to a derivation of the forward Fokker-Planck equation, describing the evolution of densities for these systems. We close with some results concerning the asymptotic stability of solutions to the Fokker-Planck equation.

11.1 One-Dimensional Wiener Processes (Brownian Motion) In this and succeeding sections of this chapter, we turn to a consideration of continuous time systems with stochastic perturbations. We are specifically interested in the behavior of the system

dx dt = b(x) + o-(x)e,

(11.1.1)

where o-(x) is the amplitude of the perturbation and e = dwjdt is known as a ''white noise" term that may be considered to be the time derivative of a Wiener process. The system (11.1.1) is the continuous time analog of the discrete time problem with a constantly applied stochastic perturbation considered in Section 10.5. The consideration of continuous time problems such as (11.1.1) will offer new insight into the possible behavior of systems, but at the expense of introducing new concepts and techniques. Even though the remainder of this

336

11. Stochastic Perturbation of Continuous Time Systems

chapter is written to be self-contained, it does not constitute an exhaustive treatment of stochastic differential equations such as (11.1.1). A definitive treatment of this subject may be found in Gikhman and Skorokhod [1969]. In this section and the material following, we will denote stochastic processes by {~(t)}, {77(t)}, ... as well as Ut}, {'7t}, ... , depending on the situation. Remember that in this notation ~(t) or ~t denote, for fixed t, a random variable, namely, a measurable function ~t: 0--. R. Thus ~(t) and ~t. are really abbreviations for ~(t,w) and ~t(w), respectively. The symbol~ will be reserved for white noise stochastic processes (to be described later), whereas 77 will be used for other stochastic processes. Let a probability space {0, F, prob) be given. We start with a definition.

Definition 11.1.1. A stochastic process {77(t)} is called continuous if, for almost all w (except for a set of probability zero), the sample path t--. 77(t,w) is a continuous function. A Wiener process can now be defined as follows. Definition 11.1.2. A one-dimensional normalized Wiener process (or Brownian motion) {w(t)}t~o is a continuous stochastic process with independent increments such that (a) w(O)

= 0; and

(b) for every s, t, 0 ~ s Gaussian density

g(t- s, x) =

< t, the random variable w(t) - w(s) has the 1

J21r(t- s)

exp[-x 2 /2(t- s)].

(11.1.2)

Figure 11.1.1a shows a sample path for a process approximating a Wiener process. It is clear that a Wiener process has stationary increments since w(t)w(s) and w(t + t') - w(s + t') have the same density function (11.1.2). Further, since w(t) w(t)-w(O), the random variable w(t) has the density

=

1 g(t,x)= ~exp(-x 2 /2t). v21Tt An easy calculation shows

1

(11.1.3)

00

E((w(t)- w(s))n) =

1 J21T(t- s)

xn exp(-x 2 /2(t- s)] dx

-oo

= { 1.3. · · (n- 1)(t- s)n/

0

2

for n even (11.1.4) for n odd

and thus, in particular,

E(w(t)- w(s))

=0

{11.1.5)

11.1. One-Dimensional Wiener Processes

337

w(t) (a)

t

w(t) (b)

·;:-.;'

0

·... l(

j ; : . i II ~ : .. ·

..

H#~#m#M~:~::~::i*;~~~~~~!~ii~!~:~::~:i~~:;~:~;;~·r.y~f:~l::~i:~:i!~·~::i~i·~·i:~i·~!:~;:~ t ;';···:;;,··',,:.::::: !!i::J!!;.:?: 0 again yields a process with properties similar to those of Definition 11.1.2, but now with the density

~ exp( -x2 /2u2 t) and with the variance u 2 t. These processes are also called Wiener processes. From this point on we will always refer to a normalized Wiener process as a Wiener process. 0 In Figure 11.1.1b we have drawn a number of sample paths for a process approximating a Wiener process. Note that as time increases they all seem to be bounded by a convex envelope. This is due to the fact that the standard deviation of a Wiener process, from (11.1.6), increases as ..fi, that is,

338


The highly irregular behavior of these individual trajectories is such that magnification of any part of the trajectory by a factor a 2 in the time direction and a in the x direction yields a picture indistinguishable from the original trajectory. This procedure can be repeated as often as one wishes, and, indeed, the sample paths of a Wiener process are fractal curves [Mandelbrot, 1977]. To obtain some insight into the origin of this behavior consider the absolute value of the differential quotient

llit I= !litllw(to E (I~~ I) = l~tl E(lw(to liw

We have

1

+lit) - w(to)l.

+lit) - w(to)l)

and, since the density of w(t0 +lit) - w(t0 ) is given by (11.1.3),

1

00

E(lw(to +lit) - w(to)l) = ~ 1 211'lit = J2lit/11' or

lxl exp( -x 2 /2lit)dx

-oo

E(l~~l)=~~·

Thus the mathematical expectation of !liwf lit! goes to infinity, with a speed proportional to (lit)- 112 , when !lit! --+ 0. This is the origin of the irregular behavior shown in Figure 11.1.1. Extending the foregoing argument, it can be proved that the sample paths of a Wiener process are not differentiable at any point almost surely. Thus, the white noise terme = dw/dt in (11.1.1) does not exist as a stochastic process. However, since we do wish ultimately to consider (11.1.1) with such a perturbation, we must inquire how this can be accomplished. AB shown in following sections, this is simply done by formally integrating (11.1.1) and treating the resulting system,

x(t)

=lot b(x(s))ds+ lot q(:z:(s))dw(s)+x

0

•

However, this approach leads to the new problem of defining what the integrals on the right-hand side mean, which will be dealt with in Section 11.3.

To obtain further insight into the nature of the process w(t), examine the alternative sequence {zn} of processes, defined by

Zn(t) = w(tf__d +

t~ -=_t;~ 1 '

t-1

[w(tf)- w(tf_d]

fortE [tf_ 1 , tf],


339

where tf = ifn, n = 1, 2, ... , i = 0, 1, 2, .... In other words, Zn is obtained by sampling the Wiener process w( t) at times tf and then applying a linear interpolation between tf and tf+l· Any sample path of the process {Zn (t)} is differentiable, except at the points tf, and the derivative = z~ is given by fortE (tf_ 1 , tf). "'n(t) = n(w(tf) - w(tf-t)J,

"'n

The process "'n(t) is piecewise constant. The heights of the individual segments are independent, have a mean value zero, and variance D 2 'f/n(t) = n. Thus, the variance grows linearly with n. If we look at this process approximating white noise, we see that it consists of a sequence of independent impulses of width (1/n) and variance n. For very large n we will see peaks of almost all possible sizes uniformly spread along the t-axis. Note that the random variable .zn(t) for fixed t and large n is the sum of many independent increments. Thus the density of Zn(t) must be close to a Gaussian by the central limit theorem. The limiting process w(t) will, therefore, also have a Gaussian density, which is why we assumed that w(t) had a Gaussian density in Definition 11.1.2. Historically, Wiener processes (or Brownian motion) first became of interest because of the findings of the English biologist Brown, who observed the microscopic movement of pollen particles in water due to the random collisions of water molecules with the particles. The impulses coming from these collisions are almost ideal realizations of the process of white noise, somewhat similar to our process "'n(t) for large n. In other applications, however, much slower processes are admitted as "white noise" perturbations, for example, waves striking the side of a large ship or the influence of atmospheric turbulence on an airplane. In the example of the ship, the reason that this assumption is a valid approximation stems from the fact that waves of quite varied energies strike both sides of the ship almost independently with a frequency much larger than the free oscillation frequency of the ship. Example 11.1.1. Having defined a one-dimensional Wiener process {w(t)}t~o, it is rather easy to construct an exact, continuous time, semidynamical system that corresponds to the partial differential equation (11.1.7) Our arguments follow those of Rudnicki (1985), which generalize results of Lasota (1981), Brunovsky (1983), and Brunovsky and Komornik (1984). The first step in this process is to construct the Wiener measure. Let X be the space of all continuous functions x: (0, 1) -+ R such that x(O) = 0. We are going to define some special subsets of X that are called cylinders. Thus, given a sequence of real numbers, 0 < St

< ••• < Bn

~

1,

340


and a sequence of Borel subsets of R,

we define the corresponding cyHnder by

C(sb ... , Bni A1, ... ,An)= {x E X:x(si) E A,,i = 1, ... ,n}.

(11.1.8)

Thus the cylinder defined by (11.1.8) is the set of all functions x E X pasing through the set A1 at s1 (see Figure 11.1.2). The Wiener measure P.w of the cylinders (11.1.8) is defined by

P.w(C(sb ... , Bni A1,. ··,An)) = prob{w(s1) E A1, ... , w(sn) E An}·

(11.1.9)

To derive an explicit formula for Jl.w, consider a transformation y = F(x) of R" into itself given by (11.1.10) and set A= A1 x · · · x An. Then the condition (w(s1), ... , w(sn)) E A is equivalent to the requirement that the random vector (w(s1), w(s2)- w(s1), ... , w(sn)- w(sn-1))

(11.1.11)

belong to F(A). Since {w(t)}t>o is a random process with independent increments, the density functio;; of the random vector (11.1.11) is given by

g(s1,yl)g(s2- B1,Y2),. · · ,g(sn- Bn-bYn), where, by the definition of the Wiener process [see equation (11.1.3)],

1 g(s, y) = rn= exp( -y2 f2s). v211's

(11.1.12)

Thus we have prob{w(sl) E A1, ... , w(sn) E An}

=

J···J

g(s1, yl)g(s2- s1, Y2) · · · g(sn- Bn-b Yn)dy1 · · · dYn·

F(A)

Using the variables defined in (11.1.10), this becomes prob{w(s1) E A1, ... ,w(sn) E An}

=

r •••jAnr

jA1

9(81 1 X1)g(s2

-

817

X2 - X1)

"· g(sn - Bn-b Xn - Xn-d dx1 · · · dxn.


341

x(s)

0

FIGURE 11.1.2. Schematic representation of implications of the cylinder definition (equation (11.1.8).]

By combining this expression with equations {11.1.9) and {11.1.12), we obtain the famous formula for the Wiener measure:

JLw(C(sl! ... 1 Bni A11. · · 1 An)) 1

(We assume, for simplicity, that so = xo = 0.) To extend the definition of JLw, we can define the u-algebra A to be the smallest u-algebra of the subsets of X that contains all the cylinders defined by (11.1.8) for arbitrary n. By definition, the Wiener measure JLw is the (unique) extension of JLw 1 given by {11.1.13) on cylinders, to the entire u-algebra A. The proof that JLw given by (11.1.13) on cylinders can be extended to the entire u-algebra is technically difficult, and we omit it. However, note that if a Wiener process {w(t)}t>O is given, then it is a direct consequence of our construction of the Wiener measure for cylinders that forEEA, JLw(E) = prob{w E E) {11.1.14) where w is the restriction of w to the interval [0, 1]. {Incidentally, from this equation, it also follows that the assumption that a Wiener process {w(t)}t;?!O exists is not trivial, but, in fact, is equivalent to the existence of the Wiener measure.) With the element of the measure space (X, A, JLw) defined, we now turn to a definition of the semidynamical system {St }t;?!o corresponding to {11.1. 7). With the initial condition

u(O,s)

= x(s),

{11.1.15)

342


equation (11.1.7) has the solution u(t, s) = et12 x(se-t). Thus, if we set (11.1.16)

this equation defines {Sth~o· We first show that {Sth~o preserves the Wiener measure J.l.w· Since the measures J.Lw on cylinders generate the Wiener measure on the entire ualgebra A, we will only verify the measure-preservation condition (11.1.17)

for cylinders. First observe that for every a E (0, 1), P.w(C(a 2 st. ... ,a2 Sni aAt, ... ,aAn)) = P.w(C(st, .. ·, Sni At, .. ·, An)).

(11.1.18)

This follows directly from equation (11.1.13) if we set Yi = ax, in the integral on the right-hand side. Further, from (11.1.16), it is clear that (Stx)(s,) E A, if and only if x(ste-t) E e-t/2 A,. Thus, s;t(C(st, ... I Sni At, ... I An)) = {x EX: (Stx)(s,) E A,,i = 1, ... ,n} = C(e-tst, ... , e-tsn; e-t/2 At. ... , e-t/2 An).

From this relation and (11.1.18) with a = e-t/2 , we immediately obtain (11.1.17), thereby verifying that {Sth~o preserves the Wiener measure P.w· To demonstrate the exactness of {St}t~ 0 , we will be content to show that lim P.w(St(C))

t--+oo

=1

if P.w(C) > 0

(11.1.19)

for cylinders. In this case we have St(C) = St(C(st, ... 1 Sni At. ... , An)) = {Stx: x E C} = {et1 2 x(se-t): x E C}.

Set y(s)

= et12 x(se-t) so this becomes St(C) = St(C(st. ... ,sn;At, ... ,An)) = {y E X:y(s) = et12 x(se-t),x(s,) E A,,i = 1, ... ,n}(11.1.20)

Since s E [0, 1], and, thus, se-t E [0, e-t], the conditions x(s,) E At are irrelevant for s, > e-t. Thus St(C(st, ... ,sn; At, ... ,An))= C(stet, ... , s~cet; et/ 2 At, ... ,et/2 A~c)


343

where k = k(t) is the largest integer k ~ n such that s1c ~ e-t. Once t becomes sufficiently large, that is, t > -logs1, then from (11.1.20) we see that the last condition x1 E A1 disappears and we are left with

St(C(sl, ... , Sni A., ... , An))= {y EX: y(s) = et12 x(se-t)}. However, since X is the space of all possible continuous functions x: [0, 1] R, the set on the right-hand side is just X and, as a consequence, for t

-+

> -log s 11

which proves equation (11.1.19) for cylinders. In the general case, for an arbitrary C E .A the demonstration that (11.1.19) holds is more difficult, but the outline of the argument is as follows. Starting with the equality

1-'w(St(C)) = J.tw(St" 18t(C)), and using the fact that the family {8t" 1 8t(C)}e>o is increasing with t, we obtain lim J.tw(St(C)) = J.tw(B), (11.1.21) t-+oo

where

(11.1.22) and to is an arbitrary nonnegative number. From (11.1.22), it follows that

B

E

.Aoo =

n 8t" (.A). 1

t2:;0

From the Blumenthal zero-one law [see Remark 11.2.1] it may be shown that the u-algebra .Aoo contains only trivial sets. Thus, since J.tw(B) ;::: J.tw(C), we must have J.tw(B) = 1 whenever J.'w(C) > 0. Thus (11.1.19) follows immediately from (11.1.21). A proof of exactness may also be carried out for equations more general than the linear version (11.1.7). The nonlinear equation

au. &t

au.

+ c(s) 88

= f(s, u),

(11.1.23)

has been used to model the dynamics of a population of cells undergoing simulaneous proliferation and maturation [Lasota, Mackey, and Wa.ZewskaCzyZewska, 1981; Mackey and Dormer, 1982], where sis the maturation variable. When the coefficients c and f satisfy some additional conditions, it can be shown that all the solutions of (11.1.23) with the initial condition (11.1.15) converge to the same limit if x(O) > 0. However, if x(O) = 0, then the solutions of (11.1.23) will exhibit extremely irregular behavior that

344


can be identified with the exactness of the semidynamical system {Sth~o corresponding to u(t, s). This latter situation [x(O) =OJ corresponds to the destruction of the most primitive cell type (maturity = 0), and in such situations the erratic behavior corresponding to exactness of {St}t~o is noted clinically. 0

11.2 d-Dimensional Wiener Processes (Brownian Motion) In considering d-dimensional Wiener processes we will require an extension of our definition of independent sets. Suppose we have a finite sequence

(11.2.1) of a-algebras. We define the independence of (11.2.1) as follows. Definition 11.2.1. A sequence (11.2.1) consists of independent a-algebras if all possible sequences of sets At, ... , An such that

are independent. Further, for every random variable ewe denote by .r(e) the a-algebra of all events of the form {w:e(w) E B}, where the Bare Borel sets, or, more explicitly, .r(e) = {Ct(B):B is a Borel set}. Having a stochastic process {TJ(t)}tea on an interval a, we denote the smallest a-algebra that contains all sets of the form {w:17(t,w) E B},t E a, B is a Borel set, by F(17 (t): t E a).

With this notation we can restate our definition of independent random variables as follows. The random variables et, ... , are independent if .r(et), ... ,.r(en) are independent. In an analogous fashion, stochastic processes {TJt(t)}tea1 , ••• , {TJn(t)hean are independent, if

en

F(TJt(t):t Eat), ... ,F(TJn(t):t E an) are independent. and n stochastic processes Finally, having m random variables et, ... , {TJt(t)heaw .. , {TJn(t)hean, we say that they are independent if the aalgebras

em

11.2. d-Dimensional Wiener Processes

345

are independent. We will also say that a stochastic process {q(t)hea and au-algebra :Fo are independent if :F(q(t):t E a) and :Fo are independent. Now it is straightforward to define a d-dimensional Wiener process. Definition 11.2.2. A d-dimensional vector valued process w(t)

= {w1(t), ... ,wd(t)},

t ~0

is a d-dimensional Wiener process (Brownian motion) if its compo-

nents {w1(t)}t;::o, ... , {wd(t)}t;::o are one-dimensional independent Wiener processes (Brownian motion). From this definition it follows that for every fixed t the random variables w1(t), ... , wd(t) are independent. Thus, it is an immediate consequence of Theorem 10.1.1 that the joint density of the random vector (w1(t), ... ,wd(t)) is given by g(t, x1, ... ,xd)

= g(t, xl) · · · g(t, xd) = (21r:)d/2 exp [-

t.

;t x~

]·

(11.2.2)

The joint density g has the following properties:

I··· I

g(t,x11 ... ,xd)dxl · · ·dxd

= 1,

(11.2.3)

Rd

I··· I

x,g(t, x11 ••• , Xd) dx1 · · · dxd

= 0,

i = 1, ... ,d,

(11.2.4)

Rd

and

I··· I

x,x;g(t,xll ... ,xd) dx1 · · · dxd

= 61;t,

i,j

= 1, ... ,d,

Rd

where 61; is the Kronecker delta (6a;

= 0, i ::/= j, 6" = 1).

(11.2.5)

Remark 11.2.1. The family :F(w(u): 0 :5 u :5 t) of u-algebras generated by the Wiener process (or d-dimensional Wiener process) has the interesting property that it is right-hand continuous. We have (modulo zero) :F(w(u):O :5 u :5 t)

=

n

:F(w(u):O :5 u :5 t+h).

h>O

(11.2.6)

346


In particular at t = 0, since w(O) = 0 and the 0'-algebra generated by w(O) is trivial, we can see from equality (11.2.6) that the product

n

.r(w(u): 0 $; u $;h)

h>O

contains only sets of measure zero or one. The last statement is referred to as the Blumenthal zero-one law (Friedman [1975]). D

11.3 The Stochastic

Ito Integral: Development

To understand what is meant by a solution to the stochastic differential equation (11.1.1), it is necessary to introduce the concept of the stochastic Ito integral. In this section we offer a simple but precise definition of this integral and calculate some specific cases so that a comparison with the usual Lebesgue integral may be made. Let a probability space (0, .r, prob) be given, and let {w(t)}t;:::o be a onedimensional Wiener process. If {17(t)he[a,~] is another stochastic process defined fortE [a, ,8], a~ 0, we wish to know how to interpret the integral

J:

17(t) dw(t).

(11.3.1)

Proceeding naively from the classical rules of calculus would suggest that (11.3.1) should be replaced by

J:

17(t)w'(t) dt.

However, this integral is only defined if w(t) is a differentiable function, which we have already observed is not the case for a Wiener process. Another possibility suggested by classical analysis is to consider (11.3.1) as the limit of approximating sums 8 of the form 1c

8=

L '7(f,)[w(t,)- w(ti-t)],

(11.3.2)

i=l

where a

= to < t1 < · · · < t1c = .B

is a partition of the interval [a, .B] and the intermediate points f, E [t,, ta+t]. This turns out to be a more fruitful idea but has the surprising consequence that the limit ofthe approximating sums 8 ofthe form (11.3.2) depends on the choice of the intermediate points f,, in sharp contrast to the situation for the Riemann and Stieltjes integrals. This occurs because w(t), at fixed w, is not a function of bounded variation.

11.3. The Stochastic Ito Integral: Development

347

With these preliminary remarks in mind, we now proceed to develop some concepts of use in the definition of the Ito integral. Definition 11.3.1. A family {Ft},

a~ t ~

{3, of 0'-algebras contained in

.r is called nonanticipative if the following three conditions are satisfied: (1) Fu C :Ft for u

~

(2) Ft :::> F(w(u): a respect to Ft;

t, soFt increases as t increases; ~

u

~

t), so w(u), a

~

u

~

t, is measurable with

(3) w(t +h)- w(t) is independent of Ft for h;::: 0, so all pairs of sets A1 , A2 such that A1 EFt and A2 E F(w(t +h)- w(t)) are independent. From this point on we will assume that a Wiener process w( t) and a family of nonanticipative 0'-algebras {Ft}, a~ t ~ {3, are given. We next define a fourth condition. Deftnition 11.3.2. A stochastic process {77{t)}, a non-anticipative with respect to {Ft} if

(4) Ft :::> .1"{77(u): a~ u

~

~

t

~

{3, is called

t}, so 77(u) is measurable with respect to :ft.

For every random process {77(t)}, a~ t ~ {3, we define the Ito sum 8 by k

8

=

E 17(ti-1)[w(t,) - w(ti-1)].

(11.3.3)

i=l

Note that in the definition of the Ito sum {11.3.3), we have specified the intermediate points fi of {11.3.2) to be the left end of each interval, f, = ti-l· For a given Ito sum 8, we define

and call a sequence of Ito sums {8n} regular if c5(8n)-+ 0 as n-+ oo. We now define the Ito integral as follows. Deftnition 11.3.3. Let {77(t)}, a~ t ~ {3, be a nonanticipative stochastic process. If there exists a random variable ( such that

( = st-lim 8n

(11.3.4)

for every regular sequence of the Ito sums {8n}, then we say that (is the Ito integral of {17(t)} on the interval [a, /3] and denote it by ( =

J:

17(t) dw(t).

{11.3.5)

348


Remark 11.3.1. It can be proved that for every continuous nonanticipative process the limit (11.3.4) always exists. D Remark 11.3.2. Definition 11.3.1 of a nonanticipative u-algebra is complicated, and the reason for introducing each element of the definition, as well as the implication of each, may appear somewhat obscure. Condition (1) is easy, for it merely means that the u-algebra Fe of events grows as time proceeds. The second condition ensures that :Ft contains all of the events that can be described by the Wiener process w(8) for times 8 E [a, t]. Finally, condition (3) says that no information concerning the behavior of the process w(u)- w(t) for u > t can influence calculations involving the probability of the events in Fe. Definition 11.3.2 gives to a stochastic process 71(u) the same property that condition (2) of Definition 11.3.1 gives to w(u). Thus, all of the information that can be obtained from 71(u) for u E [a, t] is contained in :Ft. Taken together, these four conditions ensure that the integrand 71(t) of the Ito integral (11.3.5) does not depend on the behavior of w(t) for times greater than {3 and aid in the proof of the convergence of the Ito approximating sums. Further, the nonanticipatory assumption plays an important role in the proof of the existence and uniqueness of solutions to stochastic differential equations since it guarantees that the behavior of a solution in a time interval [0, t] is not influenced by the Wiener process for times larger than t. D

Example 11.3.1. For our first example of the calculation of a specific Ito integral, we take

1T

dw(t).

=

In this case the integrand of (11.3.5) is 77(t) 1. Thus :F(77(t): 0 :5 t :5 T) is a trivial u-algebra that contains the whole space 0 and the empty set 0. To see this, note that, if 1 E B, then {w:77(t) E B} = 0 and, if 1 ¢ B then {w:77(t) E B} = 0. This trivial u-algebra {0,0} is contained in any other u-algebra, and thus condition (4) of Definition 11.3.2 is satisfied. By definition 1c 8

= L[w(ti) -

w(ti_t)]

= w(t~c) -

w(to)

i=l

and, thus,

1T

dw(t) = w(T).

0

Example 11.3.2. In this example we will evaluate

1T

w(t) dw(t),

which is not as trivial as our previous example.

= w(T)

11.3. The Stochastic Ito Integral: Development

349

In this case, 17(t) = w(t), so that condition (4) of Definition 11.3.2 follows from condition (2) of Definition 11.3.1. The Ito sum, k

=L

8

w(ti-l)[w(ti) - w(ti_t)],

i=l may be rewritten as k

8

= ! I)w

k 2

i=l

2

(ti) - w (ti-l)J-l2:!w(ti) - w(ti-1)] 2 i=l k

= lw

2

(T) -

! L 'Yi,

(11.3.6)

i=l

where 'Yi

= [w(t,) -

w(ti-1)] 2 •

To evaluate the last summation in (11.3.6), observe that, from the Chebyshev inequality (10.2.10), prob

{I! t, !t, 'Yi -

mi

~ e} ~ e~ D

2

(! t,

'Yi) (11.3.7)

where mi = E('Yi)· Further, by (11.1.4), E('Yi)

= E([w(t,) -

w(ti-lW)

= ti -ti-l

and, by equations {10.2.6) and (11.1.4), D 2 ('Y1) ~ E('Yl)

= E([w(ti)- w(ti-1)] 4 ) = 3(ti- ti-1) 2 •

Thus,

and

k

k

LD ('Yi) ~ 3 L(ti- ti-1)

~ 3Tmrx:(ti- ti-d· i=l i=l Setting 6(8) = maJCi(ti -ti-l) as before and using (11.3.7), we finally obtain 2

prob

2

{I! t, il ~ 'Yi-

e}

~ ~6(8)

350


or, from (11.3.6), prob

{I ( 8-

I}

w2(T) 3T 6(8). - - T) ~ e ::::; 4e 2

-

2

2

If {8n} is a regular sequence, then 6(8n) converges to zero as n-+ oo and

st-lim 8n

= !w2(T) - !T.

Thus we have shown that

loT w(t) dw(t) = !w2 (T) - !T, clearly demonstrating that the stochastic Ito integral does not obey the usual rules of integration. 0 This last example illustrates the fact that the calculation of stochastic integrals is, in general, not an easy matter and requires many analytical tools that may vary from situation to situation. What is even more interesting is that the sufficient conditions for the existence of stochastic integrals related to the construction of nonanticipative u-algebras are quite complicated in comparison with the Lebesgue integration of deterministic functions. Remark 11.3.3. From Example 11.3.2, it is rather easy to demonstrate how the choice of the intermediate point ti influences the value of the integral. For example, picking ti = !(ti-l+ ti), we obtain, in place of the Ito sum, the Stratonovich sum, 1c

8

=L

w (!(ti-l+ ti)} [w(ti)- w(ti-1)]

i=l 1c

= !w

2

(T) - !

1c

L 'Yi + ! ~:::>i• i=l i=l

where

'Yi = [w(ti)- w (!(ti-l+ t,))] and

2

2

Pi= [w (!(ti-l+ ti)}- w(ti-1)] • Since the variables 'Y1, ... , 'Yic are independent as are Pl, ... , Pic, we may use the Chebyshev inequality as in the previous example to show that 1c

st-lim

1c

L 'Yi = iT= st-lim L Pi· i=l

i=l Thus the Stratonovich sums {8n} converge to !w2(T), and the Stratonovich integral gives a result more in accord with our experience from calculus.

11.4. Stochastic Ito Integral: Special Cases

351

However, the use of the Stratonovich integral in solving stochastic differential equations leads to other more serious problems. 0 To close this section, we extend our definition of the Ito integral to the multidimensional case. If G(t) = ('1i;(t)), i,j = 1, ... , dis ad x d matrix of continuous stochastic processes, defined for a~ t ~ {3, and w(t) = (wi(t)), i = 1, ... , d, is a d-dimensional Wiener process, then

L{3

G(t) dw(t) =

where (i

d

1{3

j=1

a

=L

('~1)

'

(11.3.8)

'1i;(t) dw;(t)

defines the Ito integral. Thus, equation (11.3.8) is integrated term by term. In this case the family {Ft} of nonanticipative o--algebras must satisfy conditions (2) and (3) of Definition 11.3.1 with respect to all {wi(t)}, i = 1, ... , d, and condition (4) of Definition 11.3.2 must be satisfied by all {'IJi;(t)}, i,j = 1, ... ,d.

11.4 The Stochastic Ito Integral: Special Cases In the special case when the integrand of the Ito integral does not depend on w, that is to say, it is not a stochastic process, the convergence of the approximating sums is quite strong. This section is devoted to an examination of this situation and one in which we are simply integrating a stochastic process with respect tot. Before stating our first proposition, we note that, if f: [a, {3] -+ R is a continuous function, then every regular sequence {sn} of approximating sums k,.

Bn

= Lf(tf)[w(tf)- w(tf-1)], i=1

converges in the mean [i.e,, strongly in £ 2 (0)] to the integral

(=

1:

f(t) dw(t).

(11.4.1)

Although we will not prove this assertion, it suffices to say that the proof proceeds in a fashion similar to the proof of the following proposition. Proposition 11.4.1. Iff: [a,/3]-+ R is a continuous function, then

E

(!: /( t)) t) dw(

=0

(11.4.2)

352


and 2

D (J: f(t)dw(t))

= J:[f(t)] 2 dt.

(11.4.3)

Proof. Set 8

=

k

k

i=1

i=1

L f(ti-1)[w(ti)- w(ti-1)] = L f(ti-d~wi, 82 =

k

L

/(ti-1)/(t;-1)~wi~w;.

i,j=1 We have immediately that k

E(8)

=L

/(ti-1)E(~wi)

=0

i=l

and, since w(t) is a Wiener process with independent increments,

We also have k

2

D (8)

= E(8 ) = L 2

/(ti-1)/(t;-1)E(~wi~w;)

i,j=1 k

= L:!J(ti_t)] 2 (ti -ti-d· i=1 Thus for any regular sequence {8n}, lim E(sn) n-+oo

=0

(11.4.4)

and (11.4.5) Since, from the remarks preceding the proposition, {sn} converges in mean to the integral (given in equation (11.4.1), we have limn-+oo E(8n) = E(() and limn-+oo D 2 (sn) = D 2 ((), which, by (11.4.4) and (11.4.5), completes the proof. •

11.4. Stochastic Ito Integral: Special Cases

353

A second special case of the stochastic integral occurs when the integrand is a stochastic process but it is desired to have the integral only with respect

to time. Hence we wish to consider

(=

J:

{11.4.6)

11(t)dt

when {17(t)}, a$ t $ /3, is a given stochastic process. To define {11.4.6) we consider approximating sums of the form k

s=

E 11(ti)(ti -ti-d. i=l

corresponding to the partition a = to

< t1 < · · · < t~c

=

/3

with arbitrary intermediate points fiE [ti-l• ti]· We now have the following definition.

Definition 11.4.1. If every regular [6{sn) -+ OJ sequence {sn} of approximating sums is stochastically convergent and

( = st-lim Bm

{11.4.7)

then this common limit is called the integral of 17(t) on [a, /3] and is denoted by {11.4.6). Observe that, when 17{t,w) possesses continuous sample paths, that is, it is a continuous function of t, the limit lim Bn(w)

n-+oo

exists as the classical Riemann integral. Thus when {17(t)}, a $ t $ /3, is a continuous stochastic process, this limit exists for almost all w. Further, since, by Proposition 10.3.2, almost sure convergence implies stochastic convergence, the limit (11.4. 7) must exist. There is an interesting connection between the Ito integral {11.3.5) and the integral of {11.4.6) reminiscent of the classical "integration by parts" formula. It can be stated formally as follows.

Proposition 11.4.2. Iff: [a, /3] derivative f', then

1:

f(t) dw(t)

=-

J:

-+

R is differentiable with a continuous

f'(t)w(t) dt + f({J)w({J)- f(a)w(a).

{11.4.8)

354


Proof. Since the integrals in (11.4.8) both exist we may pick special approximating sums of the form len

=L

Bn

/'(~)w(~)(tf- tf-1),

(11.4.9)

i=1

where the intermediate points t, are chosen in such a way that

Substituting this expression into (11.4.9), we may rewrite Bn as len

Bn =

L[f(tf) -/(tf_t))w(~) i=1 len-1

=-

L

[w(~+l) - w(~)]/(tf) + f(t~Jw(~J

i=1

-/(ti)w(f'i).

(11.4.10)

The sum on the right-hand side of (11.4.10) corresponds to the partition f'iO·

Theorem 11.5.1 can be proved by the method of successive approximations as can the corresponding result for ordinary differential equations.

11.5. Stochastic Differential Equations

357

Thus a sequence {xi(t)}t~o of stochastic processes would be defined with x(t = 0) = x 0 and

xi(t)

= Lt b(xi- 1 (s)) ds + Lt u(xi- 1 (s)) dw(s) + x0 •

Then, using the Lipschitz conditions (11.5.4) and (11.5.5), it is possible to evaluate the series 00

x(t) = :~::)xi(t) - xi- 1(t)] + x 0 i=1 in L 2 (0) norm by a convergent series of the form,

and to prove that x(t) is, indeed, the desired solution. We omit the details as this proof is quite complicated, but a full proof may be found in Gikhman and Skorokhod [1969]. An alternative way to generate an approximating solution is to use the Euler linear extrapolation formula. Suppose that the solution x(t) is given on the interval [0, to]. Then for values to+ tit larger than, but close to, t 0 , we write x(to +tit)

= x(to) + b(x(to))tit + u(x(to))tiw,

(11.5.6)

where tiw = w(to +tit)- w(to). (Observe that for an ordinary differential equation, this equation defines a ray tangent to the solution on [0, to] at to.) In particular, when an interval [0, T] is given, we may take a partition

O=tow(s) ds + uw(t).

Using the integration by parts formula (11.4.8), this becomes

x(t)

= x0 e-bt + u it e-b(t-•)dw(s).

From (11.5.12) and (11.4.2), it follows that

E(x(t))

= e-bt E(x0 )

and, taking note of the independence of x 0 and w(t),

D 2(x(t)) = e- 2bt D 2(x0) + u 2D 2 (lot e-b(t-•>dw(s)) . With (11.4.3), this finally reduces to D2(x(t)) = e-2bt D2(xo)

+ u2fot e-2b(t-s)ds

= e-2bt D2(xo) + (u2 /2b)[1 - e-2bt].

(11.5.12)

11.6. Fokker-Planck Equation

359

Thus, for the Langevin equation,

This asymptotic property of the variance is a special case of a more general result that we will establish in the next section where we examine uses of the Fokker-Planck equation. 0

11.6 The Fokker-Planck (Kolmogorov Forward) Equation The preceding sections were aimed at obtaining an understanding of the dynamical system dx (11.6.1) dt = b(x) + u(x)e with (11.6.2)

e.

under a stochastic perturbation This required us to first introduce the abstract concept of nonanticipative a--algebras. Then we had to define the Ito integral, which is generally quite difficult to calculate. Finally we gave the solution to equations (11.6.1)-(11.6.2) in terms of a general formula, generated by the method of successive approximations, which contains infinitely many Ito integrals. In this section we extend this to a discussion of the density function of the process x(t), which is a solution of (11.6.1) and (11.6.2). This density is defined as the function u( t, x) that satisfies prob{x(t) E B} =

L

u(t, z) dz.

(11.6.3)

The uniqueness of u(t,x) follows immediately from Proposition 2.2.1, but the existence requires some regularity conditions on the coefficients b( x) and u(x), which are given in the following. We will also show how u(t,x) can be found without any knowledge concerning the solution x(t) of the stochastic differential equations (11.6.1) with (11.6.2). It will turn out that u( t, x) is given by the solution of a partial differential equation, known as the Fokker-Planck (or Kolmogorov forward) equation and that it is completely specified by the coefficients b(x) and u(x) of equation (11.6.1). Now set d

a,;(x)

= L:u,~;(x)u;~c(x). lc=l

(11.6.4)

360


From (11.6.4) it is clear that ai;

= a;i and, thus, the quadratic form,

d

L: aij(x),\i,\;,

(11.6.5)

i,j=l

is symmetric. FUrther, since

(11.6.5) is nonnegative. We are now ready to state the main theorem of this section, which gives the Fokker-Planck equation. Theorem 11.6.1. lfthefunctionsl1ij, 8l1i;/8x,ui:J2l1i;/8x~c8xz, bi, I:Jbifi:Jx;, 8uf8t, 8ufi:Jxi, and 82 uf8xi8x; are continuous fort> 0 and x E Jld, and if bi, l1ij and their first derivatives are bounded, then u(t,x) satisfies the equation t

> 0, X

E

Jld.

(11.6.6)

Equation (11.6.6) is called the Fokker-Planck equation or Kolmorgorov forward equation. Remark 11.6.1. In Theorem 11.6.1 we assumed I:Jbifi:Jx; and I:Jqi;/8x~c were bounded since this implies the Lipschitz conditions (11.5.4) and (11.5.5) which, in turn, guarantee the existence and uniqueness of the solution to the stochastic equations (11.6.1) with (11.6.2). In order to assure the existence and differentiability of u, it is sufficient, for example, that ai; and bi, together with their derivatives up to the third order, are continuous, bounded, and satisfy the uniform parabolicity condition (11.7.5). 0

Proof of Theorem 11.6.1. We will use the Euler-Bernstein approximation formula (11.5.6) in the proof of this theorem as it allows us to derive (11.6.6) in an extremely simple and transparent fashion. Thus let to > 0 be arbitrary, and let x(t) be the solution to equations (11.6.1) and (11.6.2) on the interval [0, to]. Define x(t) on [t0 , t 0 + e] by x(to +At)

= x(to) + b(x(to))At + q(x(to))[w(to +At)- w(to)],

(11.6.7)

where 0 ::; At ::; e and e is a positive number. We assume (and this is the only additional assumption needed for simplifying the proof) that x(t),

11.6. Fokker-Planck Equation

361

extended according to (11.6.7), has a density u.(t,x) for 0 ~ t ~to+ e and that fort= t 0 , Ut(t, x) exists. Observe that at the point t =to, u.(t, x) (and U.t ( t, x)) is simultaneously the density (and its derivative) for the exact and for the extended solution. Now let h: R"- _. R be a C 3 function with compact support. We wish to calculate the mathematical expectation of h(x(t0 +at)). First note that since u.(to +at, x) is the density of x(to +at), we have, by (10.2.2),

E(h(x(t 0 +at)))=

I

jRd

h(x)u.(t0 +at,x)dx.

(11.6.8)

However, using equation (11.6. 7), we may write the random variable h(x(to+ at)) in the form

h(x(to +at)) = h(Q(x(to), w(to +at)- w(to))),

(11.6.9)

where

Q(x, y) = x + b(x)at + e1(x)y. The variables x(t0 ) and aw(to) = w(to +at) - w(to) are independent for each 0 ~at~ e since x(to) is Ft0 -measurable and aw(to) is independent with respect to Fto· Thus the random vector (x(to),aw(t0 )) has the joint density

u.(to, x)g(at, y), where g is given by (11.1.3). As a consequence, the mathematical expectation of (11.6.9) is given by

I I

jRd jRd

h(Q(x,y))u.(t0 ,x)g(at,y)dxdy

= }Rd I }Rd I h(x+b(x)at+CT(x)y)u.(t0 ,x)g(at,y)dxdy. From this and (11.6.8), we obtain

I

}Rd

h(x)u.(t0 +at,x)dx

= }Rd I }Rd I h(x+b(x)at+CT(x)y)u.(t0 ,x)g(at,y)dxdy. By developing h in a Taylor expansion, we have

Ld h(x)u.(to d

+ at,x)dx =

Ld Ld {h(x)

+

t, ::i

[bi(x)at + (CT(x)y)i]

fPh

+! ..L 1 -x,.a a .[bi(x)at + (e1(x)y)i] x, I,J=

·[b;(x)at + (e1(x)y);]

+ r(at)}

·u.(t0 ,x)g(at,y)dxdy,

(11.6.10)

362


where r(~t) denotes the remainder and (u(x)y) 1 is the ith coordinate of the vector u(x)y. On the right-hand side of (11.6.10) we have a finite collection of integrals that we will first integrate with respect toy. Observe that d

(u(x)y),(u(x)y);

=L

u,~c(x)u;,(x)ykYI·

A:,l=l

By equation (11.2.3)

r u 0, u(t,x) has the same property. However, if f is discontinuous, then condition (11. 7.8) might not hold at a point of discontinuity.

11.8. Markov Operators and Parabolic Equations

369

Having a. generalized solution, we define a. family of operators {Pth~o by

Pof(x)

= f(x),

Ptf(x) =

f I'(t,x,y)f(y)dy. lnd

(11.8.2)

We will now show that, from the properties of I' stated in Theorem 11.7.1, we obtain the following corollary.

Corollary 11.8.1. The family of operators group, that is, (1) Pt(>.d1

is a stochastic semi-

+ >.2/2) = >.1Ptf1 + >.2Pth, hh E L 1;

(2) Ptf ?:. 0 for

(3)

{Pth~o

I ?:. 0;

IIPtfll = 11/11

for f ?:. 0;

(4) Ptl+t2! = Ptl(Pt2!),

IE L1 .

Proof. Properties (1) and (2) follow immediately from equation (11.8.1) since the right-hand side is an integral operator with a. positive kernel. To verify (3), first assume that f is continuous with compact support. By multiplying the Fokker-Planck equation by a. C 2 bounded function h( x) and integrating, we obtain

L Rd

h(x)utdx

=

L {! L Rd

d

h(x)

.. _

'·3-

82

. . (a,;u)8 X, 8 x3 1

} 8 L ~(b 1 u) dx d

._

,_1 uX,

and integration by parts gives

Setting h = 1, we have

! Ld Since u ?:. 0 for

u dx

=

Ld

Ut

dx

= 0.

f ?:. 0, we have !lluii=O

fort> 0.

Further, the initial condition (11.7.8), inequality (11.7.11) and the boundedness of u imply, by the Lebesgue dominated convergence theorem, that I!Pdll is continuous a.t t = 0. This proves that IIPt/11 is constant for all t ?:. 0. If f E L 1 is an arbitrary function, we can choose a. sequence {!AJ of

370


continuous functions with compact support that converges strongly to Now,

I.

I(IIPt!ll-11111>1 s; I(IIPt!II-IIPt!A:II)I+I(IIPt!A:II-IIIA:II>I+II!A:-Ill· (11.8.3} Since, as we just showed, Pt preserves the norm, the term IIPdii-IIIA:II is zero. To evaluate the first term, note that

I(IIPdii-IIPt!A:II)I s; IIPd- Pt!A:II s; f r(t,x,y)III-IA:IIdys;Mtiii-IA:II, jRtJ where Mt = SUPz,v r. Thus the right-hand side of (11.8.3} converges strongly to zero as k -+ oo. Since the left-hand side is independent of k, we have IIPdll = IIIII, which completes the proof of (3}. As we know, conditions (1}-(3} imply that IIPdll s; IIIII for all I and, thus, the operators Pt are continuous. Finally to prove (4}, again assume I is a continuous function with compact support and set ii(t, x) = u(t+t1, x). An elementary calculation shows that ii(t, x} satisfies the Fokker-Planck equation with the initial condition ii(O,x) = u(t1,x). Thus, by the uniqueness of solutions to the FokkerPlanck equation, and, at the same time,

From these it is immediate that

Pt+tJ = l't(PtJ), which proves (4) for all continuous I with compact support. If I E £ 1 is arbitrary, we again pick a sequence {!A:} of continuous functions with compact supports that converges strongly to I and for which

Pt2+tJ1c =

Pt 2 (PtJ~c)

holds. Since the Pt have been shown to be continuous, we may pass to the limit of k-+ oo and obtain (4) for arbitrary f. • Remark 11.8.1. In developing the material of Theorems 11.6.1, 11.7.1, and Corollary 11.8.1, we have passed from the description of u(t, x) as the density of the random variable x(t), through a derivation of the FokkerPlanck equation for u( t, x) and then shown that the solutions of the FokkerPlanck equation define a stochastic semigroup {Pth~o· This semigroup describes the behavior of the semi-dynamical system, equations (11.6.1} and

11.9. Asymptotic Stability of Solutions

371

(11.6.2). In actuality, our proof of Theorem 11.6.1 shows that the righthand side of the Fokker-Planck equation is the infinitesimal operator for Ptf, although our results were not stated in this fashion. Further, Theorem 11.7.1 and Corollary 11.8.1 give the construction of the semigroup generated by this infinitesimal operator. 0

Remark 11.8.2. Observe that, when the stochastic perturbation disappears (ui; = 0), then the Fokker-Planck equation reduces to the Liouville equation and {Pt} is simply the semigroup of Frobenius-Perron operators corresponding to the dynamical system i

= 1, ... ,d.

0

11.9 Asymptotic Stability of Solutions of the Fokker-Planck Equation A13 we have seen, the fundamental solution r may be extremely useful. However, since a formula for r is not available in the general case, it is not of much use in the determination of asymptotic stability properties of u(t,x). Thus, we would like to have other techniques available, and in this section we develop the use of Liapunov functions for this purpose, following Dlotko and Lasota (1983). Here, by a Liapunov function we mean any function V: Rd -+ R that satisfies the following four properties:

(1) V(x)

~

0 for all x;

(2) limlzl-+oo V(x)

= oo;

(3) V has continuous derivatives (8Vj8xi), (fPVJ8xi8x;), i,j and (4) V(x)

I I

~ pe6lzl, ~()~~:)I ~ pe61zl, and ~:~~;J ~ pe6lzl

= 1, ... , d; (11.9.1)

for some constants p, 6. Conditions (1)-(4) are not very restrictive, for example, any positive definite quadratic form (of even order m) d

V(x)

=

L

ait ... imXit ••• Xim

ilt•••tim=l

is a Liapunov function. Our main purpose will be to use a Liapunov function

372


V that satisfies the differential inequality

82V

d

:E ai;(x)

.. ,,,= 1

.+

8Xi 8X3

8V :E bi{x)O. ~ -aV(x) + f3 l= d

.

{11.9.2)

X,

1

with positive constants a and {3. Specifically, we can state the following theorem.

Theorem 11.9.1. Assume that the coefficients ai; and bi of equation {11.7.1) are regular for the Cauchy problem and that there is a Liapunov function V satisfying {11.9.2). Then the stochastic semigroup {Pth?:o defined by the generalized solution of the Fokker-Planck equation and given in {11.8.2) is asymptotically stable.

Proof. The proof is similar to that of Theorem 5.7.1. First pick a continuous density f with compact support and then consider the mathematical expectation of V calculated with respect to the solution u of equations (11.7.1) and (11.7.2). We have

E(V I u)

=f

V(x)u(t,x)dx.

}Rrl

(11.9.3)

By inequalities (11.7.11) and (11.9.1), u(t,x)V(x) and Ut{t,x)V(x) are integrable. Thus, diffemtiation of (11.9.3) with respect tot gives

dE(V I u) _ dt -

f

jR"

V( ) (

X Ut

t,x

) ..J_ ua;

d 82 d8 } = JlifRrl V(x) { ! i,j=1 :E 8-[bi(x)u] 8 x,.8x,. [~;(x)u]- :E i=1 x,

Integrating by parts and using the fact that the products u V, uVx, vanish exponentially as lxl--+ oo, we obtain

dE(V 1 u) dt =

1 { ,,,_ 1

Rrl

d

2 8 V 8 X, 8 x3

. . 2 :E ai;(x).. _

1

dx.

Uz, V,

and

d 8V} + :E~(x)a. u(t,x)dx. ._ x, ,_ 1

From this and inequality (11.9.2), we have

dE(~ I u) ~ -aE(V I u) + {3. To solve this differential inequality, multiply through by eat, which gives

!

[E(V I u)eat]

~ {Jeat.


373

Since E(V I u) at t = 0 equals E(V I /), integration on the interval [0, t] yields E(V I u)eat- E(V I f)$ (/3/o.)(eat- 1) or

E(V I u) $ e-atE(V I f)+ (/3/o.)(1- e-at).

Since E(V I f) is finite, we can find a to

E(V I u) $ (/3/o.) Now let Gq have

= {x: V(x) < q}. f

la

= to(/) such that

+1

fort 2:: to.

From the Chebyshev inequality (5.7.9), we

u(t,x)dx;::: 1- E(V I u), q

4

and taking q > 1 + (/3 /a) gives

fo

u(t,x)dx 2::

1- ~ [1 + ~] = > 0 e

9

for t ;::: to. Since V(x) -+ oo as lxl -+ oo, there is an r > 0 such that V(x) 2:: q for lxl 2:: r. Thus the set Gq is contained in the ball Br and, as a consequence,

u(t,x)=

f

r(1,x,y)u(t-1,y)dy;=::

jRd

f

r(1,x,y)u(t-1,y)dy

jBr

;::: inf r(1, x, y)

l11l~r

f

JBr

u(t- 1, y) dy;::: e inf r(1, x, y),

llli~r

fort 2:: to+ 1,x E Rd.

{11.9.4)

Since r(1, x, y) is strictly positive and continuous, the function

h(x)

= e inf

l11l~r

r(1,x,y)

is also positive. From (11.9.4), we have

Ptf(x)

= u(t,x);::: h

fort;::: to+ 1,

which shows that {Pt} has a nontrivial lower-bound function. Hence, by Theorem 7.4.1, the proof is complete. • When {Pt} is asymptotically stable, the next problem is to determine the limiting function lim Ptf(x)

t--+00

= u.(x),

JeD.

{11.9.5)

374


This may be accomplished by using the following proposition.

Proposition 11.9.1. If the assumptions of Theorem 11.9.1 are satisfied, then the limiting function u of (11.9.5) is the unique density satisfying the elliptic equation

(11.9.6)

Proof. Assume that u(x) is a solution of (11.9.6). To prove the uniqueness of u(x), note that, because u is a solution of (11.9.6), it follows that u(t, x) = u(x) is a time-independent solution of the Fokker-Planck equation (11.7.1). Thus, by Theorem 11.9.1, u(x) = lim u(t,x) = u.(x) t-+oo

and u.(x) = u(x) is unique. Next we show that u. satisfies (11.9.6). Let f E D(Rd) be a continuous function with compact support. We have u(t + s, x) = Ptu(s, x), or u(t+s,x)= Passing to the limit as s

-+

r

}Rd

r(t,x,y)u(s,y)dy.

oo, we obtain

r

u.(x) =

}Rd

r(t,x,y)u.(y)dy.

Since r is a fundamental solution of the Fokker-Planck equation, u.(x) is also a solution, and, since u.(x) is independent oft, it must satisfy equation (11.9.6). Thus the proof is complete. •

Example 11.9.1. Again consider the Langevin equation dx

dt = -bx+ue

and the corresponding Fokker-Planck equation

8u

at=

1 21:J2u

a )

2u ax2 +bax(xu.

Inequality (11.9.2) becomes

1 2a2v

av

2u ax2 - bx ax ~ -aV + {3,


375

which is satisfied with V(x) = x 2 , a = 2b, and {3 = u 2 • Thus all solutions u(t,x), such that u(O,x) = f(x) is a density, converge to the unique (nonnegative and normalized) solution u. of 2

cPu

1 20' dz2

d

+ bdx (xu) = 0.

(11.9.7)

The function

which is the Gaussian density with mean zero and variance u 2 f2b, satisfies (11.9.7), and, by Proposition 11.9.1, it is the unique solution. 0 Example 11.9.2. Next consider the system of stochastic differential equations dx

dt =Bx+u~,

where B = (b,;) and u matrix (a 1;) with

= (ui;)

(11.9.8)

are constant matrices. Assume that the d

a,;

= E O'ikUjk k=1

is nonsingular and that the unperturbed system

dx

-=Bx dt

(11.9.9)

is stable asymptotically, that is, all solutions converge to zero as t -+ oo. The Fokker-Planck equation corresponding to (11.9.8) has the form 8u

~

v~

~

d

=

d

8

~ .L. a,;8X ·8ux . - L -8 . [b,(x)u] . X 3 1 ,,,= 1 •= 1

where

1

d

b1(x)

= Lb•;x;. j=1

Since the coefficients ao; are constant and the matrix (a,;) is nonsingular, this guarantees that the uniform parabolicity condition (11.7.5) is satisfied. All of the remaining conditions appearing in Theorem 11. 7.1 in this case are obvious. Since (11.9.9) is asymptotically stable, the real parts of all eigenvalues of B are negative and from the classical results of Liapunov stability theory there is a Liapunov function V such that

ttb d

1(x)

8V x, ~ -aV(x), 8

(11.9.10)

376


where V is a positive definite quadratic form d

V(x)

L

=

(11.9.11)

k,•;•Xi•X;•·

i',j'=1

Differentiating (11.9.11) with respect to x 1 and then x;, multiplying by 1;, summing over i and j, and adding the result to (11.9.10) gives

!a

!

d

L

a2v a,;{f"7):

x, x,

i,j=1

d

av

i=1

x,

+ Lb,(x)a-:- ~ -aV(x) +

Ld a,;ki;·

i,j=1

Thus inequality (11.9.2) is satisfied. Hence the semigroup {Pt} generated by the perturbed system (11.9.8) is asymptotically stable. To summarize, if the unperturbed system (11.9.9) is asymptotically stable, then any stochastic perturbation with a nonsingular matrix (a,;) leads to a stochastic semigroup that is also asymptotically stable. In this case the limiting density is also Gaussian and can be found by the method of undetermined coefficients by substituting

u(x)

= cexp

(.t

p,;x;x;)

,,,=1

into the equation

02 d a.;--u-da(d ! .L: L:u L:b,;x; ) . 1 8x,8x; . 1 ax, . 1 '•1=

a=

=

o. o

:1=

Example 11.9.3. Consider the second-order system

tFx

dx

m dt2 +/3 dt +F(x) =

ue

(11.9.12)

with constant coefficients m, {3, and u. Equation (11.9.12) describes the dynamics of many mechanical and electrical systems in the presence of ''white noise." In the mechanical interpretation, m would be the mass of a body whose position is x, f3 is a friction coefficient, and F = 8cf>f8x is a conservative force (with a corresponding potential function cf>) acting on the body. Introducing the velocity v = dx / dt as a new variable, equation (11.9.12) is equivalent to the system

dx dt

=v

dv

and m dt

= -{3v- F(x) + ue.

(11.9.13)


377

The Fokker-Pla.nck equation corresponding to (11.9.13) is

8u &t

o-2

= 2m2

o

o2u

1

o

1Jv2 - ox (vu) + m ov {[,Bv + F(x)]u}.

(11.9.14)

Unfortunately, the asymptotic stability of the solutions of (11.9.14) cannot be studied by Theorem 11.9.1 as the quadratic form associated with the second-order term is 0 ·A~+ (o-2 /m2 ) ·A~, which is clearly not positive definite. Using some sophisticated techniques, it is possible to prove that the solutions to some parabolic equations with associated semidefinite quadratic forms are asymptotically stable. However, in this example we wish only to derive the steady-state solution to (11.9.14) a.nd to bypass the question of asymptotic stability. In a steady state, (8u/&t) = 0, so (11.9.14) becomes

o

o

o- 2 02u 1 2m2 1Jv2 -ox (vu) + m 8v {[,Bv + F(x)]u}

= 0,

which may be written in the alternate form

.!!_ - .!!_) ( p_ m ov ox

[vu

+~

au] + .!!_ [..!..F(x)u + ~ au] = 0. 8v m 2mf3 ox

2mf3 8v

Set u(x, v)

= X(x)V(v), so that the last equation becomes

13 ( m .!!_ 8v

!...) [x ox

(vv

+~

dV)] + [..!..F(x)X + ~ dX] dV = o m 2mf3 dx dv '

2m/3 dv

which will certainly be satisfied if X a.nd V satisfy dX dx

+ u 2 F(x)X = 0

(11.9.15)

dV dv

+ 2mf3 vV = 0

(11.9.16)

a.nd

2/3

o-2

'

respectively. Integrating equations (11.9.15) a.nd (11.9.16) a.nd combining the results gives u(x, v) =

cexp {-(2{3/u2) [!mv2 + 1/>(x)]}.

(11.9.17)

The constant c in (11.9.17) is determined from the normalization condition

I:I:

u(x, v) dx dv

= 1.

The velocity integration is easily carried out a.nd we have c

= c1 .j{3m/7ro-2 ,

378


where

-1 = c1

1

00

-oo

exp[(-2P/u2 )(x)] dx.

(11.9.18)

Thus (11.9.17) becomes

u(x,v)

= c1 v'Pm/1ru2 exp { -(2P/u2 ) [!mv2 + (x)]}.

(11.9.19)

The interesting feature of (11.9.19) is that the right-hand side may be written as the product of two functions, one dependent on v and the other on x. This can be interpreted to mean that in the steady state the positions and velocities are independent. Furthermore, observe that for every
0 as in Proposition 11.10.1, that is,

r = M + 1 = [p/(1 -a)] + 1. For an arbitrary

I

E

D([O, r]) and to > 0, we have

Pt0 l(x) 2::: e-toTl(to)l(x)

= e-to

to

1

To(to- r)PTo(r)l(x)dr

0

= e-to 1to [1[o,oo)(X- to+ r)

·1

00

K(x-to+r,y)

:~:-to+r

·1[o,oo)(Y- r)l(y- r)dy]dr. ~

In particular, for 0

x

~to,

Ptol(x);:::: e-to rto

[10Q K(x- to+

lto-:J:

Now set z obtain

=y-

T

and

8

=x -

T,

y)l(y- r) dy] dr.

r

to + T and remember that I

E

D([O, r]) to

Pt 0 l(x) 2::: e-to 1:~: [for K(8, z + 8 +to- x)l(z) dz] d8 2::: hr(x) 1r l(z) dz

where

hr(x)

= hr(x)

for 0

~ x ~to,

= e-to O:Sz$r inf 1:~: K(8, z + 8 +to- x) d8. 0

386


It is therefore clear that hr ~ 0, and it is easy to find a sufficient condition for hr to be nontrivial. For example, if K(8, u) = ¢(8/u)/u, as in the Chandrasekhar-Miinch equation, then hr(x)

= e-to inf

r 1/J ( z +

~ 1o

If we set q = s/ (z + 8

+ to -

hr(x)

8

8

+to - x

)

d8

z + 8 +to - x

•

x) in this expression, then

= e-to inf ~

L

z/(~+to)

1/J( ) _q_ dq

1-q

0

r/(r+to) ~ e-to 1o .,P(q) dq.

Since 1/J(q) is a density, we have z/(r+to)

lim

to-+oo

uniformly for x

1

1/J(q) dq = 1

0

e [to -1, t 0 ]. Thus, for some sufficiently large to, we obtain

r/(r+to) hr(x) ~ e-to 1o .,P(q) dq

>0

for x E [to- 1, to],

showing that hr is a nontrivial function. Therefore all the assumptions of Proposition 11.10.1 are satisfied and the semigroup {Pth~o generated by the Chandrasekhar-Miinch equation is asymptotically stable. 0

11.11 Sweeping for Solutions of the Fokker-Planck Equation As we have seen in Section 11.9, semigroups generated by the FokkerPlanck equation may, for some value of the coefficients, be asymptotically stable. The example provided was the Langevin equation. On the other hand, the heat equation, perhaps the simplest Fokker-Planck equation, generates a sweeping semigroup. In this and the next section we develop a technique to distinguish between these two possibilities. We return to equation (11.7.1) with the initial condition (11.7.2) and consider the stochastic semigroup {Pth~o given by equations (11.8.2) generated by these conditions. We say that {Pt }t~ 0 is sweeping if it is sweeping with respect to the family .Ac of all compact subsets of R!'-. Thus, {Pth~o is sweeping if lim t-+oo

lim f u(t,x)dx=O, 1fA Ptf(x)dx= t-+oo 1A

for IE D,A E .Ac. (11.11.1)

11.11. Sweeping for Solutions of the Fokker-Planck Equation

387

'In this section, we understand a Bielecki function to be any function V: Jld - R that satisfies the following three conditions: (1) V(x) > 0 for all x; (2) V has continuous derivatives

8V fPV 8xi' 8xi8x;'

i,j

= 1, ... ,d;

and

(3)

V(x)~pe6lzl, ~8~x(x,.)l~pe6lzl, u

and

lfPV(x)l< 6lzl 8xix; - pe ' (11.11.2)

for some constants p and 6. From condition (1) and the continuity of V it follows that inf V(x)

zEA

>0

for A E

.Ac,

and consequently our new definition of a Bielecki function is completely consistent with the general definition given in Section 5.9. With these preliminaries we are in a position to state an analog of Theorem 11.9.1, which gives a sufficient sweeping condition for semigroups generated by the Fokker-Planck equation. Theorem 11.11.1. Assume that the coefficients ai; and bi of equation (11.7.1) are regular for the Cauchy problem, and that there is a Bielecki function V: Jld - R satisfying the inequality

~

d

L

i,j=l

fPV

ai;(x)~ + x, x,

d

8V

L bi(x)~x, ~ -aV(x),

(11.11.3)

i=l

with a constant a> 0. Then the semigroup {.Pth~o generated by (11.7.1)(11.7.2) is sweeping. Proof. The proof proceeds exactly as the proof of Theorem 11.9.1, but is much shorter. First we pick a continuous density f with compact support and consider the mathematical expectation (11.9.3). Using inequality (11.11.3), we obtain

dE(~ I u) ~ -aE(V I u),

388


and, consequently,

E(V I Ptf)

= E(V I u) ~ e-at E(V I f).

Since e-at< 1 fort> 0, Proposition 7.11.1 completes the proof.

Example 11.11.1. Consider the stochastic equation dx dt = bx +

ue

•

(11.11.4)

e

where b and 0' are positive constants and is a white noise perturbation. Equation (11.11.4) differs from the Langevin equation because the coefficient of x is positive. The Fokker-Planck equation corresponding to (11.11.4) is

8u

= u 2 tPu _ b8(xu).

8t 2 8x2 Now the inequality (11.11.3) becomes u2

2

8x

lPV 8V 8x2 +bx 8x $ -aV.

(11.11.5)

(11.11.6)

Pick a Bielecki function of the form V(x) = e-n:~ and substitute it into (11.11.6) to obtain 2e(eu2 - b)x 2 - eu 2 $ -a. This inequality is satisfied for arbitrary positive e $ b/ u 2 and a $ eu2 • This demonstrates that forb> 0 the semigroup {~}t~o generated by equation (11.11.5) is sweeping.

11.12 Foguel Alternative for the Fokker-Planck Equation Stochastic semigroups generated by the Fokker-Planck equation are especially easy to study using the Foguel alternative introduced in Section 7.12. This is due to the fact that these semigroups are given by the integral formula {11.8.2). We have the following.

Theorem 11.12.1. Assume that the coefficients D.i; and bi of equation (11.7.1) are regular for the Cauchy problem. FUrther assume that all stationary nonnegative solutions of equation (11.7.1) are of the form cu.(x) where u. (x) > 0 a. e. and c is a nonnegative constant. Then the semigroup {Pth~o generated by equations (11.7.1)-(11.7.2) is either asymptotically stable or sweeping. Asymptotic stability occurs when I= and sweeping when I= oo.

f

}Rd

u.(x)dx 0. We already know that r is positive and that {Pth>o is stochastic. Furthermore, for each IE L 1 (R") we have

f

}Rd

l(y)dy

= }Rd f Ptf(x)dx = f f f(t,x,y)l(y)dxdy, }Rtl }Rd

and consequently

ltl [ld r(t,x,y)dx Since

-1]

l(y)dy

=

0.

IE L 1 (R") is arbitrary, this implies

f

}Rtl

r(t,x,y)dx

=1

for t > 0, y E Rd.

Thus, r is a stochastic kernel and condition (a) of Theorem 7.12.1 is satisfied. In verifying condition (b), note that according to the definition of the semigroup {Pth>o the function

u(t,x)

= Ptu.(x)

is a solution of equations (11.7.1) and (11.7.2) with I= u•. Since u. is a stationary solution and the Cauchy problem is uniquely solvable, we have fort~

u.(x) = Ptu.(x)

0.

Thus, condition (b) of Theorem 7.12.1 is satisfied for 1. = u •. To verify (c) simply observe that the positivity of r implies that Ptf (x) > 0 for every t > 0 and I E D. Thus, supp Ptf = R" and Pt is expanding for every t > 0. This completes the proof. • It is rather easy to illustrate the general theory developed above with a simple example in one dimension. Consider the stochastic differential equation dx (11.12.2) dt = b(x) + u(x)e

e

where 0' 1 b, and X are scalar functions, and is a one-dimensional white noise. The corresponding Fokker-Planck equation is of the form lJu

1 02(u 2 (x)u]

8t=2

8x2

-

8(b(x)u] 8x ·

(11.12.3)

390


Assume that a(x) and that

= u 2(x) and b(x) are regular for the Cauchy problem, ~

xb(x)

0

for

lxl 2:: r,

(11.12.4)

where r is a positive constant. This last condition simply means that the interval [-r, r] is attracting (or at least not repelling) for trajectories of the unperturbed equation x = b(x). To find a stationary solution of (11.12.3) we must solve the differential equation

! «P[u 2 (x)u] _ d[b(x)u] = 0 2

d:z:2

or

dz d:z:

where z

d:z:

'

2b(x) = u 2 (x)z + Ct

= u 2u and c1 is a constant. A straightforward calculation gives z(x) =

eB(z) {

C2

where c2 is a second constant and

+ Ct 1:~: e-B(II) dy}'

r

2b(y) B(x) = Jo u2(y) dy. The solution z(x) will be positive if and only if C2

+ Ct 1:~: e-B(II) dy > 0

for - oo < x < oo.

(11.12.5)

From condition (11.12.4) it follows that the integral

1:~: e-B(II)dy converges to +oo if x - +oo and to -oo if x - -oo. This shows that for Ct =! 0 inequality (11.12.5) cannot be satisfied. Thus, the unique (up to a multiplicative constant) positive stationary solution of equation (11.12.3) is given by

u.(x)

= _c_eB(:~:) u2(x)

with c > 0. Applying Theorem 11.12.1 to equation (11.12.3) we obtain the following. Corollary 11.12.1. Assume that the coefficient8 a= u 2 and b of equation (11.12.3} are regular for the Cauchy problem and that inequality (11.12.4) is satisfied. If

I

=

00

1

-oo

1

--eB(:~:) d:z:

u2(x)

< oo

'

Exercises

391

then the semigroup {Pt}t~o generated by equation (11.12.3) is asymptotically stable. If I= oo, then {Pt}t~o is sweeping.

Example 11.12.1. Consider the differential equation (11.12.2) with q and ~ b(x)=--1+x2' where ~ ~ 0 is a. constant. Then B(x)

and u

•

2~y

=-

1

(x)

= ce-~ln(l+,z2) =

!Z

0

=1

-1 +y2 dy = -~ ln(1 + x 2 ), c (1 +x 2 )~ •

i,

The function u. is integrable on R only for ~ > and thus the semigroup {Pth~o is asymptotically stable for ~ > and sweeping for 0 :5 ~ :5 This example shows that even though the origin x = 0 is attracting in the unperturbed system, asymptotic stability may vanish in a. perturbed system whenever the coefficient of the attracting term is not sufficiently strong.

i

i·

Remark 11.12.1. In Corollary 11.12.1, the conditions (11.12.4) may be replaced by the less restrictive assumption 00 {

Jo

e-B(,z)

dx

=

{O

Loo

e-B(,z)

dx

= 00.

(11.12.6)

Exercises 11.1. Let {w(t)}t~o be a. one-dimensional Wiener process defined on a. complete probabilistic measure space. Show that for every to ~ 0, r > 0, and M > 0 the probability of the event

{I

I

w(to + ~- w(to) :5 M for 0 < h :5

is equal to zero. Using this, show that for every fixed to of the event {w'(to) exists}

r} ~

0 the probability

is equal to zero.

11.2. Generalize the previous result and show that the probability of the event {w' (t) exists a.t least for one t ~ 0} is equal to zero.

392


11.3. Show that every regular sequence {sn} of for the integral

Ito approximation sums

loT w(t) dw(t)

tw

2 (T)converges to strongly in L (0)].

!T not only stochastically but also in the mean [i.e.,

11.4. Consider the stochastic differential equation t

> o,e E R,

e

where c and u > 0 are constant and is normalized white noise. Show that the corresponding stochastic semigroup { Pt h>o is asymptotically stable (Mackey, Longtin, and Lasota, 1990). -

11.5. Show that the stochastic semigroup {Pth~o defined in Exercise 7.8 is asymptotically stable for an arbitrary stochastic kernel K (Jama, 1986). 11.6. A stochastic semigroup {Pth~o is called weakly (strongly) mixing if, for every /t, h E D the difference Pd1 -Pd2 converges weakly (strongly) to zero in L 1 • Show that the stochastic semigroup {Tth~o given by equation (7.9.9), corresponding to the heat equation, is strongly mixing. 11.7. Consider equation (11.12.3) with b(x) = xf(l + x 2 ) and u = 1. Prove that the stochastic semigroup { Pt h>o corresponding to this equation satisfies -

j_:oo (arctan x)Pd(x) dx =constant and is not weakly mixing (Brzezniak and Szafirski, 1991).

11.8. Consider the semigroup {Pth>o defined in the previous exercise. Show that the limit lim H(Pd1 I Pth),

t-+oo

depends on the choice of tropy, cf. Chapter 9.

It

and

h, where

!l,/2ED H denotes the conditional en-

12 Markov and Foias Operators

Throughout this book we have studied the asymptotic behavior of densities. However, in some cases the statistical properties of dynamical systems are better described if we use a more general notion than a density, namely, a measure. In fact, the sequences (or flows) of measures generated by dynamical systems simultaneously generalize the notion of trajectories and the sequences (or flows) of densities. They are of particular value in studying fractals. The study of the evolution of measures related to dynamical systems is difficult. It is more convenient to study them by use of functionals on the space C0 (X) of continuous functions with bounded support. Thus, we start in Section 12.1 by examining the relationship between measures and linear functionals given by the Riesz representation theorem, and then look at weak and strong convergence notions for measures in Section 12.2. After defining the notions of Markov and Foias operators on measures (Section 12.3 and 12.4, respectively), we study the behavior of dynamical systems with stochastic perturbations. Finally, we apply these results to the theory of fractals in Section 12.8.

12.1 The Riesz Representation Theorem Let X c Jl!1 be a. nonempty closed set which, in general, is unbounded. We denote by 8 = 8(X) the a-algebra. of Borel subsets of X. A measure p: 8 -+ R+ will be called locally finite if it is finite on every bounded

394

12. Markov and Foias Operators

measurable subset of X, that is, ~t(A)

< oo

for A E 8, A bounded.

Of course, every locally finite measure I' is 0'-finite, since X may be written as a countable sum of bounded sets: 00

X=

Uxn,

(12.1.1)

n=1

where

Xn = {x EX: lxl $ n}. The space of all locally finite measures on X will be denoted by M = M(X). The subspace of M which contains only finite or probabilistic measures will be denoted by Mftn and M 1 , respectively. We say that a measure I' is supported on a set A if JJ(X \A) = 0. Observe that the set A on which I' is supported is in general not unique, since if B is measurable and con-

tains A, then X\ A ::J X\ B and consequently JJ(X \B) = 0. The elements of M 1 are often called distributions. In general, the smallest measurable set on which a measure 1£ is supported does not exist. However, this difficulty may be partially avoided. Denote by Br (x) a ball in X with center located at x E X and radius r, that is,

Br(x) = {y EX: IY- xl < r}. Let 1£ E M. We define the support of the measure 1£ by setting supp Jl.

= {x EX: JJ(Be(x)) > 0 for every e > 0}.

It is easy to verify that supp 1£ is a closed set. Observe that it also has the property that if A is a closed set and Jl. is supported on A, then A ::J supp IJ. To see this, assume that x ¢ A. Since X \ A is an open set, there exists a ball Be (x) contained in X \ A. Thus,

JJ(Be(x)) $

~t(X \A)

= 0,

and x ¢ supp IJ· This shows that x ¢ A implies x ¢ supp 1£, and consequently A ::J supp IJ· From the above arguments it follows that the support of a measure Jl. can be equivalently defined as the smallest closed set on which Jl. is supported. (The adjective closed is important here.) It should also be noted that the definition of the support of a measure J.1. does not coincide exactly with the definition of the support of an element I E £ 1 • The main difference is that supp I' is defined precisely for every single point, but supp I is not (see Remarks 3.12 and 3.13).

12.1. The Riesz Representation Theorem

395

We will often discuss measures that are supported on finite or countable sets. Perhaps the simplest of these is the o-Dirac measure defined by

xo E A, 0zo (A) -_ { 01 if if x 0 ¢ A.

(12.1.2)

Another important el8BS of measures are those absolutely continuous with respect to the standard Borel measure on X. According to Definition 3.1.4 every measure that is absolutely continuous with respect to the Borel measure is given by

J£(A)

=

L

l(x)dx

for A E 8,

(12.1.3)

where IE L1 (X) and I;?: 0. Let Co = Co(X) be the space of all continuous functions h: X -+ R with compact support. Our goal is to study the relationship between locally finite measures on X and linear functionals on Co. We start with the following.

Definition 12.1.1. A mapping 0 was arbitrary this completes the proof.

•

Remark 12.2.1. It is easy to observe that in general inequality in {12.2.6) cannot be replaced by the equality. In fact, let X= R, P.n = 6t/m p. = 6o and A= {0, 1). In this case the sequence {P.n} converges weakly top., but P.n(A) = 1, p.(A) = 0, and the inequality {12.2.6) is strong. 0

12.2. Weak and Strong Convergence of Measures

401

Now we are going to show how the lliesz representation theorem may be used to show that a given sequence of measures is convergent.

Theorem 12.2.2. Let a sequence of measures {JLn}, l'n EM, be given. If for each h E Co the sequence {(h, ~tn}} is convefYent, then there is a unique measure JL such that {JLn} convefYes weakly to I'· Proof. Define

cp(h) = lim (h, JLn} n-+oo

for hE Co.

Evidently cp is a linear positive functional. Thus, according to Theorem 12.1.1 there is a unique measure J.L such that

cp(h) = (h, JL}

for hE Co.

From this and the definition of cp, it follows that the sequence {JLn} converges to J.L weakly. • Remark 12.2.2. In the special case when the J.Ln are probabilistic measures the use of Theorem 12.2.2 can be greatly simplified. Namely, it is not necessary to verify the convergence of sequences {{h, l'nH for all h E C0 • Let C. c Co be a dense subset of Co which means that for every h E Co and e > 0 there is g E C. such that sup lg(x)- h(x)l :5 e. :z:EX

c. implies the conver-

Then the convergence of sequences { (g, l'n}} for g E gence of { (h, l'n}} for h E Co. In fact, the inequality

I{h, l'n - I'm} I :5 I(g, l'n - J.Lm} I + 2 sup lg -

hi

and the Cauchy condition for all sequences { (g, l'n}} imply the Cauchy condition for {{h, l'nH· 0 We close this section by introducing the concept of the strong convergence of measures. First we need to define the distance between two measures 1'11 1'2 E Miin· Let (Xl! ... , Xn) be a measurable partition of X, that is n

X=

UX;, X;nX;

=

0

for i

=f: j, X;

E

B.

i=l

We set

n

lll't -1'211 =sup

L l~tt(X;) -JL2(X,)I, i=l

(12.2.7)

402


where the supremum is taken over all possible measurable partitions of X (with arbitrary n). The value IIJ£1 -J£211 is the desired distance. In the special case where 1£ = 1'1 is arbitrary and 1'2 0 we have

=

n

111'11 =sup

L J£(Xi) = J£(X).

(12.2.8)

i=1

This value will be called the norm of the measure I'· It is the distance from I' to zero. The norm of a probabilistic measure is equal to 1.

Definition 12.2.2. We say that a sequence {J£n}, l'n E Man is strongly convergent to a measure I' E Man if lim

n-+oo

lll'n - I'll = 0.

(12.2.9)

Before passing to examples of strong convergence, we will calculate the norm IIJ£1 -J£211 in the case when the measures 1'1 and #£2 are absolutely continuous with Radon-Nikodym derivatives /1 and /2, respectively. We have

#'1(Xi) -J£2(Xi)

= f (/l(x) -f2(x))dx. lx,

Substituting this into (12.2.7) we obtain immediately

IIJ£1 -J£211

=sup~

IL,

(/l(x) -f2(x))dxl

:::;

sup~[, 1/l(x)- h(x)l dx

=

L

1/l(x) -f2(x)l dx.

(12.2.10)

Now let

x1

= {x: /l(x) ~ h(x)},

x2

= {x:/l(x) < h(x)}.

Then (X1, X2) is a partition of X and, consequently,

IIJ£1 -J£211 ~ IJ£1(X1) -J£2(X1)1 + IJ£1(X2) -J£2(X2)1

r (ft(x)- h(x)) dx + lx2r (h(x)- ft(x)) dx

=

lx1

=

L

1/l(x) -f2(x)l dx.

This and (12.2.10) implies

IIJ£1 -J£211

=[

1/l(x)- h(x)l dx.

(12.2.11)

12.2. Weak and Strong Convergence of Measures

403

From this equality a necessary and sufficient condition for the strong convergence of absolutely continuous measures follows immediately. Namely, if the IJ.n are absolutely continuous with densities /n, and p. is absolutely continuous with density/, then {P.n} converges strongly top. if and only if 11/n -/II - 0. Example 12.2.4. Assume X = R. Let xo E X. Denote by 11-o = 6:~:0 the 6-Dirac measure supported at xo. Further, let {P.n} be a sequence of absolutely continuous measures with densities fn· Write

Xt = {xo},

X2 =X\ {xo},

where, as usual, {xo} denotes the set that contains only the one point x = xo. We have

11-o(Xt) = 1, P.n(Xt) = {

11-o(X2)

=0

fn(x) dx = 0

J{zo}

P.n(X2)

= {

fn(x) dx

= {

Jx\{zo}

Jx

fn(x) dx

= 1.

Thus, since (X11 X 2) is a partition of X IIP.n -11-oll :?: IP.n(Xt) -~J.o(Xt)l

= IO- 11 + 11 - Ol =

+ IP.n(X2) -p.o(X2)1 2.

This shows that a sequence of absolutely continuous measures cannot converge strongly to a 6-Dirac measure. 0

=

Example 12.2.5. Assume X R and consider a probabilistic measure p. supported on the set of nonnegative integers {0, 1, ... }. The measure p.

may be written in the form 00

L:ck

= 1,

(12.2.12)

k=O

where 6k denotes the 6-Dirac measure supported at x {P.n} be a sequence of similar measures, so 00

l'n

= L:ck,.6k, k=O

= k.

Further, let

00

L:ck,. = 1,

(12.2.13)

k=O

Assume that for each fixed k (k = 0, 1, ... ) the sequence {ck,.} converges to Ck as n - oo. We are going to show that under this condition the sequence of measures {P.n} converges strongly to p.. Thus we must evaluate the distance IIPn -I'll·

404


From (12.2.12) and (12.2.13) it follows that 00

L lckn - c~cl6~c(Xi)

IJ£n(Xi)- J£(Xi)l :S;

k=O

for each measurable subset Xi of X. Consequently, m

lll'n- ~£11 =sup

L IPn(Xi)- J£(Xi)l i=l

:S; sup :S; sup

m

oo

i=l

k=O

L L lc~cn - c~cl6~c(Xi) oo

m

k=O

i=l

L lckn- c~cl L6~c(X,),

where the supremum is taken over all partitions {Xi} of X. Since for every partition m

L6~c(Xi) i=l

=1

k=0,1, ... ,

this gives 00

L lckn- c~cl·

lll'n- I'll :S;

{12.2.14)

k=O

Now fix an e > 0 and choose an integer N such that 00

L

Cfc
..1J.1.1 + >..2J.1.2) {b) PJL(X)

= >..1PJ.1.1 + >..2PJ.1.2 for >..t. >..2 ~ 0, J.l.t. J.1.2 E Man, and

= JL(X} for JL E Man·

406


Assumption (a) will often be called the linearity condition; however, it is restricted to nonnegative ..\1 only. Assumption (b) may be written in the form IIPJJ.II = IIJJ.II (see 12.2.8} and will be called the preservation of the norm. In the following we will quite often omit the qualifying phrase "on measures" if this does not lead to a misunderstanding. On the other hand, if it is necessary we will add the words "on densities" for Markov operators described by Definition 3.1.1. Our first goal is to show how these two definitions of Markov operators are related. Thus, suppose that the Borel measure of the set X is positive (finite or not) and consider one operator P: Mfin--+ Mfin· Assume that it satisfies conditions (a) and (b) of Definition 12.3.1 and that, moreover, for every absolutely continuous J.& the measure P J.' is also absolutely continuous. Take an arbitrary f E L 1 , f ~ 0, and define

J.'J(A)

=

i

f(x) dx

for A E B.

(12.3.1}

Since P J.&1 is absolutely continuous it can be written in the form

PJ.&J(A}

=

L

for A E 8,

g(x) dx

(12.3.2}

where g is a Radon-Nikodym derivative with respect to the Borel measure PJ.&J on X. In this way to every f E L 1 , f ~ 0, we adjoin a unique g E L 1 , g ~ 0, for which conditions (12.3.1} and (12.3.2} are satisfied. The uniqueness follows immediately from Proposition 2.2.1 or from the RadonNikodym theorem. Thus, f is mapped to g. Denote this mapping by P, so g = Pf. We may illustrate this situation by the diagram p --t

IF j

Ma !RN

(12.3.3}

where Ma denotes the family of absolutely continuous measures, L~ is the subspace of L 1 which contains nonnegative functions, IF denotes the integral formula (12.3.1}, and RN stands for the Radon-Nikodym derivative. The operator P is defined as a "shortcut" between L~ and L~ or, more precisely, in such a way that the diagram (12.3.3} commutes. Thus, Pis the unique operator on densities that corresponds to the operator P on measures. Substituting (12.3.1} and (12.3.2} with g = Pt we obtain for A E 8, f E L~.

(12.3.4)

12.3. Markov Operators

407

This is the shortest analytical description of P. To understand this formula correctly we must remember that on the left-hand side of the operator P is applied to the meBBure given by the integral in braces and then the new meBBure is applied to the set A. From condition (a) and formula {12.3.4) it follows immediately that P satisfies the linearity condition for nonnegative functions, that is,

Further, using {12.3.4) we obtain

and analogously

II/II = [

f(x) dx

= JLJ(X).

From condition (b) this implies

IIF/11 =II/II

for/ E L~.

{12.3.6)

Now we may extend the definition of P to the whole space L 1 that contains all integrable {not necessarily nonnegative) functions by setting

{12.3.7) Using this extension and condition {12.3.5) one can verify that Pis a linear operator. Further, from our construction, and in particular from {12.3.4), it follows that P! ~ 0 for f ~ 0. Finally, {12.3.6) shows that P preserves the norm of nonnegative functions. We may summarize this discussion with the following. Proposition 12.3.1. Let P: Man - Man be a Markov operator on measures such that for every absolutely continuous measure p. the measure P ,_, is also absolutely continuous. Then the corresponding operator P defined by fonnulas {12.3.4) and {12.3. 7) is a Markov operator on densities and the diagram {12.3.3) commutes.

The commutative property of diagram {12.3.3) hOB an important consequence. Namely, if P is the operator on densities corresponding to an operator P on meBBures, then (P)n corresponds to pn. To prove this consider the following row of n blocked diagrams {12.3.8).


408

p

Ma

------+-

p

Ma

------+-

p

Ma · · · Ma

+

Ma

~F

~F £1

------+-

------+p

L~

(12.3.8)

~ L~ ... L~ ~ L~

p

p

Since each of the blocks commutes, the total diagram (12.3.8) also commutes. This shows that (ft)n corresponds to pn. Remark 12.3.1. There is an evident asymmetry in our approach to the definition of Markov operators. In Section 3.1 we defined a Markov operator on the whole space £ 1 which contains positive and negative functions f: X -+ R. Now we have defined a Markov operator on Mfln which contains only nonnegative functions p.: B -+ R. This asymmetry can be avoided. Namely, we extend the definition of P on the set of signed measures, that is, all possible differences p.1 - J.1.2, where P.1 1 P.2 E Mfln 1 by setting P(p.1 - P.2)

= Pp.1 -

Pp.2.

Such an extension is unnecessary for our purposes and leads to some difficulties in calculating integrals, and in the use of the lliesz representation theorem which is more complicated for signed measures on unbounded regions. 0 Example 12.3.1. Let X= R+. For a given p. E Man define

Pp.(A)

= p.([O, 1))60 (A) + p.([1, oo) n A)

(12.3.9)

where, as usual, 6o denotes the 6-Dira.c measure supported at x = 0. Evidently, P satisfies the linearity condition (a) of Definition 12.3.1. Moreover,

Pp.(R+)

= p.([O, 1))60 (R+) + p.([1,oo) n R+) = p.([O, 1)) + p.([1, oo)) = p.(R+),

which shows that condition (b) is also satisfied. Thus, (12.3.9) defines a Markov operator on measures. The operator P is relatively simple, but it has an interesting property. Namely, if a measure p. E M 1 is supported on [0, 1), then Pp. is a 6-Dira.c measure. H p. is supported on [1,oo), then Pp. = p.. In other words, P shrinks all of the measures on [0, 1) down to the point x = 0 and leaves the remaining portion of the measure untouched. In particular, P does not map absolutely continuous measures into absolutely continuous ones, and the corresponding Markov operator P on densities cannot be defined. 0 Example 12.3.2. Let X = R and let t > 0 be a fixed number. For every

12.3. Markov Operators

409

p. E Man define the measure PtJ.I. by Ptp.(A) =

L{t k

2

exp (- (x ;ty)

)

JJ(dy)} dx.

(12.3.10)

Again the linearity of Pt is obvious, and to verify that Pt is a Markov operator it is sufficient to check the preservation of the norm. To do this, substitute A = R into (12.3.10) and change the order of integration to obtain

Inside the braces we have the integral of the Gaussian density, and consequently

Ptp.(R)

=

L

1p.(dy) = p.(R),

so Pt is a Markov operator. To understand the meaning of the family of operators {Pt}, first observe that for every p. E Man the measure PtJ.I. is given by the integral (12.3.10) and has the Radon-Nikodym derivative (12.3.11) If p. is absolutely continuous with density f, we may replace p.(dy) by f(y) dy and in this way obtain an explicit formula for the operator Pt on

densities corresponding to Pt. Namely,

iH(x) = 9t(x) =

~

k

2

exp {- (x ;ty)

}

f(y) dy.

The function u(t,x) = 9t(x) is the familiar solution (7.4.11), (7.4.12) of the heat equation (7.4.13)

1 8 2u 8t = 2 8x 2

8u


u(O,x)

for t > 0, x E R,

= f(x).

It is interesting that u(t, x) = 9t(x) satisfies the heat equation even in the case when JJ has no density. This can be verified simply by differentiation of the integral formula (12.3.11). (Such a procedure is always possible since JJ is a finite measure and the integrand _1_e-(z-y)2/2t

v'21it and its derivatives are bounded 0 00 functions for t

~

e > 0.)

410


Further, in the case of arbitrary I' the initial condition is also satisfied. Namely, the measures Ptl' converge weakly to I' as t - 0. To prove this choose an arbitrary hE C0 (R). Since 9t is the Radon-Nikodym derivative of Ptl' we have {h, PtJ£}

= =

L

h(x)Ptp(dx)

=

L {L k h(x)

L

h(x)gt(x) dx (x ;ty)2) p(dy)} dx

exp (

or by changing the order of integration {h,l'tp} = where

v(t,y)

=

L

(12.3.12)

v(t,y)p(dy)

kL

2

(x ;ty)

exp (

)

h(x)dx.

Observe that v(t, y) is the solution of the heat equation corresponding to the initial function h(y). Since his continuous and bounded, this is a classical solution and we have lim v(t, y)

t-+0

= h(y)

for y E R.

Evidently lv(t,y)l:::;

m~l

L

2 exp( _(x;ty) )dx=maxlhl.

Thus by the Lebesgue dominated convergence theorem (see Remark 2.2.4) lim

f

t-+O}R

v(t, y)p(dy) =

f

jR

h(y)IJ(dy).

From this and (12.3.12} it follows that Ptl' converges weakly to I'· Thus, we can say that the family of measures {Ptl'} describes the transport of the initial measure by the heat equation. From a physical point of view, if u(t, x) = 9t(x) is the temperature at timet at the point x, then

ptp(A)

=

L

9t(x}dx

is equal (up to multiplicative constraint) to the amount of heat carried by a segment A at time t. In particular, substituting I' = 6z0 (the 6-Dirac measure supported at x = x0 ) in (12.3.11) we obtain

u(t,x) = 9t(X) = ke-(z-zo)2/2t.

12.4. Foias Operators

411

This equation is identical to the fundamental solution r(t, x, xo) of the heat equation (see Section 11.7) and it gives a simple physical interpretation of this solution. Namely, r(t,x,xo) is the temperature at time t and point x corresponding to the situation in which the initial amount of heat was concentrated at a single point xo. 0

12.4 Foias Operators At the end of the previous section we have given two examples of Markov operators constructed in two different methods. The goal of the present section is to develop these methods in detail. Let X c R_d be a nonempty closed set. We start from the following

Definition 12.4.1. Let 8: X-+ X be a Borel measurable transformation. Then the operator P: Mfin -+ Mtin defined by for A E B(X)

(12.4.1)

is called the Frobenius-Perron operator on measures corresponding to S. Evidently P defined by (12.4.1) is a Markov operator. Now observe how P works on measures supported on a single point. Let x 0 E X be fixed. Then 1 P 6 (A) = 6 (s-l(A)) = { o if xo ¢ 1 (A) Zo zo 1 if Xo E (A)

ss-

or p 6 (A)= zo

Thus, P6z0

{0

if S(xo) ¢A 1 if S(xo) E A.

= 6s(zo). By induction we obtain

= 6sn(zo)·

pn6zo

This shows that the iterates of the Markov operator (12.4.1) can produce a trajectory of a transformation S. To obtain this trajectory it is sufficient to start from a 6-Dirac measure. We next show that P can also transform densities. Consider the special case when p. is absolutely continuous with density f and S is a nonsingular transformation. Then

p.(A)

=

L

f(x) dx,

and the right-hand side of (12.4.1) may be written in the form

p.(S- 1 (A)) =

f ls-l(A)

f(x)dx =

f }A

Ff(x)dx,

412


where Pis the Frobenius-Perron operator on densities corresponding to S. Now equality {12.4.1) may be explicitly written in the form

This is a special case of formula {12.3.4) and it shows that the FrobeniusPerron operator P on densities corresponds, in the sense of diagram {12.3.3), to the Frobenius-Perron operator P on measures. This correspondence was obtained under the additional assumption that S is nonsingular. For an arbitrary Borel measurable transformation S, the operator P given by {12.4.1) may transform absolutely continuous measures into measures without density.

Example 12.4.1. Let X= R+ and

S(x)

= {~..,

0$ x < 1 X~

1.

Then

s- 1 (A) = s- 1 (A n [o, 1)) u s- 1 (A u [1, oo)), where s- 1 {AU [1,oo)) =An [1,oo) and s-l(A n [O, 1))

= { [o, 1) 0

if 0 E A ifO ¢A.

From the last formula it follows that

tt(S- 1 (A n (0, 1))

= 1A{O)p{[O, 1)).

Consequently, the Frobenius-Perron operator for S is given by

Ptt(A)

= tt(S- 1 (A)) = 1A{0)p{[O, 1)) + tt(A n [1, oo))

which is identical with {12.3.9). 0 Now we are going to study a more general, and complicated, situation when the dynamical system includes random perturbations. Thus, we consider the system for n where T is a given transformation and the vectors. We make the following assumptions:

= 0, 1, ... ,

en

{12.4.2)

are independent random

(i) T is defined on the subset X x W of Jld x Rk with values in X. The set X c Jld is closed and W c Rk is Borel measurable. For every fixed y E W the function T(x, y) is continuous in x and for every fixed x EX it is measurable in y.


413

(ii) The random vectors eo,et •... , have values in Wand have the same distribution, that is, the measure v(B)

= prob(en E B)

forB E B(W)

is the same for all n.

(iii) The initial random vector Xo has values in X and the vectors Xo, eo, et. ... ' are independent. A dynamical system of the form (12.4.2) satisfying conditions (i)-(iii) will be called a regular stochastic dynamical system. We emphasize that in studying (12.4.2) it is assumed that the transformation T and the random vectors en are given. The initial vector Xo can be arbitrary, but must be such that condition (iii) is satisfied. Observe that in particular if eo, e1 are independent and Xo E X is constant (not random) then the vectors Xo, ell el •... ' are also independent. This can be easily verified using the definition of the independence of random vectors and the fact that the value of prob(xo E A) is either 0 or 1 for Xo constant. According to (12.4.2) the random vector Xn is a function of Xo and eo. 6, ... 1 en-1• From this and condition (iii) it follows that Xn and en are independent. Using this fact we will derive a recurrence formula for the measures A E B(X), J.l.n(A) = prob(xn E A), (12.4.3) which statistically describe the behavior of the dynamical system (12.4.2). Thus, choose a bounded Borel measurable function h: X -+ R and for some integer n ~ 0 consider the random vector Zn+l = h(Xn+t)· Observe that J.l.n+t(A) = prob(x;;~ 1 (B)). Using this equality and the change of variables Theorem 3.2.1, the mathematical expectation E(zn+l) can be calculated as follows:

E(zn+l)

= =

fo

i

However, since Zn+l

E(zn+t) =

h(xn+t(w))prob(tku)

h(X)J.I.n+t(dx)

=

i h(x)prob(x;;~1 (dx))

= (h,J.I.n+t}·

(12.4.4)

= h(T(xn,en)) we have

fo

=j

h(T(xn(w),en(w)))prob(tku)

{ h(T(x,y))prob((xn,en)- 1(dxdy)). lxxw

{12.4.5)

The independence of the random vectors Xn and en implies that prob((xn,en) E AxB) = prob{xn E A, en E B)= prob(xn E A)prob{en E B),

414


or

prob((xn,en)- 1 (A x B))= prob(x;- 1 (A))prob(e;- 1 (B)),

which shows that the measure prob((xn, en)- 1 (C)) is the product of measures

l'n(A)

= prob(x;- 1(A))

and v(B)

= prob(e;- 1(B)).

Thus, by the Fubini Theorem 2.2.3, equality (12.4.5) may be rewritten in the form E(ZnH)

=

L{fw

h(T(x, y))v(dy)} l'n(dx).

Equating this expression with (12.4.4) we immediately obtain (h, l'n+l} =

L{fw

h(T(x, y))v(dy)} l'n(dx).

(12.4.6)

This is the desired recurrence formula, derived under the assumption that h is Borel measurable and bounded. The boundness of h asserts that all the integrals appearing in the derivation are well defined and finite, since the measures J'n, l'n+ll prob, ... , were probabilistic. The same derivation can be repeated for unbounded h as long as all the integrals are well defined. In particular the derivation can be made for an arbitrary measurable nonnegative h. However, in this case the integrals on both sides of (12.4.5) could be infinite. Using (12.4.5) we may calculate the values of l'n+l(A) for an arbitrary measurable set A c X. Namely, setting h = 1A we obtain

l'n+l(A) =

L{fw

1A(T(x,y))v(dy)} l'n(dx).

Now we are in a position to define the Foias operator corresponding to the dynamical system (12.4.2).

Definition 12.4.2. Let a function T: X x W--+ X satisfying condition (i) and a probabilistic measure (supported on W) be given. Then the operator P: Mfln --+ Mfin given by PJ£(A)

=

L{fw

1A(T(x,y))v(dy)} l'(dx) for I' E Mfln,A E B(X)

(12.4.7) will be called the Foias operator corresponding to the dynamical system (12.4.2). Since v is a probabilistic measure, it is obvious that P is a Markov operator. Moreover, from the definition of P it follows that l'n = pnp.o, where {l'n} denotes the sequence of distributions (12.4.3) described by the dynamical system (12.4.2).


Setting Uh(x)

= fw h(T(x,y))v(dy)

for x EX,

415

(12.4.8)

we may rewrite {12.4.7) in the form

Due to the linearity of the scalar product this implies

(gn,PJ.I.} = (Ugn,J.I.}, where

n

9n

= L~i1A, i=l

is a simple function. Further, since every measurable function h can be approximated by a sequence {gn} of simple functions, we obtain in the limit (h, PJ.I.) = (Uh, J.l.) (12.4.9)

if {gn} and {U9n} satisfy the conditions of the Lebesgue dominated or Lebesgue monotone convergence theorem. In particular, (12.4.9) is valid if h is Borel measurable and bounded or nonnegative. From (12.4.9) it follows by an induction argument that for n

= 1,2, ....

(12.4.10)

Now define a sequence of functions Tn(x, y1, ... , Yn) by setting Tn(x, Y1. ... , Yn)

= T(Tn-l(x, Y1. · .. , Yn-d, Yn)·

Using this notation we obtain Unh(x)

= fw ···fw h(Tn(x, Y1, ... , Yn)v(dyl) · · · v(dyn)

{12.4.11)

from (12.4.8), or, more briefly, unh(x)

where yn =

=f

h(Tn(x, yn))vn(dyn),

(12.4.12) Jwn (Yl. ... ,yn). wn = w X ... X w is the Cartesian product of

= v(dy1) · · · v(dyn) is the corresponding product measure on wn. Equations (12.4.10) and (12.4.12) give convenient tools for studying the asymptotic behavior of the sequence {pn J.1.}. Moreover, un and Tn have a simple dynamical interpretation. Namely, from (12.4.2) and the definition of Tn it follows that n sets W and vn(dyn)

for n

= 1,2, ... ,

416


which shows that Tn describes the position of Xn as a function of the initial position x 0 and perturbations. Further, repeating the calculation of the mathematical expectation E(h(xn)) we obtain

E(h(xn))

=In h(xn(w))prob(dw) =

Lh(x)prob(:z:~ (dx)) L 1

=

h(x)JJn(dx)

or (12.4.13} In particular, if the starting point :z:o is fixed, corresponding to JJo = 6~~:01 we have (12.4.14) Thus, unh gives the mathematical expectation of h(xn) as a function of , the initial position :z:o. We close this section by discussing the relationship between the FrobeniusPerron and Foias operators. Having a continuous transformation S: X - t X we may formally write

T(x, y)

= S(x) + Oy.

In this case (12.4.7) takes the form

L{fw

Pp.(A) =

=

1A(S(x))v(dy)} JJ(dx)

L

lA(S(x))p.(dx) = p.(S- 1(A)),

and is identical with (12.4.1}. Thus, in the case when T(x, y) does not depend on y the notions of the Foias operator and the Frobenius-Perron operator coincide. Moreover, in this case

Uh(x)

= fw h(S(x))v(dy) = h(S(x)),

(12.4.15)

and U is the Koopman operator. It is evident that the operator U given by equation (12.4.8} (with v E M 1 ), or (12.4.15), maps a bounded function h into a bounded function Uh. Moreover, if S is continuous [or more generally T satisfies condition (i)], then U h is continuous for continuous bounded h. However, in general, the support of Uh is not bounded for hE C0 (X).

12.5. Stationary Measures

417

12.5 Stationary Measures: Krylov-Bogolubov Theorem for Stochastic Dynamical Systems We begin our study of the asymptotic properties of {pn J.t} by looking for stationary measures. Definition 12.5.1. A measure I'• E Mon is called invariant or stationary with respect to a Markov operator P if PJ.t• = I'•· In particular, when Pis a Foias operator corresponding to the dynamical system (12.4.2) and PJ.t. = J.t•, we say that I'• is stationary with respect to {12.4.2). A stationary probabilistic measure is called a stationary distribution. H I'• is a stationary distribution for (12.4.2) and if the initial vector xo is distributed according to J.t•, that is, prob(xo E A) then all the vectors

Xn

prob(xn E A)

= J.I,.(A)

for A E B(X),

have the same property, that is,

= J.t.(A)

for A E B(X), n

= 0, 1, ....

Our main result concerning the existence of a stationary distribution is contained in the following. Theorem 12.5.1. Let P be the Foias opemtor coTTesponding to a regular stochastic dynamical system {12.4.2). Assume that there is a J.to E M 1 having the following property. For every e > 0 there is a bounded set B E B(X) such that J.tn(B)

= pnJ.to(B) ;::: 1- e

for n

= 0, 1, 2, ....

(12.5.1)

= 1,2, ....

(12.5.2)

Then P has an invariant distribution.

Proof. Define (n

1 n-1

.

n-1

1 L P' J.to = - L J.ti n n

=-

i=O

for n

i=O

Choose a countable subset {h1, h2, ... } of Co(X) dense in C0 (X) (see Exercises 12.1 and 12.2). The sequence {(h1, (n)} is bounded since the (n are probabilistic and l(h1, (n)l:::;; max lh1l· Thus, there is a subsequence {(1n} of {(n} such that { (hl! (1n)} is convergent. Again, since { (h2, (1n)} is bounded we can choose a subsequence {(2n} of {(1n} such that { (h2, (2n)} is convergent. By induction for every integer k > 1 we may construct a sequence {(A:n} such that all sequences {(h;,(A:n}} for j = 1, ... ,k are convergent and {(A:n} is a subsequence of {(A:-1,n}· Evidently the diagonal sequence {(nn} has the property that {(h;,(nnH is convergent for every j = 1, 2, ....


418

This procedure of choosing subsequences is known as the Cantor diagonal process [Dunford and Schwartz, 1957, Chapter 1.6]. Since the set {h;} is dense in Co, then according to Remark 12.2.1 the sequence {(nn} is weakly convergent to a measure I'•· It remains to prove that J.l.• is probabilistic and invariant. Without any loss of generality we may assume that the set Bin (12.5.1) is compact. Then X\ B is open and according to Theorem 12.2.1

I'•(X \B) :5 liminf (nn(X \B) :5 1- inf J.l.n(B) :5 1- (1- e) =e. n n~~

Now we may prove that {{h, (nn}} converges to (h, J.l.•} for every bounded continuous h. Let h be given. Define he = hge where 9e E Co is such that

0:5ge:51 and 9e(z)=1

for x E B.

Then

l(h,J.i.•- (nn}l :5l(he,J.I.•- (nn}l + l(h(1- 9e),J.I.•- (nn}l :5 l(he, I'• - (nn}l +sup lhi(J.I..(X \B)+ (nn(X \B)) or

l(h,J.i..- (nn)l :5 l(he, I'•- (nn)l + 2e sup lhl. Since he E Co and {(nn} converges weakly to J.l.• this implies lim (h, (nn)

n~~

= (h, J.l.•)

for every bounded continuous h. In particular, setting h

= 1x we obtain

so J.l.• is probabilistic. Now we are ready to prove that I'• is invariant. The sequence {(nn}, as a subsequence of {(n}, may be written in the form 1

(nn =

k n

k,.-1

.

2::: P'J.i.O, i=O

where {kn} is a strictly increasing sequence of integers. Thus,

and, consequently, 1

I(Uh, (nn}- (h, (nn}l = l(h, P(nn}- (h, (nn}l :5 kn sup lhl.

12.5. Stationary Measures

419

Passing to the limit we obtain

(Uh,p..}- (h,p..} = 0, or

(h,Pp..}

= (h,p..).

The last equality holds for every bounded continuous h and in particular for hE C0 • Thus, by the Riesz representation theorem 12.1.1, Pp.. = I-'•· The proof is completed. • Condition (12.5.1) is not only sufficient for the existence of an invariant distribution I-'• but also necessary. To see this, assume that I-'• exists. Let {Bk} be an increasing sequence of bounded measurable sets such that Uk Bk =X. Then lim p..(Bk) = p..(X) = 1. k-+oo

Thus, for every e > 0 there is a bounded set Bk such that p..(Bk) Setting P.o = I-'• we have 1-'n = I-'• and, consequently, for n

~

1- e.

= 0,1, ....

Remark 12.5.1. In the case when X is bounded (and hence compact, because we always assume that X is closed), condition (12.5.1) is automatically satisfied with B = X. Thus for a regular stochastic dynamical system there always exists a stationary distribution. In particular, for a continuous transformation S: X --+ X of a compact set X there always exists an invariant probabilistic measure. This last assertion is known as the Krylov-Bogolubov theorem. It is valid not only when X is a compact subset of Rd, but also for arbitrary compact topological Hausdorff spaces. 0

Now we will concentrate on the case when XC~ is unbounded (but closed!), and formulate some sufficient conditions for (12.5.1) based on the technique of Liapunov functions. Recall from (5.7.8) that a Borel measurable function V: X --+ R is called a Liapunov function if V (x) --+ oo for lxl--+ oo. Proposition 12.5.1. Let P be the Foias operator corresponding to a regular stochastic dynamical system (12.4.2). Assume that there is an initial random vector x 0 and a Liapunov function V such that supE(V(xn))

< oo.

(12.5.3)

n

Then P has an invariant distribution. Proof. Consider the family of bounded sets

Ba

= {x EX: V(x) :5: a}

for

a~

0.

420


By Chebyshev's inequality (10.2.9) we have

IJ.n(X \ Ba)

= prob(V(xn) > a) ~ _E(,_V--'-(x_n'-'-)) a

or for n = 0, 1, ... , where K = supn E(V(xn)). Thus, for every e > 0 inequality (12.5.1) is satisfied with B = Ba and a= K- e. It follows from Theorem 12.5.1 that P has an invariant distribution and the proof is complete. • It is easy to formulate a sufficient condition for (12.5.3) related explicitly to properties of the function T of (12.4.2) and the distribution v. Thus we have the following Proposition 12.5.2. Let P be the Foias operator corresponding to a regular stochastic dynamical system (12.4.2). Assume that there exists a Liapunov function V and nonnegative constants a, {3, a < 1, such that

fw V(T(x, y))v(dy) ~ aV(x) + /3

forx EX.

(12.5.4)

Then P has an invariant distribution. Proof. By an induction argument from inequality (12.5.4), it follows that

{ V(Tn(X, yn))vn(dyn) ~ anV(x) + an-l/3 + · · · + a/3 + /3 lwn

~ V(x) + 1 ~a· Fix an Xo E X and define #J.o (12.4.12) we have

E(V(xn))

= unv(xo) = f

~n

= Dzo·

Then according to (12.4.14) and

V(Tn(xo,yn))vn(dyn)

~ V(xo) + -

/3 ,

1 -a

which implies (12.5.3), and Proposition 12.5.1 completes the proof.

•

12.6 Weak Asymptotic Stability In the previous section we developed sufficient conditions for the existence of a stationary measure IL•. Now we are going to prove conditions that ensure that this measure is asymptotically stable. Since in the space of measures there are two natural notions of convergence (weak and strong),

12.6. Weak Asymptotic Stability

421

we will introduce two types of asymptotic stability. We will start from the following.

Definition 12.6.1. Let P: Man -+ Mfin be a Markov operator. We say that the sequence {pn} is weakly asymptotically stable if P has a unique invariant distribution J.l.• and { pn J.l.} converges weakly to J.l.• for J.l. E M 1·

(12.6.1)

In the special case that P is a Foias operator corresponding to a stochastic dynamical system (12.4.2) and {pn} is weakly asymptotically stable, we say that the system is weakly asymptotically stable. It may be shown that the uniqueness of the stationary distribution J.l.• is a consequence of the condition (12.6.1). To show this, let ji. E M 1 be another stationary distribution. Then pnji. = ji. and from (12.6.1) applied to J.l. = ji. we obtain for h E Co(X). By the lliesz representation theorem 12.1.1, this gives ji. = J.l.•· On the other hand, condition (12.6.1) does not imply that J.l.• is stationary for an arbitrary Markov operator.

Example 12.6.1. Let X = [0, 1]. Consider the Frobenius-Perron operator P on measures and the Koopman operator U corresponding to the transformation S(x) = { c!x x > 0 x=O, where c E [0, 1] is a constant. Now x>O x=O. Thus, for every

J.1.

E M 1 and h E C0 (X) we have

Since h is continuous this implies lim (h,PnJJ.}

n-+oo

= JJ.(O)h(O) + JJ.((O, 1])h(O) = h(O)

and consequently {pnJJ.} converges to 6o. On the other hand, P6o = 6c and the system is weakly asymptotically stable only for c = 0 when S

422


is continuous. U c > 0 the operator P has no invariant distribution but condition (12.6.1) holds with I'• = 6o. 0 Next we give two easily proved criteria for the weak asymptotic stability of a sequence {P" }.

Proposition 12.6.1. Let P: Mfln -+ Mfln be a Markov operator. The sequence {P"} is weakly asymptotically stable if P has an invariant distribution and if lim (h, pnp.- pn[J} R-+00

=0

for hE Co; p., [J EM.

(12.6.2)

Proof. First assume that {P"} is weakly asymptotically stable. Then by the triangle inequalitY

and (12.6.1) implies (12.6.2). Alternately, if (12.6.2) holds and p.. is stationary, then substituting p, = I'• in (12.6.2) we obtain (12.6.1). • The main advantage of condition (12.6.2) in comparison with {12.6.1) is that in proving the convergence we may restrict the verification to subsets of Co and M1.

Proposition 12.6.2. Let C. c Co be a dense subset. If condition {12.6.2) holds for every h E C. and p., [J E M 1 with bounded supports, then it is satisfied for arbitrary h E Co and p., [J E M1. Proof. Choose p., [J E M1 and fix an e > 0. Without any loss of generality we may assume that e ~ 1/2. Since p. and [J are probabilistic, there is a bounded set B c X such that p.(X \B) ~ e and [J(X \B) ~ e. Define {A) p

= p.(A n B) p.(B)

an

d

_{A) p

= [J(A n B) p.(B)

for A E B(X).

Evidently p and p are probabilistic measures with bounded supports. We have p.(B) ~ 1- e ~!,and consequently lp.(A)- p(A)I ~ 2lp.(A)p.(B)- p.(A n B) I = 2lp.(A){1 - p.(X \B)) - p.(A n B) I ~ 2lp.(A) - p.(A n B) I + 2p.(A)p.(X \B) ~ 2p.{A \B)+ 2p.(X \B) ~ 4e for A E B(X).

12.6. Weak Asymptotic Stability

423

In an analogous fashion we may verify that

IJi(A) - p(A)I Now let a function g E

~

4e

for A E B(X).

c. be given. Then

l{g,Pnp- pnp)l = I{Ung,p- Ji}l ~ I(Ung,p- p)l +Be supiUngl, and, finally, l(h,Pnp- pnp)l ~ l(g,pnp- pnp)l +Be maxlgl. Since p and p have bounded supports the sequence { (g, pn p - pn p)} converges to zero. Consequently, { (g, pn p- pn Ji)} converges to zero for every g E C. and p,Ji E M 1 • Now from the inequality l(h, pnJ.&- pnp)l ~ l(g, pnp- pnii}l + 2sup In- hi

forgE C.,h E Co

and the density of C. in Co, (12.6.3) follows for all hE Co. Thus the proof is complete. • Now we may establish the main result of this section, which is an effective criterion for the weak asymptotic stability of the stochastic system (12.4.2). Theorem 12.6.1. Let P be the Foias operator corresponding to the regular stochastic dynamical system (12.4.2). Assume that

forx,z EX

(12.6.3)

and E(IT(o,en)l) ~ {3,

(12.6.4)

where E is the mathematical expectation and a, {3 are nonnegative constants with a< 1. Then the system (12.4.2) is weakly asymptotically stable. Before passing to the proof observe that conditions (12.6.3) and (12.6.4) can be rewritten, using the distribution v appearing in the definition of the stochastic system (12.4.2), in the form

and

fw IT(x, y) - T(z, y)lv(dy) ~ alx- zl

(12.6.5)

fw IT(O,y)lv(dy) ~ {3.

(12.6.6)

Proof of Theorem 12.6.1. From the inequality

fw IT(x,y)lv(dy) ~ fw IT(x,y)- T(x,O)Iv(dy) + LIT(O,y)lv(dy)

424


and conditions (12.6.5) and (12.6.6), inequality (12.5.4) follows immediately = !x!. Thus, according to Proposition 12.5.2 there exists a stationary probabilistic measure I'•. Using the definition of Tn from Section 12.4 and inequality (12.6.5) we obtain if we take V(x)

{

!Tn(x, y")- Tn(z, y")lv"(dy")

lwn = {

Jwn-1

{ { IT(Tn-1(x,y"- 1),yn) Jw

- T(Tn-1(z,y"- 1), Yn)!v(dyn) }v"- 1(dy"- 1)

::5 a { ITn-1(x,y"- 1)- Tn-1(z,y"- 1)lv"- 1(dy"- 1) Jwn-1 ::5 • · · ::5 a"lx- z!. (12.6. 7) Now consider the subset C. of Co which consists of functions h satisfying the Lipschitz condition !h(x)- h(z)l ::5 klx- zl

for x,z EX,

where the constant k depends, in general, on h. Further let p. and P, be two distributions with bounded support. Then

= I(U"h, JJ- P.}l = U"h(x)p.(dx)-

l(h, P"p.- P"P,}I

IL

L

U"h(x)P,(dx)l, (12.6.8)

where B is a bounded set such that p.(B) = p.(B) = 1. Since the measures JJ and P, are probabilistic there exist points qn, rn E B such that

IL

U"h(x)p.(dx)-

L

U"h(x)p,(dx)l ::5 !U"h(qn)- U"h(rn)l.

From this and (12.6.8) we have

l(h, P"p.- P"P,}I ::5IU"h(gn)- U"h(rn)l

::5 {

}wn !h(Tn(9m y"))- h(T"(rn, y"))lv"(dy").

Using the Lipschitz condition for hand (12.6.7) we finally obtain

l(h,P"p.- P"p,}l ::5 k {

}wn

!Tn(qn,y")- Tn(rmy")lv"(dy")

::5 ka"lqn - rnl ::5 k do.",

12.7. Strong Asymptotic Stability

425

where d = sup{lx- zl:x,z E B}. Since kdan-+ 0 as n-+ oo, this implies (12.6.3) for arbitrary h E C. and p., ji. E M1 with bounded supports. According to Propositions 12.6.1 and 12.6.2 the proof of the weak asymptotic stability is complete. • Remark 12.6.1. When T(x, y) = S(x) does not depend on y, condition {12.6.4) is automatically satisfied with {3 = IS(O)I and inequality (12.6.3) reduces to IS(x)- S(z)l $ alx- zl for x, z EX. In this case the statement of Theorem 12.6.1 is close to the Banach contraction principle. However, it still gives something new. Namely, the classical Banach theorem shows that all the trajectories {Sn(xo)} converge to the unique fixed point x. = S(x.). From Theorem 12.6.1 it follows also that the measures p.(s-n(A)) (with p. E Mt) converge to 6:r:. which is the unique stationary distribution. 0

12.7 Strong Asymptotic Stability In Example 12.2.1 we have shown that if a sequence of points {xn} converges to x., then the corresponding sequence of measures {6:r:n} converges weakly to 6:r: • • In general, this convergence is not strong since ll6:r:n - 6:r:.ll = 2 for Xn # x •. Thus, in the space of measures, weak convergence seems to be a more convenient and natural notion than strong convergence. However this is not necessarily true for stochastic dynamical systems in which the perturbations are nonsingular. To make this notion precise we introduce the following.

en

Definition 12.7.1. A measure p. E Mftn{X) is called nonsingular if there is an absolutely continuous measure J.l.a such that P.a(B) $ p.(B)

forB E B(X)

(12.7.1)

and P.a(X) > 0. It can be proved that for every measure p. E Mfin there exists a maximal absolutely continuous measure J.l.a satisfying {12.7.1}. The word maximal means that for any other continuous measure J.l.a' satisfying J.l.a•(B)::::; p.(B) for all measurable sets B, we also have P.a•(B) $ p.4 (B} for all measurable B. This maximal measure J.l.a is called the absolutely continuous part of p.. The remaining component, p.8 = p.- J.1.4 , is called the singular part. Thus, Definition 12.7.1 may be restated as follows: The measure p. E Mftn is nonsingular if its absolutely continuous part P.a is not identically equal to zero. We always denote the absolutely continuous and singular parts of any measure by subscripts a and s, respectively. The equation J.I.=J.I.a+P.s is called the Lebesgue decomposition of the measure p..

{12.7.2)

426


In this section we will exclusively consider regular stochastic dynamical systems of the form for n = 0, 1, ... ,

{12.7.3)

where 8: X -+ X is a continuous mapping of a closed set X C If'- into itself, and Xo, eo, el •... ' are independent random vectors. The values of en belong to a Borel measurable set W c If'- such that

x E X, y E W

implies x + y E X.

This condition is satisfied, for example, when X = W = Rd or X = W = R+. The dynamical system (12.7.3) with additive perturbations reduces to the general form {12.4.2) for T(x,y) = 8(x) + y. Then equations (12.4.7) and (12.4.8) for the Foias operator P and its adjoint U take the form

PJ.t(A)

=[

{fw 1A(8(x) + y)v(dy)} J.t(dx)

and Uh(x)

= fw h(8(x) + y)v(dy)

for A E B(X) (12.7.4)

for x EX.

{12.7.5)

Consequently, for the scalar product we obtain {h, pnJ.t}

= (Unh, J.t} = [

{fw h(8(x) + y)v(dy)} J.t(dx).

(12.7.6)

From Proposition 12.5.2 and Theorem 12.6.1 we immediately obtain the following result.

Proposition 12. 7.1. If in the regular stochastic dynamical system (12. 7.3) the transformation 8 and perturbations {en} satisfy the conditions l8(x)l ~ alxl + 'Y

for x EX

(12.7.7)

and

{12.7.8) where a,-y,k are nonnegative constants with a < 1, then (12.7.3) has a stationary distribution. Moreover, if (12.7.7) is replaced by the stronger condition (12.7.9) l8(x)- 8(z)l ~ alx- zl for x, z EX, then (12.7.3) is weakly asymptotically stable.

Proof. The proof is immediate. It is sufficient to verify conditions {12.5.4) and (12.6.5). First observe that (12.7.8) is equivalent to

fw lylv(dy) ~ k.


Consequently, setting T(x,y)

= S(x) + y and using (12.7.7) and

427

(12.7.8)

we obtain

fw IT(O,y)!v(dy) = fw IS(O) + y!v(dy) :5 fw IS(O)Iv(dy) + fw lylv(dy) :5 IS(O)I + k :5 I+ k. This is a special case of {12.6.6) with {3 = 1

+ k. Further, {12.7.9) yields

fw IT(x,y)- T(z,y)!v(dy) = fw IS(x)- S(z)lv(dy) = IS(x)- S(z)l :5 alx- z!, which gives {12.6.5). • We will now show that under rather mild additional assumptions the asymptotic stability guaranteed by Proposition 12.7.1 is, in fact, strong. This is related to an interesting property of the absolutely continuous part J.tna of the distribution J.tn· Namely, IIJ.tnall = J.tna(X) increases to 1 as n - oo. Our first result in this direction is the following. Proposition 12. 7.2. Let P be the Foias operator coJTesponding to a regular stochastic dynamical system (12.7.3) in which Sis a nonsingular transformation. If p. E Mftn is absolutely continuous, then Pp. is also. Proof. Let f be the Radon-Nikodym derivative of p.. Then equation (12.7.4) gives

{fw 1A(S(x) + y)v(dy)} f(x) dx = fw {[ 1A(S(x) + y)f(x) dx} v(dy).

Pp.(A) = [

For fixed yEW the function 1A(S(x)+y) is the result of the application of the Koopman operator to 1A(x+y). Denoting by Ps the Frobenius-Perron operator (acting on densities) corresponding to S, we may rewrite the last integral to obtain

Pp.(A) =

L{[

= fw

1A(x + y)Psf(x) dx} v(dy)

{[+

1A(x)Psf(x- y) 11

dx} v(dy).

Inside the braces the integration runs over all x such that x E A and x EX+ y, or, equivalently, x E A and x- y EX. Thus,

428


P~-£(A) = fw =

{L

L{fw

1x(x- y)Psf(x- y)

clx} v(dy)

1x(x- y)Psf(x- y)v(dy)} clx.

(12.7.10)

The function

q(x)

= fw 1x(x- y)Psf(x- y)v(dy)

(12.7.11)

inside the braces of (12.7.10) is the convolution of the element Psi E L 1 with the measure v. Thus we have verified that PI-£ is an absolutely continuous measure with density q. • From Proposition 12. 7.2, an important consequence concerning the behavior of the absolutely continuous part of P"l-£ follows directly. Namely, we have Corollary 12.7.1. Let P be the Foias opemtor corresponding to the regular stochastic system (12.7.3) with nonsingular S. Then (P~-£)a(X) 2:::

1-'a(X)

for 1-£ E Mt1 01

{12.7.12)

and the sequence 1-'na(X) is increasing.

Proof. By the linearity of P we have PI-£= Pl-'a +PI-£, 2::: Pl-£a·

Since (P~-£)a is the maximal absolutely continuous measure which does not exceed PI-£, we have (PI-')a 2::: PIJ.a· In particular,

and the proof is complete. • Proposition 12.7.2 also implies that when Sis nonsingular the operator P on densities corresponding to P exists. In fact the right hand side of {12.7.11) gives an explicit equation for this operator, that is,

Ff(x)

= fw 1x(x- y)Psf(x- y)v(dy).

(12.7.13)

If S and v are both nonsingular, we can say much more about the asymptotic behavior of (P"~-£) 0 • This behavior is described as follows.

Theorem 12. 7.1. Let P be the Foias opemtor corresponding to the regular stochastic system (12.7.3). If the tmnsformation S and the distribution v of mndom vectors {en} are nonsingular, then (12.7.14)


429

Proof. Let 9o be the Radon-Nikodym derivative of the measure 1/0 , Using the inequality v ~ v0 in equation (12.7.4) applied to J.L 8 , we obtain

Ptt.(A)

~ =

L{fw

L{fw

1A(S(x) + y)v0 (dy)} JJ. 8 (dx) 1A(S(x) + y)go(Y) dy} J.La(dx)

= {{{

lx lw+S(x)

1A(Y)9a(Y- S(x)) dy} J.La(dx).

The integration in the braces of the last integral runs over all y such that y E A and y E W + S(x), or equivalently all y E A and y- S(x) E W. Thus, the last inequality may be rewritten in the form

PJ.ta(A)

~ =

L{L

L{L

1w(Y- S(x))g0 (y- S(x)) dy} J.La(dx) 1w(Y- S(x))ga(Y- S(x)}J.L.(dx)} dy.

Setting r(y)

=

L

1w(Y- S(x))ga(Y- S(x))J.La(dx) and u(A)

=

L

r(y) dy

we may easily evaluate the measure PJ.L from below:

The measure P J.La + u is absolutely continuous and consequently the absolutely continuous part of PJ.L satisfies

In particular, (PJ.L)a(X) ~ PJ.ta(X). + u(X) = JJ.a(X) + u(X).

(12.7.15)

We may easily evaluate u(X) since

u(X)

=

L{L

=f { f lx

1w(Y- S(x))ga(Y- S(x)) dy} J.La{dx) 1w(y)g0 (y) dy} J.La(dx).

lx-S(x)

In the braces we integrate over ally such that y E Wand y E X- S(x) or equivalently yEW andy+ S(x) EX. Since W +XC X the condition

430


y + S(:z:) EX is always satisfied withy E Wand :z; EX. Thus, u(X)

Set va(W)

fx

= {fw ga(Y) dy} p.(d:z:) = Va(W)p.(X) = Va(W)(1 - J.&a(X)).

= e and use (12.7.15) to obtain = e + (1 -

(PJJ)a(X) ~ J.&a(X) + e(1 - J.&a(X))

e)pa(X).

From this, we obtain by an induction argument (Pnp) 4 (X) ~ e+e(1-e)+ .. ·+e(1-et- 1 +e(1-etJJa(X) ~ 1-(1-et. Since e = va(X) > 0, this completes the proof. • Now we are in a position to state our main result concerning the strong asymptotic stability of (12.7.3).

Theorem 12.7.2. Assume that (12.7.3) is a regular stochastic system and that the transformation 8 and the distribution v are nonsingular. If (12. 7.3) is weakly asymptotically stable, then it is also strongly asymptotically stable and the limiting measure J.&• is absolutely continuous. Proof. Let P be the Foias operator given by equation (12.7.4) and P the corresponding operator (12.7.13) for densities. The proof will be constructed in three steps. First we are going to show that P is constrictive. Then we will prove that r = 1 in equation (5.3.10) and {Fn} is asymptotically stable in the sense of Definition 5.6.1. Finally, using Theorem 12.7.1 we will show that {pn} is strongly asymptotically stable. Step I. Since (12. 7.3) is weakly asymptotically stable there exists a stationary measure J.'•· Choose e = v4 (X)/3 and an open bounded set B in X such that p.(B) > 1- e. Now consider an absolutely continuous J.&o E M1 with a density fo. According to the diagram (12.3.8), for each integer n ~ 1 the function pn fo is the density of J.&n = pnJ.&o. The sequence {J.&n} converges weakly top. and according to Theorem 12.2.1 there is an integer no such that

Is or

pn fo(:z:) d:z:

r

lx\B Now let F

cX

= J.&n(B) ~ 1- e

pnfo(:z:)d:z:~e

for n

for n ~no,

~no.

be a measurable set. We have

L

pn fo(:z:) d:z:

= J.&n(F) = PJ.&n-l(F),

(12.7.16)


and from (12.7.4) with

PJ.tn-1(F)

=

11

=

114

431

+ "•

L{fw + L{fw

1F(S(x) + y)v0 (dy)} J.tn-1(dx) 1F(S(x) + y)v.(dy)} J.tn-1(dx).

Since J.tn-1 is a probabilistic measure and

v.(X)

= 1-

114

(X)

= 1-

3e,

this implies

PJ.tn-1(F) :5 sup { f 1F(Y + z)va(dy)} zex lw

+ 1- 3e.

Let g0 be the Radon-Nikodym derivative of 114 so we may rewrite the last inequality in the form

PJ.tn-1 (F) :5 sup { f 1F(Y + z)go(Y) dy} zex lw

r

=sup { 9a(y)dy} zex lwn(F-z)

+1-

3e

+ 1- 3e.

The standard Borel measure of W n (F- z) is smaller than the measure of F. Thus there exists a o> 0 such that

f

9o(Y) dy :5 e

lwn(F-z)

for F E B(X), m(F) :5 o

and consequently

L

pn /o(x) dx

= PJ.tn-1(F) :5 e + (1- 3e) = 1- 2e.

From this and (12.7.16) we obtain

f

pn fo(x) dx :5 e + (1- 2e)

= 1- e

for n ~no(/)

(X\B)UF

which proves that P is a constrictive operator. According to the spectral decomposition theorem [see equation (5.3.10)] the iterates of P may be written in the form r

pn f =

L Ai(/)ga"(i) + Qnf

(12.7.17)

n=0,1, ... ,

i=1

where the densities 9i have disjoint supports and />gi =

9a(i)·

432


Step II. Now we are going to prove that r = 1 in equation (12.7.17). Let k = rl and 9i be an arbitrary density. Then ak(i) = i and consequently pkn9i = 9i for all n. Since (12. 7.3) is weakly asymptotically stable the sequence {{h,f>nkgi}} converges to (h,p..}. However, this sequence is constant so (12.7.18) for hE Co.

The last equality implies that 9i is the density of JL•. Thus, there is only one term in the summation portion of (12.7.17) and g1 is the invariant density. Step III. Consider the sequence { pn JLo} with an arbitrary JLo E M 1· Choose an e > 0. According to Theorem 12.7.1 there exists an integer k such that (PicJLo)a(X) = JL~ca(X) ~ 1- e.

= JLka(X). Since JLk = JLka + JLk• we have JLn+lc - I'• = pn #Lie - JL• = pn #Lka - fJp.. + pn l'k• -

Define fJ

(1 - fJ)p..

or

where 11·11 denotes the distance defined by equation (12.2.7). The last two terms are easy to evaluate since {12.7.20)

and (1- fJ)IIJL·II

= (1- fJ)p..(X) = 1-8 ~e.

(12.7.21)

The measure (J- 1JLka is absolutely continuous and normalized. Denote its density by Ia· pn((J- 1JLkn) clearly has density f>n Ia and from equation (12.2.11) Since {f>n} is asymptotically stable the right-hand side of this equality converges to zero as n-+ oo. From this convergence and inequalities (12.7.20) and (12.7.21) applied to (12.7.19), it follows that lim IIJLn+lc -1-'nll

n-+oo

This completes the proof.

= 0.

•

12.8 Iterated Function Systems and Fractals In the previous section we considered a special case of a regular stochastic dynamical system with additive nonsingular perturbations. As we have

12.8. Iterated Function Systems and Fractals

433

seen, these systems produce absolutely continuous limiting distributions. In this section we consider another special class in which the set W is finite. We will see that such systems produce limiting measures supported on very special sets-fractals. Intuitively a system with finite W can be described as follows. Consider N continuous transformations i= 1, ... ,N of a closed nonempty subset X C Jld. If the initial point xo E X is chosen we toss anN-sided die, and if the number io is drawn we define Xt·= sio(xo). Then we toss up the die again and if the number it is drawn we define X2 = 8i 1 (xt), and SO on. This procedure can be easily formalized. Consider a probabilistic vector N

{pt,····PN),

Pi~

o,

LPi i=l

= 1,

and the sequence of independent random variables prob(~n

= i) =Pi

~o.~t •...

such that

for i = 1, ... , N.

The dynamical system is defined by the formula for n

= 0, 1, ....

(12.8.1)

It is clear that in this case T(x, y) = Sy(x) and W = {1, ... , N}. The system (12.8.1) is called (Barnsley, 1988) an iterated function system

(IFS). Using the general equations (12.4. 7) and (12.4.8) it is easy to find explicit formulas for the operators U and P corresponding to an iterated function system. Namely,

Uh(x)

= fw h(T(x, y))v(dy) = fw h(Sy(x))v(dy)

or

N

Uh(x)

= LPih(Si(x))

for x EX.

(12.8.2)

i=l Further, N

PJL(A)

= (UlA,JL} = LPi 11A(Si(X))JL(dx) i=l

or

X

N

PJL(A)

= LPiJL(S; 1 (A)) i=l

for A E B(X).

{12.8.3)

434


Now assume that the Si satisfy the Lipschitz condition forx,zEX; i=1, ... ,N,

(12.8.4)

where Li are nonnegative constants. In this case Theorem 12.6.1 implies the following result.

Proposition 12.8.1. If N

~PiLi< 1,

(12.8.5)

i=l

then the iterated function system (12.8.1) is weakly asymptotically stable.

Proof. It is sufficient to verify conditions (12.6.3) and (12.6.4). We have N

E(IS~"(x)- S~"(z)l) = ~PiiSi(x)- Si(z)l i=l N

~ lx- zl ~PiLi i=l

and

N

E(IS~"(O)I)

= ~PiiSi(O)I. i=l

Consequently (12.6.3) and (12.6.4) are satisfied with a= :EPiLi and /3 = :EPiiSi(O)I, and by Theorem 12.6.1 the proof is complete. • Condition (12.8.5) is automatically satisfied when Li < 1 fori = 1, ... , N. An iterated function system for which

L

= m~ Li < 1 I

and Pi > 0,

i= 1, ... ,N

(12.8.6)

is called hyperbolic. Our goal now is to study the structure of the set

A. = supp I'•,

(12.8.7)

where I'• is a stationary distribution for hyperbolic systems. We will show that A. does not depend on the probabilistic vector (p11 ... , PN) as long as all the Pi are strictly positive. To show an alternative, nonprobabilistic method of constructing A., we introduce a transformation F on the subset of X such that the iterates pn approximate A•.

Definition 12.8.1. Let an iterated function system (12.8.1) be given. Then the transformation N

F(A)

= U Si(A) i=l

forA eX

(12.8.8)


435

mapping subsets of X into subsets of X is called the Barnsley operator corresponding to (12.8.1). It is easy to observe that for every compact set A C X its image F(A) is also a compact set. In fact, the S,(A) a.re compact since the images of compact sets by continuous transformations a.re compact and the finite union of compact sets is compact. To show the connection between F and the dynamical system (12.8.1) we prove the following.

Proposition 12.8.2. Let F be the Bamsley operator corresponding to {12.8.1). Moreover, let {Pn} be the sequence of distributions corresponding to {12.8.1), that is, J'n = P"JJ.o. Jfsupp Jl.o is a compact set, then supp J'n = F(supp JJ.o)

(12.8.9)

Proof. It is clearly sufficient to verify that supp J.t 1 = F(supp JJ.o) since the situation repeats. Let x E F(supp JJ.o) and e > 0 be fixed. Then x = Sj(z) for some integer j and z E supp Jl.o. Consequently, for the ball Br(z) we have JJ.o(Br(z)) > 0 for every r > 0. Further, due to the continuity of Sj there is an r > 0 such that

This gives n

J.t1(Be(x))

= LPsJJ.o(Bi 1(Be(x)) i=1

Since e > 0 was arbitrary this shows that x E supp 1'1· We have proved the inclusion F(supp JJ.o) c supp 1'1· Now, suppose that this inclusion is proper and there is a point x E supp J.t 1 such that x ¢ F(supp JJ.o). Due to the compactness of F(supp JJ.o) there must exist an e > 0 such that the ball Be(x) is disjoint with F(supp JJ.o). This implies fori= 1, ... ,N or

Si 1(Be(x)) n supp J.to = 0

fori= 1, ... ,N.

The last condition implies that N

J'1(Be(x))

= LPsJJ.o(Bi 1(Be(x)) = 0 i=1

436


which contradicts to the assumption that x E supp l't· This contradiction shows that F(supp JJo) = supp l't· An induction argument completes the proof. • Formula (12.8.9) allows us to construct the supports of J.'n from the support of JJo by purely geometrical methods without any use of probabilistic arguments. Now we will show that the set

A.

= supp I'•,

(12.8.10)

which is called the attractor of the iterated function system, can be obtained as the limit of the sequence of sets

An = supp J.'n

= F(Ao).

{12.8.11)

To state this fact precisely we introduce the notion of the Hausdorff distance between two sets. Definition 12.8.2. Let At, A2 c R"- be nonempty compact sets and let > 0 be a real number. We say that At approximates A2 with accuracy r if, for every point Xt EAt, there is a point x2 E A2 such that lxt- x2l ~ r and for every x 2 E A there is an Xt E At such that the same inequality holds. The infimum of all r such that At approximates A2 with accuracy r is called the Hausdorff distance between At and A2 and is denoted by dist(At, A2). We say that a sequence {An} of compact sets converges to a compact set A if lim dist(Am A) = 0. n-o ex> r

From the compactness of A it easily follows that the limit of the sequence {An}, if it exists, must be unique. This limit will be denoted by liron..... oo An. Example 12.8.1. Let X

= R, A= [0, 1] and

2n-1} 1 2 An= { 2n'2n''"'2ft

for n

= 1,2, ....

Clearly, An C [0, 1]. Moreover for every x E [0, 1] there is an integer k, 1 ~ k ~ 2n-t, such that X-

!_I < _!:_,

2n - 2n I Thus, An approximates A with accuracy 1/2n. Moreover, for x the nearest point in An is 1/2n. Consequently,

=0 E A

dist(An, A) = ;n. This example shows that sets which are close in the sense of Hausdorff distance can be quite different from a topological point of view. In fact, each


437

An consists of a finite number of points, whereas A = (0, 1] is a continuum. This is a typical situation in the technical reproduction of pictures; on a television screen a picture is composed of a finite number of pixels. 0 We have introduced the notion of the distance between compact sets only. We already know that for compact Ao = supp p.o all the sets An = supp P.n are compact. Now we are going to show the compactness of the limiting set A,.= supp p.,.. Proposition 12.8.3. If the iterated function system (12.8.1) is hyperbolic and p.,. is the stationary distribution, then the set A,. = supp p.,. is compact. Proof. Since the support of every measure is a closed set, it is sufficient to verify that A,. is bounded. Further, since p.,. does not depend on p.o we may assume that p.o = 6z0 for an xo E X. Define r

= max{ISi(xo) -

xol: i

= 1, ... , N}.

Then

or by induction, ISil

0 ••• 0

sin (xo) - xol $ Ln-lr + ... + Lr + r $ 1 ~ L

(12.8.12)

for every sequence of integers i1, ... , in with 0 $ ik $ N. Choose an arbitrary point z E X such that r

lz- xol :?: 1 - L

+ 1.

(12.8.13)

We are going to prove that z f/. supp p.,.. Fix an e E (0, 1). From inequality (12.2.6) and equation (12.8.3} we obtain p.,.(B~(z))

$liminf P.n(B~(z))

(12.8.14}

n-+oo

= liminf n-+oo

" L..J' Pi 1

1

Pin6z0 (Si1 o · · · o Sin (B~(z))). 1

• ..

i1, ... ,in

According to (12.8.12) and (12.8.13) we have lz- sin which implies that

x0 ~

0 ... 0

sil (xo)l :?: 1

"'

Si; 1 o · · · o Si:

1

(B~(z)).

Thus the right-hand side of (12.8.14} is equal to zero and as a consequence 0. We have proved that z f/. supp p.,. and that the support of p.,. is contained in a ball centered at xo with radius 1 + r/(1- L). •

p.,.(B~(z)) =

438


Now we formulate a convergence theorem which allows us to construct the set A. without any use of probabilistic tools. Theorem 12.8.1. Let (12.8.1) be a hyperbolic system and let F be the corT"esponding Bamsley operator. Further, let A. be the support of the invariant distribution. Then

A. = lim Fn(Ao)

(12.8.15)

n-+oo

whenever

Ao C X

is a nonempty compact set.

Prooof. We divide the proof into two steps. First we show that the limit of {FR(Ao)} does not depend on the particular choice of Ao, and then we will prove that this limit is equal to supp P.•. Step I. Consider two initial compact sets Ao, Zo c X and the corresponding sequences

An= F(Ao), Zn = F(Zo)

n=0,1, ....

We are going to show that dist(An, Zn) converges to zero. Let r > 0 be sufficiently large so A0 and Zo are contained in a ball of radius r. Now fix an integer nand a point x E An. According to the definition ofF there exists a sequence of integers kt, ... , kn and a point u E Ao such that

Now choose an arbitrary point v E Zo and define z E Zn by

Since the Si are Lipschitzean we have

We have proved that for every x E An there is a z E Zn such that lx- zl ~ 2r Ln. Since the assumptions concerning the sets A0 and Zo are symmetric this shows that the distance between An and Zn is smaller than 2r Ln. Consequently, (12.8.16} lim dist(An, Zn) = 0. R-+00

Step II. Choose an arbitrary nonempty compact sets Ao Zo

= A. = supp p.•.

Since p.. is invariant we also have

c

X and define


439

Substituting this into (12.8.16) we obtain (12.8.15) and the proof is complete. • It is worth noting that for systems which are not hyperbolic, equality (12.8.15) may be violated even if condition (12.8.5) is satisfied. In general the set lim F"(Ao) is larger than A.= supp p.•. Example 12.8.2. Let X = R, S1(x) = x and S2(x) = 0 for x E R. Evidently for every probabilistic vector (p1,P2) with Pl < 1 the condition (12.8.5) is satisfied. Thus the system is weakly asymptotically stable and there exists unique stationary distribution P.•. It is easy to guess that p.. = 60 • In fact, according to (12.8.3), P6o(A)

=Plc5o(S1 1 (A)) + P2c5o(S2 1 (A)),

where S} 1 (A) =A and

Sil(A)

= {R

ifO E A

0 ifO ¢A.

Therefore

= Plc5o(A) + P2c5o(A) = c5o(A). On the other hand, for Ao = (0, 1] we have P6o(A)

F(Ao) = S1(Ao) U S2(Ao) = (0, 1] U {0} = (0, 1] and by induction F"(Ao)

= (0, 1]

n=0,1, ....

This sequence does not converge to A. = supp p.. = {0}. 0 Now we are going to use equation (12.8.15) for the construction of attractors of hyperbolic systems. This procedure can often be simplified using the following result concerning the Barnsley operator (12.8.8). Proposition 12.8.3. Assume that the Si:X-+ X, i = 1, ... ,N appearing in equation (12.8.8) are continuous and that Ao C X is a compact set. Denote An = pn(Ao) and assume that A. = limn-+oo An exists. If Ao :,) F(Ao), then (12.8.17) Proof. The Barnsley operator F is monotonic, that is, A C B implies F(A) C F(B). Thus from A1 :,) Ao it follows F"(Al) :,) pn(A0 ) or An+l :,) An. It remains to prove that An :,:, A •. Fix an integer n and a point x E A •. Consider a sequence e; = 1/j. Since {An+lc} converges to A. as k -+ oo we can find a set An+A:U> which approximates A. with accuracy e;. There exists, therefore, x; E An+A:(j) such that lx;- xl 5 e;. Evidently x; E An

440


0

0

0

••••

1/~

0

II II

II II

0

1/~

FIGURE 12.8.1.

since, by the first part of the proof, An :::> An+k(j). The set An is closed and the conditions xi E An, xi -+ x imply x E An· This verifies the inclusion An :::> A. and completes the proof. • Our first example of the construction of an attractor deals with a onedimensional system given by two linear transformations. Despite the simplicity of the system the attractor is quite complicated. Example 12.8.3. Let X

St(x) Choose

= ix

= R and and S2(x)

= !x + j

forxeR.

Ao = (0, 1] (see Figure 12.8.1). Then At

= F(Ao) = St((O, 1]) U 82([0, 1]) = [0, i] U [j, 1].

Thus, At is obtained from Ao by taking out the middle open interval ( Now

i, j).

([0, }] u [j, 1]) u S2 ([0, }] u [j, 1]) [0, !] u [~, ~] u [t, ~] u [i, 1] .

A2 = F(At) = St =

Again A2 is obtained from At by taking out two middle open intervals (!, ~) and ( ~, i). Proceeding further we observe that this operation repeats and As can be obtained from A2 by taking out the four middle intervals. Thus, the set As consists of eight intervals of length 2~ . In general, An is


441

the sum of 2n intervals of length 1/3n. The Borel measure of An is {f} n and converges to zero as n -+ oo. The limiting set A. has Borel measure zero since it is contained in all sets An. This is the famous Cantor set-the source of many examples in analysis and topology. 0 Example 12.8.4. Let X = R 2 and Si(x)

=

(13 !0) + (ai) x

bi

i = 1,2,3,

,

where a t -- bt -- O·'

a2 -

12' ",._2-- O·'

a3 -

14' " ,._3-- 12.

Choose A0 to be the isosceles triangle with vertices (0, 0), (1, 0), (!, 1) (see Figure 12.8.2a). St(Ao) is a triangle with vertices {0,0), (},0}, (!,}).The triangles S 2(A0 } and S3(Ao) are congruent to St(Ao) but shifted to the right, and to the right and up, respectively. AB a result, the set At= F(Ao) = St(Ao) U S2(Ao) U S3(Ao) is the union of three triangles as shown in Figure 12.8.2b. Observe that At is obtained from Ao by taking out the middle open triangle with vertices (},0}, (!, }), (i, }). Analogously each set Si(At}, i = 1,2,3, consisting of three congruent triangles of height land A2 = F(At} in the union of nine triangles shown in Figure (12.8.2c). Again A2 can be obtained from At by taking out three middle triangles. This process repeats and in general An consists of 3n triangles with height (!}n, base (!}n, and total area m(An} =

! {if,

which converges to zero as n-+ oo. The limiting set A., called the Sierpinski triangle, has Borel measure zero. It is shown in Figure 12.8.2d. Unlike the Cantor set, the Sierpinski triangle is a continuum (compact connected set) and from a geometric point of view it is a line whose every point is a ramification point. The Sierpinski triangle also appears in cellular automata theory [Wolfram, 1983]. 0 In these two examples the construction of the sets An approximating A. was ad hoc. We simply guessed the procedure leading from An to An+l, taking out the middle intervals or middle triangles. In general, for an arbitrary iterated function system the connection between An and An+l is not so simple. In the next theorem we develop another way of approximating A. which is especially effective with the aid of a computer. Theorem 12.8.2. Let (12.8.1} be a hyperbolic system. Then for every x 0 E X and e > 0 there exist two numbers no= no(e) and ko = k0 (e) such that prob(dist({xn.···•Xn+fe},A.) <e)> 1-e

for n ?:

no. k ?: ko, (12.8.18)

where {Xn} denotes the tmjectory starting from Xo.

442


FIGURE 12.8.2.

In other words Theorem 12.8.2 says the following. If we cancel the first or more elements of the trajectory {xn}, then the probability that a sufficiently long segment Xn, ••• , Xn+fc approximates A. with accuracy e: is greater than 1 - e:.

no

Proof. Let e: > 0 be fixed. Choose a compact set Ao C X such that xo E Ao and F(Ao) c Ao. [From condition (12.8.4) and (12.8.6) it follows that such

a set exists.] The sequence An = F(Ao) is decreasing and Theorem 12.8.1 there is an integer no(e:) such that dist(An, A.)< e:

for n

By

~no.

From this inequality, for every value of the random vector An for which lxn- Znl < e: for n ~no. This determines the number

Xn E An.

Xn

there is

no appearing in condition (12.8.18).

Zn E

(12.8.19)


443

Now we are going to find /co. Since A. is a compact set there is a finite sequence of points ai EA., i = 1, ... , q such that q

A. c

U B~;2(ai)·

(12.8.20)

i=l

Pick a point u E An0 • The set {u}, which contains the single point u, is compact and according to Theorem 12.8.1 there exists an integer r such that dist(Fr({u}),A.) < ~· The points of Fr({u}) are given by Ba 1 o .. · o Sar(u). Thus, for every i = 1, ... , q, there exists a sequence of integers a(i, 1), ... , a(i, r) for which IBa(i,l) 0 " . 0 Ba(i,r)(u)- ail
0 fori= 1, ... , N. However, the convergence to A. may be slow if the value p = min Pi is small. Changing Pi does not change A. = supp J.l.•, but may change J.l.•· Example 12.8.5. Let X = R2 and take Si, i = 1, 2, 3 to be given by

S1(x)= S2(x)

(~ ~)x+ (3)

=(

S3(x) =

(

- r cos cp -r sin cp - rsincp

) )·

rcoscp

q cos 1/J

-r sin 1/J

q sin 1/J

r cos 1/J

where

c = 0.255,

r = 0. 75,

no =

ko = 150000,

q = 0.625,

Choosing 100,

lxol ~ 1,

P1 = P2 = P3 =

l

we obtain a representation of a tree as shown in Figure 12.8.3, as prepared for this volume by Dr. Z. Kielek. It should be noted that the "tree" has some natural asymmetry as usually appears in nature. 0 The objects shown in Figures 12.8.1, 12.8.2, and 12.8.3 are called fractals, a name derived from the Latin word fractus meaning broken or partial. One possible definition says that a fractal is a set which has a fractional dimension. To make this definition precise it is necessary to define a dimension applicable to a large class of sets. This is not an easy task and the several existing dimension definitions are, in general, not equivalent. Here we give a simplification of the Hausdorff dimension proposed by N. A. Kolmogorov. It is called the capacity or fractal dimension. To understand the ideas that lead to this notion, define a d-cube of size l by K = {(x1, ... , xd): ai ~xi ~ ai + l fori= 1, ... , d}.


445

FIGURE 12.8.3.

This set K is evidently ad-dimensional object. The question arises how to derive the number d from the intrinsic properties of the cube, neglecting the trivial fact that the index i takes on d values in the definition of K. Assume first that l = 1 and that K is subdivided into cubes of size En= 1/n. The number of these cubes is N(E) = nd. From this we obtain immediately d = lognd = logN(En) logn log(1/En)'

When the size l of the cube K is arbitrary the calculation is a little more complicated. Namely, K may be divided into cubes of size En = lfn and the number of these cubes is N(En) = nd. Consequently, d= lim n-+oo

dlogn =lim logN(En). logn -logl n-+oo log(1/En)

These calculations suggest the following. Definition 12.8.3. Let A C Rd be a compact set. For every E > 0 denote by N(E) the minimum number of cubes of size E needed to cover A. We define the dimension of K by the formula

. K d1m if this limit exists.

lim = e-+0

logN(E) log(1/E)

--.:::...,.....-"-'-

(12.8.23)

446

12. Markov

and Foias Operators

Calculation of the fractal dimension by a direct application of Definition 12.8.a is difficult. It may be simplified and the continuous variable e replaced by an appropriate sequence {en.}· Namely, if for some c > 0 and 0 < q < 1 we define En. = cq"', and if d

cq

= lim log N(en.) n-ooo log(1/en)

(12.8.24)

exists, then the limit (12.8.2a) also exists and dim K = dcq [Barnsley, 1988; Chapter 5). Using this property we may find the dimension of the attractors A. described in Examples 12.8.a and 12.8.4. First consider the case when A. is the Cantor set. We have

A.

c An = F"([O, 1))

and A. can be covered by 2"' disjoint intervals of length a-n whose sum is equal to An. Since A. contains the endpoints of these intervals the number of covering intervals cannot be made smaller. We have, therefore, N(en) = 2n for En= a-n. which gives . d1m A.cantor

log2

= log a.

For the Sierpinski triangle the situation is similar. We have

A. c An

= Fn(Ao),

where Ao is the initial triangle. The set An consists of an isosceles triangles of height 2-:-n and base length 2-n. Every such triangle can be covered by four squares of size 2-(n+l). On the other hand the vertices of these triangles belong to A.. It is necessary to use an different squares of size 2-(n+l) just to cover the top vertices of these triangles. Thus, for En = 2-(n+l) we have and consequently nloga < logN(en) < log4 + nloga (n + 1) log 2 - log(1/en) - (n + 1) log 2 which gives in the limit . dim A.sierplnski

loga

= log 2 .

Exercises

447

Exercises 12~1. Let X c R!'- be a compact set and C(X) be the space of continuous functions f: X -+ R. Using the Weierstrass approximation theorem prove that in C(X) there exists a dense countable subset of Lipschitzean functions.

12.2. Let X

c R!'- be a closed unbounded set. Using the family of functions fnw(x)

= w(x)max(l- n- 1 lxi,O),

where n is a positive integer and w: R!'- -+ R is a polynomial, show that in the space Co(X) there exists a dense countable subset of Lipschitzean functions. 12.3. Let X = [0, 211'] and let {Jtn} be the sequence of probabilistic measures with densities (1/11') sin2 nx. Find the weak limit I'• of {Jtn}· Is I'• also the strong limit of {Jtn}? 12.4. LetS: R-+ R be a continuous function such that S(x) ::/= x for x e R. Show that for the operator P~t(A) = ~t(S- 1 (A)) there does not exist an invariant probabilistic measure. 12.5. Let X = {0, 1, ... } be the set of nonnegative integers: Consider the iterated function system given by the two transformations for x EX, and the probability vector Pl asymptotically stable.

= P2 = ! . Show that this system is strongly

12.6. Generalize the previous result and consider an arbitrary iterated function system (12.8.1). Show that if 81 = 0 and P1 > 0, then this system is strongly asymptotically stable. 12.7. Let (12.8.1) be a hyperbolic dynamical system. Fix an arbitrary xo EX. Prove that for every sequence {in} with in E {1, ... , N} the limit X=

lim 8i1

n-+oo

0

8i 2

0 ••• 0

Si n (xo)

exists, and that the set of all such points x corresponding to all possible sequences {in} is equal to A. [Barnsley, 1988; Chapter 4]. 12.8. Consider the hyperbolic dynamical system given, on X eight transformations

i

= R 2 , by the

= 1, ... ,8,

1,

where (ai, bi) are all possible pairs made from the numbers (0, ~) excluding (l, l). The attractor A. of this system is called a Sierpinski carpet. Make a picture of A. and calculate dim A •.

References

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If A and Bare sets, then x E B means that "xis an element of B," whereas A c B means that "A is contained in B." For x ¢ B and A ¢. B substitute "is not" for "is" in these statements. Furthermore, A U B = {x: x E A or x E B}, AnB = {x:x E A and x E B}, A \B = {x:x E A and x ¢ B}, and Ax B = {(x,y):x E A andy E B}, respectively, define the union, intersection, difference, and Cartesian product of two sets A and B. Symbol0 denotes the empty set, and

lA(x)

1 ifxEA if x ¢A

={0

is the characteristic (or indicator) function for set A. When a < b the closed interval [a, bJ = {x: a ::::; x ::::; b}, whereas the open interval (a, b)= {x:a < x < b}. The half-open intervals [a, b) and (a, b] are similarly defined. The real line is denoted by R, and the positive half-line by R+. If A is a set and a E R, then aA = {y: x E A andy= ax}. The notation I:A-+ B means that "I is a function whose domain is A and whose range is in B," or "I maps A into B." Given two functions I: A -+ B and g: B -+ C, then g o I denotes the composition of g with I and go 1: A-+ C. If I maps R (or a subset of R) into R, and b is a positive

number, then g(x)

= l(x)

(mod b)

means that g(x) = l(x)- nb, where n is the largest integer less than or equal to l(x)fb. IIIIILP and {!,g), respectively, denote the V' norm ofthe

458


V!

function 1, and the scalar product of the functions I and g. I is used for the variation of the function I over the interval [a, b). The following is a list of the most commonly used symbols and their meaning:

a. e. A

8 d(g, :F)

d+lld.x D,D(X,A,p)

D2 <e>

1>(A)

E<e>

E(V I/) Ei(x)

{17t}

I I. :F {:Ft}

9ub(x) .p

9ij

Hn(x) H(/) H(/ I g) I

K(x,y) V,V(X,A,p)

v' p(A) J.'J(A) J.Lw

{Nth~o

w 0 (O,:F, prob) p

Pe,P {Fth~o prob Prob R>.

8

almost everywhere u-algebra Borel u-algebra L 1 distance between functions g and :F right lower derivative set of densities variance of random variable domain of an infinitesimal operator A mathematical expectation of a random variable expected value of V with respect to I exponential integral continuous time stochastic process an element of V, often a density stationary density V set of functions, u-algebra in probability space family of u-algebras Gaussian density with variance u 2 /2b Riemannian metric Hermite polynomial entropy of a density I conditional entropy of I with respect to g identity operator stochastic kernel V space space adjoint to V measure of a set A measure of a set A with respect to a density I Wiener measure counting process an element of 0; angular frequency space of elementary events probability space Markov or Frobenius-Perron operator Markov operator continuous semigroup generated by the linear Boltzmann equation probability measure probability measure on a product space resolvent operator transformation

e

e


s- 1 (A) Bm Sl {StheR, {Sth>o u(e) T Ttl {7t}t~o

u v {w(t)}t>O (X,A,~t)

e.e,

{en}, {et}

counterimage of a set A under a transformation S Chebyshev polynomial unit circle dynamical or semidynamical system standard deviation of a random variable transformation d-dimensional torus semigroup corresponding to an infinitesimal operator A Koopman operator Liapunov function, potential function Wiener process measure space random variables discrete or continuous time stochastic process

e

459

Index

Abel Inequality, 142 abstract ergodic theorem, 89 acoustic feedback system, 164 adjoint operator, 48, 49 almost everywhere (a.e.) 19, 38 almost sure convergence, 312 Anosov diffeomorphism 57, 77 arcs, equivalent, 177 asymptotic periodicity, 95, 117, 156 and constrictive operators, 99, 321, 322, 331, (333) and asymptotic stability, 105 of stochastically perturbed systems,321,322,331,(333) asymptotic stability, 104, 202 of Chandrasekhar-Miinch equation, 386 via change of variables, 165 of convex transformation, 154 of expanding mappings on manifolds, 184 of fluid flow, 156 of integral operators, 112, 115 of integro-differential equations, 379--386

of iterated function systems, 434 of Lorenz equations, 150 of measures, weak, 397, 421 of measures, strong, 402, 425, 430 of monotonically increasing transformation, 144 and overlapping support (137) proof via lower-bound function, 106, 201 of quadratic transformation, 166 relation to asymptotic periodicity, 105 relation to conditional entropy, 299 relation to exactness, 110 relation to Liapunov function, 115, 372, 378, relation to statistical stability, 105 of Renyi transformation, 145 of stochastic semigroups, 202 of stochastically perturbed sys-

462

Index

teiDS,317,323,325-326, 332, (333) of strong repellor, 154 of transformations on R, 172 automorphism, 80

baker transformation, 54, 65, 81, (83), 295 relation to dyadic transformation, 56, 295 Barnsley operator, 435 Bielecki function, 127, 387 and sweeping of stochastic semi-group, 245, 387 Birkhoff ergodic theorem continuous time, 196 discrete time, 63,64 Boltzmann equation and entropy, 295, 299 Borel measure, 19, 29-30 on manifolds, 182 Borel sets, 18 Borel u-algebra, 18 bounded variation, function of, 139-144 Brownian motion d-dimensional, 345 one-dimensional, 336

Cantor set, 441 dimension of, 446 capacity, 444 Cartesian product, 27 Cauchy condition for convergence, 34 Cauchy-Holder inequality, 27 Cauchy problem for Fokker-Planck equation, 364-365 cell proliferation, 114, 119-122, 127-129, 133-135, 343 Cesaro convergence, 31, 122 of Frobenius-Perron operator, 72 of Koopman operator, 75

Chandrasekhar-Miinch equation, 131, 240, 383 change of variables and asymptotic stability, 165171 in Lebesgue integral, 46 characteristic function, 5 Chebyshev inequality, 114, 310 Chebyshev polynomials, 169 classical solution of Fokker-Planck equation, 365 closed linear operator (248) closed linear subspace, 91 compact support, 207 comparison series, 31 complete space, 34 complete measure, 30 conditional entropy, 291-292, 299 connected manifold, 180 constant of motion, 214 constrictive Markov operator, 95, 96 and asymptotic periodicity, 98, 321-323 and perturbations, 321 continuous semigroup, 195 of contractions, 204 of contractions and infinitesimal operator, 226 and ordinary differential equations, 21G-214 continuous stochastic processes, 336 continuous stochastic semigroup, 229 continuous time stochastic process, 254 continuous time system asymptotically stable, 202 and discrete time systeiDS, 198, 251-252 ergodic, 197 exact, 199, 338-344 mixing, 198, 22G-224 sweeping, 244-245 contracting operator, 39,204

Index

convergence almost sure, 312 Cauchy condition for, 34 Cesaro, 31 comparison series for, 31 spaces, 33 in different in mean, 311 to set of functions, 160 stochastic, 311 strong, 31 strong, of measures, 402, 425 weak, 31 weak Cesaro, 31 weak convergence of measures, 397, 421 convex transformation, 153-156 counterimage, 5 counting measure, 18 counting process, 254 curvature, 225-226 cycle, 102 cyclical permutation, 102 cylinder, 221, 340

v

dense set, 97 dense subset of densities, 97 density, 5, 9, 41 of absolutely continuous measure, 41 Gaussian, 398, 409 evolution of by Fcobenius-Perron operator, 38, 241-243 of random variable, 253 stationary, 41 and sweeping, 125, 129, 386 derivative Radon-Nikodym, 41 right lower, 123 strong, 207 determinant of differential on manifold, 182 diffeomorphism, 58 differential determinant of, on manifold, 182

463

of transformation on manifold, 178 differential delay equation as semidynamical system, 190 differential equation as dynamical system, 190 dimension, 445 fractal, 444 Dirac measure, 395, 399, 403, 408, 411 discrete time stochastic process, 254 discrete time system and Poisson processes, 258261 as sampled semidynamical system, 251-252 embedded in continuous time system, 252 distance Hausdorff, 436 between measures, 401 in V spaces, 26 on manifold, 180 distribution, 394 invariant, 417 stationary, 417 dyadic transformation, 8, 66, 77, 295 related to baker transformation, 56 dynamical system, 191 and exactness, 199 ergodic, 197 mixing, 198 trace of, 193 elementary events, 253 endomorphism, 81 entropy, 284 conditional, 291 and exact transformations, 293, 295, 297 and Fcobenius-Perron operator, 292-295, 297

464

Index

and Hamiltonian systems, 293 and heat equation (300) and invertible transformations, 292 and Liouville equation, 295 and Markov operators, 289292 and quadratic map, 290 of reversible and irreversible systems, 295 equivalent arcs on manifold, 177 ergodic Birkhoff theorem, 63-64,

196 ergodic dynamical system, 197 ergodic Markov operator, 79, (83), 102 ergodic semidynamical system, 197 ergodic transformation, 59 ergodicity conditions for via FrobeniusPerron operator, 61, 72, 94, 220 conditions for via Koopman operator, 59, 75,215,220, 230 and Hamiltonian systems, 230 and linear Boltzmann equation, 273-276 illustrated, 68 of motion on torus, 216-218 necessary and sufficient conditions for, 59 relation to mixing, exactness and K -automorphisms, 80 and rotational transformation, 62, 75, 198 essential supremum, 27 Euler-Bernstein equations, 357 events elementary, 253 independent, 253 mutually disjoint, 253 exact Markov operator, 79, (83), 103 exact semidynamical system, 199

exact semidynamical system with continuous time, 339-344 exact semigroup of linear Boltzmann equation, 273 exact transformation, 66 exactness and asymptotic periodicity, 103 and entropy, 293, 295, 297 illustrated, 69 necessary and sufficient conditions for via FrobeniusPerron operator, 72,220 of r-adic transformation, 77 relation to dynamical systems, 199 relation to ergodicity, mixing, and K-automorphisms, 66, 79, 82, 199 relation to statistical stability, 110, 167 of transformations on torus, 186, (188) expanding mappings, 183 expanding Markov operator, 132 and asymptotic stability, 247 factor of transformation, 82 finite measure space, 19 first return map, 252 fixed point of Markov operator, 40 fluid flow, 156 Foias operator, 414 relation to Frobenius Perron operator, 416 relation to Koopman operator, 416 Foguel alternative to sweeping, 130 and expanding Markov operators, 133 and Fokker-Planck equation, 388 Fokker-Planck equation asymptotic stability of solutions, 372, 388, 390

Index

and Cauchy problem, 364-365 cl88Sical solution, 365 derivation of, 359-363 existence and uniqueness of solutions, 366 fundamental solution, 365 generalized solution, 368 for Langevin equation, 367, 374 and Liouville equation, 371 for second-order system, 376 stationary solutions, 374-376 for stochastic differential equations, 360 and stochastic semigroups, 369 and sweeping, 386-388 forced oscillator, 161-163 fractal, 444 dimension 444 Frobenius-Perron operator for densities, 37, 200 for Anosov diffeomorphism, 57 for baker transformation, 54 for dyadic transformation, 9 and evolution of densities, 513, 38 for Hamiltonian system, 214 and invariant measure, 52, 215, 229 for invertible transformations, 43, 45, 47 and Koopman operator, 48, 203 and Liouville equation, 230 as Markov operator, 43 and ordinary differential equations, 211-213 for quadratic map, 7, 53 for r-adic transformation, 9, 52 relation to entropy, 292-295, 297-298 relation to ergodicity, 61, 72, 94, 220

465

relation to infinitesimal operator, 211-213 relation to Koopman operator, 48, 203, 241-243 relation to mixing, 72, 220 and semidynamical systems, 199-200, 215 semigroups of, 199 support of, 44 for transformations on R, 10, 43, 172 for transformations on R 2 , 45 weak continuity of, 48 Frobenius-Perron operator for measures, 411 relation to Foias operator, 416 Fubini's theorem, 29 function Bielecki, 127 of bounded variation, 140 with compact support, 207 left lower semicontinuous, 123 Liapunov, 114, 115, 117 locally integrable, 130 lower bound, 106 lower semicontinuous, 122 subinvariant, 129 support of, 39 functional, linear, 395 fundamental solution of FokkerPlanck equation, 365 gas dynamics, 20Q-224, 277-280 Gaussian density, 286, 325, 336, 345, 398, 409, (410) Gaussian kernel, 202, 234, 239, 243, 366 generalized solution of Fokker-Planck equation, 368 geodesic, 225 flow, 225 motion on, 224-226 Gibbs canonical distribution function, 288 Gibbs inequality, 284

466

Index

gradient of function, 177 length of, 181 Hahn-Banachtheorem, 91 Hamiltonian, 213 system, 213, 218, 225, 23G231, 293 hat map, (50), 167, (188) Hausdorff distance, 436 and capacity, 444 and fractal dimension, 444 Hausdorff space, 177 heat equation, 203,234,243, (300), 409 Henon map, 56 Hille-Yosida theorem, 226, (248) homeomorphism, 176 hyperbolic iterated function system, 434, (447) ideal gas, 22G-224, 277-280 independent events, 253, (280) independent increments, 254 independent u-algebras, 344 independent random variables, 253, 304, 314 independent random vectors, 304, (333) indicator function, 5 inequality Cauchy-Holder, 27 Gibbs, 284 Jensen, 288 triangle, 26 infinitesimal operator, 206 of continuous semigroup of contractions, 226 and differential equations, 210 and ergodicity, 215 and Frobenius-Perron operator, 212-214, 229 and Hamiltonian systems, 213214 and Hille-Yosida theorem, 226

illustrated by parabolic differential equations, 234235 illustrated by heat equation, 234 and invariant measure, 229 and Koopman operator, 21G212, 230 and ordinary differential equations, 21Q-213 and partial differential equation, 206-209 as strong derivative, 206 integrable function, 22 integral Ito, 347 Lebesgue, 19-22 Riemann, 23 integro-differential equations, 238, 240, 379, 383 intermittency, 156 invariant measure, 52, 83), 196, 417 and differential equations, 230 and Frobenius-Perron operator, 52, 215 and Hamiltonian systems, 230 and infinitesimal operators, 229 and Liouville's theorem, 229230 and sweeping, 130 invariant set, 59, 197 invertibility, 56, 68, 19G-191, 292, 295 iterated function system, 433 attractor of, 436 and Cantor set, 441 hyperbolic, 434 and Sierpinski triangle, 441 weak asymptotic stability of, 434 Ito integral, 346-351 Ito sum, 347, (392)

Index

Jacobian matrix, 46 Jensen inequality, 288 joint density function, 304

K -automorphism, 80 and exactness, 82 and geodesic flows, 226 and mixing, 82 Keener map, 322-323 K-flow, 226 Kolmogorov automorphism, 80 Kolmogorov equation, see FokkerPlanck equation Koopman operator, 47-49, 203 and Anosov diffeomorphism, 77 and motion on torus, 216-218 relation to ergodicity, 59, 75, 215-216, 220, 230 relation to Frobenius-Perron operator, 48, 204, 241243 relation to infinitesimal operators, 21o-211, 230 relation to mixing, 75, 220 relation to ordinary differential equations, 21o-211 and rotational transformation, 75 Krylov-Bogolubov theorem, 419

Langevin equation, 357, 367, 374 law of large numbers strong, 314 wea.k, 313 Lebesgue decomposition of measures, 426 Lebesgue dominated convergence theorem, 22 Lebesgue integral, 19-22 on product spaces, 29 relation to Riemann integral, 23 Lebesgue measure, 30

467

Lebesgue monotone convergence theorem, 22 left lower semicontinuous function, 123 length of gradient on manifolds, 181 Liapunov function, 114, 117, 321, 325, 371, 378, 419 linear abstract Boltzmann equation, 264 linear Boltzmann equation, 261268, (280}, 299 linear functional, 395 linear operator, closed (248) linear subspace, 91 linear Tjon-Wu equation, 277, (280} linearly dense set, 31 Liouville equation, 229, 295, 371 Liouville's theorem, 229 locally finite measure, 393 locally integrable function, 130 Lorenz equations, 150 lower bound function, 106 conditions for existence, 122124, 183 relation to asymptotic stability, 106, 112, 210 lower semicontinuous function, 122 V, space adjoint to, 26 V distance, 26 V norm, 25 V space, 25 complete, 34 manifold, 175-183 connected, 180 d-dimensional, 176 geodesic flow on, 225 Markov operator, 37-38 adjoint operator to, 49 asymptotic periodicity, 95100, 117, 118, 321, 331, (333} asymptotic stability, 104, 202

468

Index constrictive, 95, 96 contractive property of, 39, 201

deterministic, (50) ergodic, 79, (83), 102 exact, 79, (83), 103 expanding, 132 fixed point of, 40 and Foias operator, 414 and Frobenius-Perron operator, 43 and linear abstract Boltzmann equation, 261-268 lower-bound function for, 106 for measures, 405 nrubdng, 79, (83), 104 and parabolic equation, 368 properties of, 38-39 relation to entropy, 289--292 semigroup of, 201 stability property of, 39, 202 stationary density of, 41 with stochastic kernel, 111, (136), 243, 270 and stochastic perturbation, 317, 320, 327, 331 s~ping, 125-129 weak continuity of, 49 mathematical expectation, 306 maximal entropy, 285-288 maximal measure, 435 Maxwellian distribution, 378 mean value of function, 139 of random variable, 306 measurable function, 19 space of, 25 measurable set, 18 measurable transformation, 41 measure, 18 absolutely continuous, 41 absolutely continuous part, 425 Borel, 19 complete, 30 continuous part of, 425

density of, 41 Dirac, 395, 399, 403, 408, 411 distance bet~n, 401 with Gaussian density, 398, 409 invariant, 52, (83), 196, 417 Lebesgue, 30 Lebesgue decomposition of, 426 locally finite, 393 maximal, 425 and Markov operator, 405 nonsingular, 425 norm of, 402 normalized, 41 preserving transformation, 52 product, 30, 259 probabilistic, 402, 403 singular part of, 425 stationary, 417 strong convergence of, 402, 425 support of, 394 uniqueness, 395 weak convergence of 397, 421 Wiener, 340-341 · measure-preserving transformation, 52, 196 measure space, 18 finite, 19 normalized, 19 probabilistic, 19 product of, 30 u-finite, 19 metric, Riemannian, 179 mixing, 65 of Anosov diffeomorphism, 77 of baker transformation, 65, (83) of dyadic transformation, 66 dynamical system, 198 illustrated, 70 Markov operator, 79, (83), 104 necessary and sufficient conditions for via FrobeniusPerron operator, 72, 220

Index

necessary and sufficient conditions for via Koopman operator, 75, 220 relation to ergodicity, exactness, and K -automorphisms, 66, 79, 82, 199 semidynamical system, 198 transformation, 65 modulo zero equality, 39 moments of solutions, 367-368 nonanticipative u-algebra, 347 nonsingular semidynamical system, 199 nonsingular transformation, 42 nontrivial lower-bound function, 106 norm in V, 25 of measure, 402 preservation of, 406 of vector on manifold, 180 normalized measure space, 19 normalized Wiener process, 336 one-dimensional Brownian motion, 336 one-dimensional Weiner process, 336 operator constrictive, 95-96 contractive, 39, 201 expanding, 132 Frobenius-Perron, 37, 200 infinitesimal, 206 Koopman, 47-49, 203 Markov, 37-38 resolvent, 227 oscillators, 161-163, 218-220 parabolic equation, 203 parabolicity condition, 364 paradox of weak repellor, 11, 150 partition function, 288 phase space, 1, 192

469

Phillip's perturbation theorem, 236, (248) piecewise convex transformations, 153-156 piecewise monotonic mappings, 144-153, 172 Poincare map, 252 Poincare recurrence theorem, 65 Poisson bracket, 213 Poisson distribution, 405 Poisson processes, 254-257, (280) Poisson's theorem, 405 probabilistic measure space, 19, 253 product measure, 30, 259 product space, 28, 259 proper cylinder, 221 quadratic transformation, 1, 7, (14), 53, 56, 166, 290, (333) r-adic transformation, 9, (15) 52, 77 Radon-Nikodym theorem, 24, 27 Radon-Nikodym derivative, 41 random number generator, 171 random variable, 253 density of, 253 independent, 253, 305, 314 mathematical expectation of, 306 mean value of, 306 standard deviation of, 309 variance of, 308 random vector, 304 randomly applied stochastic perturbation, 315-319 regular family, 129 regular Ito sum, 347 regular stochastic dynamical system, 413 Renyi transformation, 144 resolvent operator, 227 Riemann integral, relation to Lebesgue integral, 23

470

Index

Riemannian metric, 179 Riesz representation theorem, 395 right lower derivative, 123 rock drilling, 163 rotation on circle, 62, 75, (83), 198 rotation on torus, 216-218 sample path, 254 scalar product, 27 on manifolds, 180 semidynamical system, 195 and ergodicity, 197 and exactness, 199 and mixing, 198 semigroup continuous, 195 continuous stochastic, 201 of contracting operators, 204 of contractions, 204 of Frobenius-Perron operator,

199 of Koopman operator, 203 stochastic, 201 sweeping, 243 of transformations, 195 u-algebra, 18 Borel, 18 independent, 344 nonanticipative, 347 trivial, 80 u-finite measure, 394 u-finite measure space, 19 Sierpinski carpet, 447 triangle, 441 simple function, 21 space adjoint, 26 of measurable functions, 25 space and time averages, 64, 196 spectral decomposition theorem,

98 sphere bundle, 225 stability property of Markov operators, 39

standard deviation, 309 state space, 1 stationary density, 41 stationary distribution, 417 stationary independent increments,

254 stationary measure, 417 statistical stability, 105, (187) relation to asymptotic stability, 105 relation to exactness, 110 statistically stable transformation, construction of, 167 Stirling's formula, 267 stochastic convergence, 311 stochastic differential equations,

335,355 relation to Fokker-Planck equation, 359-360 stochastic integrals, 347, 353 stochastic kernel, 111, (136), 243,

274, 277 stochastic perturbation additive, 315, 320, 327, (333) and asymptotic periodicity,

321-323, 331, (333) constantly applied, 320 multiplicative, 330, (333-334) randomly applied, 315 small, 315, 327 stochastic processes, 254 continuous time, 254 and convergence of measures, 410 discrete time, 254 with independent increments,

254 with stationary independent increments, 254 stochastic semigroup, 201, (248) asymptotic stability of, 202 and Bielecki function, 245 relation to Fokker-Planck equation, 369 and sweeping, 245, (392)

Index Stratonvich sum, 350 strong asymptotic stability of measures, 425 in regular stochastic systems, 430 and weak asymptotic stability, 426, 434 strong convergence, 31 Cauchy condition for, 34 of densities, 72 of measures, 397, 402 strong law of large numbers, 314 strong precompactness, 86, (135) conditions for, 87-88 strong repellor, 153-156 subinvariant function, 129 support, 39 compact, 207 and Frobenius-Perron operator, 44 of measure, 394 sweeping 125-127, (136), 243-244, (333) and Bielecki function, 127, 245, 387 and Foguel alternative, 130, 133 and Fokker-Planck equation, 386-388 and invariant density, 130, 247 and stochastic semigroup, 243 tangent space, 178 tangent vector, 177 tent map, (50), 167, (188) time and space averages, 64, 196 torus, 186 Anosov diffeomorphism on, 57 d-dimensional, 216 exact transformation on, 186 rotation on, 216-218 trace of dynamical system, 193194 trajectory, 192 versus density, 10

471

transformation asymptotically periodic, 156165 convex, 153-156 ergodic, 59, 197 exact,66, 199 factor of, 82 Frobenius-Perron operator for, 7, 42, 199-200, 215 Koopman operator for, 47, 203 measurable, 41 measure-preserving, 52 mixing, 65, 198 nonsingular, 42 piecewise monotonic, 144, 153, 156, 165, 172 statistically stable, 105, 110 weakly mixing, 80 triangle inequality, 26 trivial set, 59, 197 trivial u-algebra, 80 uniform parabolicity, 364 unit volume function, 181 variance of function, 139 of random variable, 308 of Wiener process, 337 variation of function, 140 vector norm of, 180 scalar product of, 180 space, 26 von Neumann series, 265

weak asymptotic stability of iterated function systems, 434 of measures, 397, 421 and strong asymptotic stability, 426 weak Cesaro convergence, 31 of densities, 72

472

Index

weak continuity, 49 weak convergence, 31 of densities, 72 of measures, 397, 400, 401 weak law of large numbers, 313 weak precompactness, 86 condition for, 87-88 weak repellor, paradox of, 11, (15), 151

weakly mixing transformation, 80 Wiener measure, 34Q-341 Wiener process d-dimensional, 345 normalized, 337 one-dimensional, 336, (391) variance of, 337 Yorke inequality, 143-144

Applied Mathematical Sciences (continued from page ii) 61. Sattinger!Weaver: Ue Groups and Algebras with Applications to Physics, Geometry, and Mechanics. 62. LaSalle: The Stability and Control of Discrete Processes. 63. Grasman: Asymptotic Methods of Relaxation Oscillations and Applications. 64. Hsu: Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems. 65. Rand/Armbruster: Perturbation Methods, Bifurcation Theory and Computer Algebra. 66. Hlavdcek/Haslinger/NecasVLovfsek: Solution of Variational Inequalities in Mechanics. 67. Cercignani: The Boltzmann Equation and Its Applications. 68. Ternam: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. 69. Golubitsky/Stewart/Schae.ffer: Singularities and Groups in Bifurcation Theory, Vol. II. 70. Constantin/Foias/Nicolaenko/Ternam: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. 71. Catlin: Estimation, Control, and the Discrete Kalrnan Filter. 72. Lochak/Meunier: Multiphase Averaging for Classical Systems. 73. Wiggins: Global Bifurcations and Chaos. 74. Mawhin/Willem: Critical Point Theory and Hamiltonian Systems. 75. Abraham!Marsden!Ratiu: Manifolds, Tensor Analysis, and Applications, 2nd ed. 76. Lagerstrom: Matched Asymptotic Expansions: Ideas and Techniques. 77. Aldous: Probability Approximations via the Poisson CIUJq!ing Heuristic. 78. Dacorogna: Direct Methods in the Calculus of Variations. 79. Herndndez-Lerma: Adaptive Markov Processes. 80. Lowden: Elliptic Functions and Applications. 81. Blurnan/Kumei: Symmetries and Differential Equations. 82. Kress: Unear Integral Equations, 2nd ed. 83. Bebernes!Eberly: Mathematical Problems from Combustion Theory. 84. Joseph: Fluid Dynamics of Viscoelastic Fluids. 85. Yang: Wave Packets and Their Bifurcations in Geophysical Fluid Dynamics. 86. Dendrinos/Sonis: Chaos and Socio-Spatial Dynamics. 87. Weder: Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media. 88. BogaevskVPovzner: Algebraic Methods in Nonlinear Perturbation Theory.

89. O'Malley: Singular Perturbation Methods for Ordinary Differential Equations. 90. Meyer/Hall: Introduction to Hamiltonian Dynamical Systems and the N-body Problem 91. Straughan: The Energy Method, Stability, and Nonlinear Convection. 92. Naber: The Geometry of Minkowski Spacetime. 93. Colton/Kress: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. 94. Hoppensteadt: Analysis and Simulation of Chaotic Systems, 2nd ed. 95. Hackbusch: Iterative Solution of Large Sparse Systems of Equations. %. Marchiora!Pulvirenti: Mathematical Theory of Incompressible Nonviscous Fluids. 97. Lasota/Mackey: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, 2nd ed. 98. de Boor/Hollig/Riemenschneider: Box Splines. 99. Hale!Lunel: Introduction to Functional Differential Equations. 100. Sirovich (ed): Trends and Perspectives in Applied Mathematics. 101. Nusse!Yorke: Dynamics: Numerical Explorations, 2nded. 102. Chossatllooss: The Couette-Taylor Problem 103. Chorin: Vorticity and Turbulence. 104. Farkas: Periodic Motions. 105. Wiggins: Normally Hyperbolic Invariant Manifolds in Dynamical Systems. 106. CercignanVlllner!Pulvirenti: The Mathematical Theory of Dilute Gases. 107. Antman: Nonlinear Problems of Elasticity. 108. Zeidler: Applied Functional Analysis: Applications to Mathematical Physics. 109. Zeidler: Applied Functional Analysis: Main Principles and Their Applications. 110. Diekmannlvan Gils!Verduyn LuneVWalther: Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Ill. Visintin: Differential Models of Hysteresis. 112. Kuznetsov: Elements of Applied Bifurcation Theory. 2nd ed. 113. Hislop/Sigal: Introduction to Spectral Theory: With Applications to Schrlldinger Operators. 114. Kevorkian/Cole: Multiple Scale and Singular Pertorbation Methods. 115. Taylor: Partial Differential Equations I, Basic Theory. 116. Taylor: Partial Differential Equations II, Qualitative Studies of Unear Equations. 117. Taylor: Partial Differential Equations ill, Nonlinear Equations.

(continued on next page)

Applied Mathematical Sciences Icontinued from previous poge) 118. GodJewskVRaviart: Numerical Approximation of Hyperbolic Systems of Conservation Laws. 119. Wu: Theory and Applications of Partial Functional Differential Equations. 120. Kirsch: An Introduction to the Mathematical Theory of Inverse Problems. 121. Brokate/Sprekels: Hysteresis and Phase Transitions. 122. Glildikh: Global Analysis in Mathematical Physics: Geometric and Stochastic Methods. 123. Le/Schmitt: Global Bifurcation in Variational Inequalities: Applications to Obstacle and Unilateral Problems. 124. Polak: Optimization: Algorithms and Consistent Approximations. 125. Amold/Khesin: Topological Methods in Hydrodynamics. 126. Hoppenstead/1/zhikevich: Weakly Connected Neural Networks. 127. Isakov: Inverse Problems for Partial Differential Equations. 128. LVWiggins: Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrlldinger Equations. 129. MUller: Analysis of Spherical Synunetries in Euclidean Spaces. 130. Feintuch: Robust Control Theory in Hilbert Space. 131. Ericksen: Introduction to the Thermodynamics of Solids, Revised ed.

132. lhlenburg: Finite Element Analysis of Acoustic Scattering. 133. Vorovich: Nonlinear Theory of Sballow SheDs. 134. Vein/Dale: Detenninants and Their Applications in Mathematical Physics. 135. Drew!Passman: TheoryofMulticomponent fluids. 136. Cioranescu/Saint Jean Paulin: Homogenization of Reticulated Structures. 137. Gurtin: Configurational Forces as Basic Concepts of Continuwn Physics. 138. Haller: Chaos Near Resonance. 139. Sulem!Sulem: The Noulinear Schrl!dinger Equation: Self-Focusing and Wave Collapse. 140. Cherkaev: Variational Methods for Structural Optimization.

141. Naber: Topology, Geometry, and Gauge Fields: Interactions. 142. Schmid/Henningson: Stability and Transition in Shear flows. 143. SeiVYou: Dynamics of Evolutionary Equations. 144. Nldelec: Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems. 145. Newton: TheN-Vortex Problem: Analytical Techniques. 146. Allaire: Shape Optimization by the Homogenization Method. 147. Aubert/Komprobst: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations.

In recent years there has been an explosive growth in the study of physical , biological, and economic systems that can be profitably studied using densities. Because of the general inaccessibility of the mathematical literature to the nonspecialist, little diffusion of the applicable mathematics into the study of these "chaotic" systems has taken place. This book will help bridge that gap. To show how densities arise in simple deterministic systems , the authors give a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integra-partial-differential equations. Examples have been drawn from many fields to illustrate the utility of the concepts and techniques presented, and the ideas in this book should thus prove useful in the study of a number of applied sciences. The authors assume that the reader has a knowledge of advanced calculus and differential equations. Basic concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and stochastic integrals and differential equations are introduced as needed . Physicists, chemists , and biomathematicians studying chaotic behavior will find this book of value. It will also be a useful reference or text for mathematicians and graduate students working in ergodic theory and dynamical systems.

ISBN 0-387 -94049-9 ISBN 3-540-94049-9 www .spri nger-ny.com

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