Stability of infinite dimensional stochastic differential equations with applications

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STABILITY OF INFINITE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH APPLICATIONS

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics Main Editors H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)

Editorial Board R. Aris, University of Minnesota G.I. Barenblatt, University of California at Berkeley H. Begehr, Freie Universität Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware R. Glowinski, University of Houston D. Jerison, Massachusetts Institute of Technology K. Kirchgässner, Universität Stuttgart B. Lawson, State University of New York B. Moodie, University of Alberta L.E. Payne, Cornell University D.B. Pearson, University of Hull G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University of Adelaide

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

135

STABILITY OF INFINITE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH APPLICATIONS

Kai Liu

Boca Raton London New York Singapore

C598X_Discl.fm Page 1 Wednesday, July 13, 2005 2:45 PM

Published in 2006 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-58488-598-X (Hardcover) International Standard Book Number-13: 978-1-58488-598-6 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of T&F Informa plc.

and the CRC Press Web site at http://www.crcpress.com

To My Parents: Zhiyun Liu and Fengdi Cai

Contents

Preface

ix

1 Stochastic Diﬀerential Equations in Inﬁnite Dimensions 1.1 Notations, Deﬁnitions and Preliminaries . . . . . . . . . . . 1.2 Wiener Processes and Stochastic Integration . . . . . . . . 1.3 Stochastic Evolution Equations . . . . . . . . . . . . . . . . 1.3.1 Variational Approach and Strong Solutions . . . . . 1.3.2 Semigroup Approach and Mild Solutions . . . . . . . 1.4 Deﬁnitions and Methods of Stability . . . . . . . . . . . . . 1.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

1 1 11 16 18 22 29 37

2 Stability of Linear Stochastic Diﬀerential Equations 2.1 Stable Semigroups . . . . . . . . . . . . . . . . . . . . 2.1.1 Lyapunov Functions . . . . . . . . . . . . . . . 2.1.2 A Useful Stability Criterion . . . . . . . . . . . 2.2 Lyapunov Equations and Stability . . . . . . . . . . . 2.2.1 Characterization of Mean Square Stability . . . 2.2.2 Almost Sure Pathwise Stability . . . . . . . . . 2.3 Uniformly Asymptotic Stability . . . . . . . . . . . . 2.4 Some Examples . . . . . . . . . . . . . . . . . . . . . 2.5 Notes and Comments . . . . . . . . . . . . . . . . . .

. . . . . . . . .

39 39 42 51 54 55 63 70 73 78

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

3 Stability of Nonlinear Stochastic Diﬀerential Equations 3.1 Equivalence of Lp -Stability and Exponential Stability . . . . 3.1.1 An Extension of Linear Stability Criteria . . . . . . . 3.1.2 Comparison Approaches . . . . . . . . . . . . . . . . . 3.2 A Coercive Decay Condition . . . . . . . . . . . . . . . . . . 3.3 Stability of Semilinear Stochastic Evolution Equations . . . . 3.4 Lyapunov Functions for Strong Solutions . . . . . . . . . . . 3.5 Two Applications . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Stability in Probability . . . . . . . . . . . . . . . . . 3.5.2 Ultimate Boundedness and Invariant Measures . . . . 3.6 Further Results on Invariant Measures . . . . . . . . . . . . 3.7 Stability, Ultimate Boundedness of Mild Solutions and Invariant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Lyapunov Functions for Mild Solutions . . . . . . . . . 3.7.2 Ultimate Boundedness and Invariant Measures . . . .

81 81 83 88 94 102 110 121 121 124 131 137 137 143

vii

viii

Contents 3.8

Decay Rates of Systems . . . . . . . . . . 3.8.1 Decay in the p-th Moment . . . . . 3.8.2 Almost Sure Pathwise Decay . . . 3.9 Stabilization of Systems by Noise . . . . 3.9.1 Nonlinear Deterministic Equations 3.9.2 Nonlinear Stochastic Equations . . 3.10 Lyapunov Exponents and Stabilization . 3.11 Notes and Comments . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

150 151 154 159 160 164 166 173

4 Stability of Stochastic Functional Diﬀerential Equations 4.1 Linear Deterministic Equations . . . . . . . . . . . . . . . . 4.1.1 Stable Semigroups (Finite Delays) . . . . . . . . . . 4.1.2 Stable Semigroups (Inﬁnite Delays) . . . . . . . . . . 4.2 Stability Equivalence and Reduction of Neutral Equations 4.2.1 Stability of Retarded Stochastic Systems . . . . . . . 4.2.2 Stability of Neutral Stochastic Systems . . . . . . . 4.3 Decay Criteria of Stochastic Delay Diﬀerential Equations . 4.3.1 Nonlinear Coercive Conditions for Decay . . . . . . 4.3.2 Linear Stability Conditions . . . . . . . . . . . . . . 4.4 Razumikhin Type Stability Theorems . . . . . . . . . . . . 4.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

175 175 176 186 192 192 201 212 213 217 220 227

5 Some Related Topics of Stability and Applications 229 5.1 Parabolic Equations with Boundary and Pointwise Noise . . 229 5.2 Stochastic Stability and Quadratic Control . . . . . . . . . . 236 5.2.1 Optimal Control on a Finite Interval . . . . . . . . . . 237 5.2.2 Optimal Control on an Inﬁnite Interval . . . . . . . . 241 5.3 Feedback Stabilization of Stochastic Diﬀerential Equations . 246 5.4 Stochastic Models in Mathematical Physics . . . . . . . . . . 251 5.4.1 Stochastic Reaction-Diﬀusion Equations . . . . . . . . 251 5.4.2 Stochastic Navier-Stokes Equations . . . . . . . . . . . 254 5.5 Stochastic Systems Related to Multi-Species Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 5.6 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . 264 Appendix 267 A The Proof of Proposition 4.2.4 . . . . . . . . . . . . . . . . . 267 B Existence and Uniqueness of Strong Solutions of Stochastic Delay Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . 268 References

273

Index

297

Preface

In most dynamical systems which describe processes in engineering, physics and economics, stochastic components and random noise are included. The stochastic aspects of the models are used to capture the uncertainty about the environment in which the system is operating and the structure and parameters of the models of physical processes being studied. Stochastic diﬀerential equations in inﬁnite dimensional spaces are motivated by the development of analysis and the theory of stochastic processes itself such as stochastic partial diﬀerential equations and stochastic delay diﬀerential equations on the one hand, and by such topics as stochastic control, population biology and turbulence in applications on the other. The analysis and control of such systems then involves investigating their stability, which is a qualitative property and often regarded as the ﬁrst characteristic of the dynamical systems (or models) studied. Although the theory of stochastic diﬀerential equations in inﬁnite dimensional spaces is already an established area of research, the corresponding study of stability properties only saw a rapid growth in the last two decades. In particular, most of the existing results are scattered throughout research journals and conference proceedings. These results have been obtained by using various methods, concepts and theorems from functional analysis, stochastic partial diﬀerential equations and functional diﬀerential equations. This makes it diﬃcult for a newcomer to enter this interesting and important ﬁeld. The purpose of this book is to provide a systematic presentation of the contemporary theory of stability of stochastic diﬀerential equations in inﬁnite dimensional spaces, mainly Hilbert spaces. The applications of this theory to various concrete problems are also shown. Hopefully, the style of presentation will be helpful in making the up to date material in this ﬁeld accessible and meanwhile lay the foundation for future researches. The fundamental prerequisite for an intelligent reading of most material in this book is a knowledge of stochastic diﬀerential equations in inﬁnite dimensions, for instance, at the level of the initial seven chapters in Da Prato and Zabczyk’s book [1]. An acquaintance with stability theory of ﬁnite dimensional stochastic diﬀerential equations such as those in Khas’minskii’s classic book [1] will prove useful, although not essential. A brief description of the organization of this work follows. In Chapter 1, we recall basic concepts of the theory of stochastic diﬀerential equations in inﬁnite dimensional spaces, mainly Hilbert spaces. In this way, such notions as Q-Wiener processes, stochastic integrals, strong and mild solutions will

ix

x

Preface

be appropriately reviewed. We will also present some proofs of the results which are not available in existing books and are to be found scattered in the literature. Chapter 2 of the book is devoted to an investigation of stability for the essential classes of linear stochastic evolution equations. The central result is a formulation of the characteristic conditions of mean square exponential stability in terms of stationary Lyapunov type equations. We also deal in this chapter with the topic of almost sure pathwise stability of solution processes (strong and mild) of stochastic linear evolution systems. In Chapter 3, we proceed to the study of nonlinear stochastic diﬀerential equations. This chapter contains fundamental theory and interesting examples concerning a number of topics involved with the stability behavior of nonlinear systems. In particular, we generalize linear characteristic results in Chapter 2 in a suitable way to obtain a version for a class of nonlinear stochastic evolution systems. Motivated by the idea of reducing the stability of a nonlinear stochastic system to problems about a linear system, we explore the so-called Lyapunov function characterization method and the associated ﬁrst order approximation technique. Various interesting topics such as stabilization, Lyapunov exponents and stochastic decay rates of stochastic systems are also investigated. Chapter 4 is an extensive study of the stability theory of stochastic functional diﬀerential equations in inﬁnite dimensions. The reduction of a stochastic neutral equation to a stochastic retarded equation with inﬁnite delay and the notion of the L2 -stability in mean are developed. As a consequence, the established result that L2 -stability in mean implies asymptotic stability allows one to apply the Lyapunov function method to obtain stability criteria. For nonlinear cases, the methods of Lyapunov functionals and Razumikhin types are emphasized and contrasted, and applications are illustrated by interesting examples. In Chapter 5, selected topics and applications are presented in which the choice of the material reﬂects my own personal preference. The treatment here is somewhat sketchy and by no means the only approach or even the usual one. The purpose of our formulation here is to show some of the topics, especially applications such as stochastic optimal control and feedback stabilization, stochastic reaction-diﬀusion, Navier-Stokes equations and stochastic population dynamics, beyond the main scheme of the book, but associated with the stability theory of stochastic evolution equations. I hope also to stimulate further work in these and relevant ﬁelds. The appendix contains one proof of existence and uniqueness of strong solutions for a certain class of stochastic functional diﬀerential equations. As far as I know, there seems not to be a similar presentation in the existing literature. At this stage, I feel it appropriate to present this as a background for the arguments in the previous chapters. Notes and comments at the end of each chapter contain historical and related background material as well as references to the results discussed in that

Preface

xi

chapter. The pervading inﬂuence of a variety of authors’ work on this book is obvious. I have drawn freely on their work and hopefully I can acknowledge my scientiﬁc debt to them by some remarks shown there. The book is mainly devoted to the stability theory of stochastic diﬀerential equations in Hilbert spaces. It should be emphasized that my choice of material is highly subjective. This subject appears to be important and has recently been receiving increasing attention. As a consequence, the volume of relevant literature is rapidly expanding. In particular, the lengthy list of references at the end of the book is somewhat incomplete and only includes those titles which pertain directly to the contents. The author wishes to apologize to those people whose research might have been overlooked. It should be pointed out that this work would not have been possible without the inspiration and wisdom of my colleagues. I am indebted to many good friends who read the ﬁrst draft on which this book is based, corrected errors, and suggested improvements. In particular, I would like to express my sincere thanks to T. Caraballo, P. L. Chow, J. A. Langa, V. Mandrekar, B. Maslowski and Y. F. Shi for their valuable advice. Gratitude goes to Professor A. Truman for his constructive suggestions on the initial two chapters and Professor P. Giblin for his careful check of language errors. Thanks also go to Professor Jiezhong Zou, Professor Zhiyuan Huang and Dr. Xiaogu Zheng for their lasting concern and encouragement. The author acknowledges the various ﬁnancial supports from EPSRC, Royal Society, LMS and the University of Liverpool in UK during the preparation of this book.

Kai Liu

Chapter 1 Stochastic Diﬀerential Equations in Inﬁnite Dimensions

We begin by recalling some basic deﬁnitions and preliminaries, especially those on stochastic integration and stochastic diﬀerential equations in inﬁnite dimensional spaces. We recall important inequalities for stochastic integrals with respect to Wiener processes which are essential for the subsequent developments. We also establish two notions of solutions, strong and mild, and investigate the existence and uniqueness of these two kinds of solutions under suitable assumptions. To present the proofs of all of these results here would require preparatory background material which would considerably increase both the size and scope of this book. Therefore, we would like to adopt the approach of omitting the proofs of those results which are treated in detail in well-known standard books, such as Da Prato and Zabczyk [1], Rozovskii [1] and M´etivier [1]. However, those proofs will be presented which are not available in existing books and are to be found scattered in the literature, or which discuss ideas specially relevant to our purposes.

1.1

Notations, Deﬁnitions and Preliminaries

A measurable space is a pair (Ω, F) where Ω is a set and F is a σ-ﬁeld, also called a σ-algebra, of subsets of Ω. This means that the family F contains the set Ω and is closed under the operation of taking complements and countable unions of its elements. If (Ω, F) and (S, S) are two measurable spaces, then a mapping ξ from Ω into S such that the set {ω ∈ Ω : ξ(ω) ∈ A} = {ξ ∈ A} belongs to F for arbitrary A ∈ S is called a measurable mapping or a random variable from (Ω, F) into (S, S). A random variable is called simple if it takes on only a ﬁnite number of values. In this book, we shall only be concerned with the case when S is a complete, separable, metric space, then S = B(S), the Borel σ-ﬁeld of S which is the smallest σ-ﬁeld containing all closed (or open) subsets of S. If S is a separable Banach or Hilbert space, we shall denote its norm by · S and its topological dual by S ∗ . A probability measure on a measurable space (Ω, F) is a σ-additive function P from F into [0, 1] such that P (Ω) = 1. The triplet (Ω, F, P ) is called a

1

2

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

probability space. If (Ω, F, P ) is a probability space, we set F¯ = {A ⊂ Ω : ∃B, C ∈ F; B ⊂ A ⊂ C, P (B) = P (C)}. ¯ the probability Then F¯ is a σ-ﬁeld, called the completion of F. If F = F, space (Ω, F, P ) is said to be complete. Let (Ω, F, P ) denote a complete probability space. A family {Ft }, t ≥ 0, for which the Ft are sub-σ-ﬁelds of F and form an increasing family of σﬁelds, is called a ﬁltration if Fs ⊂ Ft ⊂ F for s ≤ t. With the{Ft }t≥0 , one can associate two other ﬁltrations by setting σ-ﬁelds: Ft− = s 0, Ft+ = s>t Fs if t ≥ 0, where s
∀A ∈ S.

The measure is called the distribution or the law of ξ. Deﬁnition 1.1.1 With respect to an increasing family {Ft }t≥0 of sub-σﬁelds of the type discussed above, we call a function τ : Ω → R+ = [0, ∞] a stopping time for {Ft }, t ≥ 0, if {ω : τ (ω) ≤ t} ∈ Ft for every t ≥ 0. A similar notion is deﬁned for a family {Ft }, t ∈ [0, T ], 0 ≤ T ≤ ∞, with the assumption that the range of τ is now in [0, T ]. Deﬁne the σ-ﬁeld of events prior to τ , denoted by Fτ , as Fτ = {A ∈ F∞ : A ∩ {τ ≤ t} ∈ Ft for every t}. Theorem 1.1.1 (a). (b). (c). (d).

For σ and τ stopping times relative to {Ft }t≥0 ,

Fτ is a σ-ﬁeld; τ is Fτ -measurable; σ ≤ τ almost surely implies Fσ ⊂ Fτ ; σ ∧ τ and σ ∨ τ are also stopping times with respect to {Ft }t≥0 .

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

3

Assume now that S is a separable Banach space with norm · S and ξ is an S-valued random variable on (Ω, F, P ). By a standard limit argument, we can deﬁne the integral Ω ξdP of ξ with respect to the probability measure P , often denoted by E(ξ). The integral deﬁned in this way is called Bochner’s integral. We denote by L1 (Ω, F, P ; S) the set of all equivalence classes of S-valued random variables with respect to the equivalence relation of almost sure equality. In the same way as for real random variables, one can check that L1 (Ω, F, P ; S), equipped with the norm ξ1 = EξS , is a Banach space. In a similar manner, one can deﬁne Lp (Ω, F, P ; S) for arbitrary p > 1 with norms ξp = (EξpS )1/p ,

p ∈ (1, ∞),

and ξ∞ = ess. sup ξ(ω)S . ω∈Ω

If Ω is an interval [0, T ], F = B([0, T ]), 0 ≤ T < ∞, and P is the Lebesgue measure on [0, T ], we also write Lp (0, T ; S), or more simply Lp (0, T ), for the spaces deﬁned above when no confusion arises. Of great interest to us will be operator-valued random variables and their integrals. Let K and H be two separable Hilbert spaces with norms · K , · H and inner products ·, ·K , ·, ·H , respectively. We denote by L(K, H) the set of all linear bounded operators from K into H, equipped with the usual operator norm · . From now on, without further speciﬁcation we always use the same symbol · to denote norms of operators regardless of the spaces involved when no confusion is possible. The set L(K, H) is a linear space and, equipped with the operator norm, becomes a Banach space. Unfortunately, if both the spaces K and H are inﬁnite dimensional, the space L(K, H) is not generally separable. A direct consequence of this inseparability is that Bochner’s integral deﬁnition cannot be applied to L(K, H)-valued random variables. One of the methods of overcoming these diﬃculties is to introduce a weaker concept of measurability. A mapping Φ(·) from Ω into L(K, H) is said to be strongly measurable if for arbitrary k ∈ K, Φ(·)k is measurable as a mapping from (Ω, F) into (H, B(H)). Let F(L(K, H)) be the smallest σ-ﬁeld of subsets of L(K, H) containing all sets of the form {Φ ∈ L(K, H) : Φk ∈ A},

k ∈ K,

A ∈ B(H),

then Φ : Ω → L(K, H) is a strongly measurable mapping from (Ω, F) into the space (L(K, H), F(L(K, H))). Elements of F(L(K, H)) are called strongly measurable. Mapping Φ is said to be Bochner integrable with respect to the

4

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

measure P if for arbitrary k, the mapping Φ(·)k is Bochner integrable and there exists a bounded linear operator Ψ ∈ L(K, H) such that Φ(ω)kP (dω) = Ψk, k ∈ K. Ω

The operator Ψ is then denoted as Ψ= Φ(ω)P (dω) Ω

and called the strong Bochner integral of Φ. This integral has many of the properties of the Lebesgue integral. For instance, it is easy to show that if K and H are both separable, then Φ(·) is a measurable function and Ψ ≤ Φ(ω)P (dω). Ω

The following operator spaces are of fundamental importance. Let K and H be two separable Hilbert spaces. We denote by L1 (K, H) the space of all nuclear operators and by L2 (K, H) the space of all Hilbert-Schmidt operators from K into H. It can be proved that the spaces L1 (K, H) and L2 (K, H) are both separable Banach spaces (L2 (K, H) is actually a Hilbert space) under the usual norms and are strongly measurable subsets of L(K, H). Assume that S is a separable Banach space with norm · S and let B(S) be the σ-ﬁeld of its Borel subsets. Let (Ω, F, P ) be a probability space. An arbitrary family X = {Xt }t≥0 of S-valued random variables Xt , t ≥ 0, deﬁned on Ω is called a stochastic process. Sometimes, we also write X(t, ω) = X(t) = Xt (ω) for all t ≥ 0 and ω ∈ Ω. The functions X· (ω) are called the trajectories of X. In our study of stochastic processes, we need some additional regularities to proceed with our programme. To this end, we now introduce several deﬁnitions of regularity for a process X on I = [0, T ), where T could be ﬁnite or inﬁnite. (a). X is measurable if the mapping X(·, ·) : I×Ω → S is B(I)×F-measurable (all stochastic processes considered in this book will be assumed to be measurable); (b). Let {Ft }, t ∈ I, be an increasing family of σ-ﬁelds. The process X is {Ft }t∈I -adapted if each Xt is measurable with respect to Ft , t ∈ I; (c). X is stochastically continuous at t0 ∈ I if, ∀ε > 0, ∀δ > 0, ∃ρ > 0 such that P {Xt − Xt0 S ≥ ε} ≤ δ,

∀t ∈ [t0 − ρ, t0 + ρ] ∩ [0, T );

(d). X is stochastically continuous in I if it is stochastically continuous at every point of I; (e). X is continuous with probability one (or continuous) if its trajectories X(·, ω) are continuous almost surely.

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

5

Deﬁnition 1.1.2 (i). The S-valued processes X = {Xt } and Y = {Yt }, t ∈ I, deﬁned on the probability space (Ω, F, P ) are called equivalent if for every {t1 , · · · , tn } ⊂ I and set Bi ∈ B(S), i = 1, 2, · · · , n, P {ω : Xt1 ∈ B1 , · · · , Xtn ∈ Bn } = P {ω : Yt1 ∈ B1 , · · · , Ytn ∈ Bn }; (ii). A stochastic process Y = {Yt }, t ∈ I, is called a modiﬁcation or a version of X = {Xt } if P {ω ∈ Ω : Xt (ω) = Yt (ω)} = 0,

∀t ∈ I;

(iii). The processes X = {Xt } and Y = {Yt }, t ∈ I, are called indistinguishable if P {ω ∈ Ω : Xt (ω) = Yt (ω); ∀t ∈ I} = 1. An immediate consequence of Deﬁnition 1.1.2 is that if Y is a modiﬁcation of X, and supposing that both of the processes have almost surely continuous sample paths, then X and Y are indistinguishable. Note that if X is a stochastic process on I = [0, T ), the function X(·, ·) need not be measurable in the product space B([0, T )) × F. However, we have the following result: Theorem 1.1.2 Let Xt , t ∈ I, be a stochastically continuous process with values in the separable Banach space S. Then X has a measurable modiﬁcation. Deﬁnition 1.1.3 Suppose X = {Xt }, t ∈ I, is an S-valued process and {Ft }t∈I is a normal increasing family of sub-σ-ﬁelds of F. X is said to be progressively measurable with respect to {Ft }t∈I if for every t ∈ I, the mapping [0, t] × Ω → S,

(s, ω) → X(s, ω),

is B([0, t]) × Ft -measurable. It is obvious that if X is progressively measurable with respect to {Ft }t∈I , then it must be both measurable and {Ft }t∈I -adapted. The converse of this statement is known to be false. However, if we are concerned only with versions of a given process, this converse can hold in the following sense: Theorem 1.1.3 Let Xt , t ∈ I, be a stochastically continuous and adapted process with values in the separable Banach space S. Then X has a progressively measurable modiﬁcation.

6

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Given an S-valued process X = {Xt }, t ∈ I, and supposing that σ : Ω → I is a real random variable, it is desirable for many applications that the mapping Xσ : Ω → S deﬁned by Xσ (ω) = Xσ(ω) (ω) is also measurable. This will probably be true if X is a measurable process, although it is not generally true otherwise. More information associated with this is provided by the following result: Theorem 1.1.4 Let X = {Xt }, t ∈ I, be an S-valued progressively measurable process with respect to {Ft }t∈I , t ∈ I, and let τ ∈ I be a ﬁnite stopping time. Then the random variable Xτ is Fτ -measurable. Using some of the notions deﬁned above for stochastic processes, we can construct interesting examples of stopping times as follows. Let Xt , t ∈ I, be a process with state space S which is continuous and {Ft }t∈I -adapted. For any Borel set B in B(S), deﬁne τB , called the hitting time for the set B, by τB =

inf{t ≥ 0 : Xt (ω) ∈ B}, ∞ if the set is empty.

Then τB is in fact a stopping time relative to the family {FtX } := σ{Xs , s ≤ t ∈ I}, the σ-ﬁelds generated by the process X up to time t. The following σ-ﬁeld P∞ of subsets of [0, ∞) × Ω plays an essential role in the construction of stochastic integrals. That is, P∞ is deﬁned as the σ-ﬁeld generated by sets of the form: (s, t] × F,

0 ≤ s < t < ∞,

F ∈ Fs and {0} × F,

F ∈ F0 .

This σ-ﬁeld is called predictable and its elements are called predictable sets. The restriction of the σ-ﬁeld P∞ to [0, T ] × Ω, 0 ≤ T < ∞, will be denoted by PT . An arbitrary measurable mapping from ([0, ∞) × Ω, P∞ ) or ([0, T ] × Ω, PT ) into (S, B(S)) is called a predictable process. A predictable process is necessarily an adapted one. Theorem 1.1.5 (i). An adapted process Xt , t ∈ I, which takes its values in (L(K, H), F(L(K, H))) such that for arbitrary k ∈ K and h ∈ H the process Xt k, hH , t ∈ I, has left continuous trajectories, is predictable; (ii). Assume that Xt , t ∈ I, is an adapted and stochastically continuous process on the interval I. Then the process X has a predictable version on I. Let E be an arbitrary sub-σ-ﬁeld of F and E(· | E) denote the conditional expectation given E. An S-valued process X = Xt , t ∈ I, deﬁned on (Ω, F, P )

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

7

and adapted to the family {Ft }t∈I is said to be a Markov process with respect to {Ft }t∈I if the following property is satisﬁed: for all s and t (T ≥ t ≥ s), E(f (Xt ) | Fs ) = E(f (Xt ) | Xs ) almost surely for every bounded real-valued Borel function f (·) on S. We say that Xt , t ∈ I, is a Markov process if it is Markov process with respect to {FtX }t∈I . Let X be a Markov process with state space S. Then it can be shown that there exists a function P (s, x, t, Γ) (s < t, x ∈ S, Γ ∈ B(S)) with the following properties: (a). For all (s, x, t), P (s, x, t, ·) is a probability measure on B(S); (b). For each (s, t, Γ), P (s, ·, t, Γ) is B(S)-measurable; (c). P (Xt ∈ Γ | FsX ) = P (s, x, t, Γ) almost surely where P (Xt ∈ Γ | FsX ) = E(χ{Xt ∈Γ} | FsX ) and χ{Xt ∈Γ} is the indicator function of {Xt ∈ Γ}; (d). The function P (s, x, t, Γ) satisfying the properties (a), (b) and (c) is called the transition probability function of the Markov process if it further satisﬁes the following Chapman-Kolmogorov equation P (s, x, t, Γ) = P (s, x, u, dy)P (u, y, t, Γ) S

for all x ∈ S, Γ ∈ B(S) and (s, u, t) such that s < u < t. For each s and t (0 ≤ s ≤ t < T ), deﬁne Pst f for Borel measurable function f : S → R1 by Pss = I (the identity operator), Pst f (x) = f (y)P (s, x, t, dy), S

provided the integral on the right exists. On the class of bounded Borelmeasurable functions, Pst deﬁnes a family of operators satisfying the semigroup property Psu f = Pst Ptu f, 0 ≤ s ≤ t ≤ u < T. The equation is essentially a restatement of the Chapman-Kolmogorov equation. The family {Pst } is called the semigroup of the Markov process X. The Markov process Xt , t ≥ 0, is said to have homogeneous transition probability functions if P (s, x, t, Γ) = P (0, x, t − s, Γ). It is clear that in this case, we have Pst = P0t−s . In particular, write P (t − s, x, Γ) = P (0, x, t − s, Γ)

and Pt−s = P0t−s ,

the semigroup is now given by {Pt }, t ≥ 0, Pt f (x) = f (y)P (t, x, dy). S

8

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Let Cb = Cb (S) be the class of real-valued, bounded continuous functions on S. Deﬁnition 1.1.4 The semigroup {Pst } has the Feller property, or is called a Feller semigroup if, for arbitrary f ∈ Cb and s ≥ 0, u > 0, the function x → Pss+u f (x) is continuous. Let S be a separable Banach space with norm · S and M = Mt , t ∈ I = [0, T ), an S-valued stochastic process deﬁned on (Ω, F, {Ft }t∈I , P ). If EMt S < ∞ for all t ∈ I, then the process is called integrable. An integrable and adapted S-valued process Mt , t ∈ I, is said to be a martingale with respect to {Ft }t∈I if E(Mt | Fs ) = Ms

P − a.s.

(1.1.1)

for arbitrary t ≥ s, t, s ∈ I. By the deﬁnition of conditional expectations, the identity (1.1.1) is equivalent to the following statement Mt dP = Ms dP, ∀F ∈ Fs , s ≤ t, s, t ∈ I. F

F

We also recall that a real-valued integrable and adapted process Mt , t ∈ I, is said to be a submartingale (resp. supermartingale) with respect to {Ft }t∈I if E(Mt | Fs ) ≥ Ms , (resp. E(Mt | Fs ) ≤ Ms ), P − a.s. for any s ≤ t, s, t ∈ I. The following is a straightforward consequence of classical martingale theory. Theorem 1.1.6

The following statements hold:

(i). If Mt is an S-valued martingale, then Mt S , t ∈ I, is a submartingale; (ii). If g(·) [0, ∞) into [0, ∞) and is an increasing convex function from Eg Mt S < ∞ for all t ∈ I, then g Mt S , t ∈ [0, T ), is a submartingale. As an immediate consequence of Theorem 1.1.6 and the maximal inequalities for real-valued submartingales, we obtain the following inequalities due to Doob.

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

9

The following statements hold:

Theorem 1.1.7

(i). If Mt , t ∈ L ⊂ [0, ∞), is an S-valued martingale, L a countable set and p ≥ 1, for arbitrary λ > 0, P sup Mt S ≥ λ ≤ λ−p sup E Mt pS ; (1.1.2) t∈L

t∈L

(ii). If, in addition, p > 1, then

E

sup Mt pS t∈L

≤

p p−1

p

sup E Mt pS ;

(1.1.3)

t∈L

(iii). The above estimates remain true if the set L is uncountable and the martingale Mt , t ∈ L, is continuous. Let [0, T ], 0 ≤ T < ∞, be a subinterval of [0, ∞). A continuous S-valued stochastic process Mt , t ∈ [0, T ], deﬁned on (Ω, F, {Ft }t∈[0,T ] , P ), is a continuous square integrable martingale with respect to {Ft }t∈[0,T ] if it is a martingale with almost surely continuous trajectories and satisﬁes, in addition, supt∈[0,T ] EMt 2S < ∞. Let us denote by M2T (S) the space of all S-valued continuous, square integrable martingales M . By a standard argument, we can prove, using Theorem 1.1.7, that: The space M2T (S), equipped with the norm

Theorem 1.1.8

1/2

M M2T (S) =

Mt 2S

E sup

,

t∈[0,T ]

is a Banach space. Denote by L1 = L1 (K) = L1 (K, K) the space of all nuclear operators from the separable Hilbert space K into itself, equipped with the usual nuclear norm. Then L1 is a separable Banach space. An L1 -valued process V· is said to be increasing if the operators Vt , t ∈ [0, T ], are nonnegative, denoted by Vt ≥ 0, i.e., for any k ∈ K, Vt k, kK ≥ 0, t ∈ [0, T ], and 0 ≤ Vs − Vt if 0 ≤ t ≤ s ≤ T . An L1 -valued continuous, adapted and increasing process Vt such that V0 = 0 is said to be a tensor quadratic variation process of the martingale Mt ∈ M2T (K) if and only if for arbitrary a, b ∈ K, the process Mt , aK Mt , bK − Vt a, bK ,

t ∈ [0, T ],

is a continuous Ft -martingale, or equivalently, if and only if the process M t ⊗ Mt − V t ,

t ∈ [0, T ],

10

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

is a continuous Ft -martingale, where (a ⊗ b)k := ab, kK for any k ∈ K and a, b ∈ K. One can show that the process Vt is uniquely determined and can be denoted therefore by Mt , t ∈ [0, T ]. On the other hand, one can also show that there exists a real-valued, increasing, continuous process

which

is uniquely determined up to probability one, denoted by Mt with M0 = 0, called the quadratic variation of Mt , such that Mt 2K − Mt is an Ft -martingale. With regard to the relation between Mt the following:

and Mt of Mt , we have

Theorem 1.1.9 For arbitrary Mt ∈ M2T (K), there exists a unique predictable, positive symmetric element QM (ω, t), or simply Q(ω, t), of L1 (K) such that t Mt = QM (ω, s)d Ms 0

for all t ∈ [0, T ]. In particular, we also call the K-valued stochastic process Mt , t ≥ 0, a QM (ω, t)-martingale process. In a similar manner, one can deﬁne the so-called cross quadratic variation for any Mt ∈ M2T (K), Nt ∈ M2T (K) as a unique continuous process, denoted by Mt , Nt , such that Mt ⊗ N t − Mt , N t

,

t ∈ [0, T ],

is a continuous Ft -martingale (cf. Da Prato and Zabczyk [1]). As a special case, we can consider a symmetric non-negative operator Q ∈ L(K) with trQ < ∞, where trA denotes the trace of operator A. Then there exists a complete orthonormal system {ek }k≥1 in K, and a bounded sequence of nonnegative real numbers λk such that Qek = λk ek ,

k = 1, 2, · · · .

A K-valued stochastic process Wt , t ≥ 0, is called a Q-Wiener process if (1). (2). (3). (4).

W0 = 0; Wt has continuous trajectories; Wt has independent increments; E(Wt ) = 0 and Cov(Wt − Ws ) = (t − s)Q for all t ≥ s ≥ 0, where Cov(X) denotes the covariance operator of X ∈ H (cf. Da Prato and Zabczyk [1]).

It is straightforward to see that the tensor quadratic variation of a Q-Wiener process in K, with trQ < ∞, is given by the formula Wt = tQ, t ≥ 0.

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

1.2

11

Wiener Processes and Stochastic Integration

Let K be a real separable Hilbert space with norm · K and inner product ·, ·K , respectively. A probability measure N on (K, B(K)) is called Gaussian if for arbitrary u ∈ K, there exist numbers µ ∈ R1 , σ ≥ 0, such that N {x ∈ K : u, xK ∈ A} = N (µ, σ 2 )(A),

A ∈ B(R1 ),

where N (µ, σ 2 ) is the usual one dimensional normal distribution with mean µ and standard deviation σ. It can be proved that if N is Gaussian, then there exist an element m ∈ K and a symmetric nonnegative trace class operator Q ∈ L(K) such that k, xK N (dx) = m, kK , ∀k ∈ K, K

k1 , xK k2 , xK N (dx) − m, k1 K m, k2 K K

= Qk1 , k2 K ,

∀k1 , k2 ∈ K,

and, furthermore the characteristic function 1 ˆ (λ) = N ei λ, xK N (dx) = ei λ, mK − 2 Qλ, λK ,

λ ∈ K.

K

Therefore, the measure N is uniquely determined by m and Q and denoted also by N (m, Q). In particular, in this case we call m the mean and Q the covariance operator of N . Recall that we always assume the probability space (Ω, F, P ) is equipped with a right continuous ﬁltration {Ft }t≥0 such that F0 contains all sets of P -measure zero. Let W (t) or Wt , t ≥ 0, be a K-valued, Q-Wiener process which is assumed to be adapted to {Ft }t≥0 and for every t > s the increments Wt − Ws are independent of Fs . Hence, Wt , t ≥ 0, is a continuous martingale relative to {Ft }t≥0 and we have the following representation of Wt : Wt =

∞

Bti ei ,

(1.2.1)

i=1

where {ei } is an orthonormal set of eigenvectors of Q, {Bti }, t ≥ 0, is a family of mutually independent real Wiener ∞ processes with incremental covariance λi > 0, Qei = λi ei and trQ = i=1 λi < ∞. It is also clear that EWt = 0 and for all t ≥ s ≥ 0, the distribution Q(Wt − Ws ) = N (0, (t − s)Q). Let K0 = Ran Q1/2 , the image of K under the operator Q1/2 , which is a Hilbert space equipped with the inner product ·, ·K0 , u, vK0 = Q−1/2 u, Q−1/2 vK ,

u, v ∈ K0 ,

12

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where Q−1/2 is the pseudo-inverse of Q1/2 , then for arbitrary g, h ∈ K0 and t, s ≥ 0, Eg, Wt K0 h, Ws K0 = h, gK0 t ∧ s. Therefore, the process Wt , t ≥ 0, is also called a cylindrical Wiener process on K0 . Its covariance operator with respect to K0 is the identity operator. Theorem 1.2.1 For an arbitrary symmetric nonnegative trace class operator Q on the real separable Hilbert space K, there exists a Q-Wiener process Wt , t ≥ 0. Moreover, the series (1.2.1) is uniformly convergent on [0, T ] almost surely for arbitrary T ≥ 0. We may also derive the following direct generalization of L´evy’s celebrated characterization result. Theorem 1.2.2 A continuous martingale Mt ∈ M2T (K), M0 = 0, is a QWiener process on [0, T ] adapted to the ﬁltration {Ft }t≥0 and with increments Mt − Ms , 0 ≤ s ≤ t ≤ T , independent of Fs , s ∈ [0, T ], if and only if Mt = tQ, t ∈ [0, T ]. t Roughly speaking, the stochastic integral 0 Φ(s, ω)dWs may be deﬁned in the following way. Deﬁne ﬁrst of all a proper space of operators ∗ 1 1 L02 (K0 , H) = G ∈ L(K0 , H) : tr G · Q 2 G · Q 2 <∞ , i.e., L02 (K0 , H) is the family of all Hilbert-Schmidt operators from K0 into H, equipped with the usual Hilbert-Schmidt norm topology. For arbitrarily given T ≥ 0, let Φ(t, ω), t ∈ [0, T ], be an Ft -adapted, L02 (K0 , H)-valued process. We deﬁne the following norm for arbitrary t ∈ [0, T ], |Φ|t :=

t 12 Φ(s, ω)2 0 ds E L 2

0

t ∗ 12 1 1 ds . = E tr Φ(s, ω) · Q 2 Φ(s, ω) · Q 2

(1.2.2)

0

In general, we denote all L02(K0 , H)-valued predictable processes Φ such that 2 |Φ|T < ∞ by W [0, T ]; L02 . In particular, if Φ(t, ω) ∈ L02 (K0 , H), t ∈ [0, T ], is an Ft -adapted, L(K, H)-valued process, (1.2.2) turns out to be |Φ|t =

t 12 E tr Φ(s, ω)QΦ(s, ω)∗ ds 0

and on this occasion, we still write Φ ∈ W 2 [0, T ]; L02 .

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

13

t The stochastic integral 0 Φ(s, ω)dWs ∈ H may be well deﬁned for all Φ(t, ω) ∈ W 2 ([0, T ]; L02 ) by

t

Φ(s, ω)dWs = L2 − lim

n→∞

0

n i=1

t

Φ(s, ω)ei dBsi ,

t ∈ [0, T ],

0

∞ i where Wt = i=1 Bt ei and the limit is taken in mean square sense with respect to the probability P . For a detailed description and relevant properties of stochastic integrations, the reader is referred to Da Prato and Zabczyk [1] or M´etivier [1]. For future purposes, we review brieﬂy the following results which are directly deduced by standard arguments from the deﬁnition of stochastic integration and carrying out a standard argument. Proposition 1.2.1 For arbitrary T ≥ 0, let Φ(·) ∈ W 2 ([0, T ]; L02 ), then the t stochastic integral 0 Φ(s, ω)dWs is a continuous, square integrable H-valued martingale on [0, T ] and t 2 E Φ(s, ω)dWs = |Φ|2t ,

t ∈ [0, T ].

H

0

As a matter of fact, the stochastic integral t Φ(s, ω)dWs ,

(1.2.3)

t ≥ 0,

0

may be generalized for any L02 (K0 , H)-valued adapted process Φ(·, ω) satisfying t 2 P Φ(s, ω)L0 ds < ∞, 0 ≤ t ≤ T = 1. 2

0

Moreover, we may deduce the following generalized relation of (1.2.3) t 2 t E Φ(s, ω)dW ≤ E Φ(s, ω)2L0 ds, s 0

H

2

0

0 ≤ t ≤ T,

(1.2.4)

with the equality holding in (1.2.4) if the right hand side is ﬁnite. By virtue of the deﬁnitions of stochastic integrals and appropriate limiting arguments, we immediately obtain some important properties. t Proposition 1.2.2 Let Φ ∈ W 2 ([0, T ]; L02 ), then 0 Φ(s, ω)dWs is a continuous square integrable martingale, and its tensor quadratic variation is of the form t

t

Φ(s, ω)dWs 0

=

QΦ (s, ω)ds, 0

14

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where

∗ QΦ (t, ω) = Φ(t, ω)Q1/2 Φ(t, ω)Q1/2 ,

t ∈ [0, T ].

Proposition 1.2.3 Assume that both Φ1 , Φ2 ∈ W 2 ([0, T ]; L02 ). Then t 2 t E Φi (s, ω)dWs = 0, E Φi (s, ω)dWs < ∞, t ∈ [0, T ], i = 1, 2. 0

H

0

Moreover, the correlation operators

s t V (s, t) = Cor Φ1 (u, ω)dWu , Φ2 (u, ω)dWu , 0

s, t ∈ [0, T ],

0

are given by the formula s∧t ∗ V (s, t) = Φ1 (u, ω)Q1/2 Φ2 (u, ω)Q1/2 du. 0

Corollary 1.2.1 Under the hypotheses of Proposition 1.2.3, we have s t E Φ1 (u, ω)dWu , Φ2 (u, ω)dWu 0

s∧t

=E

tr

0

Φ1 (u, ω)Q1/2

H

Φ2 (u, ω)Q1/2

∗

du.

(1.2.5)

0

In particular, if the processes Φ1 and Φ2 both are L = L(K, H)-valued, then we can rewrite formula (1.2.5) in a slightly more compact way: s t s∧t

E Φ1 (u, ω)dWu , Φ2 (u, ω)dWu =E tr Φ1 (u, ω)QΦ2 (u, ω)∗ du. 0

0

0

H

(1.2.6) We will quite often use the so-called maximal inequalities formulated in the following theorems. Assume T ≥ 0 and

Theorem 1.2.3 (Doob’s inequalities) E 0

T

Φ(s, ω)2L0 ds < ∞. 2

(i). For arbitrary p ≥ 1 and λ > 0, t P sup Φ(s, ω)dWs 0≤t≤T

0

H

≥λ

p T 1 ≤ p E Φ(s, ω)dWs . λ 0 H

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

15

(ii). For arbitrary p > 1,

t p sup ≤ Φ(s, ω)dWs

E

0≤t≤T

H

0

p T p Φ(s, ω)dWs E . p−1 0 H

Theorem 1.2.4 (Burkholder-Davis-Gundy) For arbitrary p > 0, there exists a constant C = Cp > 0, dependent only on p, such that for any T ≥ 0, E

t p sup ≤ Cp · E Φ(s, ω)dWs

0≤t≤T

0

H

0

T

Φ(s, ω)2 0 ds L

p/2 .

2

Assume that A is a linear operator, generally unbounded, on H and T (t), t ≥ 0, a strongly continuous semigroup of bounded linear operators with inﬁnitesimal generator A. Suppose Φ(t, ω) ∈ W 2 ([0, T ]; L02 ), t ∈ [0, T ], is an L02 (K0 , H)-valued process such that the stochastic integral

t

T (t − s)Φ(s, ω)dWs = WAΦ (t, ω),

t ∈ [0, T ],

0

is well deﬁned, then the process WAΦ (t, ω) is called the stochastic convolution of Φ. In general, the stochastic convolution is no longer a martingale. However, we have the following result which could be regarded as an inﬁnite dimensional version of Burkholder-Davis-Gundy type of inequality for stochastic convolutions. 2 0 Let p > 2, T ≥ 0 and assume Φ(s, ω) ∈ W ([0, T ]; L2 ) T is an L02 (K0 , H)-valued process such that E 0 Φ(s, ω)pL0 ds < ∞. Then 2 there exists a constant C = Cp,T > 0, dependent on p and T , such that

Theorem 1.2.5

t p

E sup T (t − s)Φ(s, ω)dWs ≤ C · E p,T t∈[0,T ]

0

H

0

T

Φ(s, ω)pL0 ds . 2

(1.2.7) In Theorem 1.2.5, there are some weak points from the practical viewpoint due to the assumption p > 2. A version including the case p = 2 is possible if A is assumed to generate a pseudo contraction C0 -semigroup T (t), i.e., T (t) ≤ eat , t ≥ 0, for some a ∈ R1 . Based on this version, the continuity of a suitable modiﬁcation of the stochastic convolution can be also derived. Theorem 1.2.6 Let p ≥ 2. Assume that A generates a contraction semigroup T (t), t ≥ 0, and Φ(t, ω) ∈ W 2 ([0, T ]; L02 ) is an L02 (K0 , H)-valued process. Then the stochastic convolution WAΦ (t) has a continuous modiﬁcation

16

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

and there exists a constant C = Cp,T > 0, dependent of p and T , such that

E

t p

sup ≤ C T (t − s)Φ(s, ω)dW · E s p,T

t∈[0,T ]

0

H

0

p/2

T

Φ(s, ω)2L0 ds

.

2

(1.2.8) As another important tool, we mention the following inﬁnite dimensional version of the classic Itˆo’s formula which will play an essential role in the remainder of this book. Suppose that V (t, x) : I × H → R1 is a continuous function with properties: (i). V (t, x) is diﬀerentiable in t and Vt (t, x) is continuous on I × H; (ii). V (t, x) is twice Fr´echet diﬀerentiable in x, Vx (t, x) ∈ H and Vxx (t, x) ∈ L(H) are continuous on I × H, where I = [0, T ], T > 0. Assume that Φ(t, ω) ∈ W 2 ([0, T ]; L02 ) is an L02 (K0 , H)-valued process, φ(t, ω) is an H-valued continuous, Bochner integrable process on [0, T ], and x0 is an F0 -measurable, H-valued random variable. Then the following H-valued process t t Xt = x0 + φ(s, ω)ds + Φ(s, ω)dWs , t ∈ [0, T ], (1.2.9) 0

0

is well deﬁned. Theorem 1.2.7 (Itˆ o’s formula) Suppose the above conditions (i) and (ii) hold, then for all t ∈ [0, T ], Z(t) = V (t, Xt ) has the stochastic diﬀerential dZ(t) = Vt (t, Xt ) + Vx (t, Xt ), φ(t)H ∗ 1 1/2 1/2 Φ(t)Q dt + tr Vxx (t, Xt ) Φ(t)Q 2 + Vx (t, Xt ), Φ(t)dWt H .

1.3

Stochastic Evolution Equations

The theory of stochastic diﬀerential equations in inﬁnite dimensional spaces, mainly Hilbert and Banach spaces, is a natural generalization of the classical Itˆo’s stochastic diﬀerential equations introduced by Itˆ o and in a slightly diﬀerent form by Gihman in the 1940s. The theory is motivated by the internal development of analysis and the theory of stochastic diﬀerential equations such as stochastic partial diﬀerential equations, stochastic ﬂows and stochastic

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

17

delay diﬀerential equations on the one hand, and by a need to analyse certain random phenomena studied in the natural sciences and engineering like stochastic Navier-Stokes equations in turbulence, stochastic models of population genetics in biology and stochastic optimal control in control theory on the other. The reader is also referred to Da Prato and Zabczyk [1] for some more motivation. On this occasion, we content ourselves by explaining brieﬂy this formulation in the case of a deterministic heat equation with external forces. Roughly speaking, a stochastic partial diﬀerential equation, like a partial diﬀerential equation, can be viewed in two diﬀerent ways. Firstly, we can consider its solution as a real-valued function of t and x, where t is the time parameter, and x (which varies typically in a domain O of Rn ) is a space n ∂ 2 parameter. Denoting as usual by ∆ the Laplace operator i=1 ∂x 2 , the heat i equation with external forces and Dirichlet boundary conditions can be formulated as ∂u (t, x) = ∆u(t, x) + f (t, x), t > 0, x ∈ O, ∂t u(0, x) = u0 (x), x ∈ O; u(t, x) = 0, t > 0,

x ∈ ∂O.

For a nonnegative integer m, we denote by C m (O) the set of all m-times continuously diﬀerentiable real-valued functions in O, and by C0m (O) the subspace of C m (O) consisting of those functions which have compact supports in O. For u ∈ C m (O) and 1 ≤ p < ∞, we deﬁne um,p =

1/p

|D u(x)| dx α

O |α|≤m

p

,

and for p = 2, u, v ∈ C m (O), u, vm,2 =

O |α|≤m

Dα u(x) · Dα v(x)dx.

Let C˜pm (O) be the subset of C m (O) consisting of those functions u for which um,p < ∞. We deﬁne W m,p (O) and W0m,p (O) to be the completions in the norm · m,p of C˜pm (O) and C0m (O). It is well known that W m,p and W0m,p are Banach spaces and W0m,p (O) ⊂ W m,p (O). We will also let H m (O) = W m,2 (O),

H0m (O) = W0m,2 (O).

The spaces H m (O) and H0m (O) are Hilbert spaces under the scalar product ·, ·m,2 deﬁned above. The spaces W m,p (O) consist of functions u ∈ Lp (O) whose derivatives Dα u, in the sense of distributions, of order |α| ≤ m are in Lp (O). The space W0m,p is the subspace of elements of W m,p (O) which vanish in some generalized sense on ∂O.

18

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

The point is that one can also consider a solution of the equation above as a function of t with values in a space of functions of x, say L2 (O). We can write ∆ as an abstract operator, say A, from the Sobolev space H 2 (O) ∩ H01 (O) into L2 (O), or from H01 (O) into H −1 (O), the dual of H 1 (O). Note that the Dirichlet boundary condition is implicit in the fact that we look for the solution in H01 (O). Assuming that the initial condition u0 is in L2 (O) and that t → f (t) is in L2 (0, ∞; L2 (O)), we have the following formulation: du(t) = Au(t) + f (t), dt u(0) = u0 ∈ L2 (O).

t > 0,

In the same way, a solution of a stochastic partial diﬀerential equation can be considered either as a real-valued random ﬁeld indexed by t and x, or as a stochastic process indexed by t with values in an inﬁnite dimensional space, e.g., a Hilbert space, of functions of x. Similar formulation of other types of stochastic systems which are diﬀerent from the heat equation shown above can be found, for instance, in Da Prato and Zabczyk [1] and Rozovskii [1]. In this book, we shall mainly adopt the second viewpoint as the basic setting for our analysis. Generally, we are concerned with two ways of giving a rigorous meaning to solutions of stochastic diﬀerential equations in inﬁnite dimensional spaces, that is, the variational one (cf. Rozovskii [1] and Pardoux [1]) and the semigroup one (cf. Da Prato and Zabczyk [1]). Correspondingly, as in the case of deterministic evolution equations, we have two notions of strong and mild solutions.

1.3.1

Variational Approach and Strong Solutions

Let V be a real reﬂexive Banach space (i.e., V ∗∗ = (V ∗ )∗ = V ), which is continuously embedded in the real separable Hilbert space H, which we identify with its dual. Hence, we have the following relation: V ;→ H ≡ H ∗ ;→ V ∗ where “ ;→ ” denotes an injection. Unless otherwise speciﬁed, we always denote by · V , · H and · V ∗ the norms in V , H and V ∗ , respectively; by ·, ·V,V ∗ the dual product between V and V ∗ , and by ·, ·H the inner product in H. Recall that K is a separable Hilbert space with norm · K and we assume Wt , t ≥ 0, is a K-valued Wiener process with covariance operator Q ∈ L1 (K). Here Wt , t ≥ 0, is supposed to be deﬁned on some complete probability space (Ω, F, P ) equipped with a normal ﬁltration {Ft }t≥0 with respect to which {Wt }t≥0 is a continuous martingale. Consider the following nonlinear stochastic diﬀerential equation in V ∗ : t t Xt = x0 + A(s, Xs )ds + B(s, Xs )dWs , (1.3.1) 0

0

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

19

where A(t, ·) : V → V ∗ and B(t, ·) : V → L(K, H), are two families of measurable nonlinear operators satisfying that t ∈ [0, T ] → A(t, x) ∈ V ∗ , t ∈ [0, T ] → B(t, x) ∈ L(K, H) are Lebesgue measurable for any x ∈ V , T ≥ 0. Deﬁnition 1.3.1 For arbitrarily given numbers T ≥ 0, p > 1 and x0 ∈ H, a stochastic process Xt , 0 ≤ t ≤ T , is said to be a strong solution of Equation (1.3.1) if the following conditions are satisﬁed: (a). For any 0 ≤ t ≤ T , Xt is a V -valued Ft -measurable random variable; (b). Xt ∈ M p (0, T ; V ), where M p (0, T ; V ) denotes the space of all V -valued processes (Xt )t∈[0,T ] which are measurable from [0, T ] × Ω into V and satisfy T EXt pV dt < ∞; 0

(c). Equation (1.3.1) in V one.

∗

is satisﬁed for every t ∈ [0, T ] with probability

In particular, if T is replaced by ∞ above, Xt , t ≥ 0, is called a global strong solution, or strong solution on [0, ∞), of (1.3.1). In order to obtain the existence and uniqueness of Equation (1.3.1), we shall impose the following conditions on A(·, ·) and B(·, ·): there exist constants α > 0, p > 1 and λ, γ ∈ R1 such that (a) (Coercivity). 2v, A(t, v)V,V ∗ + B(t, v)2L0 2

≤ −αvpV + λv2H + γ,

∀v ∈ V,

0 ≤ t ≤ T, (1.3.2)

where · L02 denotes the Hilbert-Schmidt norm B(t, v)2L0 = tr(B(t, v)QB(t, v)∗ ); 2

(b) (Growth). There exists a constant c > 0 such that A(t, v)V ∗ ≤ c(1 + vp−1 V ),

∀v ∈ V,

0 ≤ t ≤ T;

(1.3.3)

(c) (Monotonicity). −2u − v, A(t, u) − A(t, v)V,V ∗ + λu − v2H ≥ B(t, u) − B(t, v)2L0 2

for all u, v ∈ V,

0 ≤ t ≤ T; (1.3.4)

20

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

(d) (Continuity). The map θ ∈ R1 → w, A(t, u + θv)V,V ∗ ∈ R1 is continuous for arbitrary u, v, w ∈ V and 0 ≤ t ≤ T ; (e) (Lipschitz). There exists constant L > 0 such that B(t, u) − B(t, v)L02 ≤ Lu − vV

for all u, v ∈ V,

0 ≤ t ≤ T. (1.3.5)

Theorem 1.3.1 (Pardoux [1]) Under the assumptions (a)–(e) above, Equation (1.3.1) has a unique {Ft }-progressively measurable strong solution Xt , 0 ≤ t ≤ T , which satisﬁes: X· (ω) ∈ M p (0, T ; V )

for all

T ≥ 0,

and X· (ω) ∈ C([0, T ]; H) almost surely where C(0, T ; H) denotes the space of all continuous functions from [0, T ] into H. We shall omit the proof of this theorem and refer the reader to Pardoux [1]. Also, see the Appendix for a statement of uniqueness and existence of strong solutions of stochastic delay diﬀerential equations which contains as a special case Theorem 1.3.1. It can also be shown (cf. Rozovskii [1]) that under the same conditions as above, the strong solution of (1.3.1) possesses the Markov property. Theorem 1.3.2 Within the same assumptions as Theorem 1.3.1, then for each Borel set Γ in H and arbitrary s, t ∈ [0, T ] with s ≤ t, P Xt ∈ Γ | FsX = P Xt ∈ Γ | Xs , (1.3.6) where {FsX } is the σ-ﬁeld generated by the solution Xr of (1.3.1) for r ∈ [0, s]. That is, the strong solution is Markovian and moreover the corresponding semigroup has the Feller property in the sense of Deﬁnition 1.1.4. We now present an example of strong solutions to illustrate the conditions imposed above. Proposition 1.3.1 Let T (t), t ≥ 0, be a strongly continuous semigroup on the real Hilbert space H with inﬁnitesimal generator A and S(t), t ∈ R1 , a strongly continuous group on H with inﬁnitesimal generator G. Let D(A) and D(G2 ) denote the domains of A and G2 , respectively, and assume D(A) ⊂ D(G2 ) and that T (·) and S(·) commute. Then Xt = T (t)S(Bt )x0 , x0 ∈ D(A), is a strong solution of

1 dXt = A + G2 Xt dt + GXt dBt , X0 = x0 , (1.3.7) 2

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

21

where Bt , t ≥ 0, is a real standard one-dimensional Brownian motion. Proof Applying Theorem 1.2.7 to the function v(t, x) = T (t)S(x)x0 and the process Bt , t ≥ 0, then we get vt (t, Bt ) = AT (t)S(Bt )x0 , vx (t, Bt ) = GT (t)S(Bt )x0 and vxx (t, Bt ) = G2 T (t)S(Bt )x0 from which the desired result follows. Remark As to the coercive condition (1.3.2), let us consider now a linear version of Equation (1.3.7): dXt =

1 ∂ 2 Xt ∂Xt dt + θ dBt , 2 ∂x2 ∂x

t ≥ 0,

where θ is some real number. Note that the condition (1.3.2) of Theorem 1.3.1 requires θ2 < 1. In the case θ = 1 (or −1), if the initial condition x0 (·) is smooth enough, the equation has an explicit solution. Indeed, by a similar argument to that of Proposition 1.3.1, it is easy to deduce Xt (x) = x0 (x+Bt ), t ≥ 0. Hence, the case θ2 = 1 is qualitatively diﬀerent from the case θ2 < 1. In the latter case, the equation has a regularizing eﬀect: the solution is more regular at time t > 0 than at time 0. Instead, in the ﬁrst case, the regularity in x of the solution is constant in time. The second case corresponds to a parabolic behavior, while the ﬁrst one corresponds to a hyperbolic behavior. Indeed, if we rewrite the equation with θ = 1 in Stratonovich form, we obtain dXt =

∂Xt ◦ dBt , ∂x

t ≥ 0,

which is a ﬁrst order equation. Note that the equation becomes ill-posed for θ2 > 1. (Also see, for instance, Ikeda and Watanabe [1] for the exact deﬁnition and relevant properties of the Stratonovich integral.) The proofs of Theorems 1.3.1 and 1.3.2 rely in an essential way on a version of Itˆ o’s formula for strong solutions which we now present in what follows. The reader is referred to Pardoux [1] for more details of its proof. Theorem 1.3.3 (Itˆ o’s formula) Let Xt ∈ M p (0, T ; V ), p > 1, be a ∗ continuous process with values in V . Suppose there exist x0 ∈ H, φ(·) ∈ M q (0, T ; V ∗ ), 1/p + 1/q = 1, and Φ(·) ∈ W 2 ([0, T ]; L02 ) such that t t Xt = x0 + φ(s)ds + Φ(s)dWs , t ∈ [0, T ]. 0

0

Then X· ∈ C([0, T ]; H) almost surely and moreover t t Xt 2H = x0 2H + 2 Xs , φ(s)V,V ∗ ds + 2 Xs , Φ(s)dWs H 0

0

22

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations t + tr[Φ(s)QΦ(s)∗ ]ds (1.3.8) 0

for arbitrary 0 ≤ t ≤ T . Remark We specialise Itˆo’s formula to the function x → x2H . As a matter of fact, a similar result holds for a large class of functions deﬁned on H. Precisely, suppose V

and V ∗ are uniformly convex

(1.3.9)

(cf. Dunford and Schwartz [1]). Let Ψ(t, x) : R+ × H → R1 , R+ = [0, ∞), be a function satisfying: (i). Ψ(·, x), x ∈ V , is the ﬁrst order diﬀerentiable and Ψ(t, ·), t ≥ 0, twice (Fr´echet) diﬀerentiable with Ψt (·, ·), Ψx (·, ·) and Ψxx (·, ·) locally bounded on R+ × H; (ii). Ψ(·, ·), Ψx (·, ·) are continuous on R+ × H; (iii). For all trace class operators S, tr(SΨxx (·, ·)) is continuous on R+ ×H → R1 ; (iv). If x ∈ V , then Ψx (t, x) ∈ V , t ≥ 0, and Ψx (·, ·), v ∗ V,V ∗ is continuous on R+ × H → R1 for each v ∗ ∈ V ∗ ; (v). Ψx (t, x)V ≤ C · (1 + xV ), t ≥ 0, for some constant C > 0, ∀x ∈ V . It may be shown that under the conditions (i)–(v) above, Itˆ o’s formula remains true for the function Ψ(·, ·). When one is only interested in the case Ψ(x) = x2H , the condition (1.3.9) may be relaxed by the assumption that V ∗ is strictly convex. This is always true however after an equivalent change of norms because V ∗ is reﬂexive (see Pardoux [1], [2] for more details).

1.3.2

Semigroup Approach and Mild Solutions

In most situations, one ﬁnds that the concept of strong solution is too limited to include important examples. There is a weaker concept, mild solution, which is found to be more appropriate for practical purposes. Consider the following semilinear stochastic diﬀerential equation on I = [0, T ], T ≥ 0,

dXt = (AXt + F (t, Xt ))dt + G(t, Xt )dWt , X0 = x0 ∈ H,

(1.3.10)

where A, generally unbounded, is the inﬁnitesimal generator of a C0 -semigroup T (t), t ≥ 0, of bounded linear operators on the Hilbert space H. The coeﬃcients F (·, ·) and G(·, ·) are two nonlinear measurable mappings from [0, T ]×H to H and [0, T ]×H to L(K, H), respectively, satisfying the following Lipschitz

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

23

continuity conditions: F (t, y) − F (t, z)H ≤ α(T )y − zH , α(T ) > 0, y, z ∈ H, t ∈ [0, T ], G(t, y) − G(t, z) ≤ β(T )y − zH , β(T ) > 0, y, z ∈ H, t ∈ [0, T ]. (1.3.11) Deﬁnition 1.3.2 A stochastic process Xt , t ∈ I, deﬁned on (Ω, F, {Ft }t≥0 , P ) is called a mild solution of (1.3.10) if (i). Xt is adapted to Ft , t ≥ 0; t (ii). For arbitrary 0 ≤ t ≤ T , P ω : 0 Xs (ω)2H ds < ∞ = 1 and

t

t

T (t − s)F (s, Xs )ds +

Xt = T (t)x0 +

T (t − s)G(s, Xs )dWs ,

0

0

(1.3.12) for any x0 ∈ H almost surely. As a direct application of the properties of semigroup theory, it may be proved that: Proposition 1.3.2 For arbitrary x0 ∈ D(A), the domain of A, assume Xt ∈ D(A), t ∈ I, is a solution of (1.3.10) in the sense of satisfying t t Xt = x0 + (AXs + F (s, Xs ))ds + G(s, Xs )dWs , (1.3.13) 0

0

then it is also a mild solution. Bearing Proposition 1.3.2 in mind and also for purposes of future use, we would like to introduce the following concept which is actually an appropriate version of Deﬁnition 1.3.1 for the equation (1.3.10). Deﬁnition 1.3.3 A stochastic process Xt , t ∈ I, deﬁned on (Ω, F, {Ft }t≥0 , P ) is called a strong solution of (1.3.10) if (i). Xt ∈ D(A), 0 ≤ t ≤ T , almost surely and is adapted to Ft , t ∈ I; (ii). X almost surely. For arbitrary 0 ≤ t ≤ T , t is continuous in t ∈ I t 2 P ω : 0 Xs (ω)H ds < ∞ = 1 and Xt = x0 +

t

(AXs + F (s, Xs ))ds + 0

for any x0 ∈ D(A) almost surely.

t

G(s, Xs )dWs , 0

(1.3.14)

24

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

By a straightforward argument, it is possible to establish the following result. Proposition 1.3.3 Assume the condition (1.3.11) holds, then there exists at most one mild solution of (1.3.10). In other words, under the condition (1.3.11) the mild solution of (1.3.10) is unique. The following stochastic version of the classic Fubini theorem will be frequently used in the book and its proof can be found in Da Prato and Zabczyk [1]. Let I = [0, T ], T ≥ 0, and

Proposition 1.3.4

G : I × I × Ω → (L(K, H), F(L(K, H))) be strongly measurable in the sense of Section 1.1 such that G(s, t) is {Ft }measurable for each s ≥ 0 with T T G(s, t)2 dsdt < ∞ a.s. 0

0

Then

T

T

T

G(s, t)dWt ds = 0

0

T

G(s, t)dsdWt 0

a.s.

(1.3.15)

0

The following result gives suﬃcient conditions for a mild solution to be also a strong solution. Proposition 1.3.5

Suppose that the following conditions hold:

(1). x0 ∈ D(A), T (t − s)F (s, x) ∈ D(A), T (t − s)G(s, x)k ∈ D(A) for each x ∈ H, k ∈ K, and t ≥ s; (2). AT (t − s)F (s, x) ≤ f (t − s)xH , f (·) ∈ L1 (0, T ; R+ ); H (3). AT (t − s)G(s, x) ≤ g(t − s)xH , g(·) ∈ L2 (0, T ; R+ ). Then a mild solution Xt , t ∈ I, of (1.3.10) is also a strong solution with Xt ∈ D(A), t ∈ I, in the sense of Deﬁnition 1.3.3. Proof It suﬃces to prove that the mild solution Xt , t ∈ I, satisﬁes (1.3.14). By the above conditions, we have almost surely T t AT (t − r)F r, Xr drdt < ∞, 0

0

H

Stochastic Diﬀerential Equations in Inﬁnite Dimensions T t tr (AT (t − r)G(r, Xr ))Q(AT (t − r)G(r, Xr ))∗ drdt < ∞. 0

25

0

Thus by the classic Fubini’s theorem, we have t

t

s

t

AT (s − r)F (r, Xr )drds = 0

AT (s − r)F (r, Xr )dsdr t t = T (t − r)F (r, Xr )dr − F (r, Xr )dr.

0

0

r

0

0

On the other hand, in view of Proposition 1.3.4, we also have t

t

s

AT (s − r)G(r, Xr )dWr ds = 0

0

t

AT (s − r)G(r, Xr )dsdWr 0

r t

T (t − r)G(r, Xr )dWr

= 0

−

t

G(r, Xr )dWr . 0

Hence, AXt is integrable almost surely and 0

t

t t AXs ds = T (t)x0 − x0 + T (t − r)F (r, Xr )dr − F (r, Xr )dr 0 0 t t + T (t − r)G(r, Xr )dWr − G(r, Xr )dWr 0 0 t t = Xt − x0 − F (r, Xr )dr − G(r, Xr )dWr . 0

0

In other words, Xt ∈ D(A), t ∈ I, is a strong solution of (1.3.10). By the standard Picard iteration procedure or a probabilistic ﬁxed-point theorem type of argument, we can establish an existence theorem for mild solutions of (1.3.10) in the following form. Theorem 1.3.4 Assume the condition (1.3.11) holds. Let x0 ∈ H be an arbitrarily given F0 -measurable random variable with Ex0 pH < ∞ for some integer p ≥ 2. Then there exists a unique mild solution of (1.3.10) in the space C 0, T ; Lp (Ω, F, P ; H) . As we pointed out in Section 1.2, the stochastic convolution in (1.3.12) is no longer a martingale. A remarkable consequence of this fact is that we cannot employ Itˆ o’s formula for mild solutions directly on most occasions of our arguments. We can handle this problem, however, by introducing approximating systems with strong solutions to which Itˆ o’s formula can be

26

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

well applied and by using a limiting argument. In particular, by virtue of Proposition 1.3.5, we may obtain an approximation result of mild solutions, which will play an important role in the subsequent stability analysis. To this end, we introduce an approximating system of (1.3.10) as follows: dXt = AXt dt + R(l)F (t, Xt )dt + R(l)G(t, Xt )dWt , X0 = R(l)x0 , x0 ∈ H,

(1.3.16)

where l ∈ ρ(A), the resolvent set of A and R(l) := lR(l, A), R(l, A) is the resolvent of A. Proposition 1.3.6 Let x0 be an arbitrarily given random variable in H with Ex0 pH < ∞ for some integer p > 2. Suppose the nonlinear terms F (·, ·), G(·, ·) in (1.3.10) satisfy the Lipschitz condition (1.3.11). Then, for each l ∈ ρ(A), the stochastic diﬀerential equation (1.3.16) has a unique strong solution Xt (l) ∈ D(A), which lies in Lp Ω, F, P ; C(0, T ; H) for all T > 0 and p > 2. Moreover, there exists a subsequence, denoted by Xtn , such that for arbitrary T > 0, Xtn → Xt almost surely as n → ∞, uniformly with respect to [0, T ]. Proof The existence of unique strong solution Xt (l) is an immediate consequence of Theorem 1.3.4 and Proposition 1.3.5 by noting the fact that AR(l) = AlR(l, A) = l − l2 R(l, A) are bounded operators. To prove the remainder of the proposition, let us consider for any t ≥ 0, Xt − Xt (l) = T (t) x0 − R(l)x0 t + T (t − s)[F (s, Xs ) − R(l)F (s, Xs (l))]ds 0 t T (t − s)[G(s, Xs ) − R(l)G(s, Xs (l))]dWs . + 0

Since R(l) ≤ 2 for l > 0 large enough, we have for any T ≥ 0, p > 2, p E sup Xt − Xt (l) H

0≤t≤T

t p p ≤ 3 E sup T (t − s)R(l)[F (s, Xs ) − F (s, Xs (l))]ds 0≤t≤T

H

0

t p + 3p E sup T (t − s)R(l)[G(s, Xs ) − G(s, Xs (l))]dWs

+

H

0

t E sup T (t)(x0 − R(l)x0 ) + T (t − s)[I − R(l)]F (s, Xs )ds

+ 3p

0≤t≤T

0≤t≤T

t

0

p T (t − s)[I − R(l)]G(s, Xs )dWs

0

:= 3p [I1 + I2 + I3 ].

H

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

27

The condition (1.3.11) and H¨ older’s inequality imply that

t p

I1 ≤ E sup T (t − s)R(l) F (s, Xs ) − F (s, Xs (l)) ds 0≤t≤T

0

t

≤ C1 (T )E sup 0≤t≤T

0

F (s, Xs ) − F (s, Xs (l))p ds H

H

T

sup Xr − Xr (l)pH ds,

≤ C2 (T )E

0≤r≤s

0

where C1 (T ), C2 (T ) are two positive numbers, dependent of T ≥ 0. On the other hand, by virtue of Theorem 1.2.5, we have for l > 0 large enough, there exists a real number C3 (T ) > 0 such that t p I2 ≤ E sup T (t − s)R(l)[G(s, Xs ) − G(s, Xs (l))]dWs 0≤t≤T

H

0 T

p sup Xr − Xr (l)H ds,

≤ C3 (T )E

0≤r≤s

0

and

p I3 ≤ 3p E sup T (t)(x0 − R(l)x0 )H 0≤t≤T

t p + E sup T (t − s)[I − R(l)]F (s, Xs )ds 0≤t≤T

0

0≤t≤T

0

(1.3.17)

H

t p + E sup T (t − s)[I − R(l)]G(s, Xs )dWs . H

We now estimate each term in (1.3.17), p E sup T (t)(x0 − R(l)x0 )H ≤ C4 (T ) · Ex0 − R(l)x0 pH → 0,

l → ∞,

0≤t≤T

where C4 (T ) > 0 is some positive number. On the other hand, using the Lipschitz condition (1.3.11), we get t p E sup T (t − s)[I − R(l)]F (s, Xs )ds 0≤t≤T

0

≤ C5 (T )E

H

T

I − R(l)p 1 + Xs pH ds

0

≤ I − R(l)2 C6 (T ) → 0,

as

l → ∞,

for some C5 (T ) > 0, C6 (T ) > 0. In a similar manner, by using Theorem 1.2.5, it is easy to deduce that there exists a number C7 (T ) > 0 such that t p E sup T (t − s)[I − R(l)]G(s, Xs )dWs 0≤t≤T

0

≤ C7 (T )I − R(l)p → 0,

H

as l → ∞.

28

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Hence, we can get that there exist numbers C(T ) > 0 and ε(l) > 0 such that T p p E sup Xt − Xt (l)H ≤ C(T ) E sup Xr − Xr (l)H ds + ε(l), 0≤t≤T

0

0≤r≤s

where liml→∞ ε(l) = 0. By the well-known Gronwall’s inequality, we deduce p E sup Xt − Xt (l)H ≤ ε(l)C(T )T → 0, as l → ∞. (1.3.18) 0≤t≤T

Now we are in a position to construct the desired sequence by a diagonal sequence trick. Indeed, for the positive integer n = 1, by virtue of (1.3.18), there exists a positive integer sequence {m1 (i)}∞ i=1 in ρ(A) such that Xt (lm1 (i) ) → Xt almost surely as i → ∞, uniformly with respect to t ∈ [0, 1]. Now for the positive integer n = 2, consider the sequence Xt (lm1 (i) ), we can ﬁnd a subsequence Xt (lm2 (i) ), {lm2 (i) } ⊂ {lm1 (i) }, such that Xt (lm2 (i) ) → Xt almost surely as i → ∞, uniformly with respect to t ∈ [0, 2]. Proceeding inductively, we ﬁnd successive subsequences, Xt (lmn (i) ) so that (a): Xt (lmn+1 (i) ) is a subsequence of Xt (lmn (i) ), {lmn+1 (i) } ⊂ {lmn (i) }, and (b): Xt (lmn (i) ) → Xt almost surely as i → ∞, uniformly with respect to t ∈ [0, n]. To get a subsequence converging for each n, one may take the diagonal sequence l(n) = lmn (n) . Then we can obtain the sequence {Xt (l(n))}∞ n=1 , more simply denoted by {Xtn }∞ , which has the desired properties. n=1 It is worth pointing out that, in general, we cannot conclude directly from Theorem 1.3.4 that the mild solution of (1.3.10) has continuous paths, a fact which makes it justiﬁable to consider asymptotic stability of its sample paths. However, Proposition 1.3.6 allows us to have a modiﬁcation with continuous sample paths of the mild solution of (1.3.10). In particular, unless otherwise stated, we will always suppose the mild solutions considered have continuous sample paths in the sequel. Suppose p ≥ 2, 0 ≤ t ≤ T , and let Lpt (Ω, F, P ; H), simply Lpt (Ω; H), be the subspace of Lp (Ω, F, P ; H) which consists of all Ft -measurable random variables. For arbitrary 0 ≤ s ≤ T , let Ca ([s, T ]; Lp (Ω, F, P ; H)) be the subspace of C([s, T ]; Lp (Ω, F, P ; H)) which consists of {Ft }-adapted processes. For arbitrary 0 ≤ t ≤ T , consider the stochastic evolution equation (1.3.12), however, with initial datum xs ∈ Lps (Ω; H), s ≤ t ≤ T , t t Xt = T (t − s)xs + T (t − u)F (u, Xu )du + T (t − u)G(u, Xu )dWu , s

Xs = xs ∈ Lps (Ω; H).

s

(1.3.19)

In many situations, we ﬁnd it convenient to restate Theorem 1.3.4 in the following form. Theorem 1.3.5 For any 0 ≤ s ≤ t ≤ T , there exists a unique map U (t, s) : Lps (Ω; H) → Lpt (Ω; H) with properties:

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

29

(i). For any s ≤ t ≤ T , xs ∈ Lps (Ω; H), U (t, s)xs is B([s, T ])×F measurable; (ii). U (t, s)xs ∈ Ca ([s, T ]; Lp (Ω, F, P ; H)) for any 0 ≤ T < ∞ and it satisﬁes the equation (1.3.19); (iii). U (s, s)xs = xs for all xs ∈ Lps (Ω; H) and 0 ≤ s ≤ T ; (iv). U (t, r)U (r, s)xs = U (t, s)xs for any s ≤ r ≤ t and xs ∈ Lps (Ω; H); W (v). For each ﬁxed s < t, the map (x, ω) ∈ H × Ω → U (t, s)x is B(H) × Fs,t W measurable, where Fs,t is the σ-ﬁeld generated by Wr − Ws , s ≤ r ≤ t. In particular, if the equation (1.3.19) is linear, the map U (t, s) is a linear operator. Proof We only sketch the proof because it is quite similar to the ﬁnite dimensional case. Deﬁne U (t, s) by U (t, s)x = Xt (x), x ∈ H, Xs (xs ) = xs ∈ Lps (Ω; H). The properties (i)–(iii) follow from those of Xt (xs ) and (iv) follows from uniqueness. By arguments similar to those in Gihman and Skorohod [1], W it is possible to show that Xt (x) has a modiﬁcation which is B(H) × Fs,t measurable as a function of (x, ω) ∈ H × Ω. So we deﬁne U (t, s)x by this particular choice to obtain the required (v). Let Mb (H) be the space of all bounded measurable functions on H with its Borel σ-ﬁeld. Deﬁne a map Pst , 0 ≤ s ≤ t ≤ T , on Mb (H) by (Pst f )(x) = E[f (U (t, s)x)],

x ∈ H,

f ∈ Mb (H),

0 ≤ s ≤ t ≤ T.

By Theorem 1.3.5(v), it is B(H)-measurable and hence lies in Mb (H). Deﬁne furthermore P (s, x, t, Γ) = E[χΓ (Xt (x))], Xs = x ∈ H, Γ ∈ B(H) and let χΓ be the indicator function of the set Γ. Using Theorem 1.3.5, we can show that P (s, x, t, Γ) is the transition probability function of the mild solution of (1.3.19) and Pst is the semigroup associated with it. In particular, we have the following result: Theorem 1.3.6 Under the same conditions as in Theorem 1.3.4, the mild solution of (1.3.19) is a Markov process (in fact, a strong Markov process). Moreover, the semigroup Pst , 0 ≤ s ≤ t ≤ T , associated with it has the Feller property in the sense of Deﬁnition 1.1.4.

1.4

Deﬁnitions and Methods of Stability

Stability of a system is the ability of the system to resist a small inﬂuence or disturbance unknown beforehand. A system or process is said to be stable if such disturbance does not essentially change it. Indeed, an individual

30

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

predictable process can be physically realized only if it is stable in the corresponding natural sense. For instance, we know from classical control theory that, before we can consider the design of a regulatory or tracking control system, we need to make sure that such a system is stable from input to output. Problems like this naturally suggest that we should formulate stability concepts not only from a strictly mathematical viewpoint but also with practical applications in mind. To motivate our stability ideas, let us investigate the following situation. Consider solutions Yt (y0 ), t ≥ 0, of a deterministic diﬀerential equation on the Hilbert space H, dYt = f (t, Yt )dt, t ≥ 0, (1.4.1) Y0 = y0 ∈ H, where f (·, ·) is some given function. Let Y˜t , t ≥ 0, be a particular solution of (1.4.1); the corresponding system is thought of as describing a process without perturbations. The systems associated with other solutions Yt (y0 ) are regarded as perturbed ones. When one talks about stability, or stability in the sense of Lyapunov, of the solution Y˜t , t ≥ 0, it means that the norm Yt − Y˜t H could be made small enough if some reasonable conditions are imposed, for instance, that the initial perturbation scale Y0 − Y˜0 H is very small or time t is large enough. In practice, it is enough to investigate the stability problem for the null solution of some relevant system. Indeed, let Xt = Yt − Y˜t , then the equation (1.4.1) could be changed into dXt = dYt − dY˜t = (f (t, Yt ) − f (t, Y˜t ))dt = (f (t, Xt + Y˜t ) − f (t, Y˜t ))dt := F (t, Xt )dt, (1.4.2) where F (t, 0) = 0, t ≥ 0. Therefore, we could content ourselves with deﬁning and studying stability for the null solution of (1.4.2). Deﬁnition 1.4.1 The null solution of (1.4.2) is said to be stable if for arbitrarily given ε > 0, there exists δ = δ(ε) > 0 such that if x0 H < δ, then Xt (x0 )H < ε for all t ≥ 0. Deﬁnition 1.4.2 The null solution of (1.4.2) is said to be asymptotically stable if it is stable and there exists δ > 0 such that x0 H < δ guarantees lim Xt (x0 )H = 0.

t→∞

Deﬁnition 1.4.3 The null solution of (1.4.2) is said to be exponentially stable if it is asymptotically stable and there exist numbers α > 0 and β > 0

Stochastic Diﬀerential Equations in Inﬁnite Dimensions such that

31

Xt (x0 )H ≤ βx0 H e−αt

for all t ≥ 0. There are at least three times as many deﬁnitions for the stability of stochastic systems as there are for deterministic ones. This is certainly because in a stochastic setting, there are three basic types of convergence: convergence in probability, convergence in mean and convergence in almost sure (sample path, probability one) sense. The above deterministic stability deﬁnitions can be translated into a stochastic setting by properly interpreting the notion of convergence. During the initial development (mainly, ﬁnite dimensional cases) of the theory and methods of stochastic stability, some confusion about stability concepts, their usefulness in applications and the relationship among the diﬀerent concepts of stability existed. Kozin [1], Khas’minskii [1] and Arnold [1] (among others) clariﬁed some of the confusion and provided a good foundation for further work. In what follows, we have no intention of listing all the possible deﬁnitions, but prefer to conﬁne ourselves to those which are in our view of the greatest practical interest. We are interested in the stability of the equations (1.3.1) and (1.3.12). To this end, we assume that A(t, 0) = B(t, 0) = 0 and F (t, 0) = G(t, 0) = 0 for any t ≥ 0. Deﬁnition 1.4.4 (Stability in Probability) The null solution of (1.3.1) or (1.3.12) is said to be stable in probability if for arbitrarily given ε, ε > 0, there exists δ = δ(ε, ε ) > 0 such that if x0 H < δ, then P Xt (x0 )H > ε < ε for all t ≥ 0. Deﬁnition 1.4.5 (Stability in p-th Moment) The null solution of (1.3.1) or (1.3.12) is said to be stable in p-th moment, p > 0, if for arbitrarily given ε > 0, there exists δ = δ(ε) > 0 such that x0 H < δ guarantees EXt (x0 )pH < ε for all t ≥ 0. Deﬁnition 1.4.6 (Almost Sure Stability) The null solution of (1.3.1) or (1.3.12) is said to be almost surely stable if for each ε > 0, there exists a δ = δ(ε) > 0 such that x0 H < δ guarantees P Xt (x0 )H < ε for all t ≥ 0 = 1. Similar statements can be made for asymptotic stability and exponential stability.

32

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Deﬁnition 1.4.7 (Asymptotic Stability in Probability) The null solution of (1.3.1) or (1.3.12) is said to be asymptotically stable in probability if it is stable in probability and for each ε > 0, there exists δ = δ(ε) > 0 such that x0 H < δ guarantees lim P Xt (x0 )H > ε = 0. t→∞

Deﬁnition 1.4.8 (Asymptotic Stability in p-th Moment) The null solution of (1.3.1) or (1.3.12) is said to be asymptotically stable in p-th moment, p > 0, if it is stable in p-th moment and there exists δ > 0 such that x0 H < δ guarantees lim EXt (x0 )pH = 0. t→∞

Deﬁnition 1.4.9 (Asymptotic Almost Sure Stability) The null solution of (1.3.1) or (1.3.12) is said to be asymptotic almost surely stable if it is stable in probability and there exists δ > 0 such that x0 H < δ guarantees P lim Xt (x0 )H = 0 = 1. t→∞

Deﬁnition 1.4.10 (Stability in the Large) The null solution of (1.3.1) or (1.3.12) is said to be stable in probability in the large if it is stable in probability and furthermore for each x0 ∈ H, ε > 0 and δ > 0, there exists a number T = T (x0 , ε, δ) > 0 such that P Xt (x0 )H > ε < δ for all t ≥ T > 0. A similar deﬁnition can be derived for asymptotic stability in probability and p-th moment stability in the large. Deﬁnition 1.4.11 (p-th Moment Exponential Stability) The null solution of (1.3.1) or (1.3.12) is said to be exponentially stable in p-th moment, p > 0, if there exist positive numbers α > 0 and β > 0 such that EXt (x0 )pH ≤ βx0 pH e−αt for all t ≥ 0. Deﬁnition 1.4.12 (Almost Sure Exponential Stability) The null solution of (1.3.1) or (1.3.12) is said to be exponentially stable in almost sure sense if there exist numbers α > 0 and β > 0 such that P Xt (x0 )H ≤ βx0 H e−αt = 1

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

33

for all t ≥ 0. Remark It is worth pointing out that we ﬁnd it useful on some occasions to remove the condition A(t, 0) = 0, B(t, 0) = 0 in (1.3.1) and F (t, 0) = 0, G(t, 0) = 0 in (1.3.12), while consider exactly the same behavior of solutions as Deﬁnitions 1.4.4 to 1.4.12 (e.g., see Section 3.2). This treatment will recapture the above stability deﬁnitions when this condition is assumed to hold. As far as this extension is concerned, we intend to ﬁnd conditions under which the solutions of (1.3.1) or (1.3.12) decay to zero, and say in this case that the solutions or systems are decayable. It is clear that (asymptotic) stability in p-th moment of the null solution of (1.3.1) or (1.3.12) for any value of p > 0 implies its (asymptotic) moment stability for every smaller value than p and stability in probability. On the other hand, one can easily show that a null solution could be (asymptotically) p-th moment stable for some p > 0 and not (asymptotically) q-th moment stable for q > p. The case most often discussed in the literature is (asymptotic) p-th moment stability with p = 2. Henceforth, we shall also refer to this case as (asymptotic) stability in mean square. Although the stability Deﬁnitions 1.4.8 and 1.4.11 above do not appear to be as strong a restriction on systems as that given in the deﬁnitions such as Deﬁnitions 1.4.4 and 1.4.7, there are signiﬁcant implications in Deﬁnitions 1.4.8 and 1.4.11 for sample stability behavior. However, it is worth pointing out that stability of the moment alone does not always provide a satisfactory intuitive basis upon which to judge the stability characteristics of the systems of interest. This can be illustrated by the following simple example. Example 1.4.1

Consider a one-dimensional Itˆ o equation dXt = aXt dt + bXt dBt ,

(1.4.3)

where Bt , t ≥ 0, is a one dimensional real Brownian motion, X0 = x0 ∈ R1 , and a, b are two real numbers. A direct computation yields that the solution process Xt , t ≥ 0, is given by Xt = exp bBt + (a − b2 /2)t x0 , t ≥ 0. (1.4.4) Hence, by using the law of iterated logarithm (cf. Revuz and Yor [1]), it is easy to deduce that the asymptotically exponential growth rate of the solution is log |Xt | b2 λ = lim sup =a− (1.4.5) t 2 t→∞ almost surely. We then conclude that the null solution is almost surely exponentially stable in the sense that P lim Xt = 0 at an exponential decay = 1 t→∞

34

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

if and only if a < b2 /2. On the other hand, using the standard exponential martingale properties of Brownian motions, it is also easy to see for any n ∈ N, N = {1, 2, · · ·}, b2 n 2 n n 2 EXt = x0 · exp (a − b /2)nt + t , 2 and we conclude that the null solution is exponentially stable in n-th moment 2 if and only if a < b2 (1−n). Therefore, unlike deterministic systems, for a < 0, the ﬁrst moment is exponentially stable, but higher moments are probably unstable. For a < −b2 /2, the ﬁrst and second moments are exponentially stable, and higher moments are probably unstable, etc. It seems diﬃcult to associate a physical meaning to the behavior of a system, knowing only that the ﬁrst moment is stable but the second moment is unstable, or that the ﬁrst n moments are stable and all higher moments are unstable. To make matters even more interesting, it is clear from (1.4.5) that the stability of the sample trajectories are determined by the algebraic sign of a − b2 /2 only. That is, the condition a < b2 /2 is necessary and suﬃcient for the null solution to be almost surely asymptotically stable. It is also interesting to notice that for a < b2 /2, the sample solution possesses almost surely asymptotic stability but it is possible that all moments will diverge exponentially. Hence, we see in the example that unlike deterministic systems, even though stability in mean square implies almost sure stability, almost sure stability need not imply the moment stability of the system. Remark If a system is almost surely asymptotically stable, then it is also asymptotically stable in probability. From the analogy of deterministic stability, it seems reasonable to assume in Deﬁnition 1.4.9 almost sure stability rather than stability in probability. However, it is worth pointing out this requirement is actually too strong. In fact, let us consider Example 1.4.1 once again. By (1.4.4) and the properties of Brownian motion, it is easy to see that for no positive constant ε > 0 does there exist a number δ > 0 such that almost all the sample trajectories of the solutions originating at x0 = 0, |x0 | < δ, remain in an ε-neighborhood of zero (i.e., not almost surely stable) even if the unperturbed term is very stable (i.e., a < 0) and |b| is very small. In the history of the study of stability properties, there are two main approaches: the Lyapunov function method and the Lyapunov exponent method. The Lyapunov function method is probably the most eﬀective tool to handle stochastic stability of systems. This approach, also called Lyapunov’s second (direct) method, provides a powerful tool for the study of stability properties of (random) dynamical systems because the technique does not require one to solve the system equation explicitly. We shall take ﬁnite dimensional deterministic systems as an example and interpret brieﬂy the main ideas as follows.

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

35

Consider a nonnegative continuous function Λ(x) on Rn with Λ(0) = 0 and Λ(x) > 0 for x = 0. Suppose for some m ∈ R1 , the set Dm = {x ∈ Rn : Λ(x) < m} is bounded and Λ(x) has continuous ﬁrst order derivatives in Dm . Let Xt = Xt (x0 ) be the unique solution of the initial value problem: dXt = f (Xt )dt, t ≥ 0, f (0) = 0, X0 = x0 ∈ Dm ⊂ Rn ,

(1.4.6)

for a suitable function f (·) ∈ Rn . Since Λ(x) is continuous, the open set Dr for r ∈ (0, m] deﬁned by Dr = {x ∈ Rn : Λ(x) < r} contains the origin and monotonically decreases to the singleton set {0} as r → 0+ . If the total ˙ derivative Λ(x) of Λ(x) (along the solution trajectory Xt (x0 )), which is given by dΛ(x) ∂Λ ˙ Λ(x) = = f T (x) · := −k(x), (1.4.7) dt ∂x satisﬁes −k(x) ≤ 0 for all x ∈ Dm , where k(x) is continuous and f T (x) the transpose of f (x), then Λ(Xt ) is a non-increasing function of t, i.e., Λ(x0 ) < m implies Λ(Xt ) < m for all t ≥ 0. Equivalently, x0 ∈ Dm implies that Xt ∈ Dm for all t ≥ 0. This establishes the stability of the null solution of (1.4.6) in the sense of Lyapunov and Λ(x) is called a Lyapunov function for Equation (1.4.6). If we further assume that k(x) > 0 for x ∈ Dm \ {0}, k(0) = 0. Then Λ(Xt ), as a function of t, is strictly monotone decreasing. In this case, Λ(Xt ) → 0 as t → ∞ from (1.4.7), which implies that Xt → 0 as t → ∞. This fact can be seen through an integration of the equation (1.4.7), i.e., 0 < Λ(x0 ) − Λ(Xt ) =

t

k(Xs )ds < ∞

for t ∈ [0, ∞).

0

It is evident from the above that Xt → {0} = {x ∈ Dm : k(x) = 0} as t leads to inﬁnity. This establishes the asymptotic stability for the system (1.4.6). The Lyapunov function Λ(x) may be regarded as a generalized energy function of the system equation (1.4.6). The above argument illustrates the physical intuition that if the energy of a physical system is always decreasing near an equilibrium state, then the equilibrium state is stable. Since Lyapunov’s original work, this direct method for stability study has been extensively investigated. The main advantage of the method is that one can obtain considerable information about the stability of a given system without explicitly solving the system equation. One major drawback of this method is that for general classes of nonlinear systems a systematic way does not exist to construct or generate a suitable Lyapunov function Λ(·), and the stability criterion with this method, which usually provides only a suﬃcient condition for stability, depends critically on the Lyapunov function chosen. The ﬁrst attempts to generalize the Lyapunov function method to stochastic stability for ﬁnite dimensional stochastic diﬀerential equations were made at least sixty years ago. Since then, a comprehensive study has been carried

36

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

out by various researchers. A systematic presentation of this theory and applications can be found in the existing literature, for example, Kushner [1], [2] and Khas’minskii [1]. The other contribution to the study of stochastic stability (mainly in ﬁnite dimensional spaces) during the past three decades is the application of the Lyapunov exponent concept to stochastic systems. This method uses sophisticated mathematical techniques to study the solution behavior of stochastic systems and often yields necessary and suﬃcient conditions for stability. We present a brief summary of Lyapunov’s original ideas for this method. Consider the linear initial value problem deﬁned by dXt = AXt dt, t ≥ 0, (1.4.8) X0 = x0 ∈ Rn , where A is some n × n matrix. Let Xt (x0 ) denote the unique solution of ¯ 0 ) determined by (1.4.8) initially from x0 ∈ Rn . The Lyapunov exponent λ(x x0 is deﬁned by the term ¯ 0 ) = lim sup log Xt (x0 )Rn , λ(x t t→∞

(1.4.9)

for (1.4.8). In particular, Lyapunov [1] proved the following fundamental ¯ 0 ) for the system: results of the exponent λ(x ¯ 0 ) is ﬁnite for all x0 ∈ Rn \{0}; (i). λ(x (ii). The set of real numbers which are Lyapunov exponent for some x0 ∈ Rn \{0} is ﬁnite with cardinality m, 1 ≤ m ≤ n, −∞ < λ1 < · · · < λm < ∞,

λi ∈ R1 , i = 1, · · · , m;

¯ 0 ) for arbitrary x0 ∈ Rn \{0} and constant c ∈ R1 \{0}. ¯ 0 ) = λ(x (iii). λ(cx n 1 ¯ ¯ ¯ λ(αx + βy 0 0 ) ≤ max{λ(x0 ), λ(y0 )} for x0 , y0 ∈ R \{0} and α, β ∈ R ¯ 0 ) < λ(y ¯ 0 ) and β = 0. The sets Li = {x ∈ Rn \{0} : with equality if λ(x ¯ λ(x) = λi }, i = 1, 2, · · · , m, are linear subspaces of Rn , and {Li }m i=1 is a ﬁltration of Rn , i.e., {0} := L0 ⊂ L1 ⊂ L1 ⊂ · · · ⊂ Lm = Rn where ni := dim(Li ) − dim(Li−1 ) is called the multiplicity of the exponent λi for i = 1, · · · , m, and the collection {(λi , ni )}m i=1 is referred to as the Lyapunov spectrum of the system (1.4.8). ¯ 0 ) < 0, then the solution For the system, the relation (1.4.9) implies that if λ(x ¯ 0 )|, and if λ(x ¯ 0 ) > 0, Xt (x0 ) will converge to zero at the exponential rate |λ(x then the solution cannot remain in any bounded region of Rn . From this we see that {λi }, i = 1, 2, · · · , m, contains information about stability of the system. For ﬁnite dimensional stochastic systems, a similar scheme can be

Stochastic Diﬀerential Equations in Inﬁnite Dimensions

37

carried out and under some circumstances a necessary and suﬃcient condition for stability can be obtained. At this stage, we have no intention of going through any further details in this respect but refer the reader to the existing literature such as Arnold [5], Arnold and Wihstutz [1], Arnold, Kliemann and Oeljeklaus [1], Khas’minskii [1], Mohammed [2] and Oseledec [1] for more details. Although Lyapunov exponent theory produces remarkable results in the developments of stochastic stability in ﬁnite dimensional spaces, it is at present far from being a mature subject area in inﬁnite dimensional cases. In particular, this method uses sophisticated tools from various ﬁelds such as stochastic analysis, functional analysis and stochastic partial diﬀerential equations and few satisfactory results have been obtained until now. From the next chapter on, we shall mainly focus on the Lyapunov function approach for analyzing and establishing stability theory of inﬁnite dimensional stochastic diﬀerential equations.

1.5

Notes and Comments

Much material in Section 1.1 is classical and taken mainly from ChojnowskaMichalik and Goldys [1], Da Prato and Zabczyk [1], M´etivier [1] and Rozovskii [1]. For more details of Sections 1.2 and 1.3, see also Curtain and Pritchard [1], Da Prato and Zabczyk [1], [2], Gihman and Skorohod [1], Hille and Phillips [1], Ikeda and Watanabe [1], Kallianpur [1], Kallianpur and Xiong [1], Karazas and Shreve [1], M´etivier [1], Pardoux [1], Rozovskii [1] and Skorohod [1]. Theorems 1.2.5, 1.2.6 are taken from Da Prato and Zabczyk [1] and Tubaro [1]. A systematic study of the variational method for inﬁnite dimensional (stochastic) systems was carried out by Lions [1], Pardoux [1], Krylov and Rozovskii [1] and Rozovskii [1]. As for applications of semigroup approaches to inﬁnite dimensional stochastic systems, more detailed material can be found in the existing literature such as Da Prato and Zabczyk [1], M´etivier [1] and M´etivier and Pellaumail [1]. Theorems 1.3.1 and 1.3.3 are adapted from Pardoux [1], [2]. Much material in Section 1.3.2 is taken from Ichikawa [3], [5]. As to stability of diﬀerential equations, some systematic statements can be found in the existing literature, for instance, in Hahn [1], LaSalle and Lefschetz [1], Hale [1] and Hale and Lunel [1] for ﬁnite dimensional deterministic systems, in Arnold [1], Khas’minskii [1], Kolmanovskii and Myshkis [1], Kolmanovskii and Nosov [1], Kushner [1], Mao [1], Mohammed [2] and Skorohod [1] for ﬁnite dimensional stochastic systems and in Luo, Guo and Morgul [1], Pazy [1], Curtain and Pritchad [1], Curtain and Zwart [1] and Wu [1] for inﬁnite dimensional deterministic systems.

Chapter 2 Stability of Linear Stochastic Diﬀerential Equations

The purpose of this chapter is to establish the stability of systems deﬁned by stochastic linear evolution equations. We mainly concern ourselves with exploring some characteristic results which are a natural extension of Lyapunov’s classical work in ﬁnite dimensional spaces. We begin our statements with the deterministic case in which a linear unbounded operator satisfying appropriate conditions generates a stable C0 -semigroup of bounded linear operators. Under suitable circumstances, the characterization of mean square exponential stability is established and applied at the end of the chapter to various stochastic (partial or delay) diﬀerential equations. Subsequently, we shall establish almost sure pathwise stability of stochastic systems, a case which can be most closely related to their deterministic counterparts. In some sense, it is this kind of stability that one really likes to have in practical situations.

2.1

Stable Semigroups

We start our discussion by considering the following linear system on the n-dimensional Euclidean space Rn : dXt (x0 ) = AXt (x0 ), dt

X0 (x0 ) = x0 ∈ Rn ,

where A is some n × n constant matrix. Clearly, the equation has a unique solution which is given by Xt (x0 ) = eAt x0 . It is well known (cf. Hahn [1]) that for ﬁnite dimensional linear systems as above, the following statements are equivalent: (P1) the null solution is asymptotically stable, (P2) the null solution is exponentially stable, ∞ (P3) for some real number 1 ≤ p < ∞, we have 0 Xt (x0 )pRn dt < ∞. 39

40

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Furthermore, if we deﬁne T (t) = eAt , the transition matrix, it is easy to see that (P2) is also equivalent to (P4) T (t) is exponentially stable, i.e., there exist positive constants M and µ such that T (t) ≤ M e−µt , where T (t) denotes the matrix norm of T (t) : Rn → Rn . In this case, T (t) is actually a uniformly continuous semigroup of bounded linear operators on Rn . For inﬁnite dimensional systems, the stability analysis becomes much more complicated. For instance, the equivalence of (P1) and (P4) generally does not hold. To illustrate this, let us consider the following deterministic linear Cauchy problem dXt (x0 ) = AXt (x0 ), dt

X0 (x0 ) = x0 ∈ S,

(2.1.1)

where the unbounded operator A is the inﬁnitesimal generator of some C0 semigroup T (t), t ≥ 0, on a real separable Banach space S with norm · S . If x0 ∈ D(A), the domain of A, then T (t)x0 ∈ D(A), and d (T (t)x0 ) = AT (t)x0 = T (t)Ax0 , dt

t ≥ 0.

Hence, Xt = T (t)x0 is a solution of the linear system (2.1.1). Deﬁnition 2.1.1 Let T (t), t ≥ 0, be a strongly continuous semigroup of bounded linear operators on the Banach space S. It is said to be asymptotically stable if limt→∞ T (t)x = 0 for all x ∈ S. Clearly, the semigroup T (t) of (2.1.1) is asymptotically stable if and only if the null solution of (2.1.1) is asymptotically stable. The following example shows that when the null solution or T (t) of (2.1.1) is asymptotically stable, it remains possible for it to be exponentially unstable in inﬁnite dimensions. Example 2.1.1 Let l2 be ∞the Hilbert space of all square summable sequences with norm a2l2 = i=1 a2i < ∞, a = (a1 , · · · , an , · · ·) ∈ l2 . On l2 we deﬁne the semigroup of operators T (t)a = (e−t a1 , e−t/2 a2 , · · · , e−t/n an , · · ·) for any a = (a1 , a2 , · · · , an , · · ·) ∈ l2 . Then, T (t) is a C0 -semigroup on l2 , and for each t in [0, ∞), T (t) = sup T (t)a = sup al2 =1

l2

al2 =1

∞ n=1

e−2t/n a2n

1/2

= lim e−t/n = 1. n→∞

Stability of Linear Stochastic Diﬀerential Equations

41

However, for each a in l2 we have lim T (t)a2l2 = lim

t→∞

t→∞

∞

e−2t/n a2n = 0

n=1

which indicates that T (t) is asymptotically stable. Remark Although it is generally not true that asymptotic stability of semigroups implies exponential stability in inﬁnite dimensional spaces, there is however an important category of semigroups in which asymptotic stability is equivalent to exponential stability. In fact, it may be shown (Datko [5]) that if for some t0 > 0, the operator T (t0 ) is compact, i.e., T (t0 ) maps any bounded sets in S into subsets of compact sets (thus T (t), t ≥ t0 , is compact), then the asymptotic stability of T (t) implies its exponential stability. In the following subsection, we shall carry out a Lyapunov equation and function type of argument to generalize signiﬁcant results from ﬁnite to inﬁnite dimensional spaces. In particular, we shall show that the equivalence of (P2)–(P4) remains true. These results are especially useful in dealing with stochastic stability of inﬁnite dimensional systems. There are some useful criteria for the exponential stability of C0 -semigroups of the following form. Proposition 2.1.1 Let T (t) be a C0 -semigroup with inﬁnitesimal generator A on S. The following statements are equivalent: (i). T (t) is exponentially stable, i.e., T (t) ≤ M e−µt , for M ≥ 1, µ > 0; (ii). limt→∞ T (t) = 0; (iii). there exists some t0 > 0 such that T (t0 ) < 1. Proof It is obvious that (i) implies (ii), and (ii) implies (iii). Suppose (iii) is true, then by the well-known results of semigroup theory, the upper stability indice (cf. Yosida [1]) satisﬁes log T (nt0 ) < 0, n→∞ nt0

Γ(A) := lim

and so there exist M ≥ 1 and µ > 0 such that T (t) ≤ M e−µt , Hence, (iii) implies (i).

t ≥ 0.

42

2.1.1

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Lyapunov Functions

We start in this subsection by an investigation of the exponential stability of (2.1.1) since it guarantees the convergence rate of systems. Actually, it is immediate to see, as in the ﬁnite dimensional case, that the null solution of (2.1.1) is exponentially stable if and only if the semigroup T (t), t ≥ 0, is exponentially stable. It was shown above that, if the state space S is ﬁnite dimensional, there are a few equivalent conditions for the null solution of (2.1.1) to be exponentially stable. The same conditions can also be based either on properties of the spectrum of the matrix A or on the existence of an appropriate Lyapunov function. More precisely, the following proposition holds (cf. Hahn [1]). Proposition 2.1.2 Let S be ﬁnite dimensional, for instance, Rn , n ≥ 1. The null solution of (2.1.1) is exponentially stable if and only if one of the following conditions holds: (i). All eigenvalues of the matrix A have negative real parts, max{Re λ : det(λI − A) = 0} < 0;

(2.1.2)

(ii). There exists a nonnegative deﬁnite matrix, denote it by P ≥ 0, such that P A + AT P = −I,

(2.1.3)

where AT is the transpose of A. In the latter case, the function Λ(x) = P x, xRn , x ∈ Rn , is the Lyapunov function for (2.1.1) in the sense that for every Xt , t ≥ 0, of (2.1.1) dΛ(Xt ) d = P Xt , Xt Rn < 0, dt dt where ·, ·Rn denotes the standard Euclidean inner product in Rn , n ≥ 1. If the Banach space S is inﬁnite dimensional, the above Proposition 2.1.2 is only partially true. For instance, the condition (2.1.2) does not imply stability of the Cauchy problem (see Example 2.1.2 below) unless some extra conditions on A are imposed and in that case, (2.1.2) is, of course, replaced by sup{Re λ : λ ∈ σ(A)} < 0

(2.1.4)

where σ(A) is the spectrum of the linear operator A. This behavior is certainly a consequence of the fact that linear operators in ﬁnite dimensional Banach spaces have only pure point spectra. Since this is generally not the case in inﬁnite dimensional spaces, one does not expect this result to remain true. As a matter of fact, in this setting, (i) of Proposition 2.1.2 may be formulated as follows:

Stability of Linear Stochastic Diﬀerential Equations

43

Proposition 2.1.3 Deﬁne for the generator A of T (t), t ≥ 0, the lower and upper stability indices: γ(A) = sup Re λ : λ ∈ σ(A) , Γ(A) = inf µ : T (t) ≤ M · eµt f or some M ≥ 1 and all t ≥ 0 . Then γ(A) ≤ Γ(A),

(2.1.5)

and therefore, if the system (2.1.1) is exponentially stable, γ(A) < 0.

(2.1.6)

Moreover, if (i) the semigroup T (t) is diﬀerentiable on t ∈ [0, ∞), i.e., for every x ∈ S, t → T (t)x is diﬀerentiable for t ≥ 0, or (ii) for some t0 > 0, T (t0 ) is a compact operator, then γ(A) = Γ(A),

(2.1.7)

and consequently (2.1.6) implies exponential stability. In particular, if S is ﬁnite dimensional, the equality (2.1.7) always holds. Proof The ﬁrst part of Proposition 2.1.3 is an immediate consequence of the celebrated Hille-Yosida Theorem. The proof of (i) is contained in Pazy [1] and part (ii) can be found in Hille and Phillips [1]. We shall not go into further details here because these results are well-established. It is worth mentioning that unlike ﬁnite dimensional systems, the equality of (2.1.5) is generally not true in inﬁnite dimensional spaces, a fact which is illustrated by the example below. Example 2.1.2

For a measurable function f deﬁned on [0, ∞), set ∞ f exp = es |f (s)|ds, 0

and let E be the space of all measurable functions f (·) on [0, ∞) for which f exp < ∞. Let S = E ∩ Lp (0, ∞), 1 < p < ∞. It is easy to see that S, endowed with the norm f e,p = f exp + f p , is a Banach space. In S we deﬁne a semigroup {T (t)} by: T (t)f (x) = f (x + t)

for

t ≥ 0.

(2.1.8)

44

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

It follows readily from its deﬁnition that {T (t)} is a C0 -semigroup on S and that T (t) ≤ 1. Choosing f ∈ S to be the indicator function of the interval [t, t + εp ], ε > 0, and letting ε ↓ 0 shows that T (t) ≥ 1 and thus T (t) = 1 for t ≥ 0, i.e., Γ(A) = 0 in Proposition 2.1.3. The inﬁnitesimal generator A of {T (t)} is given by du(t) D(A) = u : u(t) is absolutely continuous, ∈ S almost surely , dt (2.1.9) and du(t) Au(t) = for u(·) ∈ D(A). dt Let f ∈ S and consider the equation λu(t) − Au(t) = λu(t) −

du(t) = f (t), dt

A simple computation shows that ∞ u(t) = e−λs f (t + s)ds = eλt 0

∞

t ≥ 0.

e−λs f (s)ds

(2.1.10)

(2.1.11)

t

is a solution of (2.1.10). We will show that if λ satisﬁes Re λ > −1, then u, given by (2.1.11), is in D(A) and thus {λ : Re λ > −1} ⊂ ρ(A), the resolvent set of A. To show that u ∈ D(A), it suﬃces by (2.1.10) to show that u ∈ S and this follows from ∞ ∞ (Re λ)t −(Re λ+1)s s −t |u(t)| ≤ e e e |f (s)|ds ≤ e es |f (s)|ds ≤ e−t f exp t

t

which implies that u ∈ Lp (0, ∞), and ∞ ∞ uexp ≤ e(Re λ+1)(t−s) es |f (s)|dsdt 0 t ∞ s = e(Re λ+1)(t−s) dt es |f (s)|ds 0 0 ∞ = (Re λ + 1)−1 1 − e−(Re λ+1)s es |f (s)|ds 0

≤ (Re λ + 1)−1 f exp . The set {λ : Re λ > −1} is therefore a subset of ρ(A), γ(A) = sup{Re λ : λ ∈ σ(A)} ≤ −1 while Γ(A) = 0. In the case where S is inﬁnite dimensional, although it is not generally guaranteed that (2.1.4) yields stable semigroups, the Lyapunov type of the stability condition (2.1.3) has its complete inﬁnite dimensional counterpart.

Stability of Linear Stochastic Diﬀerential Equations

45

Indeed, we may show the following characteristic theorem about stable semigroups which also justiﬁes the equivalence of (P2)–(P4) at the beginning of this section. Theorem 2.1.1 Consider the system (2.1.1) deﬁned on a separable real Hilbert space H. Then the following relations are equivalent: (i). There exist constants M ≥ 1, µ > 0 such that T (t) ≤ M · e−µt ,

t ≥ 0.

(2.1.12)

(ii). There exists a nonnegative self-adjoint operator P ≥ 0 such that 2P Ax, xH = −x, xH for each x ∈ D(A). (iii). For every x ∈ H,

∞

T (t)x2H dt < ∞.

(2.1.13)

(2.1.14)

0

Proof If T (t) ≤ M e−µt , t ≥ 0, for some M ≥ 1, µ > 0, then for every ∞ x in H the integral 0 T (s)x2H ds is convergent. Moreover, if we deﬁne a ∞ mapping P from H into itself which is given by P x = 0 T ∗ (t)T (t)xdt, we easily see that P is a self-adjoint operator on H and P ≥ 0. The equation (2.1.13) follows from the fact that if x and y both are in D(A), d ∞ AT (t)x, P T (t)yH + P T (t)x, AT (t)yH = T (t + s)x, T (t + s)yH ds dt ∞0 d = T (t + s)x, T (t + s)yH ds ds 0 s=∞ = T (t + s)x, T (t + s)yH . s=0

Expanding the right hand side of the above equality and noting limt→∞ T (t)x = 0 for all x ∈ H, we obtain the relation (2.1.13). Hence (i) =⇒ (ii). In order to show (ii) =⇒ (iii), suppose that there exists a self-adjoint operator P on H with P ≥ 0 such that for all x in D(A), 2P Ax, xH = −x2H . For each x in H, t ≥ 0, deﬁne the function Λ(x, t) = P T (t)x, T (t)xH . Since P is nonnegative, this implies that Λ(x, t) ≥ 0 for all x ∈ H, t ≥ 0. Let x ∈ D(A), then T (t)x ∈ D(A), t ≥ 0, and Λ(x, t) is diﬀerentiable with derivative dΛ(x, t) = 2P AT (t)x, T (t)xH = −T (t)x2H . dt Hence,

0 ≤ Λ(x, t) = Λ(x, 0) −

t

T (s)x2H ds, 0

46

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

which means that t Λ(x, 0) ≥ T (s)x2H ds

for all t ≥ 0,

x ∈ D(A).

(2.1.15)

0

On the other hand, by the properties of C0 -semigroups, there exist C ≥ 1, λ ∈ R1 such that T (t)(xn − x)H ≤ Ceλt xn − xH for arbitrary xn , x ∈ H which shows that if xn → x, then T (t)xn → T (t)x uniformly on compact intervals as n → ∞. Hence, the inequality (2.1.15) is valid for all x in H since D(A) is dense in H. This means ∞ T (s)x2H ds ≤ Λ(x, 0) = P x, xH < ∞ (2.1.16) 0

for all x in H, which implies the desired (iii). Finally, let us show (iii) =⇒ (i). We ﬁrst prove that there exists a positive constant C > 0 such that for all x ∈ H, ∞ T (t)x2H dt ≤ Cx2H . (2.1.17) 0

˜ : H → L2 (R+ ; H) by Qx ˜ = T (t)x, To see this, let us deﬁne a mapping Q ˜ is deﬁned on all elements of H. It is x ∈ H. From (2.1.14), it follows that Q ˜ not diﬃcult to see by the standard principle of uniform boundedness that Q is bounded, i.e., ∞

T (t)x2H dt ≤ Cx2H

(2.1.18)

0

where C is some positive constant. Let L ≥ 1, λ > 0 be such numbers that T (t) ≤ Leλt

for all t ≥ 0.

Then, since for all x ∈ H, 1 − e−2λt T (t)x2H = 2λ

t

e−2λs T (t)x2H ds

0

t

e−2λs T (s)2 T (t − s)x2H ds t ≤ L2 T (t − s)x2H ds ≤ L2 Cx2H ,

≤

0

0

we have for some θ > 0 and all t ≥ 0, T (t) ≤ θ. Thus

t

T (t)x2H ds ≤

tT (t)x2H = 0

t

T (s)2 T (t − s)x2H ds ≤ θ2 Cx2H 0

Stability of Linear Stochastic Diﬀerential Equations for all x ∈ H. Therefore,

47

√ θ C T (t) ≤ , t

which immediately implies T (t0 ) < 1 for some t0 > 0 suﬃciently large. Then, by Proposition 2.1.1, we have (iii) =⇒ (i). The proof is complete. A C0 -semigroup T (t), t ≥ 0, satisfying (2.1.14) is called L2 -stable. By analogy with the ﬁnite dimensional case, the condition (2.1.14) in Theorem 2.1.1 which implies stability of semigroups may be appropriately relaxed in the following manner. Theorem 2.1.2 Let T (t), t ≥ 0, be a strongly continuous semigroup of bounded linear operators. If for some positive number p, 1 ≤ p < ∞, ∞ T (t)xpH dt < ∞ for every x ∈ H, (2.1.19) 0

then there exist constants M ≥ 1 and µ > 0 such that T (t) ≤ M e−µt , t ≥ 0. Proof We begin by showing that (2.1.19) implies the boundedness of t → T (t). Let T (t) ≤ Leλt , t ≥ 0, where L ≥ 1 and λ > 0. From (2.1.19), it follows that T (t)x → 0 as t → ∞ for every x ∈ H. Indeed, if this were false, we could ﬁnd x ∈ H, δ > 0 and tj → ∞ such that T (tj )xH ≥ δ. Without loss of generality, we can assume that tj+1 − tj > λ−1 . Set ∆j = [tj − λ−1 , tj ], then dis(∆j ) = λ−1 > 0 and the intervals ∆j do not overlap. For t ∈ ∆j , we have T (t)xH ≥ δ(Le)−1 and therefore 0

∞

T (t)xpH dt ≥

∞ j=0

∆j

T (t)xpH dt ≥

∞ δ p dis(∆j ) = ∞ Le j=1

contradicting (2.1.19). Thus T (t)x → 0 as t → ∞ for every x ∈ H and the principle of uniform boundedness implies T (t) ≤ R for some R ≥ 1 and all t ≥ 0. ˜ : H → Lp (R+ ; H) deﬁned by Qx ˜ = T (t)x. From Consider the mapping Q ˜ is deﬁned on all elements of H. It is not diﬃcult to (2.1.19), it follows that Q ˜ is closed and therefore, by the closed graph theorem, Q ˜ is bounded, see that Q i.e., ∞ T (t)xpH dt ≤ C p xpH (2.1.20) 0

for some constant C > 0. Let 0 < ρ < R−1 and deﬁne tx (ρ) by tx (ρ) = sup t : T (s)xH ≥ ρxH for 0 ≤ s ≤ t .

48

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Since T (t)xH → 0, as t → ∞, tx (ρ) is ﬁnite and positive for every x ∈ H. Moreover, tx (ρ)ρp xpH ≤ 0

tx (ρ)

T (t)xpH dt ≤

∞

T (t)xpH dt ≤ C p xpH ,

0

and therefore tx (ρ) ≤ (C/ρ)p = t0 . For t > t0 , we have T (t)xH ≤ T (t − tx (ρ)) · T (tx (ρ))xH ≤ RρxH = βxH , where 0 < β = Rρ < 1. Finally, let t1 > t0 be ﬁxed and t = nt1 +s, 0 ≤ s < t1 . Then T (t) ≤ T (s) · T (nt1 ) ≤ RT (t1 )n ≤ Rβ n ≤ M e−µt , where M = Rβ −1 and µ = −(1/t1 ) log β > 0. Remark The arguments in the proof cannot go through for 0 < p < 1. This is because p-integrable functions in general do not form a Banach space if p < 1. However, by a diﬀerent approach, we shall show in Section 3.1 that the result in Theorem 2.1.2 remains valid for 0 < p < 1. If S is ﬁnite dimensional, it is known from Proposition 2.1.2 that there exist necessary and suﬃcient conditions for exponential stability which are stated in terms of Lyapunov functions. The following theorem extends these results to inﬁnite dimensional Banach spaces. Theorem 2.1.3 Let T (t), t ≥ 0, deﬁned in (2.1.1) be exponentially stable. Then there exists a unique continuous mapping Λ(·) : S → [0, ∞) such that for each x ∈ S: (i) the mapping ¯ x) t → Λ(T (t)x) = Λ(t,

(2.1.21)

¯ x) = 0; from [0, ∞) → [0, ∞) has the property limt→∞ Λ(t, (ii) ¯ x)/dt = −T (t)x2 ; dΛ(t, S

(2.1.22)

(iii) there exists a positive constant C such that the inequality Λ(x) ≤ Cx2S

(2.1.23)

holds. Conversely, if such a mapping Λ(·) : S → [0, ∞) satisfying (2.1.21), (2.1.22) and (2.1.23) exists, then the semigroup T (t), t ≥ 0, is exponentially stable.

Stability of Linear Stochastic Diﬀerential Equations

49

Proof Suppose T (t), t ≥ 0, is exponentially stable and hence the mapping Λ deﬁned by ∞ Λ(x) = T (t)x2S dt (2.1.24) 0

is well-deﬁned for any x ∈ S, and by Theorem 2.1.1 it actually satisﬁes an inequality of the form (2.1.23). Moreover, since T (t) is exponentially stable and satisﬁes (2.1.23), it follows that Λ(T (t)x) ≤ CT (t)x2S ≤ CM 2 e−2µt x2S ¯ x) → 0 as t → ∞ for any x ∈ S. for some C > 0, M ≥ 1, µ > 0. Hence, Λ(t, On the other hand, from (2.1.24) we see that if t ≥ 0,

∞

Λ(T (t)x) =

T (s)T (t)x2S ds =

0

∞

T (s)x2S ds,

t

and hence ¯ x)/dt = −T (t)x2 , dΛ(t, S which establishes (2.1.22). Uniqueness is a consequence of the equality ¯ x) = Λ(x) + Λ(t, 0

t

¯ x) dΛ(s, ¯ x) − ds = Λ(0, ds

t

T (s)x2S ds

(2.1.25)

0

¯ x) → 0 as t → ∞. and the fact that Λ(t, The converse statement is due to (2.1.23) and (2.1.25). Since from (2.1.25) and (2.1.23)

∞

¯ x) = Λ(x) ≤ Cx2 , T (t)x2S dt = Λ(0, S

0

and by Theorem 2.1.1, this is suﬃcient for exponential stability. Corollary 2.1.1 A necessary and suﬃcient condition that a C0 -semigroup T (t), t ≥ 0, is exponentially stable is the existence of a mapping Λ(·) : S → [0, ∞) such that Λ(x) ≤ Cx2S for some positive constant C and all x ∈ S, and that dΛ(T (t)x) ≤ −T (t)x2S (2.1.26) dt for all x in S and t ≥ 0. Proof If T (t) is exponentially stable, then by Theorem 2.1.3, such a Λ(·) satisfying the hypotheses of the corollary exists and is given by (2.1.24).

50

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

If, on the other hand, some Λ(·) satisfying the hypotheses of the corollary exists, then for each x ∈ S t dΛ(T (s)x) 0 ≤ Λ(T (t)x) = Λ(x) + ds ds 0 t t ≤ Λ(x) − T (s)x2S ds ≤ Cx2S − T (s)x2S ds. 0

Hence,

0

∞

T (s)x2S ds ≤ Cx2S .

0

By Theorem 2.1.1, this implies exponential stability of T (t). Corollary 2.1.2 Let S = H be a separable real Hilbert space. Then T (t) deﬁned on H will be exponentially stable if and only if there exists an operator P ∈ L(H) such that P ≥ 0 and for all x ∈ H, t ≥ 0, dP T (t)x, T (t)xH = −T (t)x2H . dt

(2.1.27)

Proof If T (t), t ≥ 0, is exponentially stable, then T (t) ≤ M e−µt for some M ≥ 1, µ > 0 and all t ≥ 0. Let T ∗ (t) denote the adjoint of T (t). Clearly, T ∗ (t) ≤ M e−µt for all t ≥ 0. Thus the mapping ∞ P = T ∗ (s)T (s)ds 0

in L(H) is well deﬁned. Moreover, for each x ∈ H and t ≥ 0, ∞ P T (t)x, T (t)xH = T (s)x2H ds. t

Consequently, dP T (t)x, T (t)xH = −T (t)x2H . dt On the other hand, assume P satisﬁes the hypotheses of the corollary and let x ∈ H, then t dP T (s)x, T (s)xH 0 ≤ P T (t)x, T (t)xH = P x, xH + ds ds 0 t = P x, xH − T (s)x2H ds, t ≥ 0. 0

Thus,

∞

T (t)x2H dt ≤ P x, xH ≤ P x2H

0

which, by Theorem 2.1.1, implies that T (t), t ≥ 0, is exponentially stable.

Stability of Linear Stochastic Diﬀerential Equations

2.1.2

51

A Useful Stability Criterion

Proposition 2.1.3 supplements conditions on the spectral set of A to yield an exponentially stable semigroup T (t). In most situations, it is found convenient to give conditions on the inﬁnitesimal generators themselves. In the case where S is inﬁnite dimensional, the following theorem turns out to be quite useful to yield exponentially stable semigroups. Proposition 2.1.4 Let A be a closed, densely deﬁned linear operator on the real separable Hilbert space H with norm · H and inner product ·, ·H , respectively. Then there exists a real number α ∈ R1 such that x, AxH ≤ αx2H and

x, A∗ xH ≤ αx2H

x ∈ D(A),

for all for all

x ∈ D(A∗ ),

(2.1.28) (2.1.29)

∗

where A is the adjoint of A if and only if A generates a strongly continuous semigroup T (t) such that T (t) ≤ eαt ,

t ≥ 0,

(2.1.30)

for some number α ∈ R1 . In particular, if α < 0, i.e., both A and its adjoint A∗ are strictly dissipative (cf. Pazy [1]), the semigroup T (t), t ≥ 0, is then exponentially stable. Proof First of all, suppose (2.1.28) and (2.1.29) both are valid. Letting Aα = A − αI, it is obvious that Aα is a densely deﬁned, closed linear operator satisfying x, Aα xH ≤ 0 for x ∈ D(Aα ) = D(A) and x, A∗α xH = x, A∗ xH − αx, xH ≤ 0 for all x ∈ D(A∗α ) = D(A∗ ). Therefore, it follows from Corollary 1.4.4 in Pazy [1] that Aα generates a C0 -semigroup of contractions S(t), t ≥ 0, and hence, A generates a semigroup T (t) = S(t)eαt , t ≥ 0, in H with T (t) ≤ eαt . Conversely, assume A generates a semigroup T (t), t ≥ 0, with T (t) ≤ eαt . Now, A is a densely deﬁned, closed linear operator (cf. Pazy [1]) and the semigroup S(t) = T (t)e−αt , t ≥ 0, corresponding to the generator Aα = A − αI, is contractive. Hence, x, Aα xH ≤ 0 for x ∈ D(Aα ). On the other hand, since H is a Hilbert space, its adjoint S ∗ (t) is generated by A∗α and it is also contractive as S ∗ (t) = S(t) ≤ 1. Thus x, A∗α xH ≤ 0 for x ∈ D(A∗α ). That is, x, AxH ≤ αx2H for x ∈ D(A) and x, A∗ xH ≤ αx2H for x ∈ D(A∗ ). This completes the proof. As an illustration of the technique described above, we point out that many parabolic partial diﬀerential equations can be formulated as dz(t) = Az(t), dt

52

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where A = A∗ ≤ 0 is some properly interpreted self-adjoint operator. The next example is typical for wave equations. Let α ∈ R1 and consider d2 v(t) dv(t) dv +α· + Av(t) = 0, v(0) = v0 ∈ H, (0) = v1 ∈ H, (2.1.31) dt2 dt dt where A is a positive, self-adjoint operator on a real separable Hilbert space H with domain D(A), so that the so-called coercive condition Ax, xH ≥ Cx2H ,

∀x ∈ D(A),

C > 0,

holds. From this it is clear that A is injective, and so we obtain that its algebraic inverse exists. For each y ∈ Ran A, the range of A, we have A−1 y2H ≤

1 1 AA−1 y, A−1 yH ≤ yH A−1 yH . C C

This implies that A−1 yH ≤ C1 yH , and A−1 is bounded on its range. If Ran A is dense in H, then 0 is in the resolvent set of A and A−1 ∈ L(H). Indeed, let x be in the orthogonal complement to the range of A, i.e., for all y ∈ D(A) the following holds: y, xH = 0. By the deﬁnition of adjoint operators this implies that x ∈ D(A∗ ) and A∗ x = 0. Since A is self-adjoint, we conclude that Ax = A∗ x = 0. The positivity of A shows that this can only happen if x = 0, and so Ran A is dense in H. We intend to show that the space H = D(A1/2 ) ⊕ H, equipped with the mapping ·, ·H : H × H → R1 , z, z˜H := A1/2 z1 , A1/2 z˜1 H + z2 , z˜2 H ,

where z=

z1 z2

,

z˜ =

z˜1 z˜2

is actually a Hilbert space. Indeed, since A1/2 is positive and ·, ·H is the inner product on H, it is easy to see that ·, ·H deﬁnes an inner product on H. Thus it remains to show that H with the norm z = z, zH is complete. H To this end, let zn = zz1,n be a Cauchy sequence in H. This implies that 2,n A1/2 [z1,n − z1,m ]2H + z2,n − z2,m 2H = zn − zm 2H → 0

as

n, m → ∞.

Hence, z2,n is a Cauchy sequence in H, and since H is a Hilbert space we know that z2,n converges to some z2 ∈ H. Similarly, we have that A1/2 z1,n converges to some x ∈ H. Since A is boundedly invertible, so is A1/2 and (A1/2 )−1 = (A−1 )1/2 = A−1/2 . So z1,n = A−1/2 [A1/2 z1,n ] → A1/2 x as n → ∞, and z1 = A−1/2 x ∈ D(A1/2 ). Thus zn − z2H → 0, where z = zz12 and H is complete. Deﬁne linear operators on H

0 I A= with domain D(A) = D(A) ⊕ D(A1/2 ), −A −αI

Stability of Linear Stochastic Diﬀerential Equations and

Q=

−αA−1 −A−1 I 0

53

,

then Q is a bounded linear operator on H with Ran Q = D(A) and AQ = I. Thus we see that A is closed. It is also easy to show that the domain of A is dense in H. Hence, (2.1.31) may be rewritten as a ﬁrst order diﬀerential equation on H, dz(t) = Az(t), z(0) = z0 ∈ H, (2.1.32) dt where

z(t) =

v(t) dv(t)/dt

,

z0 =

v0 v1

.

Note that it is immediate to deduce Az, zH = Az1 , z2 H + z2 , −Az1 − αz2 H = −αz2 2H ≤ 0 for any z ∈ D(A) ⊕ D(A1/2 ). Similarly, the adjoint operator of A with respect to the Hilbert space H is easily shown to be

z1 0 −I z1 ∗ A = , D(A∗ ) = D(A), A −α z2 z2 which immediately implies z, A∗ zH = −αz2 2H ≤ 0. Therefore, by virtue of Proposition 2.1.4, we conclude that A generates a strongly continuous semigroup T (t), t ≥ 0, on H. In particular, we may obtain the following stability results by means of the Lyapunov equation method in Theorem 2.1.1 and Proposition 2.1.3. Example 2.1.3 γ(A) =

(1). For every number α ∈ R1 , − α2 +

α2 4 − α2

2

+ γ(−A) if α4 + γ(−A) ≥ 0, otherwise,

(2.1.33)

where γ(·) is the lower stability index deﬁned in Proposition 2.1.3. Moreover, γ(A) ≥ − |γ(−A)|;

(2.1.34)

(2). If γ(−A) = 0, then for any α, the system (2.1.31) is not exponentially stable. (3). If γ(−A) < 0, then for any positive α > 0, the operator

P=

I−

α2 −1 2 (−A) α 2I

− α2 (−A)−1 I

54

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

is the unique nonnegative solution of the following Lyapunov equation, 2

z1 z1 z1 −1/2 2 PA , = −α ). z2 , z1 ∈ D(A), z2 ∈ D(A z2 z2 H H (4). If γ(−A) < 0 and α > 0, then Γ(A) ≤ −

2α|γ(−A)| , 4|γ(−A)| + α(α + α2 + 4|γ(−A)|)

where Γ(·) is the upper stability index deﬁned in Proposition 2.1.3 and consequently, the system (2.1.31) is exponentially stable. Proof From the very deﬁnition of the resolvent sets ρ(−A) and ρ(A), we easily obtain that λ ∈ ρ(A) if and only if λ(λ + α) ∈ ρ(−A). Now a straightforward computation yields (2.1.33). Property (2) follows from (1) and Proposition 2.1.3. Property (3) follows from the easily checked identities: α2 z1 z1 P , z1 2H + αz1 , z2 H + z2 2H = A1/2 z1 2H + z2 z2 2 H 1 = A1/2 z1 2H + (αz1 + z2 2H + z2 2H ), 2

d v(t) v(t) = −α(dv(t)/dt2H + A−1/2 v(t)2H ), P , dv(t)/dt dv(t)/dt dt H (2.1.35) provided dv v(0) ∈ D(A), (0) ∈ D(A1/2 ). dt Also, a direct calculation gives the following estimate: 2 1 z1 ≤ P z1 , z1 z2 z2 2 z2 H H 2 4|γ(−A)| + α(α + α2 + 4|γ(−A)|) z1 .(2.1.36) ≤ z 4|γ(−A)| 2 H Combining (2.1.35) with (2.1.36), we ﬁnally obtain Property (4).

2.2

Lyapunov Equations and Stability

Assume T (t), t ≥ 0, is a strongly continuous semigroup of bounded linear operators on the real Hilbert space H with inﬁnitesimal generator A. In

Stability of Linear Stochastic Diﬀerential Equations

55

Theorem 2.1.1, we have shown that the following statements which actually play a key role in deducing stability characteristics in terms of Lyapunov functions are equivalent: (i). T (t), t ≥ 0, is an exponentially stable semigroup. ∞ (ii). 0 T (t)x2H dt < ∞ for each x ∈ H. (iii). There exists a nonnegative, self-adjoint operator P in L(H) such that 2P Ax, xH = −x, xH for each x ∈ D(A). One of the main objectives in this section is to establish a stochastic version of the above result in the mean square sense. Based on this, we shall study and derive suﬃcient conditions which are more eﬀective and easily checked to ensure the mean square and almost sure stability of associated linear stochastic systems.

2.2.1

Characterization of Mean Square Stability

Assume a complete probability space (Ω, F, P ), equipped with a normal ﬁltration {Ft }t≥0 with respect to which {Wt }t≥0 , t ≥ 0, is some given QWiener process with trQ < ∞ in the Hilbert space K. Consider the following linear stochastic integral equation on the Hilbert space H t Xt = T (t)x0 + 0 T (t − s)B(Xs )dWs , (2.2.1) X0 = x0 ∈ H, where T (t), t ≥ 0, is a strongly continuous semigroup with its inﬁnitesimal generator A on the Hilbert space H and B ∈ L(H, L(K, H)). It is immediate from Theorem 1.3.4 that the equation (2.2.1) has a unique (mild) solution Xt ∈ C(0, ∞; L2 (Ω; H)), t ≥ 0. Our arguments to establish stability equivalences below are basically involved with a calculation of the term T EM Xt , Xt H dt + EGXT , XT H , T ≥ 0, (2.2.2) 0

where M ≥ 0, G ≥ 0 both are in L(H). To this end, consider the following backward linear operator diﬀerential equation d P (t)x, xH + 2Ax, P (t)xH + [M + ∆(P (t))]x, xH = 0, dt x ∈ D(A), 0 ≤ t ≤ T, P (T ) = G, or its integral version (see Proposition 2.2.1 below), T P (t)x = T ∗ (r − t)[M + ∆(P (r))]T (r − t)xdr t

+T ∗ (T − t)GT (T − t)x,

∀x ∈ H,

0 ≤ t ≤ T,

(2.2.3)

(2.2.4)

56

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where ∆(P )x, yH := tr{B ∗ (y)P B(x)Q}, x, y ∈ H, P ∈ L(H). Lemma 2.2.1 There exists a unique solution P (t) ∈ L(H), 0 ≤ t ≤ T , satisfying (2.2.4) in the class of linear nonnegative, self-adjoint strongly continuous operators on H. Proof Deﬁne a sequence of strongly continuous, self-adjoint, nonnegative operators Pn (t), n ≥ 0, 0 ≤ t ≤ T , by the equations: T Pn (t)x = T ∗ (r − t)[M + ∆(Pn−1 (r))]T (r − t)xdr t

+T ∗ (T − t)GT (T − t)x,

x ∈ H,

n ≥ 1,

P0 (t) = 0. Then for any n ≥ 2 and T ≥ t ≥ 0, we have the estimate T Pn (t) ≤ p1 (T ) + a Pn−1 (r)dr, t

where p1 (T ) = sup0≤t≤T P1 (t) and a = a(T ) > 0. By induction, we have Pn (t) ≤ p1 (T )ea(T −t)

for any n ≥ 1.

Thus, Pn (t) is uniformly bounded in t ∈ [0, T ] and n ≥ 0. Since ∆(P )x, xH =

∞

λi P B(x)ei , B(x)ei H ,

i=1

where {ei } is the orthonormal sequence of H consisting of eigenvectors of Q ∞ with Qei = λi ei , λi ≥ 0, trQ = i=1 λi < ∞. Now, Pn (t) is monotonically increasing, i.e., Pn (t) ≤ Pn+1 (t) for each t ≥ 0. Therefore, there exists a strong limit P (t), t ≥ 0, which satisﬁes (2.2.4) and has the desired properties. The uniqueness follows from that of the null solution of the integral equation T P (t)x = T ∗ (r − t)∆(P (r))T (r − t)xdr, t

0 ≤ t ≤ T . Indeed, we have

P (t) ≤ a

T

P (r)dr,

∀T ≥ t ≥ 0,

t

for some a = a(T ) > 0, which immediately implies P (t) = 0, 0 ≤ t ≤ T . Proposition 2.2.1 The equations (2.2.3) and (2.2.4) are equivalent. Moreover, there exists a unique solution P (t), 0 ≤ t ≤ T , satisfying (2.2.3) in the class of linear, self-adjoint, nonnegative strongly continuous operators on H.

Stability of Linear Stochastic Diﬀerential Equations

57

Proof Obviously, it suﬃces by Lemma 2.2.1 to show the equivalence of (2.2.3) and (2.2.4). Suppose ﬁrst that P (t), t ≥ 0, satisﬁes (2.2.4). Then diﬀerentiating P (t)x, xH , x ∈ D(A), yields (2.2.3). Conversely, suppose P (t), t ≥ 0, satisﬁes (2.2.3). Let x ∈ D(A) and 0 < s ≤ t, then P (t)T (t − s)x, T (t − s)xH is diﬀerentiable in t and d P (t)T (t − s)x, T (t − s)xH = −2AT (t − s)x, P (t)T (t − s)xH dt −[M + ∆(P (t))]T (t − s)x, T (t − s)xH +2P (t)T (t − s)x, AT (t − s)xH = −[M + ∆(P (t))]T (t − s)x, T (t − s)xH . Integrating this from s to T , we obtain P (s)x, xH = GT (T − s)x, T (T − s)xH T + [M + ∆(P (t))]T (t − s)x, T (t − s)xH dt. s

Since D(A) is dense in H, (2.2.4) follows easily. Let Ki , i = 1, 2, be two real separable Hilbert spaces. To calculate (2.2.2), we intend to consider the following stochastic diﬀerential equation in H, dXt = AXt dt + B(Xt )dWt1 + F dWt2 , (2.2.5) X0 = x0 ∈ H, where A is the inﬁnitesimal generator of a C0 -semigroup T (t), t ≥ 0, on H, Wti are two Ki -valued Q-Wiener processes independent mutually with covariance operators Qi , trQi < ∞, i = 1, 2, respectively, and B ∈ L(H, L(K1 , H)), F ∈ L(K2 , H). Using the same notations as in (2.2.2) and (2.2.3), we have: Proposition 2.2.2 Let Xt , t ≥ 0, be the mild solution of (2.2.5), then the following relation holds: T EM Xs , Xs H ds + EGXT , XT H = EP (t)Xt , Xt H t T (2.2.6) + tr{F ∗ P (s)F Q2 }ds t

for any 0 ≤ t ≤ T . Proof

To prove (2.2.6), let us introduce the approximating system of (2.2.5): dXt = AXt dt + nR(n, A)[B(Xt )dWt1 + F dWt2 ], X0 = nR(n, A)x0 ,

(2.2.7)

58

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where 0 < n0 ≤ n ∈ ρ(A) for some n0 ∈ N, and R(n, A) is the resolvent of A. Take x0 ∈ D(A) and since nAR(n, A) = n − n2 R(n, A) is bounded, the conditions in Proposition 1.3.5 may be proved to hold so that there exists a unique strong solution of (2.2.7), denote it by Xtn , t ≥ 0. Then applying Itˆ o’s formula to P (t)Xtn , Xtn H , we obtain

T

EM Xsn , Xsn H ds + EGXTn , XTn H t

=

EP (t)Xtn , Xtn H

+

T

tr{nR(n, A)F ∗ P (s)nR(n, A)F Q2 }ds

t

which, passing to the limit n → ∞ and using Proposition 1.3.6, immediately yields (2.2.6). But since D(A) is dense in H and Xt (x0 ) depends continuously on x0 , (2.2.6) holds for any x0 ∈ H. The proof is complete. We are now in a position to establish the main results in this subsection. Theorem 2.2.1 Suppose Xt (x0 ), t ≥ 0, is the unique solution of (2.2.1) with initial datum x0 ∈ H. Then the following statements are equivalent: (i). The solution Xt (x0 ), t ≥ 0, satisﬁes ∞ EXt (x0 )2H dt < ∞

for each x0 ∈ H.

(2.2.8)

0

(ii). There exists a nonnegative, self-adjoint operator P ∈ L(H) such that 2Ax, P xH + ∆(P )x, xH = −x, xH

for any x ∈ D(A), (2.2.9)

where ∆(P )x, xH = tr{B ∗ (x)P B(x)Q}. (iii). There exist positive numbers M ≥ 1, µ > 0 such that for all t ≥ 0, EXt (x0 )2H ≤ M · e−µt x0 2H . Proof

Suppose (i) holds and let PT (·) be the solution of d PT (t)x, xH + 2Ax, PT (t)xH + [I + ∆(PT (t))]x, xH = 0, dt 0 ≤ t ≤ T, x ∈ D(A), (2.2.10) PT (T ) = 0.

Then, by using Proposition 2.2.1, (2.2.4) and (2.2.6), we obtain

T

T (t)x0 2H dt ≤ PT (0)x0 , x0 H = 0

T

EXt 2H dt ≤ 0

0

∞

EXt 2H dt < ∞.

Stability of Linear Stochastic Diﬀerential Equations

59

Hence, by virtue of Theorem 2.1.1, T (t), t ≥ 0, is exponentially stable which implies PT (0) is monotonically increasing in T and uniformly bounded (Banach-Steinhaus theorem). Thus, there exists a limit P ≥ 0, and in view of (2.2.10), we can conclude that P satisﬁes (2.2.9). This proves that (i) implies (ii). Suppose (ii) holds. With the aid of Itˆ o’s formula, for Λ(x) = P x, xH , x ∈ H, we can establish by using (2.2.6) that t P x0 , x0 H = EP Xt , Xt H + EXs (x0 )2H ds, t ≥ 0. (2.2.11) 0

Hence,

∞

T (t)x0 2H dt ≤

0

∞

EXt (x0 )2H dt ≤ P x0 , x0 H < ∞,

0

which implies (i) as well as the exponential stability of T (t), t ≥ 0. On the other hand, from (2.2.11) we obtain d EP Xt , Xt H = −EXt 2H ≤ −P −1 EP Xt , Xt H , dt which immediately yields EP Xt , Xt H ≤ e−pt P x0 , x0 H ,

p = P −1 .

Since T (t), t ≥ 0, is exponentially stable, T (t) ≤ C · e−γt for some C ≥ 1, γ > 0. Now carrying out an approximating argument as in Proposition 1.3.6 and applying Itˆ o’s formula yield that for some number C¯ > 0, t 2 2 EXt (x0 )H = T (t)x0 H + E tr{T (t − r)B(Xr )QB ∗ (Xr )T ∗ (t − r)}dr 0 t 2 −2γt 2 ¯ ≤C e x0 H + C e−2γ(t−r) EXr 2H dr 0 t d ≤ C 2 e−2γt x0 2H − C¯ e−2γ(t−r) EP Xr , Xr H dr dr 0 ¯ ¯ −2γt P x0 , x0 H ≤ C 2 e−2γt x0 2H − CEP Xt , Xt H + Ce C¯ t −2γ(t−r) + e EP Xr , Xr H dr 2γ 0 ¯ −2γt P x0 , x0 H ≤ C 2 e−2γt x0 2H + Ce C¯ t −2γ(t−r) −pr + e e P x0 , x0 H dr 2γ 0 ¯ −2γt P x0 , x0 H ≤ C 2 e−2γt x0 2H + Ce C¯ 1 + (p = 2γ) (e−pt − e−2γt )P x0 , x0 H 2γ 2γ − p

60

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations ¯ −2γt P x0 , x0 H ≤ C 2 e−2γt x0 2H + Ce C¯ e−pt P x0 , x0 H , + (p = 2γ). 2γ(2γ − p)

−2γt ¯ If p = 2γ, the last term above is replaced by (C/(2γ))te P x0 , x0 H . This implies (iii), i.e.,

EXt (x0 )2H ≤ M · e−µt x0 2H

for some

M ≥ 1, µ > 0.

Note that it is obvious (iii) implies (i), so the proof is complete. Remark When the relation (2.2.8) holds for solutions of stochastic differential equations such as (2.2.1), it is said that the systems are L2 -stable in mean, a concept which seems weaker than exponential stability in mean square. Theorem 2.2.1 shows however that these two concepts are actually equivalent for the equation (2.2.1). Corollary 2.2.1

The equation (2.2.9) has at most one solution.

Proof If there exists a solution P of (2.2.9), then the null solution of (2.2.1) is exponentially stable in mean square. Therefore EP Xt (x0 ), Xt (x0 )H → 0,

as

Hence, by virtue of (2.2.11), it is easy to deduce ∞ P x0 , x0 H = EXt (x0 )2H dt,

t → ∞.

x0 ∈ H,

0

which immediately implies the uniqueness. Corollary 2.2.2 Assume P is a self-adjoint, nonnegative operator solution of (2.2.9), then we have ∞ P x0 , x0 H = EXs (x0 )2H ds, x0 ∈ H, 0

and

P =

∞

T ∗ (t)[I + ∆(P )]T (t)dt.

0

Proof This is immediate by the proof of Corollary 2.2.1, the construction of P and (2.2.4).

Stability of Linear Stochastic Diﬀerential Equations

61

Remark A similar characterization in terms of Lyapunov functions to those in Theorem 2.1.3 and its corollaries may be established based on the results obtained above. However, we prefer to present hesitantly at the moment and leave its study to Sections 3.4–3.7. Over there, a detailed investigation of this topic with its application to nonlinear stochastic systems will be carried out. If the null solution of (2.2.1) is stable, the average of second moment for the mild solution of (2.2.5) is ﬁnite. Precisely, assume Xt (x0 ), t ≥ 0, is the unique mild solution of Equation (2.2.5) with initial datum x0 ∈ H. The system (2.2.5) is said to be mean square stable in average if 1 T lim EXt (x0 )2H dt < ∞. T →∞ T 0 Corollary 2.2.3 Let Xt (x0 ), t ≥ 0, be the mild solution of (2.2.5). Then the null solution of (2.2.1) is exponentially stable in mean square if and only if the system (2.2.5) is mean square stable in average. Moreover, in that case, the solution Xt , t ≥ 0, of (2.2.5) satisﬁes 1 T lim EXt (x0 )2H dt = tr{F ∗ P F Q2 } < ∞ T →∞ T 0 where P is the solution of (2.2.9). Proof Let PT (·) be the solution of (2.2.10), then in view of Proposition 2.2.2, we have T T EXt (x0 )2H dt = PT (0)x0 , x0 H + tr{F ∗ PT (t)F Q2 }dt. 0

0

However, we know by Theorem 2.2.1 that PT (T ) converges monotonically to P as T → ∞. Hence the assertion easily follows. Under some suitable conditions on the operators A and B, the results in this subsection allow an extension to the case when the operator B is unbounded. Also, it is worth pointing out that in many practical situations, it is not always easy to check the hypothesis (i) or (ii) of Theorem 2.2.1 straightforwardly. At the moment we do not try to go into further details about this and refer the reader to Da Prato and Ichikawa [1] or Da Prato and Zabczyk [1]. In the following subsection, by employing Theorem 2.2.1 we shall establish some useful and easily checked suﬃcient conditions to obtain stability of Equation (2.2.1). Lastly, to close this subsection we present an interesting example due to Da Prato and Zabczyk [1] (also see Zabczyk [2]) for a system of the socalled Lurie type whose stability can be completely characterized by Theorem 2.2.1.

62

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations Consider the stochastic system

Example 2.2.1

dXt = AXt dt +

n

bi hi , Xt H dBti ,

X0 = x0 ,

(2.2.12)

i=1

where A is the generator of an exponentially stable C0 -semigroup T (·). bi , hi , i = 1, · · · , n are some given non-zero elements in H, and Bti , i = 1, · · · , n, are independent real-valued Wiener processes. We have here K = Rn , Q = I and B(·) in (2.2.1) is given by B(x)k =

n

bi hi , xH ki ,

k = (k1 , · · · , kn ) ∈ K.

(2.2.13)

i=1

Moreover, note that ∆(P )x, yH := tr{B ∗ (y)P B(x)Q}, x, y ∈ H, and we have in this case ∆(P ) =

n

P hi , hi H bi ⊗ bi .

i=1

We remark that since T (t), t ≥ 0, is exponentially stable, Equation (2.2.9) is equivalent to the following one: ∞ ∞ P = T (t)∆(P )T ∗ (t)dt + T (t)T ∗ (t)dt 0

=

n

0

∞

P hi , hi H

(T (t)hi ) ⊗ (T (t)hi )dt +

0

i=1

∞

T (t)T ∗ (t)dt,

0

and in order to solve it, it suﬃces to ﬁnd a vector ξ ∈ Rn , ξ = {P hi , hi H : i = 1, · · · , n}, with nonnegative components such that ξ = M ξ + η,

(2.2.14)

where M = (Mij ) is the n × n matrix deﬁned as ∞ Mij = T (t)bi , hj 2H dt, i, j = 1, · · · , n, 0

and η is the vector ηj =

∞

T ∗ (t)hj 2H dt,

j = 1, · · · , n.

0

Since the entries of matrix M are all nonnegative and the components of η are strictly positive, Equation (2.2.14) has a nonnegative solution, and so the system (2.2.12) is stable if and only if the eigenvalues of M are all of modulus less than 1.

Stability of Linear Stochastic Diﬀerential Equations

2.2.2

63

Almost Sure Pathwise Stability

As mentioned in the last subsection, it is not always easy to use Theorem 2.2.1 in a straightforward way. Based on Theorem 2.2.1, in this subsection we shall establish and analyse suﬃcient conditions for sample path stability of processes in the almost sure sense and apply them to some examples in the section 2.4. Consider the linear stochastic evolution equation (2.2.1) again. We wish to show that under some conditions the null solution of (2.2.1) is exponentially stable in mean square, and from this fact we further deduce that the sample paths of solutions decay to zero almost surely as t → ∞. To this end, we impose the following conditions which are easy to verify in most situations: (H1). The semigroup T (t), t ≥ 0, is exponentially stable, i.e., there exist −γt constants ∞ ∗ C ≥ 1, γ > 0 such that T (t) ≤ C · e , t ≥ 0; ∗ (H2). 0 T (t)∆(I)T (t)dt < 1, where ∆(P )x, yH := tr[B(x) P B(y)Q] for any P ∈ L(H) and x, y ∈ H. Theorem 2.2.2 Assume (H1) and (H2) hold. Then there exist positive constants M , µ such that for any solution Xt (x0 ) of Equation (2.2.1), EXt (x0 )2H ≤ M · x0 2H e−µt ,

t ≥ 0.

(2.2.15)

Proof By virtue of Theorem 2.2.1, it suﬃces to prove that there exists a nonnegative, self-adjoint operator P in L(H) which is the solution of (2.2.9). Indeed, P can be constructed by P = limn→∞ Pn , where ∞ 0 < P1 := T ∗ (t)T (t)dt ∈ L(H), 0

and Pn+1 := P1 + S(Pn ),

where S(G) =

∞

T ∗ (t)∆(G)T (t)dt

0

for any G ∈ L(H). Note that S(G) ≥ 0 if G ≥ 0, so that Pn is nondecreasing as n tends to inﬁnity. On the other hand, ∞ S(G) = sup T ∗ (t)∆(G)T (t)x, xH dt xH =1

=

0

sup xH =1

∞

tr[B(T ∗ (t)x)GB(T (t)x)Q]dt ≤ GS(I).

0

According to (H2), S(I) < 1 which immediately implies P = limn→ Pn exists, and ∞ P = T ∗ (t)[I + ∆(P )]T (t)dt, (2.2.16) 0

64

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

or 2P Ax, xH + ∆(P )x, xH + x2H = 0

for all x ∈ D(A),

which, together with Theorem 2.2.1, immediately yields the required result. Remark There exists a condition which is usually stronger but easier to use than (H1) and (H2). It implies (2.2.15) by an approximation type of argument as in Proposition 1.3.6. This condition may be described as follows: there exists a constant ν > 0 such that for arbitrary x ∈ D(A), 2Ax, xH + ∆(I)x, xH ≤ −νx2H .

(2.2.17)

We shall not go into further details at present because some more general conditions which contain (2.2.17) as a special case will be derived to deal with nonlinear stochastic systems in Chapter 3. In the early stages (1940s to 1960s) of the study of stability of ﬁnite dimensional stochastic systems, investigators were primarily concerned with moment stability and stability in probability. The mathematical theory for the study of almost sure (sample path) stability was not yet fully developed then. Subsequently, work along these lines gradually appeared in the literature. Over the past twenty years, pathwise stability with probability one studies of inﬁnite dimensional stochastic systems have attracted increasing attention from researchers. This is not surprising because sample paths rather than moments or probabilities associated with trajectories are observed in real systems and the stability properties of sample paths can be most closely related to their deterministic counterpart. In what follows, we shall carry out an investigation of sample path stability in the almost sure sense. First of all, we mention the following conditions imposed on (2.2.1): (H3). {T (t)}, t ≥ 0, is an analytic semigroup; (H4). There exists a real function f (·) > 0 such that for all t < ∞, t f (s)2 ds < ∞ 0

and for all x ∈ H, AT (t)B(x) ≤ f (t)xH . In particular, the following stability result was derived in Haussmann [1]: Theorem 2.2.3 Assume Conditions (H3) and (H4) above hold. Suppose Xt , t ≥ 0, is the solution of (2.2.1) satisfying (2.2.15), then there exist constants M ≥ 1, µ > 0, and random variable 0 ≤ T (ω) < ∞ such that for all

Stability of Linear Stochastic Diﬀerential Equations

65

t ≥ T (ω), x0 ∈ H, Xt (x0 )H ≤ M x0 H e−µt

a.s.

(2.2.18)

We prefer to omit the proof here and refer the reader to Haussmann [1]. The reason is that the conditions (H3) and (H4) are somewhat restrictive so that it is not quite satisfactory for them to be applied to practical situations. In fact, a more powerful criterion is (2.2.17) which will be generalized to deal with sample path almost sure stability of nonlinear stochastic evolution systems in Chapter 3. Another eﬀective method to treat almost sure stability of Equation (2.2.1) is to consider its strong solution. Let V be a densely embedded subspace of the real Hilbert space H and a separable Banach space under some norm · V . Then V ;→ H ;→ V ∗ with · H ≤ β · V for some β > 0. Consider the following linear stochastic system dXt = AXt + B(Xs )dWs , (2.2.19) X0 = x0 ∈ H, where A : V → V ∗ is a bounded mapping which is coercive in the following sense: there exist constants α > 0, λ ∈ R1 such that for any x ∈ V , x, AxV,V ∗ ≤ −αx2V + λx2H ,

(2.2.20)

and B is supposed to be an element of L(H, L(K, H)). Then, Theorem 1.3.1 immediately tells us that for any T < ∞, there is a unique strong solution X ∈ L2 (Ω × (0, T ); V ) ∩ L2 (Ω; C(0, T ; H)). Note that under the above condition (2.2.20), A generates a strongly continuous semigroup T (t), t ≥ 0, and the strong solution is also a mild solution since (2.2.19) is linear and D(A) is dense in H. Hence, if we assume (H1) and (H2) hold, then, according to Theorem 2.2.2, there exist positive constants M , µ such that for the strong solution Xt (x0 ), t ≥ 0, of the equation (2.2.19), EXt (x0 )2H ≤ M · x0 2H e−µt ,

t ≥ 0.

(2.2.21)

Lemma 2.2.2 Suppose (2.2.21) holds. Then there exists positive constant C > 0 such that E sup Xt (x0 )2H ≤ C · x0 2H (2.2.22) 0≤t≤T

for any T ≥ 0 and x0 ∈ H. Proof Xt 2H

Applying Itˆ o’s formula to Xt (x0 ), t ≥ 0, yields t t = x0 2H + 2 Xs , AXs V,V ∗ ds + 2 Xs , B(Xs )dWs H 0

0

66

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations t + ∆(I)Xs , Xs H ds 0 t t ≤ x0 2H + (2|λ| + ∆(I)) Xs 2H ds + 2 Xs , B(Xs )dWs H . 0

0

(2.2.23) Hence, E sup 0≤t≤T

Xt 2H

≤

x0 2H

T

+ (2|λ| + ∆(I))

EXs 2H ds 0

t +2E sup Xs , B(Xs )dWs H . 0≤t≤T

(2.2.24)

0

However, by using Burkholder-Davis-Gundy inequality, we have t 2E sup Xs , B(Xs )dWs H 0≤t≤T

0

≤ 6E

T

Xt 2H ∆(I)Xs , Xs H ds

1/2

0

≤ 3E 2 sup Xt H 0≤t≤T

≤ 3lE

T

∆(I)Xs , Xs H ds

(2.2.25)

0

sup Xt 2H + 3l−1

0≤t≤T

1/2

T

E∆(I)Xs , Xs H ds

0

for any l > 0. If we take l = 1/6 in (2.2.25) and substitute it into (2.2.24), we obtain after using (2.2.21) E sup Xt 2H ≤ 2x0 2H + (4|λ| + 38∆(I))M µ−1 x0 2H (1 − e−µT ) 0≤t≤T

≤ 2 + M µ−1 (4|λ| + 38∆(I)) x0 2H .

The proof is complete. We are now in a position to obtain the desired pathwise exponential decay of strong solutions. Theorem 2.2.4 Assume the coercive condition (2.2.20) holds. If Xt , t ≥ 0, is a strong solution of (2.2.19) satisfying (2.2.21), then there exist positive constants M , µ, and a random variable 0 ≤ T (ω) < ∞ such that for all t ≥ T (ω), Xt (x0 )H ≤ M · x0 H e−µt

a.s.

Stability of Linear Stochastic Diﬀerential Equations Proof Xt 2H

67

From (2.2.20) and (2.2.23), it follows that for arbitrary t ≥ N > 0, t t = XN 2H + 2 Xs , AXs V,V ∗ ds + 2 Xs , B(Xs )dWs H N N t + ∆(I)Xs , Xs H ds N t t ≤ XN 2H + (2|λ| + ∆(I)) Xs 2H ds + 2 Xs , B(Xs )dWs H . N

N

Hence, for any positive constant εN > 0 to be determined later, we have P sup Xs H ≥ εN N ≤t≤N +1

≤

P {XN 2H

≥

ε2N /3}

N +1

+P

! Xt 2H dt ≥ ε2N 3(2|λ| + ∆(I))

N

+P

sup

N ≤t≤N +1

t Xs , B(Xs )dWs H ≥ ε2N /6 .

(2.2.26)

N

Now, from (2.2.21), (2.2.22) and (2.2.25) it follows t P sup Xs , B(Xs )dWs H ≥ ε2N /6 N ≤t≤N +1

N

≤ 6ε−2 E N ≤ 18ε−2 N E

≤ ≤

sup

N ≤t≤N +1

t Xs , B(Xs )dWs H

sup

N ≤t≤N +1

N

Xt H

N +1

E∆(I)Xs , Xs H ds

1/2

N

1/2 18ε−2 x0 2H ∆(I)1/2 M 1/2 e−µN/2 N K ! k1 x0 2H e−µN/2 ε2N ,

where k1 is some positive constant. Therefore, from (2.2.26) it follows after using (2.2.21) that there exist some positive constants k2 , k3 , k4 > 0 such that P sup Xs H ≥ εN N ≤t≤N +1

" " ≤ k2 x0 2H e−µN ε2N + k3 x0 2H · e−µN/2 + x0 2H · e−µN ε2N

≤ k4 e−µN/4 if we take ε2N = 2x0 2H e−µN/4 . The well-known Borel-Cantelli lemma now implies that there exists N1 (ω) > 0 such that if N ≥ N1 (ω), then sup

N ≤t≤N +1

Xt 2H ≤ 2x0 2H e−µN/4

a.s.

68

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

and the required result follows. It is possible to relax our conditions on B in the way that B will only be required to lie in L(V, L(K, H)), a case which may be used to treat unbounded operators B(·). However, for our stability purpose the coercive condition (2.2.20) will be strengthened as 2x, AxV,V ∗ + x, ∆(I)xV,V ∗ ≤ −αx2V + λx2H ,

∀x ∈ V.

(2.2.27)

On this occasion, for arbitrary P ∈ L(H, H), ∆(P ) ∈ L(V, V ∗ ) is deﬁned by y, ∆(P )xV,V ∗ = tr[B(x)∗ P B(y)Q],

x, y ∈ V.

It immediately follows that in view of Theorem 1.3.1, a unique strong solution exists in L2 (Ω × (0, T ); V ) ∩ L2 (Ω; C(0, T ; H)). In particular, we also have the following stability result similar to Theorems 2.2.2 and 2.2.4. Theorem 2.2.5 Assume B ∈ L(V, L(K, H)) satisfying (2.2.27). If the conditions (H1) and (H2) hold, then there exist constants M ≥ 1, µ > 0 such that for any strong solution Xt , t ≥ 0, of (2.2.19) EXt (x0 )2H ≤ M · x0 2H e−µt ,

t ≥ 0.

Moreover, under the same conditions the null solution is also pathwise exponentially stable with probability one, i.e., there exist constants L ≥ 1, θ > 0, and random variable 0 ≤ T (ω) < ∞ such that for all t ≥ T (ω), Xt (x0 )H ≤ L · x0 H e−θt

a.s.

Proof Following a similar argument as in Theorems 2.2.1 and 2.2.2, it is easy to show that there exists a symmetric nonnegative operator P ∈ L(H) such that for all x0 ∈ D(A), ∞ P = T ∗ (t)[I + ∆(P )]T (t)dt, 0

and EP Xt (x0 ), Xt (x0 )H = EP Xs , Xs H −

t

EXr 2H dr,

∀0 ≤ s ≤ t.

s

(2.2.28) On the other hand, Itˆ o’s formula, (2.2.23) and (2.2.27) yield EXt 2H ≤ EXs 2H + λ

t

EXr 2H dr − α s

t

EXr 2V dr. (2.2.29) s

Stability of Linear Stochastic Diﬀerential Equations

69

Let gt = λEP Xt , Xt H + EXt 2H and from (2.2.28) and (2.2.29), we obtain

t

gt ≤ gs − α

t

EXr 2V dr ≤ gs − αC1

gs ds,

s

0 ≤ s ≤ t,

s

for some constant C1 > 0 since EXr 2V ≥ β −2 EXr 2H ≥ β −2 gr (1 + λP )−1 . It follows that

dgt ≤ −αC1 gt := −νgt , dt

which immediately implies gt ≤ g0 · e−νt ,

t ≥ 0,

(2.2.30)

and EXt 2V ≤ −α−1

dgt , dt

t ≥ 0.

(2.2.31)

Note that Xt , t ≥ 0, also satisﬁes (2.2.1). By using (H1) and (2.2.30), (2.2.31), we deduce 2 t EXt 2H ≤ 2C 2 e−2γt x0 2H + 2E T (t − s)B(Xs )dWs H

0

and t 2 E T (t − s)B(Xs )dWs H 0 t ≤ trQ ET (t − s)B(Xs )2H ds 0 t 2 2 ≤ trQC BL(V,L(K,H)) e−2γ(t−s) EXs 2V ds ≤ C2 · e−C3 t , 0

where T (t) ≤ C · e−γt , t ≥ 0, C2 , C3 are two positive constants. Hence, there exist positive constants M , µ such that EXt 2H ≤ M · x0 2H e−µt ,

t ≥ 0,

if x0 ∈ D(A). But since D(A) is dense in H the result holds for all x0 ∈ H. Lastly, in order to complete our proof, note that the proofs of Lemma 2.2.2 and Theorem 2.2.4 go through with (2.2.23) and (2.2.25) changed into Xt 2H ≤ x0 2H + |λ|

t

Xs 2H ds + 2 0

t

Xs , B(Xs )dWs H , 0

70

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

and for some C4 > 0, t 2E sup Xs , B(Xs )dWs H 0≤t≤T

0

t 1 2 ≤ E sup Xs H + 18 EXs , ∆(I)Xs V,V ∗ ds 2 0≤t≤T 0 1 ≤ E sup Xs 2H + C4 x0 2H , 2 0≤t≤T

where we use (2.2.30) together with EXs , ∆(I)Xs V,V ∗ = E tr[B(Xs )∗ B(Xs )Q] ≤ trQB2L(V,L(K,H)) EXs 2V ≤ −α−1 trQB2L(V,L(K,H))

dgs ds

by (2.2.31).

2.3

Uniformly Asymptotic Stability

In the section, we intend to study a stability of the equation (2.2.1), uniformly asymptotic stability, which is usually stronger than asymptotic stability but equivalent to exponential stability. To this end, we prefer to take the viewpoint of Theorem 1.3.5 to consider a stochastic version of (2.2.1): t U (t, s)ξ = T (t − s)ξ + s T (t − r)B(U (r, s)ξ)dWr , 0 ≤ s ≤ t, (2.3.1) U (s, s)ξ = ξ, ξ ∈ L2s (Ω; H), where L2s (Ω; H) is the subspace of L2 (Ω; H) which consists of all Fs -measurable random variables. This is certainly an operator integral equation version of (2.2.1), and by Theorem 1.3.5 the following is immediate: Proposition 2.3.1 There exists a unique linear operator family U (t, s) : L2s (Ω; H) → L2t (Ω; H), 0 ≤ s ≤ t, such that W (i). U (t, s)ξ is adapted to Fs,t = σ{Wr − Ws , s ≤ r ≤ t} for each ξ ∈ L2s (Ω; H); (ii). U (s, s) = I, s ≥ 0; (iii). U (t, r)U (r, s) = U (t, s), 0 ≤ s ≤ r ≤ t; W (iv). E{U (t, s)ξ | Fs,r } = T (t − r)U (r, s)ξ, 0 ≤ s ≤ r ≤ t, ξ ∈ L2s (Ω; H);

Stability of Linear Stochastic Diﬀerential Equations

71

(v). U (t, s)ξ is mean square continuous in t, 0 ≤ s ≤ t, ξ ∈ L2s (Ω; H); (vi). U (t, s)ξ satisﬁes the equation (2.3.1). Example 2.3.1 Consider the following stochastic heat equation ∂2 dXt (x) = ∂x µ > 0, 2 Xt (x)dt + µ · Xt (x)dBt , Xt (0) = Xt (1) = 0,

t ≥ 0;

X0 (x) = x0 (x) ∈ R1 ,

(2.3.2) x ∈ [0, 1],

where Bt , t ≥ 0, is a one dimensional standard Brownian motion. In this case H = L2 (0, 1), the C0 -semigroup T (t), t ≥ 0, is generated by d2 dx2 ,

A=

D(A) = u(·) ∈ L2 (0, 1) :

2

du(x) d u(x) dx , dx2

(2.3.3) ∈ L2 (0, 1), u(0) = u(1) = 0 ,

and B(X) = µ · X in (2.2.1). It is not diﬃcult to see that the unique mild solution is given by Xt = e−(µ

2

/2)t+µBt

T (t)x0 ,

∀t ≥ 0,

which has continuous sample paths, and for arbitrary ξ ∈ L2s (Ω; H), 0 ≤ s ≤ t, U (t, s)ξ = e−(µ

2

/2)(t−s)+µ(Bt −Bs )

T (t − s)ξ.

It is easy to see from the very deﬁnitions of stability in Section 1.4 that for the null solution of (2.2.1), the concept of exponential stability in mean square implies asymptotic stability in mean square which further implies stability in mean square. It is also known from Section 2.1 that for deterministic ﬁnite dimensional linear systems, the concept of exponential stability is equivalent to asymptotic stability although Example 2.1.1 shows that this is generally not true for linear systems in inﬁnite dimensions. In spite of this, it is worth pointing out however that for the evolution process U (t, s)ξ in (2.3.1), there exists a stability concept shown below which is somewhat stronger than asymptotic stability but equivalent to exponential stability in the mean square sense. Deﬁnition 2.3.1 The null solution of (2.3.1) is said to be uniformly asymptotically stable in mean square, if for arbitrary ξ ∈ L2s (Ω; H) and number ε > 0, there exists T (ε) ≥ s ≥ 0 such that EU (t, s)ξ2H < ε · Eξ2H whenever t ≥ T (ε) ≥ s. Before moving to our main conclusion, let us ﬁrst derive a result which is also important for its own sake.

72

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Lemma 2.3.1 Suppose (i)–(vi) of Proposition 2.3.1 about the equation (2.3.1) hold, then there exist constants M ≥ 1, τ > 0 such that for any ξ ∈ L2s (Ω; H), t ≥ s ≥ 0, EU (t, s)ξ2H ≤ M · Eξ2H · eτ (t−s) . Proof Since T (t), t ≥ 0, is a strongly continuous semigroup, T (t) ≤ C·eλt , t ≥ 0, for some C ≥ 1, λ ∈ R+ . Now it follows from (2.3.1) that there exists some constant C¯ > 0 such that for any t ≥ s ≥ 0, EU (t, s)ξ2H ≤ 2ET (t − s)ξ2H t

+2 E tr T (t − r)B(U (r, s)ξ)QB ∗ (U (r, s)ξ)T ∗ (t − r) dr s t 2 2λ(t−s) 2 ¯ ≤ 2C e EξH + C e2λ(t−r) EU (r, s)ξ2H dr s

which immediately implies e−2λt EU (t, s)ξ2H ≤ 2C 2 e−2λs Eξ2H + C¯

t

e−2λr EU (r, s)ξ2H dr,

s

for all 0 ≤ s ≤ t. Hence, by virtue of the well-known Gronwall’s inequality, it follows e−2λt EU (t, s)ξ2H ≤ 2C 2 e−2λs Eξ2H · eC(t−s) , ¯

i.e., ¯

EU (t, s)ξ2H ≤ 2C 2 Eξ2H · e(2λ+C)(t−s) = M · Eξ2H · eτ (t−s) where M = 2C 2 ≥ 2 and τ = 2λ + C¯ > 0. We are now in a position to show the main result in the section. Theorem 2.3.1 Let U (t, s) be the unique evolution operator family of (2.3.1) satisfying all the properties in Proposition 2.3.1. Then, the null solution of (2.3.1) is uniformly asymptotically stable in mean square if and only if it is exponentially stable in mean square. Proof Suppose the null solution of (2.3.1) is uniformly asymptotically stable in mean square and choose ε0 = 1/e. Then there exists T (ε0 ) ≥ s such that for arbitrary ξ ∈ L2s (Ω; H), EU (t, s)ξ2H < ε0 Eξ2H for all t ≥ T (ε0 ). Hence, for any ﬁxed t ∈ [s, ∞), t − s = nT (ε0 ) + θ,

Stability of Linear Stochastic Diﬀerential Equations

73

where n is some positive integer and 0 ≤ θ ≤ T (ε0 ). By virtue of (iii) in Proposition 2.3.1 and using Lemma 2.3.1, it follows that for any ξ ∈ L2s (Ω; H), EU (t, s)ξ2H = E U (nT (ε0 ) + s + θ, nT (ε0 ) + s) ·

n−1 #

2 U (n − k)T (ε0 ) + s, (n − k − 1)T (ε0 ) + s ξ

H

k=0

nT (ε0 ) Eξ2H ≤ M eτ θ (1/e)n Eξ2H = M exp τ θ − T (ε0 ) nT (ε0 ) + θ θ = M exp τ θ − + Eξ2H T (ε0 ) T (ε0 )

= M exp[τ θ + θ/T (ε0 )] exp[−(t − s)/T (ε0 )]Eξ2H ≤ M exp[τ T (ε0 ) + 1] exp[−(t − s)/T (ε0 )]Eξ2H . Hence, let C = M exp[τ T (ε0 ) + 1]

and

− µ = −1/T (ε0 ),

then EU (t, s)ξ2H ≤ Ce−µ(t−s) Eξ2H .

(2.3.4)

The converse statement is obvious, and the proof is now complete.

2.4

Some Examples

In this section, we shall investigate several examples to illustrate the theory established in the preceding sections. Example 2.4.1 Consider a one-dimensional rod of length π whose ends are maintained at 0◦ and whose sides are insulated. Assume that there is an exothermic reaction taking place inside the rod with heat being produced proportionally to the temperature. The temperature, denoted by Xt (x), in the rod may be modelled in the following way  ∂X (x) 2 Xt (x) t  dt = ∂ ∂x + rXt (x), t > 0, 0 < x < π, 2 (2.4.1) Xt (0) = Xt (π) = 0, t ≥ 0,  X0 (x) = x0 (x), 0 ≤ x ≤ π, where r ∈ R1 depends on the rate of reaction. If we assume r = r0 , a constant, then we can solve the equation in an explicit way Xt (x) =

∞ n=1

an e−(n

2

−r0 )t

sin nx,

74

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

∞ where x0 (x) = n=1 an sin nx. Hence, we obtain exponential stability if n2 > r0 for all n ∈ N, or equivalently, r0 < 1. This is exactly the condition (H1) in Subsection 2.2.2. Observe that, in general, for r0 ≥ 1 the null solution is not stable. Suppose now that r is random, and assume it is modelled as r = r0 + r1 B˙ t , so that (2.4.1) becomes dXt (x) =

∂2 + r0 Xt (x)dt + r1 Xt (x)dBt , 2 ∂x

(2.4.2)

where Bt is a standard one-dimensional Brownian motion. We substitute this ∂2 into our formulation (2.2.1) by letting K = R1 , H = L2 (0, π), A = ∂x 2 + r0 , B(Xt ) = r1 Xt . Now, by a simple computation ∆(P ) deﬁned in (2.2.3) can be shown to be r12 P so that (H2) in Subsection 2.2.2 becomes r12 < 2(1 − r0 ). This is exactly (2.2.17). Hence, if the unperturbed system (2.4.1) is very stable, i.e., r0 is suﬃciently less than one, then the perturbations (i.e., r1 ) can be fairly large and according to Theorem 2.2.2, we still have EXt (x0 )2H ≤ M · x0 2H e−µt , t ≥ 0, for some constants M ≥ 1, µ > 0. In order to deduce pathwise almost sure stability, note that the conditions in Theorem 2.2.3 are not satisﬁed in this situation. However, we can apply the theory of strong solutions. Precisely, we set for any u, v ∈ V , π ∂u(x) ∂v(x) V = W01,2 = H01 , − u, AvV,V ∗ = + r0 u(x)v(x) dx. ∂x ∂x 0 Then u, AuV,V ∗ = −u2V + r0 u2H , so α = 1 and λ = r0 in (2.2.20). Then by Theorem 2.2.4, it follows that there exist constants M > 0, µ > 0 and a random variable 0 ≤ T (ω) < ∞ such that for all t ≥ T (ω), Xt H ≤ M · x0 H e−µt Example 2.4.2

a.s.

For the next example we suppose (2.4.2) is replaced by

dXt (x) =

∂2 ∂ ∂Xt (x) Xt (x)dt + γ(x) dBt , + r 0 2 ∂x ∂x ∂x

(2.4.3)

where γ(x) ∈ L∞ (0, π; R1 ), i.e., we are observing heat diﬀusion in a rod relative to an origin moving with velocity r0 + γ(·)B˙ t . Let K, V and H be formulated as in Example 2.4.1, then π ∂u ∂u ∂u u, AuV,V ∗ = + r0 u dx, ∀u ∈ V, − ∂x ∂x ∂x 0 π ∂u ∂u u, ∆(I)uV,V ∗ = γ(x)2 dx, ∀u ∈ V. ∂x ∂x 0 Hence, (2.2.27) becomes 2u, AuV,V ∗ + u, ∆(I)uV,V ∗ ≤ −2u2V + γ(·)2∞ u2V ,

Stability of Linear Stochastic Diﬀerential Equations

75

which immediately yields that for arbitrary r0 and γ(·)2∞ < 2, the null solution is exponentially stable both in the mean square and almost sure sense. Example 2.4.3 Also as an application of Theorem 2.2.2, let us investigate the stability of a class of stochastic second order equations generalizing (2.1.31). To this end, let us recall the formulation of the deterministic second order system (2.1.31) with α > 0, d2 v dv +α· + Av = 0, 2 dt dt

dv (0) = v1 , dt

v(0) = v0 ,

(2.4.4)

where A is a strictly positive, self-adjoint operator on some real separable Hilbert space H with domain D(A), so that Ax, xH ≥ Cx2H ,

∀x ∈ D(A),

C > 0.

Then H = D(A1/2 ) ⊕ H is a real separable Hilbert space under the inner product z, z˜H = A1/2 z1 , A1/2 z˜1 H + z2 , z˜2 H , where

z=

z1 z2

,

z˜ =

z˜1 z˜2

.

Deﬁne the closed, linear operator on H

0 I A= with domain D(A) = D(A) ⊕ D(A1/2 ). −A −αI Thus (2.4.4) may be rewritten as a ﬁrst order diﬀerential equation on H, dz(t) = Az(t), dt where

z(t) =

v(t) dv(t)/dt

z(0) = z0 ,

,

z0 =

v0 v1

(2.4.5) .

We have already known from Section 2.1 that A generates a strongly continuous, exponentially stable semigroup T (t), t ≥ 0, on H. We would like to consider the second order stochastic partial diﬀerential equation ∂v

∂2v ∂v dt + c dBt = 0, + αv − 2 ∂t ∂x ∂x v(0, t) = v(1, t) = 0, t ≥ 0, d

t > 0,

x ∈ [0, 1],

76

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where Bt , t ≥ 0, is a standard one-dimensional Brownian motion and α, c are real constants with α > 0. Let H = L2 (0, 1),

2

∂ A = − ∂x 2,

D(A) = u ∈ H : ux , uxx ∈ H and u(0) = u(1) = 0 . Suppose

A=

0 I −A −αI

with domain D(A) = D(A) ⊕ D(A1/2 ).

Then, by virtue of Example 2.1.3, |γ(−A)| = π 2 and T (t) ≤ e−µt , where µ≥ Now

∞

2απ 2 √ . 4π 2 + α(α + α2 + 4π 2 )

T (t)∆(I)T (t)dt ≤ ∗

0

∞

T (t) ∆(I)dt ≤ 2

0 2

=

∞

e−2µt c2 dt

0

c . 2µ

Hence, by Theorem 2.2.2 we have the mean square stability of the null solution if c2 <

4απ 2 √ . 4π 2 + α(α + α2 + 4π 2 )

Example 2.4.4 Consider the following stochastic second order model ∂v ∂4v ∂2v ∂2v d dt + θ dt + c dBt = 0, t > 0, x ∈ [0, 1], + αv + ∂t ∂x4 ∂x2 ∂x2 v(0, t) = v(1, t) = 0, t ≥ 0, where Bt , t ≥ 0, is a standard one-dimensional Brownian motion and α, θ, c are real constants with α > 0. Let D(A) =

∂4 ∂2 ∂x4 + θ ∂x2 , h ∈ H : hx , hxx , h xxx , hxxxx ∈ H and h(0) = h(1) = 0, hxx (0) = hxx (1)

H = L2 (0, 1),

A=

=0

.

2 < π 2 and For A to be self-adjoint and positive, we require θ

obtain by 0 I 2 2 2 Example 2.1.3 that |γ(−A)| = π (π − θ ), and A = is stable if −A −αI θ2 < π 2 and α > 0, with T (t) ≤ e−µt where

µ≥

2απ 2 (π 2 − θ2 ) . 4π 2 (π 2 − θ2 ) + α α2 + 4π 2 (π 2 − θ2 )

Stability of Linear Stochastic Diﬀerential Equations

77

As in the previous example, ∞ T ∗ (t)∆(I)T (t)dt ≤ c2 /2µ 0

and the suﬃcient condition for the mean square stability is c2 < 2µ. Example 2.4.5 Consider an n-dimensional stochastic delay diﬀerential equation of the form 0 dXt + −r Xt+s dN (s)dt = G(Xt )dBt , t ≥ 0, (2.4.6) Xt = 0, −r ≤ t ≤ 0, r > 0, where Xt ∈ Rn , Bt = (Bt1 , · · · , Btn ), t ≥ 0, is a given n-dimensional standard Brownian motion, and N (·) is a left continuous function of bounded variation deﬁned on [−r, 0] into the space of n×n matrices. Let H = Rn ×L2 (−r, 0; Rn ) and D(A) = W 1,2 (−r, 0; Rn ). Then D(A) can be embedded in H as D(A) = {(f (0), f (·)) ∈ H : f ∈ W 1,2 (−r, 0; Rn )} and moreover D(A) is dense in H. Note that f ∈ D(A) is continuous and we write f˙ for its generalized derivative. Deﬁne 0 Af = f (s)dN (s), −f˙(·) , f ∈ D(A). −r

We shall assume that −A generates a strongly continuous semigroup T (t), t ≥ 0, in H. In Delfour, McCalla and Mitter [1], it was shown that this assumption is natural and satisﬁed for many delay systems. Now if we set X t = (Xt , Xt+· ) ∈ H, G(X t ) = (G(Xt ), 0), then (2.4.6) implies dX t + AX t dt = G(X t )dBt . Now, (H1) in Subsection 2.2.2 becomes 0 sup Re λ : det eλs dN (s) + λI = 0 < 0.

(2.4.7)

−r

In (H2), let us be more speciﬁc and assume K = Rp , G(x)k = porder to justify j j j=1 Gj xk , k = {k } ∈ K, where Gj is an n × n matrix. Then ∆(I)x, xH =

p ' j=1

( G∗j Gj x, x

H

and (H2) becomes sup f H =1

0

∞

p j=1

Gj yt (f )2H dt < 1,

(2.4.8)

78

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where yt (f ) is the solution at time t of equation dyt + dt

0

−r

yt+s dN (s) = 0,

t ≥ 0,

with yt (f ) = f (t) for t ≤ 0, f ∈ D(A). Note that A is not usually coercive. However, Theorem 2.2.2 gives the exponential decay of the second moment, and (2.4.6) is actually a ﬁnite dimensional equation, i.e., Xt ∈ Rn . Hence, we can proceed to estimate t 0 P sup Xh+s dN (s) dh > ε N ≤t≤N +1

≤P

N N +1

N N +1

≤

−r 0

E

−r 0

−r

N

H

Xt+s dN (s) dt > ε H

2 " Xt+s dN (s) dt ε2 . H

Now we assume that H = Rn × L2 (−r, 0; Rn ) where either L2 is taken under the measure d|N (·)|, or L2 is taken under the usual Lebesgue measure and dN (s) =

m

ci δsi (s) + n(s) ds,

i=1

where δsi (s) is the delta function at si , and

N +1

N

E

0

−r

2 Xt+s dN (s) dt ≤ C¯ H

sup

0 −r

N −r≤t≤N +1

|n(s)|2 ds < ∞. In either case, EX t 2H ≤ CαEX 0 2H e−µN

for some positive constants C and µ, so that the null solution is exponentially stable in the almost sure sense.

2.5

Notes and Comments

There exist various concepts of stability for inﬁnite-dimensional (stochastic) systems, and the one in which we are especially interested in this chapter is that of exponential stability. Its relation for the deterministic case to the existence of a nonnegative self-adjoint operator solution to the Lyapunov equation (Theorem 2.1.1) has been shown in Datko [1]. The generalization from L2 stability to Lp -stability, i.e., Theorem 2.1.2, was derived in Pazy [3] but the arguments here are taken from Pazy [1]. The idea of Example 2.1.2 is mainly based on Greiner, Voigt and Wolﬀ [1]. The contents of Lyapunov functions

Stability of Linear Stochastic Diﬀerential Equations

79

in inﬁnite dimensional spaces (Theorem 2.1.3 and subsequent corollaries) are mainly taken from Datko [2]. The material in Section 2.3 seems new, but the corresponding idea for the deterministic case goes back at least to Datko [2]. See also Benchimol [1], Daletskii and Krein [1], Ichikawa and Pritchard [1], Pazy [1] and Yosida [1] for related topics in the deterministic situation. There is an extensive literature in an attempt to establish stability theory for stochastic diﬀerential equations in inﬁnite dimensions now. Theorem 2.2.1 is well known in ﬁnite dimensional spaces (cf. Wonham [1]). If the semigroup T (t), t ≥ 0, is generated by a bounded inﬁnitesimal operator on Banach spaces, this result in the deterministic case could be found in Daletskii and Krein [1]. An inﬁnite dimensional stochastic version of Theorem 2.1.1 was ﬁrst derived by Zabczyk [2] in a slightly diﬀerent form from Theorem 2.2.1. The main proofs of Theorem 2.2.1 employed here are taken from Ichikawa [1], [2] and Haussmann [1]. The material in Section 2.2.2 is principally due to Haussmann [1]. But it is worth pointing out that the corresponding ideas of almost sure stability study in ﬁnite dimensional spaces should go back at least to, for instance, Kozin [1], [2]. See also Da Prato and Ichikawa [1], Ichikawa [1], [7], and Skorohod [1] for some related topics on stability of linear stochastic evolution equations. Examples 2.4.3 and 2.4.4 are taken from Curtain [1]. In connection with this, the reader is also referred to Curtain and Pritchard [1], Curtain and Zwart [1] and Pritchard and Zabczyk [1] to ﬁnd some more detailed research for deterministic systems.

Chapter 3 Stability of Nonlinear Stochastic Diﬀerential Equations

The purpose of this chapter is to investigate stability of nonlinear stochastic diﬀerential equations. We ﬁrst formulate this problem as a comparison of the quadratic functionals of two stochastic diﬀerential equations. Then, coercivity types of criteria, actually, Lyapunov’s function methods are presented to deal with stability properties for strong and mild solutions. A Lyapunov function programme is also carried out to handle, apart from stability, ultimate boundedness and associated existence and uniqueness of invariant measures. We investigate the decay rate of mild solutions for a class of nonlinear stochastic evolution equations. Lastly, based on the viewpoint of perturbations of inﬁnite dimensional deterministic systems, the so-called stabilization by white noise sources of systems is studied.

3.1

Equivalence of Lp -Stability and Exponential Stability

Let T (t), t ≥ 0, be a strongly continuous semigroup of bounded linear operators on a real separable Hilbert space H with norm · H and inner product ·, ·H . Let the operator A be the inﬁnitesimal generator of T (t), t ≥ 0. Then recall that T (t)x0 , x0 ∈ D(A), is the unique solution of the diﬀerential equation in H dXt = AXt , dt

X0 = x0 ∈ D(A).

(3.1.1)

It was shown in Theorems 2.1.1 and 2.1.2 that the two statements below are equivalent: (a). T (t) ≤ M · e−µt , t ≥ 0, for some M ≥ 1 and µ > 0; ∞ (b). 0 T (t)xpH dt ≤ CxpH , x ∈ H, for some p ≥ 1 and C > 0. It is natural therefore to ask whether the equivalence of (a) and (b) above still holds for some nonlinear semigroups, or more generally, nonlinear stochastic

81

82

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

systems. To illustrate this more precisely, consider the following autonomous semilinear stochastic diﬀerential equation: dXt = [AXt + F (Xt )]dt + G(Xt )dWt ,

X0 = x0 ∈ H,

where both F : H → H, G : H → L(K, H) are nonlinear, and they satisfy F (0) = 0, G(0) = 0 and the usual Lipschitz and linear growth conditions. By Theorem 1.3.4, it immediately follows that the equation has a unique mild solution Xt = Xt (x0 ), t ≥ 0, for arbitrary x0 ∈ H. In other words, Xt , t ≥ 0, satisﬁes t t Xt = T (t)x0 + T (t − s)F (Xs )ds + T (t − s)G(Xs )dWs . (3.1.2) 0

0

The question is whether or not the two statements below are equivalent for each p ≥ 1: p p −µt (a ). EX ∞ t (x0 )H ≤pM e x0 Hp, x0 ∈ H, for some M ≥ 1 and µ > 0; (b ). 0 EXt (x0 )H dt ≤ Cx0 H , x0 ∈ H, for some C > 0.

It is worth pointing out that the question above is also important even in the deterministic case, i.e., G(·) = 0 in (3.1.2), as one may expect from the viewpoint of Lyapunov’s function methods. The point is that there exist some situations where one may easily ﬁnd Lyapunov functions Λ(·) which are not strict positive deﬁnite (i.e., Λ(·) ≥ c · pH for any c > 0) but ensure p-th moment integrality, i.e., (b) or (b ) holds. To be speciﬁc, let P ≥ 0 satisfy (2.1.13). One may ﬁnd conditions on the nonlinear perturbation F (·) such that 2P Ax, Ax + F (x)H ≤ −λx2H , x ∈ D(A) for some λ > 0.

(3.1.3)

Then, similarly to ﬁnite dimensional case, applying the usual Lyapunov function type of arguments to (3.1.2) with G(·) = 0 and Λ(x) = P x, xH , it is possible to obtain Λ(Xt (x0 )) = P Xt (x0 ), Xt (x0 )H ≤ M e−µt x0 2H for some M > 0 and µ > 0. However, it does not generally follow immediately that Xt (x0 )H ≤ C · e−νt x0 H for some C ≥ 1 and ν > 0, although it is indeed true if Λ(·) is strictly positive deﬁnite in the sense Λ(·) ≥ c · 2H for some constant c > 0 (also see Example 3.1.1 below). The reason is that for some A, for instance, if A is the generator of an analytic, exponentially stable semigroup, P x, xH cannot be equivalent to x2H . Indeed, suppose the contrary is true. It can be deduced that there exist t0 > 0 and a constant C > 0 such that T (t0 )xH ≥ CxH , x ∈ H (see Pazy [2]). Therefore,

Stability of Nonlinear Stochastic Diﬀerential Equations

83

if x ∈ D(A), we have T (t0 )AxH ≥ CAxH . But since T (t), t ≥ 0, is analytic, AT (t0 ) is a bounded operator, and therefore, AxH ≤

1 AT (t0 )xH C

for every x ∈ D(A).

Since D(A) is dense, this holds for every x ∈ H, and A is thus bounded, a conclusion which is generally untrue. In spite of the diﬃculty mentioned above, fortunately, we can deduce in most situations from (3.1.3) the following L2 -stability ∞ Xt (x0 )2H dt ≤ Cx0 2H < ∞, x0 ∈ H for some C > 0. (3.1.4) 0

In what follows, we shall show that under certain circumstances, the condition (3.1.4) or (b ) above actually implies exponential stability of some nonlinear diﬀerential equations in the square or mean square sense, which is a direct nonlinear generalization of Theorem 2.2.1. At the moment, we will not go into further details of Lyapunov’s function methods, especially those related to the above arguments. In Sections 3.4 and 3.5, based on the so-called ﬁrst order linear approximation approaches, a more detailed investigation of the construction of proper Lyapunov’s functions will be presented to deal with stochastic stability of nonlinear diﬀerential equations.

3.1.1

An Extension of Linear Stability Criteria

We intend to begin our discussions by considering a general two-parameter evolution process similar to Theorem 1.3.5 in Section 1.3. Let Z be the space of random variables in the Hilbert space H deﬁned in (Ω, F, P ) and S(t, s), t ≥ s ≥ 0, be a family of nonlinear operators with domain Ds ⊂ Z satisfying the properties: (1). (2). (3). (4).

S(t, s)Ds ⊂ Dt , t ≥ s; S(s, s)ξ = ξ, s ≥ 0, ξ ∈ Ds ; S(t, u)S(u, s) = S(t, s) on Ds , s ≤ u ≤ t; S(·, s)ξ, for each s ≥ 0 and ξ ∈ Ds , is measurable on [s, ∞) × Ω.

We need to use the following fundamental lemma. Lemma 3.1.1 Let 0 < r < 1, L > 0 and n be a nonnegative integer. Then nL ≤ t ≤ (n + 1)L implies e−at ≤ rn ≤ (1/r)e−at , a = −(log r)/L > 0. Proof

Note that rn = en log r and log r < 0, then the result is easily deduced.

The theorem below could be regarded as a nonlinear version of Theorem 2.2.1.

84

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Theorem 3.1.1 Let g(·) be a positive continuous function on [0, ∞) and p > 0. Suppose S(t, s), t ≥ s ≥ 0, deﬁned above satisﬁes the following: ES(t, s)zpH ≤ g(t − s)EzpH ,

z ∈ Ds (p) := {z ∈ Ds : EzpH < ∞}. (3.1.5) Then two conditions below are equivalent: ∞ (i). s ES(t, s)zpH dt ≤ C · EzpH , z ∈ Ds (p), for some C > 0; (ii). ES(t, s)zpH ≤ M · e−µ(t−s) EzpH , z ∈ Ds (p), for some M ≥ 1 and µ > 0. Proof It is suﬃcient to show that (i) implies (ii). This can be obtained by generalizing the proof of Theorem 2.2.1. For any 0 ≤ s < t and z ∈ Ds (p), we have by Condition (3) above and the relation (3.1.5) t t ES(t, s)zpH g −p (t − u)du = g −p (t − u)ES(t, s)zpH du s s t = g −p (t − u)ES(t, u)S(u, s)zpH du s t ≤ g −p (t − u)g p (t − u)ES(u, s)zpH du s t = ES(u, s)zpH du, s

which, together with the condition (i), immediately implies t p ES(t, s)zH g −p (t − u)du ≤ C · EzpH .

(3.1.6)

s

˜ > 0 be an arbitrary but ﬁxed number. Deﬁne J > 0 by Let L J=

˜ L

g −p (u)du.

0

˜ we have from (3.1.6) that ES(t, s)zp ≤ Then J > 0 and for any t − s ≥ L, H p (C/J)EzH . This, together with (3.1.5), implies the existence of some constant C¯ > 0 such that ES(t, s)zpH ≤ C¯ · EzpH

for any t ≥ s ≥ 0

and z ∈ Ds (p). (3.1.7)

Now let t > s, z ∈ Ds (p). By virtue of the condition (3) at the beginning of this subsection and (3.1.7), it follows that t t (t − s)ES(t, s)zpH = ES(t, s)zpH du ≤ C¯ ES(u, s)zpH du s p ¯ ≤ (CC)Ez H.

s

Stability of Nonlinear Stochastic Diﬀerential Equations

85

p ¯ Hence, it follows that ES(t, s)zpH ≤ (CC)Ez H /(t − s), t > s, z ∈ Ds (p). So for each 0 < r < 1, we can choose a number L = L(r) > 0 such that

ES(t, s)zpH ≤ r · EzpH ,

z ∈ Ds (p)

whenever

t − s ≥ L.

(3.1.8)

Let t − s ≥ L, then there is an integer n ≥ 1 such that nL ≤ t − s ≤ (n + 1)L. Using the semigroup property (3) of S(·, ·) and (3.1.8) n times and then (3.1.7), we obtain ES(t, s)zpH ≤ rn REzpH , z ∈ Ds (p) for some constant R > 0. Now Lemma 3.1.1 yields ˜ · e−µ(t−s) Ezp , ES(t, s)zpH ≤ M H

z ∈ Ds (p)

for any t − s ≥ L,

˜ = R/r and µ = −(log r)/L > 0. Combining this with (3.1.7), we where M conclude that ES(t, s)zpH ≤ M · e−µ(t−s) EzpH ,

z ∈ Ds (p),

∀t ≥ s,

˜ , ReµL }. where M = max{M If we apply Theorem 3.1.1 to speciﬁc (deterministic or stochastic) situations, we shall obtain some important stability results. For instance, we may derive: Corollary 3.1.1 Suppose that Ds (p) = Lps (Ω, F, P ; H) for any s ≥ 0, p > 0, as in the notations of Section 1.3. It is known that S(t, s) = U (t−s, 0), t ≥ s ≥ 0, deﬁned as there satisﬁes the conditions at the beginning of this subsection. Then two statements below are equivalent: ∞ (i). 0 EU (t, 0)x0 pH dt ≤ C · x0 pH , x0 ∈ H for some C > 0; (ii). EU (t, 0)x0 pH ≤ M · e−µt x0 pH , x0 ∈ H for some M ≥ 1 and µ > 0. Remark This result improves some of those in Theorem 2.1.2 in which the positive constant p was supposed to be greater than or equal to one. Moreover, the continuity assumption on S(t, s) and g can be relaxed. We may only assume measurability and local boundedness. A typical example of g(t), t ≥ 0, is Ceνt , C > 0, ν > 0, as in the linear case. Corollary 3.1.2 Suppose S(t, s) : Ds (q) → Dt (q) for some q ≥ 2 and let 0 < p ≤ q. Then the conclusion of Theorem 3.1.1 holds provided that Ds (p) is replaced by Ds (q). In particular, (i) of Theorem 3.1.1 implies ES(t, s)zpH ≤ M · e−µ(t−s) [EzqH ]p/q , z ∈ Ds (q) for some M ≥ 1 and µ > 0. As an application of Theorem 3.1.1, we intend to consider Equation (3.1.2). First of all, it is clear that U (t, 0)x0 = Xt (x0 ) is the unique solution of (3.1.2). Proposition 3.1.1 Let F : H → H and G : H → L(K, H) in (3.1.2) both be Lipschitz continuous with F (0) = 0 and G(0) = 0. Suppose that there

86

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

exists a nonnegative twice Fr´echet diﬀerentiable (except possibly at the zero) function Λ(x) on H such that for some c > 0 and p ≥ 2, Λ(x) + xH Λ (x)H + x2H Λ (x) ≤ cxpH ,

for arbitrary x ∈ H, (3.1.9)

and for some d > 0, 1 (LΛ)(x) := Λ (x), Ax + F (x)H + tr[G(x)QG∗ (x)Λ (x)] ≤ −dxpH , 2 (3.1.10) for arbitrary x ∈ D(A). Then for arbitrary x0 ∈ H, EXt (x0 )pH ≤ M · x0 pH e−µt

for some M ≥ 1, µ > 0.

For the proof, we need to consider the following approximation systems of strong solutions: dXtn = AXtn + R(n)F (Xtn )dt + R(n)G(Xtn )dWt , X0n = R(n)x0 ∈ D(A),

(3.1.11)

where 0 < n0 ≤ n ∈ ρ(A) for some n0 ∈ ρ(A), the resolvent set of A, and R(n) = nR(n, A), R(n, A) is the resolvent of A. Proof Applying Itˆ o’s formula to the function Λ(x), x ∈ H, and the strong solution Xtn of (3.1.11) yields Λ(Xtn ) − Λ(X0n ) t t = Λ (Xsn ), AXsn + R(n)F (Xsn )H ds + Λ (Xsn ), R(n)G(Xsn )dWs H 0 0 1 t + tr R(n)G(Xsn )Q[R(n)G(Xsn )]∗ Λ (Xsn ) ds. (3.1.12) 2 0 Therefore, by virtue of (3.1.10) we can deduce that t t n n n p EΛ(Xt ) ≤ Λ(X0 ) − d EXs H ds + E Λ (Xsn ), (R(n) − I)F (Xsn )H 0

0

∗ 1 + tr R(n)G(Xsn )Q R(n)G(Xsn ) Λ (Xsn ) 2 n n ∗ n − G(Xs )QG(Xs ) Λ (Xs ) ds.

(3.1.13)

By virtue of Proposition 1.3.6, there exists a subsequence of {n} ∈ ρ(A), still denote it by {n}, such that Xtn → Xt in C(0, T ; H) almost surely for any T ≥ 0, as n → ∞. Consequently, letting n → ∞ in (3.1.13), together with (3.1.9), immediately yields that for arbitrary t > 0 t EΛ(Xt ) ≤ Λ(x0 ) − d EXs pH ds. 0

Stability of Nonlinear Stochastic Diﬀerential Equations This implies

∞

EXs (x0 )pH ds ≤

0

87

c · x0 pH < ∞. d

Hence, by Corollary 3.1.1 it follows immediately that EXt (x0 )pH ≤ M · x0 pH e−µt ,

x0 ∈ H, t ≥ 0,

for some M ≥ 1, µ > 0. For the purpose of establishing stability, Proposition 3.1.1 is quite useful in constructing less restrictive Lyapunov functions, for instance, those without being strictly positive deﬁnite as mentioned at the beginning of this section. In some situations, this may produce very good results. To see this, let us apply it to the following example. Example 3.1.1 Consider the following stochastic heat equation dy(x, t) = ∂ 2 /∂x2 y(x, t)dt + b(x)f (y(·, t))dBt , t ≥ 0, 0 < x < 1, y(0, t) = y(1, t) = 0, t ≥ 0; y(x, 0) = y0 (x), 0 ≤ x ≤ 1, (3.1.14) where Bt , t ≥ 0, is a real standard Brownian motion, b(·) ∈ L2 (0, 1) and f is a real Lipschitz continuous function on L2 (0, 1) satisfying |f (u)| ≤ cuH , u ∈ L2 (0, 1), c > 0. In this example, we take H = L2 (0, 1) and A = d2 /dx2 with D(A) = H01 (0, 1) ∩ H 2 (0, 1). Then u, AuH ≤ −π 2 u2H

for any u ∈ D(A).

For the sake of simplicity, we assume b(·)H = 1. Let Λ(u), u ∈ H, be a twice Fr´echet diﬀerentiable function on H and deﬁne L by (LΛ)(u) = Λ (u), AuH + (1/2)Λ (u)b, bH |f (u)|2 ,

∀u ∈ D(A).

Then Lu2H = 2u, AuH + b2H |f (u)|2 ≤ −(2π 2 − c2 )u2H ,

∀u ∈ D(A).

Hence, if c2 < 2π 2 , the null solution of (3.1.14) is exponentially stable in mean square by Proposition 3.1.1. On the other hand, consider P ∈ L(H) given by P =

∞

(1/2n2 π 2 )en ⊗ en

n=1

√

where en = 2 sin nπx and for each g, h ∈ H, g ⊗ h ∈ L(H) is deﬁned by (g ⊗ h)u = gh, uH ∈ H, u ∈ H. Then P is a self-adjoint positive nuclear

88

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

operator and in fact the solution of (2.1.13). Obviously, P is not strictly positive. But we have 2P u, AuH ≤ −u2H ,

∀u ∈ D(A).

Now, let Λ(u) = P u, uH and then (LΛ)(u) = 2P u, AuH + P b, bH |f (u)|2

≤ − 1 − P b, bH c2 u2H , ∀u ∈ D(A). Therefore, in view of Proposition 3.1.1 we obtain the region of exponential stability in mean square: c2 < 1/P b, bH . This is larger than {c2 < 2π 2 } if P b, bH < 1/2π 2 . If b = em , m > 1, then P b, bH = 1/2m2 π 2 . The system (3.1.14) is exponentially stable in the mean square sense if c2 < 2m2 π 2 . In other words, the function Λ(u) = P u, uH now plays the role of a Lyapunov function even though P u, uH is not equivalent to u2H .

3.1.2

Comparison Approaches

In this subsection, we shall employ Theorem 3.1.1 or Corollary 3.1.1 to deal with stability of nonlinear stochastic evolution equations. In particular, it will be shown that mean square stability of a class of nonlinear stochastic evolution equations is equivalent to the same stability of some linear stochastic evolution equations provided noise terms in the former are dominated by those of the latter. To this end, we ﬁrst establish the following results on the trace of nuclear operators. Lemma 3.1.2 Let Q ∈ L(K) and P ∈ L(H) be two self-adjoint nonnegative operators and assume G ∈ L(K, H). If Q is a nuclear (or trace class) operator, then tr[GQG∗ P ] = tr[G∗ P GQ] ≥ 0. Proof

Note that Q has a representation Q=

∞

λi ei ⊗ ei

i=1

∞ for some λi ≥ 0 with i=1 λi < ∞ and an orthonormal sequence {ei } in the real separable Hilbert space K, where (g ⊗h)k = gh, kK for any g, h, k ∈ K. Let {fi } be any orthonormal basis in H. Then tr[GQG∗ P ] =

∞ j=1

fj , GQG∗ P fj K =

∞ ∞ j=1 i=1

λi fj , G(ei ⊗ ei )G∗ P fj K

Stability of Nonlinear Stochastic Diﬀerential Equations =

=

∞ ∞

λi fj , Gei K P Gei , fj K =

j=1 i=1 ∞

∞

89

λi Gei , P Gei K

i=1

ei , G∗ P GQei K = tr[G∗ P GQ].

i=1

On the other hand, since P ≥ 0, we have tr[GQG∗ P ] = tr[G∗ P GQ] =

∞

λi Gei , P Gei K ≥ 0

i=1

as required. The proof is now complete. Lemma 3.1.3 Let T ∈ L(H) be a self-adjoint nuclear operator. Then T ≥ 0 if and only if tr(T P ) ≥ 0 for any self-adjoint nonnegative operator P ∈ L(H). Proof Since the necessity is shown in Lemma 3.1.2 (take K = H and G = IH , the identity operator on H), we need only prove the converse part. Let h ∈ H be an arbitrary vector in H and P = h ⊗ h. Then 0 ≤ tr(T P ) = tr[T (h ⊗ h)] = T h, hH . Hence, T is nonnegative. We intend to consider the following stochastic diﬀerential equations dXt = AXt dt + Gi (Xt )dWti ,

X0 = x0 ∈ H,

i = 1, 2,

(3.1.15)

where A generates a strongly continuous semigroup T (t), t ≥ 0, on H, Gi : H → L(Ki , H) is Lipschitz continuous with Gi (0) = 0 and Wti is a Wiener process deﬁned on some real separable Hilbert space Ki with incremental covariance operator Qi , trQi < ∞, i = 1, 2. Then Theorem 1.3.4 is still valid for (3.1.15). For our stability analysis, we hope to compare quadratic functionals of mild solutions under the following condition: G1 (x)Q1 G∗1 (x) ≥ G2 (x)Q2 G∗2 (x)

for any x ∈ H.

(3.1.16)

By Lemmas 3.1.2 and 3.1.3, this is equivalent to tr[G∗1 (x)P G1 (x)Q1 ] ≥ tr[G∗2 (x)P G2 (x)Q2 ]

(3.1.17)

for any self-adjoint nonnegative operator P ≥ 0 in L(H) and x ∈ H. Let Xti (x0 ), t ≥ 0, i = 1, 2, be the mild solutions of (3.1.15). Assume F and M are two self-adjoint nonnegative operators in L(H).

90

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Theorem 3.1.2 Suppose one of Gi is linear, i.e., Gi ∈ L(H, L(Ki , H)), i = 1 or 2. Then the condition (3.1.16) implies the inequality

T

EF XT2 , XT2 H +

EM Xt2 , Xt2 H dt 0 T 1 1 ≤ EF XT , XT H + EM Xt1 , Xt1 H dt

(3.1.18)

0

for any T ≥ 0. Proof (i). Let G1 ∈ L(H, L(K1 , H)). Then by virtue of Proposition 2.2.1, there exists a unique strongly continuous, self-adjoint solution P1 (t) ≥ 0 to the following: d P (t)x, xH + 2P (t)x, AxH + M x, xH + tr[G∗1 (x)P (t)G1 (x)Q1 ] = 0, dt x ∈ D(A), P (T ) = F, 0 ≤ t ≤ T. (3.1.19) Moreover, by Proposition 2.2.2 we have

T

P1 (0)x0 , x0 H = EF XT1 , XT1 H +

EM Xt1 , Xt1 H dt. 0

On the other hand, by (3.1.17) and (3.1.19) we have d P1 (t)x, xH + 2P1 (t)x, AxH + M x, xH dt +tr[G∗2 (x)P1 (t)G2 (x)Q2 ] ≤ 0,

x ∈ D(A).

(3.1.20)

Hence, by a similar argument to the proof of Proposition 2.2.2 (using the approximation procedure), applying Itˆ o’s formula to (3.1.15) and P1 (t)x, xH , we get P1 (0)x0 , x0 H ≥

EF XT2 , XT2 H

T

EM Xt2 , Xt2 H dt.

+ 0

(ii). Similarly, we can show (3.1.18) when G2 is linear. Corollary 3.1.3

Under the same conditions as in Theorem 3.1.2, we have

T

EXt2 (x0 )2H dt ≤ 0

T

EXt1 (x0 )2H dt

for all

0

EXt2 (x0 )2H ≤ EXt1 (x0 )2H

for all

t ≥ 0.

T ≥ 0,

Stability of Nonlinear Stochastic Diﬀerential Equations

91

Proof These follow immediately by taking F = 0, M = I and F = I, M = 0 in (3.1.18), respectively. Corollary 3.1.4 Suppose G1 is linear and the null solution of the corresponding (3.1.15) is exponentially stable in mean square. Then the condition (3.1.16) implies that the null solution of the other (3.1.15) is exponentially stable in mean square. Moreover, for any x0 ∈ H, EXt2 (x0 )2H ≤ M · e−µt x0 2H Proof

for some M ≥ 1

and

µ > 0.

This is immediate by using Corollary 3.1.3.

Sometimes, it is more convenient to restate our results in a slightly diﬀerent manner. Let U be a class of objects satisfying certain required conditions. We say that a stochastic system is uniformly stable in the class U if it is stable for any g ∈ U . For instance, in this notation we can restate Corollary 3.1.4 in the following way. Let U be the set of triples (K2 , Wt2 , G2 ), which satisﬁes the condition (3.1.16). Corollary 3.1.4* Let G1 be linear and the null solution of the corresponding (3.1.15) be exponentially stable in mean square. Then the stochastic system dXt = AXt dt + G2 (Xt )dWt2 ,

X0 = x0 ,

(3.1.21)

is uniformly stable in the class U in the mean square sense. In other words, the mild solution, Xt2 (x0 ), t ≥ 0, of (3.1.21) satisﬁes EXt2 (x0 )2H ≤ M · e−µt x0 2H

for some M ≥ 1

and

µ > 0.

In order to illustrate our theory derived above, let us now consider a class of stochastic diﬀerential equations in which Ki = R1 , Wti = Bt , i = 1, 2, G1 (x) = kba, xH , k > 0, b, a ∈ H and G2 (x) = bg(a, xH ) where g(·) : R1 → R1 , g(0) = 0, is some real continuous function. Then (3.1.15) turns into two stochastic diﬀerential equations: dXt = AXt dt + kba, Xt H dBt , dXt = AXt dt + bg(a, Xt H )dBt ,

X0 = x0 , X0 = x0 .

(3.1.22) (3.1.23)

Here Bt , t ≥ 0, is a one-dimensional standard Brownian motion. Now, let Lk be the set of all real Lipschitz continuous functions g(·) satisfying |g(x)| ≤ k|x|, k > 0, for any x ∈ R1 . If g ∈ Lk , then (3.1.17) turns out to be tr[G∗1 (x)P G1 (x)] = k 2 P b, bH a, x2H ≥ P b, bH |g(a, xH )|2 = tr[G∗2 (x)P G2 (x)],

x ∈ H.

92

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Hence, the nonlinear stochastic system (3.1.23) is uniformly stable in the class Lk if and only if the linear system (3.1.22) is stable. And if the system (3.1.22) is stable in mean square, the mild solution of (3.1.23), of course, satisﬁes EXt (x0 )2H ≤ Ce−γt x0 2H ,

x0 ∈ H

C ≥ 1 and γ > 0. (3.1.24) Therefore, all we need to do is to ﬁnd conditions to assure the stability of (3.1.22). Since (3.1.22) is a linear equation, we have already derived some stability results such as those in Chapter 2. However, owing to the special structure of (3.1.22), we can actually ﬁnd some more delicate suﬃcient conditions for the stability of (3.1.22) and hence of (3.1.23). In particular, by virtue of Example 2.2.1 we have the following: for some

Proposition 3.1.2 The null solution of (3.1.22) is exponentially stable in mean square if and only if the following statements hold: (i). The semigroup T (t), t ≥ 0, is exponentially stable, i.e., there exist num−γt bers C ≥ 1, γ >2 0 such that T (t) ≤ C · e , t ≥ 0; 2 ∞ (ii). k 0 T (t)b, aH dt < 1. Proof

It is immediate by virtue of Example 2.2.1.

Corollary 3.1.5 Suppose that A is stable, i.e., A generates an exponentially stable, strongly continuous semigroup T (t), t ≥ 0, and T (t)b, a2H ≤ M e−2µt ,

M > 0, µ > 0

with

k 2 M < 2µ.

Then the system (3.1.23) is uniformly stable in the class Lk in mean square and moreover (3.1.24) holds. If, in particular, T (t)b, aH = 0, t ≥ 0, then it is uniformly stable in mean square in the class of all Lipschitz continuous functions g(·) with g(0) = 0 and (3.1.24) holds. Proof The ﬁrst part is immediate by (ii) of Proposition 3.1.2. The second part also follows since we can take an arbitrarily large µ > 0. Proposition 3.1.3 Suppose that A is stable and that a ∈ D(A∗ ), A∗ a = −pa, p > 0. If 2p − k 2 b, a2H > 0, then the system (3.1.23) is stable in mean square for any g ∈ Lk and (3.1.24) holds. If, in particular, b, aH = 0, then the same is true for any Lipschitz continuous function g(·) with g(0) = 0. Proof

Note that

2a, xH a, AxH + k 2 b, a2H a, x2H = −(2p − k 2 b, a2H )a, x2H .

Stability of Nonlinear Stochastic Diﬀerential Equations

93

Thus Itˆ o’s formula applied to (3.1.22) and the function a, x2H yields Ea, Xt 2H ≤ e−ct a, x0 2H ,

Example 3.1.2

c = 2p − k 2 b, a2H .

Consider the stochastic heat equation

dy(x, t) = ∂ 2 /∂x2 y(x, t)dt + b(x)g(a(·), y(·, t))dBt ,

0 < x < 1, t ≥ 0,

y(0, t) = y(1, t) = 0, t ≥ 0; y(x, 0) = y0 (x), 0 ≤ x ≤ 1, 1 where a(·), y(·, t) = 0 a(x)y(x, t)dx, t ≥ 0, and Bt is a one-dimensional standard Brownian motion. In this example, we take H = L2 (0, 1) and A = d2 /dx2 with domain D(A) = {y ∈ H : y, y are absolutely continuous, = 0}. ∞ y ∈ H,2 y(0) = y(1) If we take b(x) = sin πx and a(·) = 1, then 0 T (t)b, aH dt = 2/π 4 . Thus by Proposition 3.1.2, we have exponential stability in mean square for any g ∈ Lk with k 2 < π 4 /2. If we take b(x) = cos πx and a(x) = sin πx, then T (t)b, aH = 0. Hence, we have the same stability for any Lipschitz continuous functions g(·) with g(0) = 0. Example 3.1.3 Consider the stochastic delay diﬀerential equation in R1 , dx(t) = [−d1 x(t) + d2 x(t − r)]dt + bg(cx(t))dBt , (3.1.25) x(0) = x0 , x(s) = 0, −r ≤ s < 0, where d1 > 0, r > 0 and d2 , b, c ∈ R1 . Bt , t ≥ 0, is a one-dimensional standard Brownian motion. In this case, we take H = R1 × L2 (−r, 0) with inner product ·, ·H and deﬁne A with D(A) = {(f (0), f (·)) ∈ H : f ∈ W 1,2 (−r, 0)} by

f (0) −d1 f (0) + d2 f (−r) A = f (·) df (·)/ds It can be shown (cf. Delfour, McCalla and Mitter [1]) that A generates a strongly continuous semigroup T (t), t ≥ 0 on H, and then the inﬁnite dimensional model for (3.1.25) is

b c x0 dy(t) = Ay(t)dt + g ,y dBt , y(0) = . (3.1.26) 0 0 0 H Then the system (3.1.26) is exponentially stable in mean square for any g ∈ Lk if and only if T (t), t ≥ 0, is exponentially stable and k2

∞

T (t) 0

2 b c , dt < 1. 0 0 H

94

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

This inequality can be rewritten as ∞ 2 k c2 z 2 (t)dt < 1,

(3.1.27)

0

where z(t) is the solution of dz(t)/dt = −d1 z(t) + d2 z(t − r),

z(0) = b,

z(s) = 0,

−r ≤ s < 0.

On the other hand, it is well known by the theory of ordinary functional diﬀerential equations that if d1 > |d2 | > 0, then T (t), t ≥ 0, is exponentially stable and further by the Lyapunov method (cf. Hale [1], p. 108) that ∞ 2(d1 − |d2 |) z 2 (t)dt ≤ b2 . 0

Thus

k2 0

∞

c2 z 2 (t)dt ≤

k 2 b2 c2 . 2(d1 − |d2 |)

If k < 2(d1 − |d2 |)/b c , then (3.1.27) is satisﬁed and we have exponential stability in mean square of (3.1.26) and hence of (3.1.25) for any g ∈ Lk . 2

3.2

2 2

A Coercive Decay Condition

In the remainder of this chapter, we intend to study decay and relevant large time properties of general non-autonomous, nonlinear stochastic diﬀerential equations. In this section and Section 3.3, we are mainly interested in the exponential decay of strong and mild solutions. Particularly, we will no longer impose the restrictions A(t, 0) = 0, B(t, 0) = 0 in (1.3.1) and F (t, 0) = 0, G(t, 0) = 0 in (1.3.10). The reason for removing this restriction will become clear when we consider some large time properties such as ultimate boundedness of solutions of non-autonomous stochastic diﬀerential equations. In the previous sections, we have shown in Theorems 2.2.1 and 3.1.1 that for linear stochastic diﬀerential equations (2.2.1) or a class of nonlinear stochastic diﬀerential equations (3.1.2), the L2 -stability in mean of systems is equivalent to their exponential stability in mean square. When ones focus on non-autonomous, nonlinear stochastic systems, the situation becomes quite diﬀerent. For instance, ones should not expect that a similar generalization to Theorem 3.1.1 remains true even in ﬁnite dimensional cases. Indeed, to see this, let us consider a one-dimensional stochastic diﬀerential equation dXt = −

p Xt dt + (1 + t)−p dBt , 1+t

t ≥ 0,

Stability of Nonlinear Stochastic Diﬀerential Equations

95

with initial datum X0 = x0 ∈ R1 , where p > 1 is a positive constant and Bt , t ≥ 0, is a one-dimensional standard Brownian motion. It is easy to obtain its explicit solution Xt = (1 + t)−p (x0 + Bt ),

t ≥ 0,

which, by standard properties of Brownian motion, immediately yields that the Lyapunov exponent log E|Xt (x0 )|2 = 0. t→∞ t lim

That is, the solution is not exponentially decayable in mean square. However, by a direct computation we have ∞ 1 + x20 1 E|Xt (x0 )|2 dt = + < ∞. 2p − 1 2p −2 0 In the following two sections, we shall investigate the exponential decay of strong and mild solutions of general non-autonomous stochastic diﬀerential equations in a diﬀerent way from that in the preceding sections. First of all, recall that V is a real, separable Banach space and H, K are two real, separable Hilbert spaces such that V ;→ H ≡ H ∗ ;→ V ∗ , where the injections “ ;→ ” are continuous and dense such that xH ≤ βxV , x ∈ V, for some constant β > 0. Consider the following nonlinear stochastic diﬀerential equation: t t Xt = x0 + A(s, Xs )ds + B(s, Xs )dWs (3.2.1) 0

0

∗

where A(·, ·) : R+ ×V → V is assumed to be a measurable family of nonlinear operators and B(·, ·) : R+ × V → L(K, H), is the measurable family of all nonlinear operators from V into L(K, H), satisfying: B(t, y) − B(t, x) ≤ Ly − xV ,

∀x, y ∈ V,

∀t ≥ 0,

(3.2.2)

for some constant L > 0. Being interested in stability analysis, we always assume that for each x0 ∈ H, there exists a global strong solution to (3.2.1). In particular, we assume the conditions (a)–(e) in Subsection 1.3.1 to ensure the existence and uniqueness of the strong solution of (3.2.1). The following coercive condition will play the role of a stability criterion. (H5). There exist constants α > 0, λ ∈ R1 , and a nonnegative continuous function γ(t), t ∈ R+ , such that 2v, A(t, v)V,V ∗ + B(t, v)2L0 ≤ −αvpV + λv2H + γ(t), 2

v ∈ V,

96

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations where B(t, v)2L0 = tr(B(t, v)QB(t, v)∗ ), p ≥ 2, and γ(t), t ≥ 0, satis2 ﬁes that there exists µ > 0 such that γ(t)eµt is integrable on [0, ∞).

Note that this assumption is compatible with the existence of the strong solution to (3.2.1) formulated in Section 1.3. Theorem 3.2.1 Assume the condition (H5) holds. Let Xt (x0 ), t ≥ 0, be a global strong solution of the equation (3.2.1), then there exist constants τ > 0, C = C(x0 ) > 0 such that EXt (x0 )2H ≤ C(x0 ) · e−τ t ,

∀t ≥ 0,

(3.2.3)

if either one of the following hypotheses holds (i). λ < 0, (∀p ≥ 2). On this occasion, τ can be taken as (−λ) ∧ µ; (ii). λβ 2 − α < 0, (particularly, for p = 2). On this occasion, τ can be taken as ν ∧ µ, where ν = (α − λβ 2 )/β 2 . Proof We omit this proof and refer the reader to Theorem 4.3.1 since a similar formulation which contains the above theorem as a special case is presented there. The following result shows that the condition (H5) not only justiﬁes exponential decay of the equation (3.2.1) in mean square, but it also ensures pathwise exponential decay with probability one. Theorem 3.2.2 Assume the hypotheses in Theorem 3.2.1 hold. Then there exist a subset Ω0 ⊂ Ω with P (Ω0 ) = 0 and random variable T (ω) ≥ 0 such that for each ω ∈ Ω\Ω0 , Xt (x0 )H ≤ M (x0 ) · e−γt ,

∀t ≥ T (ω),

(3.2.4)

for some positive constants M = M (x0 ) > 0 and γ > 0. Proof We only prove the case (ii). Case (i) can be shown in a similar manner. Firstly, applying Itˆ o’s formula to the strong solution Xt (x0 ), t ≥ 0, of (3.2.1) immediately yields for any t ≥ N > 0,

t

Xt (x0 )2H − XN (x0 )2H = 2

t

Xs , A(s, Xs )V,V ∗ ds + N

N

B(s, Xs )2L0 ds 2

t

Xs , B(s, Xs )dWs H ,

+2 N

(3.2.5)

Stability of Nonlinear Stochastic Diﬀerential Equations

97

where N is a positive integer, which, together with the coercive condition (H5), implies

t

Xt 2H ≤ XN 2H + |λ|

t

Xs 2H ds + N

t γ(s)ds + 2 Xs , B(s, Xs )dWs H .

N

N

(3.2.6) Let IN denote the interval [N, N + 1], N > 0, then from (3.2.6) we can get N +1 N +1 E sup Xt 2H ≤ EXN 2H + |λ| EXs 2H ds + γ(s)ds t∈IN N N ) * (3.2.7) t +E sup 2 Xs , B(s, Xs )dWs H . t∈IN

N

Now, according to the Burkholder-Davis-Gundy lemma, we can estimate the last term in (3.2.7) (in what follows, K1 , K2 , · · · denote some proper positive constants). ) * t E supt∈IN 2 N Xs , B(s, Xs )dWs H ) *1/2 N +1 2 2 ≤ K1 E N Xs H B(s, Xs )L0 ds 2

) 1/2 * N +1 2 ≤ K1 E supt∈IN Xt H N B(s, Xs )L0 ds 2 N +1 1 2 ≤ 2 E supt∈IN Xt H + K2 N EB(s, Xs )2L0 ds, 2 (3.2.8) which, by virtue of (3.2.7) and Markov’s inequality, immediately implies for any MN > 0, 2 P sup Xt 2H ≥ M2N ≤ M−2 E sup X t H N t∈IN

t∈IN

2 ≤ K3 M−2 EX + N H N

N +1

+

N N +1

γ(s)ds + N

N +1

N

EXs 2H ds

(3.2.9)

EB(s, Xs )2L0 ds . 2

Note that there exists a positive constant K4 > 0 such that for any N > 0,

N +1

γ(s)ds ≤ e−τ N

N

N +1

γ(s)eµs ds < K4 · e−τ N ,

N

which immediately implies that there exists a constant K5 > 0 such that

N +1

EXN 2H +

N +1

EXs 2H ds + N

N

γ(s)ds ≤ K5 · e−τ N .

(3.2.10)

98

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

For the last term on the right hand side of (3.2.9), assume that the following claim which will be proved below holds. Claim: There exists a positive constant K6 > 0 such that for N > 0 large enough, N +1 EB(s, Xs )2L0 ds ≤ K6 · e−τ N/2 . (3.2.11) 2

N

Then (3.2.9), (3.2.10) and (3.2.11), together with letting M2N = e−τ N/4 , imply that there exists K7 > 0 such that −τ N/2 P sup Xt 2H ≥ M2N ≤ K7 · M−2 = K7 · e−τ N/4 , (3.2.12) N e t∈IN

and a Borel-Cantelli’s lemma type argument completes the proof. Let us ﬁnally prove our claim (3.2.11). Indeed, for the parameter τ > 0 in Theorem 3.2.1, using (H5) we obtain from Itˆ o’s formula that there exists an increasing sequence of stopping times {τn }, n ≥ 1, exactly as in Theorem 3.2.1 such that τ

Ee 2 (t∧τn ) Xt∧τn 2H

t∧τn t∧τn τ τ τ ≤ x0 2H + E e 2 s Xs 2H ds − αE e 2 s Xs 2V ds 2 0 t∧τn 0 t∧τn τ τ s 2 2 +λE e Xs H ds + E γ(s)e 2 s ds, 0

0

which, by virtue of (3.2.3), implies that there exists K8 > 0 such that

t∧τn

E 0

τ

e 2 s Xs 2V ds ) * t t τ τ τ 1 2 s 2 s 2 2 ≤ x0 H + + |λ| e EXs H ds + γ(s)e ds α 2 0 0 ≤ K8 < ∞.

Consequently, letting n tend to inﬁnity, it is easy to deduce t t τ 2 EXu V du ≤ e 2 (u−s) EXu 2V du s

s

≤ K8 · e− 2 s τ

for any

0 ≤ s ≤ t.

Now, taking into account (3.2.2) and (H5), we can get

N +1

N

EB(u, Xu )2L0 du 2 N +1 N +1 ≤ 2 N EB(u, Xu ) − B(u, 0)2L0 du + 2 N EB(u, 0)2L0 du 2 2 N +1 N +1 ≤ 2L2 trQ N EXu 2V du + 2 N γ(u)du ≤ K6 · e−τ N/2

Stability of Nonlinear Stochastic Diﬀerential Equations

99

for some K6 > 0. Now the proof is complete. Remarks (1). In the case that B(t, ·) : H → L(K; H), and Condition (3.2.2) turns out to be B(t, u) − B(t, v) ≤ Lu − vH ,

∀u, v ∈ V,

L > 0, t the last claim follows immediately since the upper bound on s EXu 2H du follows easily from (3.2.3). (2). The exponential decay imposed on γ(t), t ≥ 0, in (H5) is essential. Indeed, to interpret this, let us study the following one-dimensional linear equation (note that on this occasion V = H = R1 ): Example 3.2.1 equation

Assume Xt , t ≥ 0, satisﬁes the stochastic diﬀerential dXt = −pXt dt + (1 + t)−q dBt ,

t ≥ 0,

with initial datum X0 = 0, where p, q > 0 are two positive constants and Bt , t ≥ 0, is a one-dimensional standard Brownian motion. Clearly, the corresponding coercive condition (H5) now becomes 2v, A(t, v)R1 + B(t, v)2L0 = −2pv 2 + (1 + t)−2q , 2

∀v ∈ R1 .

However, under this condition the solution does not decay exponentially. Indeed, it is easy to obtain the explicit solution t Xt = e−pt eps · (1 + s)−q dBs =: e−pt Mt , t ≥ 0, 0

which immediately implies that for arbitrarily given q > 0, the Lyapunov exponent log E|Xt |2 lim = 0. t→∞ t Similarly, by virtue of the well known iterated logarithmic law lim sup t→∞

Mt 2[Mt ] log log[Mt ]

=1

a.s.

where [Mt ] is the quadratic variation of Mt , and t log 0 e2ps (1 + s)−2q ds lim sup = 2p, t t→∞ we therefore get the almost sure Lyapunov exponent lim sup t→∞

log |Xt | =0 t

a.s.

100

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

In other words, despite the obvious stability of the ordinary diﬀerential equation dXt = −pXt dt, the polynomial decay type of additive noises is not suﬃcient to guarantee the exponential decay of its stochastically perturbed system. In fact, we will show in Section 3.8 that the solution has a slower, polynomial, decay than the exponential one. As an immediate consequence of Theorems 3.2.1 and 3.2.2, we may formulate the following fractional power type of coercive condition which is quite useful in dealing with certain nonlinear stochastic systems. (H5∗ ). There exist constants α > 0, λ ∈ R1 , 0 ≤ σ ≤ 1 and nonnegative continuous functions γ(t), ζ(t), t ∈ R+ , such that 2u, A(t, u)V,V ∗ + B(t, u)2L0 2

≤ − αupV + λu2H + ζ(t)u2σ H + γ(t),

u ∈ V, (3.2.13)

where p ≥ 2, and γ(t), ζ(t) satisfy that there exist θ > 0, µ > 0 such that γ(t)eµt and ζ(t)eθt both are integrable on [0, ∞). Corollary 3.2.1 Suppose that (H5∗ ) and (3.2.2) hold. Let Xt (x0 ), t ≥ 0, be a global strong solution to the equation (3.2.1). Then there exist constants τ > 0, C = C(x0 ) > 0 such that EXt (x0 )2H ≤ C(x0 ) · e−τ t ,

∀t ≥ 0,

(3.2.14)

if either one of the following hypotheses holds: (i). λ < 0, (∀p ≥ 2). On this occasion, τ can be taken as (−λ) ∧ µ ∧ θ; (ii). λβ 2 −α < 0, (particularly, f or p = 2). On this occasion, τ can be taken as ν ∧ µ ∧ θ, where ν = (α − λβ 2 )/β 2 . Moreover, there exist positive numbers M = M (x0 ) > 0, r > 0, subset Ω0 ⊂ Ω with P (Ω0 ) = 0 and a positive random variable T (ω) such that for each ω ∈ Ω\Ω0 , Xt (x0 )H ≤ M (x0 ) · e−rt , ∀t ≥ T (ω). Proof Observe that the case σ = 0 or σ = 1 is trivial by using Theorems 3.2.1 and 3.2.2. For 0 < σ < 1, by virtue of Young’s inequality a·b≤

ap bq + p q

for any a ≥ 0, b ≥ 0, p, q > 1 with 1/p + 1/q = 1,

Stability of Nonlinear Stochastic Diﬀerential Equations

101

we have for arbitrary ε > 0, the third term on the right hand side of (3.2.13) turns out to be 1

1

1/σ ζ(t)u2σ u2H + (1 − σ)ε 1−σ ζ(t) 1−σ , H ≤ σε

which, together with (3.2.13), implies that 2u, A(t, u)V,V ∗ + B(t, u)2L0 ≤ − αupV + λu2H + σε1/σ u2H 2

1

1

+ γ(t) + (1 − σ)ε 1−σ ζ(t) 1−σ ,

u ∈ V.

In the case (ii), by virtue of Theorems 3.2.1 and 3.2.2, it is easy to deduce that if ν > σε1/σ , the solution is exponentially decayable in the mean square and almost sure senses. Note that ε > 0 is an arbitrary number, then the proof of (ii) is complete letting ε → 0. The result (i) can be shown similarly.

Lastly, let us investigate an example to close this section. Example 3.2.2 Let O be an open, bounded subset in Rn , n ≥ 1, with regular boundary and let 2 < p < ∞. Consider the Sobolev space V = W01,p (O), H = L2 (O) with their usual norms, and the monotone operator A : V → V ∗ is deﬁned as for any u, v ∈ V , n ∂u(x) p−2 ∂u(x) ∂v(x) v, AuV,V ∗ = − dx − a(x)u(x)v(x)dx, ∂xi ∂xi ∂xi O i=1 O where a ∈ L∞ (O; R1 ) satisﬁes a(x) ≥ a ˜ > 0, x ∈ O. Here a ˜ is some constant. We also consider B(t, u) = g(u(x)), u ∈ V , in (3.2.1) where g : R1 → R1 is some Lipschitz continuous function satisfying g(x) − g(y) ≤ L|x − y|, ∀x, y ∈ R1 , for some constant L > 0 such that L2 < 2˜ a and g(0) = 0. Let Bt , t ≥ 0, be a standard real Brownian motion (so, K = R1 and Q = I). In this case, we can claim that there exists a unique strong solution of the equation (3.2.1). In particular, (H5) holds with γ(·) = 0, λ = −(2˜ a − L2 ) < 0, p > 2, α = 2. Consequently, using Theorems 3.2.1 and 3.2.2, we may obtain the exponential decay both in the mean square and almost sure senses of the equation which is interpreted as:

) * n ∂ ∂Xt (x) p−2 ∂Xt (x) dXt (x) = − − a(x)Xt (x) dt ∂xi ∂xi ∂xi i=1 +g(Xt (x))dBt ,

t > 0, x ∈ O;

X0 (x) = x0 (x) ∈ H in O; Xt (x) = 0 almost surely in (0, ∞) × ∂O.

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

3.3

Stability of Semilinear Stochastic Evolution Equations

In this section, we shall consider the following stochastic evolution equation: for any 0 ≤ t < ∞,  t t  dXt = T (t)x0 + T (t − s)F (s, Xs )ds + T (t − s)G(s, Xs )dWs , 0 0  X0 = x0 ∈ H, (3.3.1) where T (t), t ≥ 0, is some C0 -semigroup of bounded linear operators on H with its inﬁnitesimal generator A. Also, Wt , t ≥ 0, is some given K-valued, QWiener process with trQ < ∞, and F (t, ·) and G(t, ·) are in general nonlinear measurable mappings from H to H and H to L(K, H), respectively, satisfying the following Lipschitz and linear growth conditions F (t, y) − F (t, z)H + G(t, y) − G(t, z) ≤ Ly − zH , F (t, y)H + G(t, y) ≤ L(1 + yH ),

t ∈ [0, ∞),

t ∈ [0, ∞),

(3.3.2)

for some constant L > 0 and arbitrary y, z ∈ H. Then by virtue of Theorem 1.3.4, there exists a unique solution Xt , t ≥ 0, of (3.3.1) for arbitrary x0 ∈ H. It is also known by Proposition 1.3.6 that one can always ﬁnd a modiﬁcation with continuous sample paths of the solution. We now consider the exponential decay of (3.3.1) by means of a Lyapunov function type of argument. Theorem 3.3.1

Let Λ(x) : H → R1 satisfy that:

(i). Λ(x) is twice Fr´echet diﬀerentiable and Λ (x), Λ (x) are continuous in H and L(H), respectively, and |Λ(x)| + xH Λ (x)H +x2H Λ (x) ≤ cxpH for some p ≥ 2 and c > 0;

(3.3.3)

(ii). There exist constant α > 0 and a nonnegative continuous function γ(t), t ∈ R+ , such that (LΛ)(t, x) ≤ −αΛ(x) + γ(t),

x ∈ D(A),

(3.3.4)

where (LΛ)(t, x) := Λ (x), Ax + F (t, x)H + 1/2 · tr(Λ (x)G(t, x)QG∗ (t, x)), x ∈ D(A), t ≥ 0, and γ(t) satisﬁes that there exists a constant µ > 0 such that γ(t)eµt is integrable on [0, ∞).

Stability of Nonlinear Stochastic Diﬀerential Equations

103

Then there exist constants τ > 0, C = C(x0 ) > 0 such that for the solution Xt (x0 ), t ≥ 0, of (3.3.1), EΛ(Xt (x0 )) ≤ C(x0 ) · e−τ t ,

∀t ≥ 0.

(3.3.5)

In particular, τ can be taken as α ∧ µ. To prove this theorem, we introduce the following approximating systems of (3.3.1) and use Proposition 1.3.6: dXt = AXt dt + R(n)F (t, Xt )dt + R(n)G(t, Xt )dWt , X0 = R(n)x0 ∈ D(A),

(3.3.6)

where 0 < n0 ≤ n ∈ ρ(A) for some n0 ∈ N, the resolvent set of A and R(n) = nR(n, A), R(n, A) is the resolvent of A. Proof Applying Itˆ o’s formula to the function v(t, x) = eµt Λ(x) and the n strong solution Xt of (3.3.6) yields that for any t ≥ 0, eµt Λ(Xtn ) − Λ(X0n ) t t eµs Λ(Xsn )ds + eµs Λ (Xsn ), AXsn + R(n)F (s, Xsn )H ds = µ 0 0 t + eµs Λ (Xsn ), R(n)G(s, Xsn )dWs H (3.3.7) 0 t 1 + · eµs tr R(n)G(s, Xsn )Q[R(n)G(s, Xsn )]∗ Λ (Xsn ) ds. 2 0 Therefore, by virtue of (3.3.3), (3.3.4) and taking expectations in (3.3.7), we can deduce t t n µs n µt n e EΛ(Xt ) ≤ Λ(X0 ) + (µ − α) e EΛ(Xs )ds + γ(s)eµs ds 0 0 t + eµs E Λ (Xsn ), (R(n) − I)F (s, Xsn )H 0 (3.3.8) 1 n n ∗ n + tr R(n)G(s, Xs )Q R(n)G(s, Xs ) Λ (Xs ) 2 n n ∗ n −G(s, Xs )QG(s, Xs ) Λ (Xs ) ds. Consequently, letting n → ∞ in (3.3.8) and using Proposition 1.3.6 immediately yields t t eµt EΛ(Xt ) ≤ Λ(x0 ) + (µ − α) eµs EΛ(Xs )ds + γ(s)eµs ds. (3.3.9) 0

0

Then, we can carry out a similar argument to that in the proof of Theorem 4.3.1 to obtain that there exist positive constants τ > 0 and C = C(x0 ) > 0

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

such that EΛ(Xt ) ≤ C(x0 ) · e−τ t . In particular, the constant τ can be taken as α ∧ µ. The proof is complete now. As an immediate application of Theorem 3.3.1, we may consider stability of the following stochastic diﬀerential equation to obtain an autonomous version dXt = [AXt + F (Xt )]dt + G(Xt )dWt ,

X0 = x0 ∈ H,

(3.3.10)

where F (·), G(·) both satisfy the corresponding Lipschitz continuous conditions as in (3.3.2) and F (0) = 0, G(0) = 0. Corollary 3.3.1

Let Λ(x) : H → R1 satisfy the following:

(i). Λ(x) is twice Fr´echet diﬀerentiable and Λ (x), Λ (x) are continuous in H and L(H), respectively, and |Λ(x)| + xH Λ (x)H +x2H Λ (x) ≤ cxpH for some p ≥ 2 and c > 0; (ii). There exists a constant α > 0 such that (LΛ)(x) ≤ −αΛ(x),

x ∈ D(A),

where (LΛ)(x) := Λ (x), Ax + F (x)H + 1/2 · tr(Λ (x)G(x)QG∗ (x)), x ∈ D(A). Then the mild solution Xt (x0 ), t ≥ 0, of (3.3.10) satisﬁes EΛ(Xt (x0 )) ≤ cx0 pH · e−αt ,

∀t ≥ 0.

Moreover, if kΛ(x) ≥ xpH for some k > 0, then EXt (x0 )pH ≤ k·cx0 pH e−αt . Next, we will prove that under the same assumptions as in Theorem 3.3.1, the solution of (3.3.1) does decay exponentially in the almost sure sense. To this end, we ﬁrst derive a lemma as follows. Lemma 3.3.1 Suppose that there exists a function Λ(x) satisfying (i)–(ii) in Theorem 3.3.1 and kΛ(x) ≥ xpH for some k > 0. Let Xt (x0 ), t ≥ 0, be the solution of (3.3.1) satisfying (3.3.2). Then for any 0 ≤ t ≤ T , t (a). Λ(Xt ) ≤ C1 (x0 ) + 0 Λ (Xr ), G(r, Xr )dWr H ; (b). E sup0≤t≤T Λ(Xt ) ≤ C2 (x0 ), where C1 (x0 ) > 0, C2 (x0 ) > 0 are independent of T .

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105

∞ Proof Note that (LΛ)(t, x) ≤ −αΛ(x) + γ(t), t ≥ 0, and 0 γ(t)dt < ∞. Now we apply Itˆ o’s formula to the function Λ(x) and the process Xtn given by (3.3.6), then t n Λ(Xt ) ≤ Λ(R(n)x0 ) + Λ (Xrn ), R(n)F (r, Xrn ) − F (r, Xrn )H dr 0 ∞ 1 t + γ(r)dr + tr(R(n)G(r, Xrn )QG∗ (r, Xrn )R(n)∗ Λ (Xrn )) 2 0 0 −tr(G(r, Xrn )QG∗ (r, Xrn )Λ (Xrn )) dr t + Λ (Xrn ), R(n)G(r, Xrn )dWr H . 0

In terms of (3.3.3), we can let n tend to inﬁnity as in Proposition 1.3.6 and pass on the limit in the inequality above to obtain (a). To prove (b), note that ) t * E sup0≤t≤T Λ (Xr ), G(r, Xr )dWr H 0 ) *1/2 T ≤ E trQ Λ (Xt )2H G(t, Xt )2 dt 0 T 1 ≤ E sup Λ(Xt ) + c EΛ(Xt )dt 2 0≤t≤T 0 for some c > 0. Thus from (a) and Theorem 3.3.1, it follows that T E sup Λ(Xt ) ≤ 2C1 (x0 ) + 2c C(x0 )e−τ t dt = C2 (x0 ) 0≤t≤T

0

for some C2 (x0 ) > 0. Theorem 3.3.2 Under the same conditions as in Lemma 3.3.1, there exists a random variable 0 ≤ T (ω) < ∞ and a constant M = M (x0 ) > 0 such that for all t > T (ω), Xt (x0 )H ≤ M (x0 ) · e−τ t/4p

a.s.

Proof We only sketch the proofs because they are quite similar to those in Theorem 3.2.2. By Lemma 3.3.1 and some obvious modiﬁcations, we can deduce for any t ≥ N (an arbitrary positive integer), t t Λ(Xt ) ≤ Λ(XN ) + γ(s)ds + Λ (Xs ), G(s, Xs )dWs H . N

N

By the Burkholder-Davis-Gundy inequality, we can deduce for any εN > 0, t P sup Λ (Xs ), G(s, Xs )dWs H ≥ εN /2 ≤ (C1 (x0 )/εN )e−τ N/2 N ≤t≤N +1

N

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

for some C1 (x0 ) > 0. Hence, a similar argument to Theorem 3.2.2 yields P sup Λ(Xt ) ≥ εN ≤ (C2 (x0 )/εN )e−τ N/2 N ≤t≤N +1

for some C2 (x0 ) > 0. If we take εN = C2 (x0 )e−τ N/4 and carry out a BorelCantelli lemma type of argument, it can be concluded that there exists a random variable 0 ≤ T (ω) < ∞ such that if t ≥ T (ω), Xt (x0 )H ≤ M (x0 ) · e−τ t/4p

a.s.

for some M (x0 ) > 0. Example 3.3.1 Let us consider the following semilinear stochastic partial diﬀerential equation: ∂2 −µt dYt (x) = ∂x α Yt (x) dBt , t > 0, x ∈ (0, 1), 2 Yt (x)dt + e (3.3.11) Y0 (x) = y0 (x), 0 ≤ x ≤ 1; Yt (0) = Yt (1) = 0, t ≥ 0, where Bt , t ≥ 0, is a real standard Wiener process (so, K = R1 ). α(·) : R1 → R1 is some bounded, Lipschitz continuous function, α(0) = 0, and µ a certain positive number. We can specialise our formulation to this example by taking ∂2 H = L2 [0, 1], and A = ∂x 2 with D(A) = y ∈ H : y, y are absolutely continuous with y , y ∈ H, y(0) = y(1) = 0 , and F (t, y) = 0, G(t, y) = e−µt α(y). Clearly, the operator G(t, ·) satisﬁes Condition (3.3.2). On the other hand, let Λ(x) = x2H , then it is easy to deduce that for arbitrary y ∈ D(A), 2y, Ay + F (t, y)H + G(t, y)2 ≤ −2πy2H + M e−2µt ,

(3.3.12)

where M is some positive constant. Therefore, since the hypotheses in Theorems 3.3.1 and 3.3.2 are fulﬁlled, we may deduce that the null solution of the equation is exponentially stable in mean square, that is, there exist positive constants τ > 0, C > 0 such that EYt (y0 )2H ≤ Cy0 2H · e−τ t ,

∀t ≥ 0,

and meanwhile it is also exponentially stable in the almost sure sense. Example 3.3.2 Consider the semilinear stochastic heat equation ∂2 Yt (x) σYt (x) dYt (x) = Y (x) − dt + dBt , t > 0, x ∈ (0, 1), t ∂x2 1 + |Yt (x)| 1 + |Yt (x)| Y0 (x) = y0 (x), 0 ≤ x ≤ 1; Yt (0) = Yt (1) = 0, t ≥ 0, (3.3.13)

Stability of Nonlinear Stochastic Diﬀerential Equations

107

where σ > 0 is some constant. The space H and process Bt , t ≥ 0, are deﬁned exactly as in Example 3.3.1, K = R1 , F (y) = −

G(y) y , =− σ 1 + yH

2

∂ and A = ∂x ∈ 2 with D(A) = {y ∈ H : y, y are absolutely continuous, y , y 2 H, y(0) = y(1) = 0}. Then y, Ay + F (y)H ≤ −yH , y ∈ D(A), and

1 LypH ≤ −p 1 − σ 2 (p − 1) ypH . 2 Thus if 2 ≤ p < 1 + 2/σ 2 , by virtue of Corollary 3.3.1 the null solution is exponentially stable in p-th moment. In this case, we also have exponential stability of sample paths with probability one. There are some occasions where we only need a weaker notion of stability. In these situations, it is often possible to relax conditions such as those in Corollary 3.3.1 to ensure stability. Theorem 3.3.3 ties:

Suppose there exists a function Λ(x), x ∈ H, with proper-

i). |Λ(x)| + Λ (x)H + Λ (x) ≤ c(1 + xpH ) for some c > 0 and p > 0; ii). Λ(0) = 0, Λ(x) > 0 for 0 < xH < R < ∞, R > 0, and b(r) = inf xH =r Λ(x) > 0 for any r > 0; iii). (LΛ)(x) ≤ u(x), for any x ∈ D(A), where u(·) is a continuous function such that u(x) ≤ 0 for 0 ≤ xH ≤ R and |u(x)| ≤ k(1 + xqH ) for some k > 0 and q > 0. Then the null solution of (3.3.10) is stable in probability. Proof Carrying out a similar argument to Lemma 3.3.1 and using (iii), we may deduce that Λ(Xt (x0 )) ≤ Λ(x0 ) +

t

u(Xs (x0 ))ds + 0

t

Λ (Xs (x0 )), G(Xs (x0 ))dWs H .

0

Now let 0 < x0 H < r < R

and

τ = τ (r) = inf{t > 0 : Xt (x0 )H > r}.

Then Λ(Xt∧τ (x0 )) ≤ Λ(x0 ) + 0

t∧τ

Λ (Xs (x0 )), G(Xs (x0 ))dWs H .

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Therefore, EΛ(Xt∧τ (x0 )) ≤ Λ(x0 ). This yields

P τ
inf

xH =r

Λ(x) ≤ Λ(x0 ).

But b(r) > 0 for any r > 0, and Λ(x0 ) P τ
Λ(x0 ) P τ <∞ ≤ . b(r)

Since Λ(0) = 0 and Λ(x) is continuous, for each ε > 0 there exists a δ = δ(r, ε) > 0 such that Λ(x0 )/b(r) < ε if x0 H < δ. Thus P Xt (x0 )H > r for some t ≥ 0 ≤ P τ < ∞ < ε if x0 H < δ. The proof is now complete. Corollary 3.3.2 Suppose the conditions of Theorem 3.3.3 hold for R = ∞. Assume also Λ(x) → ∞ as xH → ∞. Then Xt (x0 ) is bounded almost surely for any x0 ∈ H, i.e., P sup Xt (x0 )H < ∞ = 1. 0≤t<∞

Proof P

Note that

sup Xt (x0 )H < ∞ = P 0≤t<∞

+ ∞

n=1

sup Xt (x0 )H > n

t

≤ P τ (n) < ∞ ≤ Λ(x0 )/b(n) → 0, as n → ∞.

Under somewhat restricted conditions, we may also consider asymptotic almost sure stability of (3.3.10) described in the following manner. Theorem 3.3.4

Suppose there exists a function Λ(x) with properties:

i). |Λ(x)| + Λ (x)H + Λ (x) ≤ c(1 + xpH ) for some c > 0 and p > 0; ii). Λ(0) = 0, Λ(x) > 0 for x = 0; iii). (LΛ)(x) ≤ −αΛ(x) for any x ∈ D(A) and some constant α > 0;

Stability of Nonlinear Stochastic Diﬀerential Equations

109

iv). b(r) = inf xH =r Λ(x) > 0 for any r > 0. Then, the null solution of (3.3.10) is asymptotically stable in the almost sure sense. Proof Note that the stability in probability follows from Theorem 3.3.3. We can show as in Theorem 3.3.2 that for 0 < τ < α, Λ(Xt (x0 )) ≤ e−τ t Λ(x0 )

for

t > T0 (ω)

almost surely for some random variable 0 ≤ T0 (ω) < ∞. Using iv) we conclude Xt (x0 ) → 0

almost surely

as

t → ∞.

Remark The results derived in Theorem 3.3.3 of course are weaker than exponential stability. However, the restriction on the Lyapunov function here is more relaxed. For instance, we do not require its strict positive deﬁniteness, a case which is useful in many practical situations such as Example 3.1.1. Lastly, we take here the diﬀerence of two solutions of (3.3.10) with diﬀerent initial conditions and consider their asymptotic behavior. The results below, whose proofs are fairly straightforward and therefore omitted here, will play an important role in the investigation of uniqueness of invariant measures later on. Proposition 3.3.1 x ∈ D(A),

Suppose that x, AxH ≤ −αx2H , α > 0, for any

F (y) − F (z)H ≤ L1 y − zH ,

G(y) − G(z) ≤ L2 y − zH ,

y, z ∈ H,

for two positive constants L1 , L2 > 0, and let τ = 2α − 2L1 − L22 trQ > 0. Then for the mild solution of (3.3.10), we have that for some C > 0, Xt (x1 ) − Xt (x2 )2H ≤ Cx1 − x2 2H e−τ t

for ∀t ≥ T (ω)

almost surely for some random variable 0 ≤ T (ω) < ∞. We also deﬁne the following operator (Ld Λ)(y − z) = Λ (y − z), A(y − z) + F (y) − F (z)H +(1/2)tr [G(y) − G(z)]∗ Λ (y − z)[G(y) − G(z)]Q ,

y, z ∈ D(A).

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Proposition 3.3.2 ties: i). ii). iii). iv).

Suppose that there exists a function Λ(x) with proper-

|Λ(x)| + Λ (x)H + Λ (x) ≤ c(1 + xpH ) for some c > 0 and p > 0; Λ(0) = 0, Λ(x) > 0, x = 0; (Ld Λ)(y − z) ≤ −αΛ(y − z) for any y, z ∈ D(A) and some α > 0; inf xH ≥r Λ(x) = b(r) > 0 for all r > 0.

Then, for the mild solution Xt (·) of (3.3.10), we have Xt (x1 ) − Xt (x2 ) → 0, as t → ∞, almost surely.

3.4

Lyapunov Functions for Strong Solutions

In this section, using the same notions and notations as in Section 1.3, we intend to consider strong solutions of the following inﬁnite-dimensional stochastic diﬀerential equation in V ∗ :   Xt (ω) ∈ M 2 (t0 , T ; V ), dXt = A(t, Xt )dt + B(t, Xt )dWt , t ∈ [t0 , T ], (3.4.1)  Xt0 = x0 ∈ H, where T ≥ t0 ≥ 0 and M 2 (t0 , T ; V ) denotes the space of all V -valued, Ft T adapted processes Xt (·) satisfying E t0 Xt (ω)2V dt < ∞. The main objective is to provide a necessary and suﬃcient condition for exponential decay of strong solutions in the mean square sense in terms of the existence of Lyapunov functions. It was shown in Proposition 2.1.2 that the usefulness of Lyapunov’s theorem in linear ordinary diﬀerential equations is that it allows for an explicit representation of a Lyapunov function as a positive deﬁnite quadratic form. Using this representation one may then, for instance, study the effects of perturbations about stability on linear ordinary diﬀerential equations (cf. Section 3.1). It is essential, as we mentioned in Section 2.1, that for ∞ P x, xH = Λ(x) = 0 T (t)x2H dt to be a Lyapunov function, we have an estimate of the form P x, xH ≥ cx2H for some c > 0. In other words, the form P has to deﬁne an equivalent norm on H. If H is ﬁnite dimensional, this is always the case. However, if H is inﬁnite dimensional, this could be false as shown by the example at the beginning of Section 3.1. One of the methods of overcoming this diﬃculty is to ﬁnd conditions to ensure L2 -stability and then use Theorem 3.1.1 to obtain exponential stability as shown in Section 3.1. However, it is still possible to construct Lyapunov functions with the desired norm equivalence by a diﬀerent method. This method provides suﬃcient and necessary conditions, thus is quite powerful in dealing with nonlinear systems by means of the so-called ﬁrst order linear approximation procedure.

Stability of Nonlinear Stochastic Diﬀerential Equations

111

In this and next sections, using a diﬀerent method from the above we shall establish Lyapunov functions for strong and mild solutions. To this end, let us ﬁrst study the following linear partial diﬀerential equation which motivates the construction of a desirable Lyapunov function, ∂Yt (x) ∂ 2 Yt (x) ∂Yt (x) +b = a2 + cYt (x) 2 ∂t ∂x ∂x

(3.4.2)

where a, b and c are real numbers, with the initial conditions y0 (·) ∈ L2 (−∞, ∞) ∩ L1 (−∞, ∞).

Y0 (x) = y0 (x),

Let y˜0 (λ) be the Fourier transform of y0 (x) and Y˜t (λ) of the solution Yty0 (x). Then (3.4.2) has the form, for each λ, dY˜t (λ) = −a2 λ2 Y˜t (λ) + (ibλ + c)Y˜t (λ) dt = Y˜t (λ)(c + ibλ − a2 λ2 ). By the well-known Plancherel theorem, we get Yty0 (·)2H = Y˜t (·)2H with H = L2 (−∞, ∞), which immediately yields ∞ Yty0 (·)2H = ˜ y0 (λ)2H exp{(−2a2 λ2 + 2c)t}dλ. −∞

Assume c < 0, then Yty0 (·)2H ≤ y0 2H · e2ct . If we further deﬁne the Lyapunov function as we usually do in ﬁnite dimensional cases in the following manner, ∞ ∞ ∞ Λ(y0 ) := Yty0 (·)2H dt = ˜ y0 (λ)2H exp{(−2a2 λ2 + 2c)t}dλdt 0

0 ∞

= −∞

−∞

˜ y0 (λ)2H dλ, 2a2 λ2 − 2c

then Λ(·) does not satisfy the condition Λ(y) ≥ dy2H , y ∈ H, for some number d > 0. However, let V ;→ L2 (−∞, ∞) be the usual Sobolev spaces such that the injection ;→ is continuously embedded, then if a = 0, we may get t → Yty0 (·)2V is continuous and in this case ∞ Λ(y0 ) = Yty0 (·)2V dt ≥ d y0 2H for some d > 0. 0

Therefore, this example reminds us that in inﬁnite dimensional spaces, we might need · 2V in the deﬁnition of the Lyapunov function Λ(·).

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

In this section, in addition to those in Subsection 1.3.1 ensuring existence and uniqueness of strong solutions, we shall impose the following coercive condition, which is compatible with (1.3.2), for our stability purpose, i.e., we shall assume that there exist constants α > 0, γ ≥ 0, µ ≥ 0 and λ ∈ R1 such that: 2u, A(t, u)V,V ∗ +B(t, u)2L0 ≤ −αu2V +λu2H +γ ·e−µt ,

∀u ∈ V, t ≥ 0. (3.4.3) In order to obtain our main results, we need to use the following fundamental result. 2

Lemma 3.4.1 Assume function g(·) ≥ 0 almost surely, and g(·) ∈ L1 [0, T ] for arbitrary T ≥ 0, then T t+∆ T g(s)ds t lim g(t)dt. dt = ∆→0 0 ∆ 0 Proof By virtue of Fubini theorem, it is easy to deduce T t+∆ g(s)ds 1 T t+∆ t dt = g(s)ds dt ∆ ∆ 0 0 t T T ∆ s 1 = g(s)dt ds + g(s)dt ds ∆ 0 0 ∆ s−∆ T +∆ T + g(s)dt ds T

=

1 ∆

s−∆

∆

0

T

sg(s)ds +

T +∆

g(s)(T + ∆ − s)ds

g(s)∆ds + ∆

T

∆ T T +∆ 1 ≤ ∆ g(s)ds + ∆ g(s)ds + ∆ g(s)ds ∆ T 0 ∆ ∆ T T +∆ = g(s)ds + g(s)ds + g(s)ds. 0

∆

T

The ﬁrst and third terms go to zero as ∆ → 0, so T t+∆ T g(s)ds t lim g(t)dt. dt ≤ ∆→0 0 ∆ 0 The other direction of the inequality follows easily from Fatou’s lemma. This proves the lemma. Theorem 3.4.1 Suppose Xt , t ≥ t0 ≥ 0, is the strong solution of (3.4.1) and assume the coercive condition (3.4.3) holds. If there exists a function Λ(·, ·) : R+ × H → R1 which satisﬁes the following:

Stability of Nonlinear Stochastic Diﬀerential Equations

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(a). Λ(·, ·) satisﬁes all the conditions in Remark at the end of Subsection 1.3.1; (b). c1 x2H − k1 e−µ1 t ≤ Λ(t, x) ≤ c2 x2H + k2 e−µ2 t , ∀x ∈ V, t ≥ 0; (c). (LΛ)(t, x) ≤ −c3 Λ(t, x) + k3 e−µ3 t , ∀x ∈ V, t ≥ 0, where ci > 0, ki ≥ 0, µi ≥ 0, i = 1, 2, 3, are some constants and L is the inﬁnitesimal generator of the Markov process Xt , i.e., (LΛ)(t, x) := Λt (t, x) + Λx (t, x), A(t, x)V,V ∗ + 1/2 · tr(Λxx (t, x)B(t, x)QB(t, x)∗ ), x ∈ V , t ≥ 0, then Xt , t ≥ t0 ≥ 0, satisﬁes EXt (x0 )2H ≤ α1 x0 2H ·e−β1 (t−t0 ) +α2 ·e−β2 t ,

x0 ∈ H,

t ≥ t0 ,

where α1 > 0, α2 ≥ 0, β1 > 0 and β2 ≥ 0 are constants. Conversely, suppose (3.4.4) holds and deﬁne T +t0 u Λ(t0 , x0 ) = EXs (x0 )2V ds du for all x0 ∈ H, t0

Xt0 = x0 , (3.4.4)

t0 ≥ 0,

t0

(3.4.5) where Xt0 = x0 and T is some proper positive constant and assume Λ(·, ·) satisﬁes all the conditions in Remark at the end of Subsection 1.3.1, then there exist constants ci > 0, ki ≥ 0, µi ≥ 0, i = 1, 2, 3, such that Conditions (b) and (c) above hold. Proof Firstly, suppose the conditions (a), (b) and (c) hold. A direct application of Itˆ o’s formula to the strong solution Xt (x0 ), t ≥ t0 ≥ 0, and the nonlinear functional Λ(·, ·) and then taking expectations immediately yield (∀t ≥ s ≥ t0 ) t EΛ(t, Xt (x0 )) − EΛ(s, Xs (x0 )) = E (LΛ)(u, Xu (x0 ))du s t − c3 EΛ(u, Xu (x0 )) + k3 · e−µ3 u du. ≤ s

Let φ(t) = EΛ(t, Xt (x0 )) and note that φ(t) is continuous in t, we get dφ(t) ≤ −c3 φ(t) + k3 · e−µ3 t dt which immediately yields that φ(t) ≤ φ(t0 )ec3 t0 · e−c3 t + k3 e−c3 t

t

e(c3 −µ3 )u du.

(3.4.6)

t0

If c3 = µ3 , it is easy to deduce from (3.4.6) and (b) that φ(t) ≤ c2 x0 2H · e−c3 (t−t0 ) + k2 e−(µ2 −c3 )t0 −c3 t + k3 · (t − t0 )e−c3 t ≤ c2 x0 2H · e−c3 (t−t0 ) + (k2 + 2k3 /c3 ) · e−(

c3 2

∧µ2 )t

.

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

On the other hand, if c3 = µ3 , a straightforward computation yields from (3.4.6) that k3 · e−(µ3 −c3 )t0 −c3 t k3 e−µ3 t φ(t) ≤ φ(t0 )e−c3 t0 + + e |c3 − µ3 | |c3 − µ3 | 2k 3 ≤ c2 x0 2H · e−c3 (t−t0 ) + k2 + e−(µ2 ∧µ3 ∧c3 )t . |c3 − µ3 | Combining the above results, we can derive that there exist constants c˜ > 0, ˜ ≥ 0, β˜1 > 0 and β˜2 ≥ 0 such that M ˜ ˜ · e−β˜2 (t−t0 ) , φ(t) ≤ c˜x0 2H e−β1 (t−t0 ) + M

which, together with (b) in Theorem 3.4.1, immediately implies that 1 k1 φ(t) + e−µ1 t ≤ α1 x0 2H e−β1 (t−t0 ) + α2 · e−β2 t c1 c1 ˜ + k1 /c1 ≥ 0, β1 = β˜1 = c3 > 0 and where α1 = c˜ = c2 /c1 > 0, α2 = M ˜ β2 = β2 ∧ µ1 ≥ 0 are four constants. The ﬁrst part is proved. Now suppose (3.4.4) holds. Deﬁne T +t0 u Λ(t0 , x0 ) = EXs (x0 )2V ds du, x0 ∈ H, t ≥ t0 , (3.4.7) EXt (x0 )2H ≤

t0

t0

where T is some positive constant to be determined later. Applying Itˆ o’s formula to Xt (x0 )2H , t ≥ t0 , and using the coercive condition (3.4.3). For the sake of simplicity, suppose at present µ > 0, β2 > 0 (the results in other situations may be proved similarly), we then obtain for arbitrary u ≥ t0 , EXu (x0 )2H −x0 2H u = ELXs (x0 )2H ds t0 u ≤ λ EXs (x0 )2H ds − α t0

u

t0

u

EXs (x0 )2V ds + γ

e−µs ds,

t0

which, in addition to (3.4.4), immediately implies that u EXs (x0 )2V ds t0 u u 1 ≤ EXs (x0 )2H ds + γ e−µs ds x0 2H + λ α t0 t0 u u 1 ≤ x0 2H + α1 |λ|x0 2H e−β1 (s−t0 ) ds + |λ|α2 e−β2 s ds α t0 t0 u +γ e−µs ds t

0 1 α1 |λ|x0 2H |λ|α2 −β2 t0 γ −µt0 ≤ . + e + e x0 2H + α β1 β2 µ

(3.4.8)

Stability of Nonlinear Stochastic Diﬀerential Equations

115

Therefore, by virtue of (3.4.7) it follows that * ) |λ|α T 1 γT −(β2 ∧µ)t0 α1 T |λ| 2 Λ(t0 , x0 ) ≤ . (3.4.9) x0 2H + + T+ ·e α β1 β2 µ On the other hand, for arbitrary v ∈ V , t ≥ t0 , (Lv2H )(t, v) = 2v, A(t, v)V,V ∗ + tr(B(t, v)QB(t, v)∗ ) which, together with (1.3.3), (1.3.5) and (3.4.3), immediately implies that |(Lv2H )(t, v)| ≤ (4β 2 + 4c2 )v2V + B(t, v)2L0 2

≤ (4β 2 + 4c2 )v2V + 2B(t, v) − B(t, 0)2L0 + 2B(t, 0)2L0 2

2

≤ (4β 2 + 4c2 )v2V + 2L2 v2V + 2γe−µt = θv2V + 2γe−µt

where θ = 4β 2 + 4c2 + 2L2 > 0. Hence (Lv2H )(t, v) ≥ −θv2V − 2γe−µt . Applying now Itˆ o’s formula to Xt (x0 )2V , t ≥ t0 , and taking expectations, we get that for all u ≥ t0 , u EXu (x0 )2H − x0 2H = E(LXs (x0 )2H )(s, Xs (x0 ))ds t0 u u ≥ −θ EXs (x0 )2V ds − 2γ e−µs ds, t0

t0

which, together with (3.4.4), immediately implies that for arbitrary u ≥ t0 , u u θ EXs (x0 )2V ds ≥ x0 2H − EXu (x0 )2H − 2γ e−µs ds t0

≥

t0

1 − α1 · e−β1 u+β1 t0 x0 2H − α2 · e−β2 u −

2γ −µt0 e . µ

Therefore, we get

T +t0

u

Λ(t0 , x0 ) = t0

t0 T +t0 )

EXs (x0 )2V ds du

* 1 2γ −µt0 −β1 u+β1 t0 2 −β2 u ≥ x0 H − α2 · e − 1 − α1 · e du e θ t0 µ 1 1 α2 2γT −(β2 ∧µ)t0 α1 ≥ x0 2H − + . (3.4.10) T− e θ β1 θ β2 µ

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

On the other hand, note that T +t0 EΛ(t, Xt (x0 )) = E t0

u

E Xs (Xt (x0 ))2V Xt (x0 ) ds du.

t0

However, by the Markov property of the solution of (3.4.1), this equals T +t0 u E E Xs (Xt (x0 ))2V FtX dsdu. t0

t0

where FtX = σ{Xu (x0 ), t0 E Xs (Xt (x0 ))2V

≤ u ≤ t}. The uniqueness X Ft = E Xs+t (x0 )2V

of the solution implies X (3.4.11) Ft .

Therefore, we have EΛ(t0 + r, Xt0 +r (x0 )) − EΛ(t0 , x0 ) r T +t u EXs (Xt0 +r (x0 ))2V dsdu − t0 0 t0 EXs (x0 )2V dsdu t0 +r

(LΛ)(t0 , x0 ) = lim r→0 T +t0 +r u t0 +r

= lim

r→0

t0

= lim

r→0

T +t0 u−t0 t0

= lim

r→0

r T +t u 2 EX (X (x )) dsdu − t0 0 t0 EXs (x0 )2V dsdu s+r t0 +r 0 V t0

T +t0 u

0

T +t0

u−t0 +r r

= lim

r→0

t0

T +t0

u−t0 +r

= lim

r→0

EXs+r+t0 (x0 )2V

u−t0

r T +t u dsdu − t0 0 t0 EXs (x0 )2V dsdu

r u EXs+t0 (x0 )2V ds − t0 EXs (x0 )2V ds r EXs+t0 (x0 )2V ds −

r 0

du

EXs+t0 (x0 )2V ds

r

t0

du

which, together with xH ≤ βxV , β > 0, for any x ∈ V , Lemma 3.4.1 and the continuity of t → EXt (x0 )2H , immediately implies (LΛ)(t0 , x0 ) ≤

T +t0

EXu (x0 )2V du −

t0

T x0 2H . β2

Now, substituting (3.4.8) into the above yields (LΛ)(t0 , x0 ) ≤

|λ|α 1 γ −(β2 ∧µ)t0 T α1 |λ| 2 x0 2H + + − 2 x0 2H . 1+ e α β1 β2 µ β

Hence, noting (3.4.10) and choosing T > 0 large enough so that T >

β2 α1 |λ| α1 ∨ , 1+ α β1 β1

Stability of Nonlinear Stochastic Diﬀerential Equations

117

we are then in a position to obtain the desired results (b) and (c) in Theorem 3.4.1. If µ = 0 or β2 = 0, by a similar argument to the above, it can be also proved that Conditions (b) and (c) remain true. The proof is now complete.

This result is powerful in studying stability and invariant measures of solutions by means of a ﬁrst-order approximation argument (see Section 3.5 below). Before proceeding to this material, we ﬁrst derive some corollaries which will play an important role in the subsequent investigation. Consider the following nonlinear autonomous stochastic diﬀerential equation in V ∗ : dXt = A(Xt )dt + B(Xt )dWt , t ∈ [0, ∞), (3.4.12) X0 = x0 . Suppose A(·) : V → V ∗ and B(·) : V → L(K, H) satisfy A(u)V ∗ ≤ a1 uV

and B(u) ≤ b1 uV

for all u ∈ V,

(3.4.13)

for some constants a1 , b1 > 0, with A(0) = 0 and B(0) = 0. Furthermore, assume the following coercive condition holds: there exist α > 0 and λ ∈ R1 such that 2u, A(u)V,V ∗ + B(u)2L0 ≤ −αu2V + λu2H , 2

∀u ∈ V.

(3.4.14)

The following result is immediate from Theorem 3.4.1: Corollary 3.4.1 Assume the coercive condition (3.4.14) holds. Suppose Xt , t ≥ 0, is the strong solution of (3.4.12). If there exists a function Λ : H → R1 which satisﬁes the following: (a). Λ(·) satisﬁes all the conditions in Remark at the end of Subsection 1.3.1; (b). c1 x2H ≤ Λ(x) ≤ c2 x2H , ∀x ∈ V ; (c). (LΛ)(x) ≤ −c3 Λ(x), ∀x ∈ V, where ci > 0, i = 1, 2, 3, are some constants and L is the inﬁnitesimal generator of the Markov process Xt , i.e., (LΛ)(x) := Λ (x), A(x)V,V ∗ + 1/2 · tr(Λ (x)B(x)QB(x)∗ ), x ∈ V , then Xt (x0 ), t ≥ 0, satisﬁes that EXt (x0 )2H ≤ α1 x0 2H · e−β1 t ,

x0 ∈ H,

t ≥ 0,

X0 = x0 ,

(3.4.15)

for some α1 > 0 and β1 > 0. Conversely, suppose (3.4.15) holds and deﬁne

T

Λ(x0 ) = 0

0

u

EXs (x0 )2V ds du

for all x0 ∈ H,

t ≥ 0,

(3.4.16)

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where X0 = x0 and T is some proper positive constant and assume Λ(·) satisﬁes all the conditions in Remark at the end of Subsection 1.3.1, then there exist constants ci > 0, i = 1, 2, 3, such that Conditions (b) and (c) above hold. Generally speaking, it is not quite obvious whether Λ(·, ·) deﬁned in (3.4.5) satisﬁes the conditions of using Itˆ o’s formula in Subsection 1.3.1. However, this is always the case if V is a real separable Hilbert space with inner product ·, ·V and Equation (3.4.12) is linear. Precisely, consider the linear system given by dXt = AXt dt + BXt dWt with X0 = x0 , (3.4.17) where A ∈ L(V, V ∗ ), B ∈ L(V, L(K, H)) satisfy the coercive condition: there exist α > 0 and λ ∈ R1 such that 2u, AuV,V ∗ + tr[(Bu)Q(Bu)∗ ] ≤ −αu2V + λu2H

(3.4.18)

for all u ∈ V . In particular, we have: Corollary 3.4.2 Assume the coercive condition (3.4.18) holds. Suppose Xt , t ≥ 0, is the strong solution of (3.4.17). If there exists a function Λ : H → R1 which satisﬁes the following: (a). Λ(·) satisﬁes all the conditions in Remark at the end of Subsection 1.3.1; (b). c1 x2H ≤ Λ(x) ≤ c2 x2H , ∀x ∈ V ; (c). (LΛ)(x) ≤ −c3 Λ(x), ∀x ∈ V, where ci > 0, i = 1, 2, 3, are some constants and L is the inﬁnitesimal generator of the Markov process Xt , i.e., (LΛ)(x) = Λ (x), AxV,V ∗ + 1/2 · tr(Λ (x)(Bx)Q(Bx)∗ ), x ∈ V , then Xt (x0 ), t ≥ 0, satisﬁes EXt (x0 )2H ≤ α1 x0 2H · e−β1 t ,

x0 ∈ H,

t ≥ 0,

for some α1 > 0 and β1 > 0. Conversely, suppose (3.4.19) holds and deﬁne ∞ Λ(x0 ) = EXs (x0 )2V ds for all x0 ∈ H,

X0 = x0 ,

(3.4.19)

t ≥ 0,

(3.4.20)

0

where X0 = x0 , then there exist constants ci > 0, i = 1, 2, 3, such that Λ(·) deﬁned above satisﬁes the conditions (a), (b) and (c). If, in addition, we assume t → EXt (x0 )2V is continuous at zero for any x0 ∈ V , then (LΛ)(x0 ) = −x0 2V , x0 ∈ V . Proof All we need to prove is the second part. By a similar argument to ∞ Theorem 3.4.1, it is immediate that Λ(·) = 0 EXs (·)2V ds satisﬁes all the

Stability of Nonlinear Stochastic Diﬀerential Equations

119

conditions in Corollary 3.4.2 except for Condition (a). To verify (a), note that Xt (x) is linear in x and hence ∞ T (x, y) := EXt (x), Xt (y)V dt, x, y ∈ V, 0

is a bilinear form on V × V . By a straightforward computation, we can obtain from (3.4.18) and (3.4.19) that ∞ EXt (x)2V dt ≤ M x2H , 0

for some constant M > 0. Using this and Schwartz inequality, we can get |T (x, y)| ≤ M xH yH for all x, y ∈ V . Considering the fact that V is densely embedded into H, there exists a unique continuous extension T˜ of T to H × H. Hence, there exists a continuous linear operator P˜ on H → H such that T˜(x, y) = (P˜ x, y). Now Λ (x) = 2P˜ x, Λ (x) = 2P˜ . Hence, Λ, Λ , Λ are locally bounded on H and Λ, Λ are continuous on H as |Λ(x)| ≤ P˜ x2H and

Λ (x)H = 2P˜ xH ≤ 2P˜ xH ,

∀x ∈ H.

For trace class Q, QΛ (x) = 2QP˜ ; also, tr(QP˜ ) being constant, it is continuous. To prove Condition (iv) of the Remark at the end of Subsection 1.3.1, we observe that T (x, y) is bilinear on V ×V ; the fact that |T (x, y)| ≤ M xH yH and the continuity of the injection V → H imply that T (x, y) is a continuous bilinear form on V × V . It follows that there exists a continuous operator P on V such that T (x, y) = P x, yV for x, y ∈ V . Hence Λ (x) = 2P x ∈ V for x ∈ V and x → P x is continuous on V → V . Since Λ (x)V = 2P xV ≤ 2P L(V ) (1 + xV ) for all x ∈ V , we get the desired result. Lastly, to conclude our proof, we notice that by (3.4.11) it follows that for any x0 ∈ V , ∞ ∞ EΛ(Xt (x0 )) − Λ(x0 ) = EXt+s (x0 )2V ds − EXs (x0 )2V ds 0 ∞ ∞0 2 = EXs (x0 )V ds − EXs (x0 )2V ds t 0 t =− EXs (x0 )2V ds 0

which, using the continuity of the mapping: t → EXt (x0 )2V at zero for x0 ∈ V and letting t → 0, immediately yields (LΛ)(x0 ) = −x0 2V

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

for any x0 ∈ V . Now the proof is complete. Remark (1). Note that if A is bounded, it is certainly coercive with ∞ V = H and hence the Lyapunov function is Λ(x) = 0 EXt (x)2H dt exactly as in the ﬁnite dimensional case. It is thus possible to extend the theory in Khas’minskii [1] to give analogues of results, for instance, in Daletskii and Krein [1] for the stochastic setting. (2). In general, we don’t know the exact conditions for the continuity of mapping: t → EXt (x)2V . However, if we choose a new Gelfand triplet (V1 , V, V1∗ ) and operators A : V1 → V1∗ and B : V1 → L(K, V ) satisfying a coercive condition corresponding to the norms · V and · V1 , then it can be proved (e.g., see Pardoux [1] or Rozovskii [1]) that for any x ∈ V , t → EXt (x)2V is a continuous function. Typical examples of these are Sobolev spaces of higher order and A, B smooth operators. For instance, Krylov and Rozovskii [1] contains several interesting examples in this case. The following result will play an important role in the study of invariant measures of strong solutions. Corollary 3.4.3 Assume the coercive condition (1.3.2) holds with p = 2. Let Xt (x0 ), t ≥ t0 , denote the strong solution of (3.4.1). If there exists a function Λ : R+ × H → R1 which satisﬁes the following: (a). Λ(·, ·) satisﬁes all the conditions in Remark at the end of Subsection 1.3.1; (b). c1 x2H − k1 ≤ Λ(t, x) ≤ c2 x2H + k2 , ∀x ∈ V, t ≥ 0; (c). (LΛ)(t, x) ≤ −c3 Λ(t, x) + k3 , ∀x ∈ V, t ≥ 0, where ci > 0, ki > 0, i = 1, 2, 3, are some constants and operator L is the inﬁnitesimal generator of the Markov process Xt , then Xt , t ≥ t0 ≥ 0, satisﬁes EXt (x0 )2H ≤ α1 x0 2H · e−β1 (t−t0 ) + α2 , x0 ∈ H, t ≥ t0 , Xt0 = x0 , (3.4.21) for some α1 > 0, β1 > 0 and α2 > 0. Conversely, suppose (3.4.21) holds and deﬁne

T +t0

u

Λ(t0 , x0 ) = t0

EXs (x0 )2V ds du

for all x0 ∈ H,

t0 ≥ 0,

t0

(3.4.22) where Xt0 = x0 and T is some proper positive constant. Assume Λ(·, ·) satisﬁes all the conditions in Remark at the end of Subsection 1.3.1, then there exist constants ci > 0, ki > 0, i = 1, 2, 3, such that Conditions (b) and (c) above hold.

Stability of Nonlinear Stochastic Diﬀerential Equations

121

In general, if (3.4.21) is true, we also call the process Xt (x0 ), t ≥ t0 , is exponentially ultimately bounded in mean square, or simply, ultimately bounded in mean square.

3.5

Two Applications

In this section, we shall apply the results derived in the last section to investigate stability and invariant measures of strong solutions of nonlinear stochastic diﬀerential equations through the method of the so-called ﬁrst order approximation by linear systems. In ﬁnite dimensional spaces, this method proves to be an eﬀective tool in treating large time behavior of nonlinear deterministic systems (cf. Hahn [1]). Its generalization to ﬁnite dimensional stochastic diﬀerential equations has been presented in Miyahara [1], [2].

3.5.1

Stability in Probability

Consider the following nonlinear stochastic diﬀerential equation in V ∗ : dXt = A(Xt )dt + B(Xt )dWt , t ∈ [0, ∞), (3.5.1) X0 = x0 ∈ H, where A(·) : V → V ∗ is a measurable nonlinear mapping satisfying A(u)V ∗ ≤ L1 uV , L1 > 0, and B(u) ∈ L(K, H), B(u) ≤ L2 uV , L2 > 0, for any u ∈ V . We ﬁrst present a result whose proof can be derived by a similar argument to that in Theorem 3.3.3. Theorem 3.5.1 Let Λ(x) be a function deﬁned on the set {x ∈ H : xH < δ} satisfying the following properties: (i). (ii). (iii). (iv).

All the conditions in Remark at the end of Subsection 1.3.1 hold; Λ(x) → 0 as xH → 0; inf xH >ε Λ(x) = λε > 0 for any ε > 0; (LΛ)(x) ≤ 0 for x ∈ V with xH < δ.

Suppose {Xt (x0 ), t ≥ 0} is the strong solution of (3.5.1), then lim P sup Xt (x0 )H > ε = 0 for each ε > 0, x0 H →0

t≥0

i.e., the null solution is stable in probability. Indeed, it is strongly stable in probability (cf. Khas’minskii [1]). Obviously, the function Λ(·) constructed in Corollary 3.4.2 of the linear system (3.4.17) under Condition (3.4.18) satisﬁes the conditions in Theorem

122

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

3.5.1. In particular, as an immediate consequence we can obtain the following result by using Corollary 3.4.2. Theorem 3.5.2 Consider the equation (3.4.17) with A, B satisfying (3.4.18) and let {Xt (x0 ), t ≥ 0} be its strong solution. If the null solution of (3.4.17) is exponentially stable in mean square, then it is also (strongly) stable in probability. Now we may present theorems on stability of nonlinear systems through the ﬁrst order approximation in the following manner. Theorem 3.5.3 Consider Equation (3.4.17) satisfying the condition (3.4.18) and let {Xt (x0 ), t ≥ 0} be its strong solution with the property t → EXt (x0 )2V continuous. Assume that the null solution of (3.4.17) is exponentially stable ˜ t (x0 ), t ≥ 0} be the strong solution of in mean square. Let {X ˜ t )dt + B(X ˜ t )dWt , dXt = A(X t ∈ [0, ∞), (3.5.2) X0 = x0 ∈ H, ˜ ˜ ˜ ˜ where A˜ : V → V ∗ with A(v) V ∗ ≤ L1 vV , L1 > 0, and B(v) ∈ L(K, H), ˜ ˜ ˜ B(v) ≤ L2 vV , L2 > 0, for all v ∈ V . If, further,

˜ ˜ ˜ ∗ − (Bv)Q(Bv)∗ ≤ εv2 A(v) − Av2V ∗ + tr B(v)Q B(v) (3.5.3) V for ε > 0 small enough in a suﬃciently small neighborhood of v = 0 in · V . Then the null solution of (3.5.2) is stable in probability (indeed, the null solution is also exponentially stable in mean square). Proof For any twice Fr´echet diﬀerentiable function Φ satisfying the conditions of using Itˆ o’s formula in Subsection 1.3.1, deﬁne for any v ∈ V , ˜ ˜ ˜ ∗ ˜ (LΦ)(v) = Φ (v), A(v) V,V ∗ + 1/2 · tr(Φ (v)B(v)QB(v) ).

Consider Λ(·) as in Corollary 3.4.2, then for any v ∈ V , ˜ (LΛ)(v) − (LΛ)(v)

˜ ˜ ˜ ∗ − (Bv)Q(Bv)∗ ) . = Λ (v), A(v) − AvV,V ∗ + 1/2 · tr Λ (v)(B(v)Q B(v) By Corollary 3.4.2 and using the conditions of this theorem, we have that for any v ∈ V and the operator P deﬁned in the proof of Corollary 3.4.2, ˜ (LΛ)(v) ≤ −v2V + 4εP L(V ) v2V . ˜ ˜ Since ε > 0 is suﬃciently small, we get (LΛ)(v) ≤ 0 (indeed, (LΛ)(v) ≤ −1 2 −β (1 − 4εP L(V ) )vH when 4εP L(V ) < 1). Hence, by virtue of Theorem 3.5.1 or Corollary 3.4.1, we get the desired result.

Stability of Nonlinear Stochastic Diﬀerential Equations

123

Theorem 3.5.4 Consider the equation (3.4.17) satisfying (3.4.18) and let {Xt (x0 ), t ≥ 0} be its strong solution. Assume that the null solution of ˜ t (x0 ), t ≥ 0} be the (3.4.17) is exponentially stable in mean square. Let {X strong solution of (3.5.2). If, further,

˜ ˜ ˜ ∗ − (Bv)Q(Bv)∗ ≤ εv2H A(v) − Av2V ∗ + tr B(v)Q B(v) (3.5.4) for ε > 0 small enough in a suﬃciently small neighborhood of v = 0 in · H . Then the null solution of (3.5.2) is stable in probability (indeed, the null solution is also exponentially stable in mean square). Proof We follow the ideas of the proof of Theorem 3.5.3, using Corollary 3.4.2 and the operator P˜ deﬁned in the proof of Corollary 3.4.2, to obtain ˜ (LΛ)(x) ≤ −M x2H + 4εP˜ L(H) x2H for some number M > 0. Since ε > 0 is suﬃciently small, we can get ˜ ˜ (LΛ)(x) ≤ 0 (indeed, (LΛ)(x) ≤ −(M − 4εP˜ L(H) )x2H when 4εP˜ L(H) < M ). Hence, by virtue of Theorem 3.5.1 or Corollary 3.4.1, we get the desired result. Let us study two examples to illustrate how to apply the above results to deal with nonlinear stochastic systems. Example 3.5.1 Let O ⊂ Rn be a bounded open domain with smooth boundary ∂O. Assume H = L2 (O) and V is the Sobolev space H01 (O). Also suppose {Wq (t, x); t ≥ 0, x ∈ O} is an H-valued Wiener process with associated covariance operator Q, trQ < ∞, given by a positively deﬁnite kernel q(x, y) ∈ L2 (O × O),

q(x, x) ∈ L2 (O).

Let A be the linear diﬀerential strictly elliptic operator of the second order on O and B(u) be the operator of multiplication by u, i.e., B(u)f (·) = u(·)f (·) for f ∈ L2 (O). It can be shown that the coercive condition (3.4.18) is fulﬁlled (Garding’s inequality, e.g., see Pazy [1]). Then the problem (3.4.17) turns out to be du(t, x) = Au(t, x)dt + u(t, x)dWq (t, x). In this case, let Λ(u) = u2H , then we get for any u ∈ H01 (O),

L(u2H ) = 2u, AuV,V ∗ + tr (Bu)Q(Bu)∗ = 2u, AuV,V ∗ + q(x, x)u(x)2 dx. O

124

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

On the other hand, let λ0 =

sup u∈H01 (O) u2H =0

L(u2H ) = u2H

sup u∈H01 (O) u2H =0

2u, AuV,V ∗ + Qu, uH . u2H

Then we have with the help of Corollary 3.4.2 that the null solution is exponentially stable in mean square if λ0 < 0. Example 3.5.2 in Example 3.5.1

Consider the nonlinear equation in O which is deﬁned as

˜ u(t, x))dt + B(x, ˜ u(t, x))dWq (t, x) du(t, x) = A(x,

(3.5.5)

with the boundary condition u(t, x)|∂O = 0. Assume that ˜ u(t, x)) = Au(t, x) + α1 (x, u(t, x)), A(x, ˜ u(t, x)) = u(t, x) + α2 (x, u(t, x)), B(x, αi (x, u) : O × R1 → R1 , i = 1, 2, are two functions satisfying the following conditions: for i = 1, 2, there exists a constant c > 0 such that sup |αi (x, u2 ) − αi (x, u1 )| ≤ c|u2 − u1 |,

∀u1 , u2 ∈ R1 ,

x∈O

αi (x, 0) = 0,

x ∈ O,

sup |αi (x, u)| = o(|u|) as

|u| → 0.

(3.5.6)

x∈O

Also assume that the operator A is strictly elliptic in O such that the conditions of Example 3.5.1 hold. These conditions guarantee the fulﬁllment of the assumptions of coercivity and monotonicity for (3.5.5). So there exists a unique strong solution of the problem. Furthermore, the last condition (3.5.6) is suﬃcient for (3.5.3). We can conclude, with the help of Theorem 3.5.4 and Example 3.5.1, that the null solution is stable in probability if λ0 < 0.

3.5.2

Ultimate Boundedness and Invariant Measures

In this subsection, by using the results in Section 3.4, we will establish conditions for ultimate boundedness of strong solutions of nonlinear stochastic diﬀerential equations and study the associated problems of existence and uniqueness of invariant measures. As we did in the above subsection, we will deal with nonlinear systems by the ﬁrst order linear approximation. In particular, by means of Corollary 3.4.3, the following results can be derived in a similar manner to Theorems 3.5.3 and 3.5.4.

Stability of Nonlinear Stochastic Diﬀerential Equations

125

Theorem 3.5.5 Consider the equation (3.4.17) satisfying (3.4.18) and let {Xt (x0 ), t ≥ 0} be its strong solution. Assume that the system (3.4.17) is ˜ t (x0 ), t ≥ 0} be the exponentially ultimately bounded in mean square. Let {X ˜ strong solution of (3.5.2). Moreover, we suppose A(v)−Av ∈ H for all v ∈ V , and

˜ ∗ 2 ˜ ˜ ∗ 2vH A(v)−Av v ∈ V, H + tr B(v)QB(v) −(Bv)Q(Bv) ≤ εvH +k, for ε > 0 small enough and some number k > 0. Then the strong solution of (3.5.2) is exponentially ultimately bounded in mean square. Corollary 3.5.1 Consider the equation (3.4.17) satisfying (3.4.18) and let {Xt (x0 ), t ≥ 0} be its strong solution. Assume that this solution is ex˜ t (x0 ), t ≥ 0} be the ponentially ultimately bounded in mean square. Let {X strong solution of (3.5.2). Furthermore, we suppose that for any v ∈ V , ˜ A(v) − Av ∈ H, and

˜ ˜ ˜ ∗ − (Bv)Q(Bv)∗ ≤ L(1 + v2 ) A(v) − Av2H + tr B(v)Q B(v) H for some L > 0. If for arbitrary v ∈ V , as vH → ∞,

˜ ∗ 2 ˜ ˜ ∗ A(v)−Av H = o(vH ) and tr B(v)QB(v) −(Bv)Q(Bv) = o(vH ). Then the strong solution of (3.5.2) is exponentially ultimately bounded in mean square. Example 3.5.3

Consider the following stochastic diﬀerential equation: du(t) = Au(t)dt + F (u(t))dt + B(u(t))dWt

(3.5.7)

with initial condition u(0) = u0 ∈ H. Suppose A, F and B satisfy the following conditions: (i). A : V → V ∗ is coercive, that is, there exist constants α > 0 and λ ∈ R1 such that for arbitrary v ∈ V , 2v, AvV,V ∗ ≤ −αv2V + λv2H ; (ii). F : H → H and B : H → L(K, H) satisfy that for arbitrary v ∈ H, F (v)2H + B(v)2 ≤ L 1 + v2H for some L > 0; (iii). For arbitrary u, v ∈ H,

F (u) − F (v)2H + tr (B(u) − B(v))Q(B(u)∗ − B(v)∗ ) ≤ λu − v2H .

126

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

If the solution {uu0 (t), t ≥ 0} of du(t) = Au(t)dt is exponentially decayable (or even exponentially ultimately bounded), and as vH → ∞, F (v)H = o(vH ),

B(v) = o(vH ),

then the strong solution {uu0 (t), t ≥ 0} of (3.5.7) is exponentially ultimately bounded in mean square. ˜ Indeed, let A(v) = Av + F (v) for v ∈ V . Since F (v) ∈ H, ∗ ˜ 2v, A(v) V,V ∗ + tr[B(v)QB(v) ] = 2v, AvV,V ∗ + 2F (v), vH + tr[B(v)QB(v)∗ ] ≤ −αv2V + λv2H + 2vH F (v)H + B(v)2 trQ

≤ −αv2V + λ v2H + γ for some constants λ ∈ R1 and γ ≥ 0. Hence, the equation (3.5.7) is coercive. Moreover, under the additional assumptions (ii) and (iii), the strong solution {uu0 (t), t ≥ 0} of (3.5.7) exists. By the assumption (ii), F (v)2H + tr[B(v)QB(v)∗ ] ≤ F (u)2H + B(v)2 trQ ≤ (1 + trQ)L(1 + v2H ) and since F (v)H = o(vH ),

tr[B(v)QB(v)∗ ] ≤ B(v)2 trQ = o(v2H )

as vH → ∞, the assertion follows from Corollary 3.5.1. Example 3.5.4 Let S 1 be the unit circle and B(·, ·) a standard Brownian sheet on [0, ∞) × S 1 . Consider the following stochastic heat equation: ∂Xt ∂2B ∂ 2 Xt (ξ) − αX (ξ) + f (X (ξ)) + g(X (ξ)) (ξ) = , t t t ∂t ∂ξ 2 ∂t∂ξ

(3.5.8)

with initial condition X0 (·) = x0 (·) ∈ L2 (S 1 ), where α is a constant and f (·), g(·) are two real-valued functions with f (0) = g(0) = 0. Suppose H = L2 (S 1 ),

V = W 1,2 (S 1 ),

A(x) =

∂2 − α x, 2 ∂ξ

and the mappings F and B are given for ξ ∈ S 1 and x(·), y(·) ∈ L2 (S 1 ) by F (x)(ξ) = f (x(ξ)), and let xH =

S1

B(x)[y](ξ) = g(x(ξ))y(ξ),

x2 (ξ)dξ

1/2

for x ∈ H,

Stability of Nonlinear Stochastic Diﬀerential Equations ,

xV =

2

x (ξ) +

∂x(ξ) 2 ∂ξ

S1

127

-1/2 dξ

for x ∈ V,

then we have 2x, AxV,V ∗ = −2x2V + (−2α + 2)x2H ≤ −2x2H + (−2α + 2)x2H = −2αx2H . Therefore, the null solution of dx(t) = Ax(t)dt is exponentially stable if α > 0. Moreover, if we further assume f (·) and g(·) are both Lipschitz continuous and bounded, then from Example 3.5.3, the strong solution of (3.5.8) is exponentially ultimately bounded in mean square. Suppose that Xt , t ≥ 0, is a time-homogeneous, stochastically continuous Markov process (in norm · H ) deﬁned on some (Ω, F, Ft , P ) with state space H. Denote the transition probability function by P (x, t, ·) deﬁned by the conditional probability: P (x, t, Γ) = P Xt ∈ Γ X0 = x , for x ∈ H and Γ ∈ B(H), the Borel σ-ﬁeld of H. A probability measure µ on (H, B(H)) is called invariant for the given transition probability function if it satisﬁes the equation µ(Γ) = P (x, t, Γ)µ(dx) H

for any t ≥ 0, or equivalently, the following relation holds: Ψ(y)µ(dy) = (Pt Ψ)(y)µ(dy) H

H

for any Ψ ∈ Cb (H), t ≥ 0, where the transition semigroup Pt , t ≥ 0, is deﬁned by (Pt Ψ)(x) =

Ψ(y)P (x, t, dy), H

and Cb (H) denotes the space of all bounded continuous functions on H. We also recall that P (x, t, ·) has the Feller property if, for any bounded continuous function Ψ on H, (Pt Ψ)(x) is continuous in x for any t ≥ 0. It is clear that exponential decay of the process Xt , t ≥ 0, implies that the system has invariant measure degenerate at zero. One of the remarkable consequences in the section is that we shall show the concept of exponentially ultimate boundedness has a close relationship with the existence of non trivial invariant measures. Consider the following nonlinear stochastic diﬀerential equation in V ∗ : dXt = A(Xt )dt + B(Xt )dWt , t ∈ [0, ∞), (3.5.9) X0 = x0 ∈ H,

128

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where A(·) : V → V ∗ and B(·) : V → L(K, H) are both nonlinear measurable mappings which satisfy all the conditions in Theorem 1.3.1. In particular, it can be shown that the strong solution process Xt , t ≥ 0, is a time-homogeneous, stochastically continuous Markov process and the associated transition probability function P (x, t, ·) with its transition semigroup Pt , t ≥ 0, has the Feller property. We now state and prove a theorem concerning the existence of invariant measures which extends to inﬁnite dimensional spaces the corresponding ﬁnite dimensional results (see, e.g. Khas’minskii [1]). Theorem 3.5.6 Let Xt (x0 ), t ≥ 0, be a strong solution of (3.5.9), and for some sequence Tn ↑ ∞, deﬁne 1 µn (Γ) = Tn

Tn

P (x0 , t, Γ)dt,

Γ ∈ B(H).

(3.5.10)

0

If µ is the weak limit of a subsequence of {µn }, then µ is an invariant measure of P (x, t, ·). In other words, the existence of invariant measures is equivalent to the tightness of {µn (·)}. w

Proof Since the sequence {Tn } was arbitrary, we may as well assume µn =⇒ µ, i.e., µn converges weakly to µ on H. By the Feller property, for each t ≥ 0 and Ψ ∈ Cb (H), the function (Pt Ψ)(·) is in Cb (H). It follows from the weak convergence of µn that (Pt Ψ)(x)µ(dx) = lim (Pt Ψ)(x)µn (dx) H

n→∞

H

1 = lim n→∞ Tn 1 n→∞ Tn

Tn

(Pt Ψ)(x)P (x0 , s, dx)ds (3.5.11)

0

= lim

H Tn

(Pt+s Ψ)(x0 )ds, 0

(3.5.12) where we have made use of the Fubini theorem and the Markov property. Now, for any ﬁxed t ≥ 0 and x0 ∈ H, we can write Tn 1 (Pt+s Ψ)(x0 )ds n→∞ Tn 0 Tn +t t 1 Tn = (Ps Ψ)(x0 )ds + (Ps Ψ)(x0 )ds − (Ps Ψ)(x0 )ds . Tn 0 Tn 0 (3.5.13) lim

Since Ps ΨH ≤ ΨH < ∞ for any s ≥ 0, by virtue of (3.5.11) and (3.5.12),

Stability of Nonlinear Stochastic Diﬀerential Equations

129

we get

Tn 1 (Pt Ψ)(x)µ(dx) = lim (Ps Ψ)(x0 )ds = lim Ψ(x)µn (dx) n→∞ Tn 0 n→∞ H H = Ψ(x)µ(dx) H

by Fubini’s theorem and weak convergence. Hence, µ is an invariant measure.

If H is compact, then the tightness of {µn (·)} in Theorem 3.5.6 is true and existence follows. For non-compact H, one needs additional conditions which lead to the following corollary to get the existence of invariant measures. Corollary 3.5.2 Assume the embedding V ;→ H is compact. Suppose that the strong solution Xt (x0 ) of the equation with X0 = x0 satisﬁes the condition that for some x0 ∈ H, there exists a real number sequence Tn ↑ ∞ such that 1 Tn

Tn

P Xt (x0 )V > R dt → 0

uniformly in n

as

R → ∞.

0

(3.5.14) Then Equation (3.5.9) has an invariant measure on (H, B(H)). Proof It suﬃces to show {µn } deﬁned in (3.5.10) is weakly compact. For any R > 0, let VR = {y ∈ V : yV ≤ R} and VRcH = {y ∈ H : y ∈ / VR }. It is clear that VR is compact in H and therefore by Condition (3.5.13), for any ε > 0, there exists a compact set VR ⊂ H such that µn {H \VR } = µn (VRcH ) < ε

for all

n ≥ 1.

By the well-known Prokhorov theorem (see, e.g. Da Prato and Zabczyk [1]), the family {µn } is weakly compact. As we pointed out before, there exists a close relationship between ultimate boundedness and invariant measures. As a matter of fact, we have the following result. Theorem 3.5.7 Assume the embedding V ;→ H is compact. Suppose the strong solution {Xt ; t ≥ 0} of (3.5.9) under the coercive condition (1.3.2) with p = 2 is ultimately bounded (in · H norm), then there exists an invariant measure µ for {Xt ; t ≥ 0}. Proof

Applying Itˆ o’s formula to Λ(x) = x2H , x ∈ H, taking expectation

130

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

and using the coercive condition (1.3.2) with p = 2 there, we get EXt (x0 )2H

−

x0 2H

t

ELXs (x0 )2H ds

= 0

t

EXs (x0 )2H ds

=λ

−α

0

t

EXs (x0 )2V ds + γt. 0

Hence

t

EXs (x0 )2V 0

1 λ ds ≤ α

t

EXs (x0 )2H ds + x0 2H + γt ,

0

which, by using Markov’s inequality, immediately implies that for any T > 0, 1 T

0

T

1 T EXt (x0 )2V P Xt (x0 )V > R dt ≤ dt T 0 R2 1 |λ| T x0 2H 2 ≤ EX (x ) dt + + γ . t 0 H αR2 T 0 T

Since {Xt (x0 ); t ≥ 0} is ultimately bounded, there exist two constants T0 > 0 and M > 0 such that EXt (x0 )2H ≤ M

for

t ≥ T0 ,

which immediately yields 1 lim lim inf R→∞ T →∞ T

T

P Xt (x0 )V > R dt

0

|λ| 1 ≤ lim lim inf R→∞ T →∞ αR2 T

T

EXt (x0 )2H dt 0

T |λ| 1 T0 2 2 EX (x ) dt + EX (x ) dt t 0 t 0 H H R→∞ T →∞ αR2 T 0 T0 |λ| 1 T0 ≤ lim lim inf EXt (x0 )2H dt + M (T − T0 ) = 0. 2 R→∞ T →∞ αR T 0

= lim lim inf

Therefore, the assertion of the theorem follows. As a consequence of Theorem 3.5.7, we may easily get a result on the existence of invariant measures of the stochastic diﬀerential equations (3.5.7) and (3.5.8). For instance, as seen in Example 3.5.4, the solution of the stochastic heat equation is ultimately bounded in mean square of · H , and since V ;→ H is compact by the Sobolev embedding theorem, the existence of an invariant measure follows.

Stability of Nonlinear Stochastic Diﬀerential Equations

3.6

131

Further Results on Invariant Measures

In the last section, we have investigated the existence of invariant measures. In inﬁnite dimensional spaces, the uniqueness question is more subtle. Throughout this section, unless otherwise stated, we always suppose the embedding V ;→ H is compact. The following condition for the uniqueness of invariant measures is somewhat stringent but seems to be natural. Let Yt (y0 ) denote the family of Markov processes such that Y0 = y0 and P {Yt (y0 ) ∈ Γ} = P (y0 , t, Γ), ∀t > 0, y0 ∈ H and Γ ∈ B(H). Recall that VR = {y ∈ V : yV ≤ R} and VRcH = {y ∈ H : y ∈ / VR } for any R > 0. Theorem 3.6.1 Suppose that the following condition holds: for arbitrary ε > 0, δ > 0 and R > 0, there exists T0 (ε, δ, R) > 0 such that 1 T

T

P Yt (ξ) − Yt (η)V ≥ δ dt < ε

(3.6.1)

0

for any ξ, η ∈ VR and T ≥ T0 (ε, δ, R). If there exists an invariant measure µ of Yt (y0 ) with support in V , then it is unique. Proof Suppose that µ and ν both are invariant measures with supports in V . It suﬃces to show that for any bounded, uniformly continuous function Ψ on H (see, e.g. Ikeda and Watanabe [1]), Ψ(x)µ(dx) = Ψ(x)ν(dx). H

H

Now deﬁne for arbitrary Γ ∈ B(H), µξT (Γ) =

1 T

T

P (ξ, t, Γ)dt,

ξ ∈ H,

T > 0,

0

and, by the invariant properties of measures µ and ν, we have Ψ(x)µ(dx) − Ψ(x)ν(dx) H H = Ψ(x) µξT (dx)µ(dξ) − µηT (dx)ν(dη) H H ≤ Ψ(x)µξT (dx) − Ψ(x)µηT (dx)µ(dξ)ν(dη) H H×H H = |F (ξ, η)|µ(dξ)ν(dη), V ×V

(3.6.2)

132

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where

Ψ(x)µξT (dx)

F (ξ, η) =

Ψ(x)µηT (dx).

−

H

(3.6.3)

H

Let VRcV = V \VR and choose R > 0 such that µ(VRcV ) + ν(VRcV ) < ε.

(3.6.4)

Note (3.6.3) and (3.6.4), then the inequality (3.6.2) yields Ψ(x)µ(dx) − Ψ(x)ν(dx) ≤ |F (ξ, η)|µ(dξ)ν(dη) + 4bε + 2bε2 , H

VR ×VR

H

(3.6.5) where b = supx∈H |Ψ(x)|. On the other hand, we have for some suitable real number δ > 0, |F (ξ, η)|µ(dξ)ν(dη) VR ×VR

≤

VR ×VR

1 T

1 ≤ 2b sup T ξ,η∈VR

T

0 T

E|Ψ(Yt (ξ)) − Ψ(Yt (η))|dt µ(dξ)ν(dη) P Yt (ξ) − Yt (η)V ≥ δ dt +

sup

|Ψ(ξ) − Ψ(η)|

ξ,η∈VR ξ−ηV <δ

0

< 2bε + ε,

(3.6.6)

for T > 0 suﬃciently large, by Condition (3.6.1) and the uniform continuity of Ψ. In terms of (3.6.5) and (3.6.6), the uniqueness result follows. In the above proof, the property of having support in V of the measure µ(·) is essential. Suppose that, for each n, µn (·), deﬁned in Theorem 3.5.6 has a support in the space V or µn (V ) = 1, n = 1, 2, · · ·. Then one expects the same is true for their weak limit µ. This is indeed the case as established by the following theorem. Theorem 3.6.2 Let all the conditions in Corollary 3.5.2 hold. If the family {µn } of probability measures is supported in V such that for any ε > 0, there exists R0 > 0 such that sup µn (VRcH ) < ε, n

∀R > R0 ,

(3.6.7)

then any invariant measure, as the weak limit of a subsequence of {µn }, has support in V . Proof Let {µnk } be a subsequence converging weakly to µ so that µnk (V ) = 1 for any k ∈ N+ . Since HR = {x ∈ H : xV ≤ R} is compact, we have

Stability of Nonlinear Stochastic Diﬀerential Equations

133

HR = i(VR )H , where i(VR )H denotes the closure of i(VR ) = {x ∈ V : xV < R} in H. Let ρ(x, i(VR )), x ∈ H, be the distance from x to set i(VR ) given by ρ(x, i(VR )) = inf y − xV : y ∈ i(VR ) , and deﬁne function φ : R+ → R+ by φ(t) = t for 0 ≤ t ≤ 1 and φ(t) = 1 for t > 1. Now we introduce 1 ΨR (x) = φ )) , δ ∈ (0, 1). ρ(x, i(V R δ δ Then ΨR δ ∈ Cb (H) and R ΨR (x)µ (dx) = Ψ (x)µ (dx) ≤ χV cH (x)µnk (dx) nk nk δ δ H

V

V

= µnk (VRcH ),

R

(3.6.8)

where χΓ (·) denotes the indicator function of set Γ ∈ B(H). It follows from (3.6.7), (3.6.8) and the weak convergence that cH ΨR (x)µ(dx) = lim ΨR δ δ (x)µnk (dx) ≤ lim µnk (VR ) < ε, ∀R > R0 . k→∞

H

k→∞

H

Also, by the well-known dominated convergence theorem, we have lim ΨR (x)µ(dx) = χV cH (x)µ(dx) ≤ ε, δ δ↓0

H

R

H

or Since V =

∞ n=1

µ(H\VR ) = µ(VRcH ) ≤ ε,

∀R > R0 .

Vn , we deduce easily that ∞ . / µ(H\V ) = µ H Vn = 0 n=1

as required. One of the most important consequences of the concept of invariant measure is that it is closely connected with some stationary properties of solution processes. Proposition 3.6.1 If µ is an invariant measure for (3.5.9) and x0 is an H-valued, F0 -measurable random variable such that the law of x0 equals µ, then the solution process X· (x0 ) is stationary in the sense that for every ﬁnite sequence of numbers t1 , · · · , tn , the joint distribution of the random variables Xt1 +h , · · · , Xtn +h is independent of h ≥ 0.

134

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Proof Let ψ1 , · · · , ψn be a family of bounded, measurable functions on H, and 0 ≤ t1 ≤ t2 ≤ · · · ≤ tn . We have to show that the expectation I = E ψ1 (Xt1 +h ) · · · ψn (Xtn +h ) is independent of h ≥ 0. This is certainly true if n = 1. Note that E ψ1 (Xt1 +h ) · · · ψn (Xtn +h ) ) * = E ψ1 (Xt1 +h ) · · · ψn−1 (Xtn−1 +h )E ψn (Xtn +h ) FtXn−1 +h . By the Markov property, I = E ψ1 (Xt1 +h ) · · · ψn−1 (Xtn−1 +h )Ptn −tn−1 ψn (Xtn−1 +h ) . So if the result is true for n − 1, it is true for n, and consequently it holds by induction. The following theorem, which provides conditions to ensure the existence and uniqueness of invariant measures for Equation (3.5.9), is the main result of this section. Theorem 3.6.3 Let all the assumptions (1.3.2)–(1.3.5) on the strong solutions of the equation (3.5.9) hold. Further assume that the strong solution Xt (x0 ) with X0 = x0 satisﬁes the condition (3.5.13). Then the equation (3.5.9) has an invariant measure µ with its support in V , and the solution Xt with initial distribution µ is stationary. Moreover, if the condition (3.6.1) holds, the invariant measure µ is unique. Proof deﬁne

Clearly, all we need to do is to show supp{µ} ⊂ V . To this end, µT (E) =

1 T

T

P Xt (x0 ) ∈ E dt

(3.6.9)

0

for arbitrary T > 0, E ∈ B(H), which is a probability measure on (H, B(H)). In view of Theorem 3.6.2, it suﬃces to show that, µT {H \ V } = 0,

∀T > 0.

First, by virtue of Theorem 1.3.1, we have for some p > 1, T E Xt (x0 )pV dt ≤ MT < ∞, x0 ∈ H, 0

so that Xt (x0 ) ∈ V

a.s.

(t, ω) ∈ ΩT := [0, T ] × Ω.

Stability of Nonlinear Stochastic Diﬀerential Equations Let

135

N = (t, ω) ∈ ΩT : Xt (x0 ) ∈ H \ V .

Then

T

P {N } =

χN (t, ω)dtP (dω) = 0, 0

Ω

where χN (·) is the indicator function of set N . Thus, by the Fubini theorem, we get for any T > 0, t P {N } = χN (t, ω)P (dω) = 0, t ∈ J ⊂ [0, T ], (3.6.10) Ω

N t = ω ∈ Ω : Xt (x0 ) ∈ H \ V ,

where

and J = [0, T ] \ J has Lebesgue measure zero. Now, by (3.6.9) and (3.6.10), we have for any T > 0, 1 T 1 T µT {H \ V } = P {N t }dt = χJ (t)P {N t }dt = 0 T 0 T 0 as was to be shown. As an immediate consequence of the theorems derived above, we can obtain the following useful criterion which ensures a unique invariant measure of strong solutions by imposing some monotonicity conditions on Equation (3.5.9). Corollary 3.6.1 Let all the assumptions (1.3.2)–(1.3.5) on the strong solution of the equation (3.5.9) hold. If there exist positive constants T0 > 0 and M > 0 such that 1 T sup EXt (x0 )2H dt ≤ M, (3.6.11) T >T0 T 0 then the equation (3.5.9) has a stationary solution with the initial invariant measure supported in V . This invariant measure is unique provided the following condition holds: there exist c > 0 and δ > 0 such that 2u − v, A(u) − A(v)V,V ∗ + B(u) − B(v)2L0 ≤ −cu − vδV , 2

where B(u) − B(v)2L0 = tr (B(u) − B(v))Q(B(u) − B(v))∗ .

∀u, v ∈ V, (3.6.12)

2

Proof

In view of Itˆ o’s formula, we have the following energy inequality t EXt (x0 )2H ≤ x0 2H + 2E Xs (x0 ), A(Xs (x0 ))V,V ∗ ds 0

136

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations t +E B(Xs (x0 ))2L0 ds 2

0

for any t ≥ 0, which, in addition to the coercive condition (1.3.2), yields

t

t

Xs (x0 )pV ds ≤ (x0 2H + γt) + λE

EXt (x0 )2H + αE 0

Xs (x0 )2H ds. 0

It follows that for any T0 > 0, supT >T0

1 T

T 0

EXs (x0 )pV ds ≤

|γ|+x0 2H /T0 α 1 + |λ| sup T >T0 T α

T

EXs (x0 )2H ds. (3.6.13) On the other hand, µT (·) deﬁned by (3.6.9) is supported in V . Hence, by the well-known Markov’s inequality, we have for any T > 0, R > 0, 1 T

0

T

0

1 1 T P Xs (x0 )V > R ds ≤ p · EXs (x0 )pV ds , R T 0

which, together with (3.5.13) and (3.6.13), shows that the condition (3.6.11) is suﬃcient for the existence of an invariant measure µ on V by Theorem 3.6.3. By Proposition 3.6.1, the existence of a stationary solution follows. To show uniqueness, let Xt (x1 ), Xt (x2 ) denote two solutions with the initial states x1 , x2 ∈ H. We set ∆Xt = Xt (x1 ) − Xt (x2 ). Then, by Itˆ o’s formula, we have t 2 2 E∆Xt H ≤ x1 − x2 H + 2E ∆Xs , A(Xs (x1 )) − A(Xs (x2 ))V,V ∗ ds 0 t +E B(Xs (x1 )) − B(Xs (x2 ))2L0 ds. 2

0

In view of Condition (3.6.12), we obtain E∆Xt 2H ≤ x1 − x2 2H − c

t

E∆Xs δV ds, 0

or

t

E∆Xs δV ds ≤ 0

x1 − x2 2H . c

This implies that the condition (3.6.1) for uniqueness is satisﬁed by applying Markov’s inequality: ! P Xt (x1 ) − Xt (x2 )V ≥ ρ ≤ EXt (x1 ) − Xt (x2 )δV ρδ for arbitrary ρ > 0.

Stability of Nonlinear Stochastic Diﬀerential Equations

3.7

137

Stability, Ultimate Boundedness of Mild Solutions and Invariant Measures

In the previous sections, for instance, in the special case A(v) = Av + f (v) of Equation (3.5.9) with A being coercive, Corollary 3.6.1 seems to have wider applications. However, the corresponding results for mild solutions are also useful and could be applied to a certain class of stochastic evolution equations without coercive and monotone conditions, such as stochastic wave equations. The existence, uniqueness and some associated problems for invariant measures of mild solutions recently received increasing attention. For instance, the reader may ﬁnd a systematic presentation of recent developments in this area in the books Da Prato and Zabczyk [1], [2]. In this section, we ﬁrst establish the Lyapunov function characteristic theorem of mild solutions for a class of semilinear stochastic evolution equations. By analogy with those in Section 3.5.2, we also study the existence and uniqueness of invariant measures, but concentrating on those results in connection with the ultimate boundedness of solutions in mean square and the use of Lyapunov function characteristic techniques.

3.7.1

Lyapunov Functions for Mild Solutions

An example was constructed in Section 3.4 to show that the usual Lyapunov functions in ﬁnite dimensional spaces are not strictly positive deﬁnite for strong solutions in the inﬁnite dimensional setting. Examples may also be constructed to show that this is the case for mild solutions of semilinear stochastic evolution equations. In this section, we will construct Lyapunov functions for mild solutions and prove that the existence of such a Lyapunov function is a necessary and suﬃcient condition for mild solutions to be exponentially decayable in mean square. To this end, consider the semilinear stochastic diﬀerential equation (3.3.1). In particular, throughout this section we will impose the following condition: there exist constants γ ≥ 0, λ > 0 and µ ≥ 0 such that for any u ∈ D(A), 2Au + F (t, u), uH + G(t, u)2 ≤ λu2H + γ · e−µt ,

t ≥ 0.

(3.7.1)

Theorem 3.7.1 Assume the condition (3.7.1) holds and Xt (x0 ), t ≥ t0 ≥ 0, is a solution of (3.3.1). If there exists a function Λ(·, ·) ∈ C 1,2 (R+ × H; R1 ) which satisﬁes the following: (a) Λ(·, ·) satisﬁes all the conditions of using Itˆ o’s formula in Theorem 1.2.7; (b) |Λt (t, x)| + xH Λx (t, x)H + x2H Λxx (t, x) ≤ cx2H , x ∈ H, for some number c = c(T ) > 0, t ∈ [t0 , T ], T ≥ 0; (c) c1 x2H − k1 e−µ1 t ≤ Λ(t, x) ≤ c2 x2H + k2 e−µ2 t , ∀x ∈ H, t ≥ 0;

138

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

(d) (LΛ)(t, x) ≤ −c3 Λ(t, x) + k3 e−µ3 t ,

∀x ∈ D(A),

t ≥ 0,

where ci > 0, ki ≥ 0, µi ≥ 0, i = 1, 2, 3, are some constants and L is deﬁned by (LΛ)(t, x) = Λt (t, x) + Ax + F (t, x), Λx (t, x)H (3.7.2) + 12 · tr(Λxx (t, x)G(t, x)QG(t, x)∗ ), where x ∈ D(A), t ≥ t0 , then Xt , t ≥ t0 ≥ 0, satisﬁes EXt (x0 )2H ≤ α1 x0 2H ·e−β1 (t−t0 ) +α2 ·e−β2 t ,

x0 ∈ H,

t ≥ t0 ,

Xt0 = x0 , (3.7.3)

for some α1 > 0, α2 ≥ 0, β1 > 0 and β2 ≥ 0. Conversely, suppose (3.7.3) holds and deﬁne

T +t0

Λ(t0 , x0 ) =

EXs (x0 )2H ds + αx0 2H ,

for all x0 ∈ H,

t0 ≥ 0,

t0

(3.7.4) where Xt0 = x0 and T , α are two proper positive constants, independent on t0 ∈ R+ , x0 ∈ H, and assume Λ(·, ·) lies in the domain of the inﬁnitesimal generator of Xt and satisﬁes all the conditions of Theorem 1.2.7 and (b), then there exist constants ci > 0, ki ≥ 0, µi ≥ 0, i = 1, 2, 3, such that Conditions (c) and (d) above hold. Proof To prove (3.7.3), ﬁrst applying Itˆ o’s formula to Λ(t, x) and Xtn , the strong solution of (3.3.6), then taking expectations and using Condition (d), we can deduce that for any t ≥ s ≥ t0 , t EΛ(t, Xtn ) ≤ EΛ(s, Xsn ) + s − c3 EΛ(u, Xun ) + k3 · e−µ3 u du t + s E Λx (u, Xun ), (R(n) − I)F (u, Xun )H (3.7.5) ∗ + 12 tr R(n)G(u, Xun )Q R(n)G(u, Xun ) Λxx (u, Xun ) −G(u, Xun )QG(u, Xun )∗ Λxx (u, Xun ) du. Consequently, letting n → ∞ in (3.7.5) and using the condition (b) above and Proposition 1.3.6, it follows t EΛ(t, Xt ) ≤ EΛ(s, Xs ) + − c3 EΛ(u, Xu ) + k3 · e−µ3 u du (3.7.6) s

which, by carrying out a similar argument to the proof of Theorem 3.4.1 and using the condition (a), immediately yields that EXt (x0 )2H ≤ α1 x0 2H ·e−β1 (t−t0 ) +α2 ·e−β2 t , for some α1 > 0, α2 ≥ 0, β1 > 0 and β2 ≥ 0.

x0 ∈ H,

t ≥ t0 ,

Xt0 = x0 ,

Stability of Nonlinear Stochastic Diﬀerential Equations Now suppose (3.7.3) is true and let T +t0 Λ(t0 , x0 ) = EXs (x0 )2H ds + αx0 2H ,

139

(3.7.7)

t0

where T and α are two proper positive constants to be determined later. Substituting (3.7.3) into the above equality yields T +t0 Λ(t0 , x0 ) ≤ α1 x0 2H · e−β1 (s−t0 ) + α2 · e−β2 s ds + αx0 2H t0 (3.7.8) α1 ≤ + α x0 2H + α2 T · e−β2 t0 . β1 Let

T +t0

EXs (x0 )2H ds

ψ(t0 , x0 ) = t0

and

T +t0

n

EXsn (R(n)x0 )2H ds

ψ (t0 , R(n)x0 ) = t0

where Xtn (·, ·) is the approximation solution of (3.3.1) in the form of (3.3.6), R(n) = R(n, A), R(n, A) is the resolvent of A, and by the same arguments as in the proofs of Theorem 3.4.1, we have T +r+t0 n n n Eψ (r + t0 , Xr+t0 (R(n)x0 )) = EXsn (Xr+t (R(n)x0 ))2H ds 0 r+t0 T +r+t0

EXsn (R(n)x0 )2H ds.

= r+t0

Hence, by the continuity of s → EXsn (R(n)x0 )2H , we get

= = = =

(Lψ n )(t0 , x0 ) d n (R(n)x )) + ε(n) Eψ n (r + t0 , Xr+t 0 0 dr r=0 n Eψ n (r + t0 , Xr+t (R(n)x0 )) − Eψ n (t0 , x0 ) 0 lim + ε(n) r→0 r T +r+t0 T +t EXsn (R(n)x0 )2H ds − t0 0 EXsn (R(n)x0 )2H ds lim r+t0 + ε(n) r→0 r

r+t0 T +r+t0 1 1 n 2 n 2 EXs (R(n)x0 )H ds + EXs (R(n)x0 )H ds lim − r→0 r t0 r T +t0 +ε(n)

where ε(n) → 0 as n → ∞. Letting n → ∞ and using (3.7.3) immediately implies 2 2 (Lψ)(t0 , x0 ) = −x +t0 (x0 )H 0 H + EXT

≤ − 1 − α1 · e−β1 T x0 2H + α2 · e−β2 t0 .

(3.7.9)

140

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

On the other hand, by virtue of the condition (3.7.1), it is easy to deduce that for any x ∈ D(A), Lx2H = 2Ax+F (t, x), xH +G(t, x)2L0 ≤ λ(1+trQ)x2H +γ(1+trQ)e−µt 2

which, together with (3.7.9), immediately yields (LΛ)(t0 , x0 ) = (Lψ)(t0 , x0 ) + α(Lx2H )(t0 , x0 ) ≤ − 1 − α1 e−β1 T − αλ(1 + trQ) x0 2H

+ α2 + γ(1 + trQ) e−(β2 ∧µ)t0 .

(3.7.10)

Therefore, if T > ln α1 /β1 , then we can choose α > 0 small enough such that Λ(·, ·) satisﬁes (d) above. By (3.7.8) and the very deﬁnition of Λ(·, ·), it is clear that Λ(·, ·) satisﬁes the condition (c) and the proof is complete. As in Section 3.4, it is generally diﬃcult to know whether or not Λ(·, ·) in (3.7.4) belongs to C 1,2 (R+ × H; R1 ). However, this may be the case if Equation (3.3.1) is linear. Consider the following linear system dXt = AXt dt + BXt dWt

with

X0 = x0 ,

(3.7.11)

where B ∈ L(H, L(K, H)) and A is the inﬁnitesimal generator of the C0 semigroup T (t), t ≥ 0, on H satisfying T (t) ≤ eµt , t ≥ 0, for some µ ∈ R1 . In particular, by similar arguments to the proof of Corollary 3.4.2, we can derive: Corollary 3.7.1 Suppose Xt (x0 ), t ≥ 0, is the mild solution of (3.7.11). If there exists a function Λ(·) ∈ C 2 (H; R1 ) which satisﬁes the following: (a) Λ(·) satisﬁes all the conditions of using Itˆ o’s formula in Theorem 1.2.7; (b) c1 x2H ≤ Λ(x) ≤ c2 x2H , ∀x ∈ H; (c) (LΛ)(x) ≤ −c3 Λ(x), ∀x ∈ D(A), where ci > 0, i = 1, 2, 3, are some constants and L is the inﬁnitesimal generator of the Markov process Xt (x0 ), i.e., (LΛ)(x) = Ax, Λ (x)H + 1/2 · tr(Λ (x)(Bx)Q(Bx)∗ ), x ∈ D(A), then Xt , t ≥ 0, satisﬁes EXt (x0 )2H ≤ α1 x0 2H · e−β1 t ,

x0 ∈ H,

t ≥ 0,

X0 = x0 ,

for some α1 > 0 and β1 > 0. Conversely, suppose (3.7.12) holds and deﬁne ∞ Λ(x0 ) = EXs (x0 )2H ds + αx0 2H for all x0 ∈ H,

(3.7.12)

(3.7.13)

0

where X0 = x0 and α is a proper positive constant, independent on x0 ∈ H, then Λ(·) deﬁned by (3.7.13) has the property that there exist constants ci > 0, i = 1, 2, 3, such that Conditions (a), (b) and (c) above hold.

Stability of Nonlinear Stochastic Diﬀerential Equations

141

Proof The ﬁrst part may be proved by carrying out a similar argument to Theorem 3.7.1. Now we suppose (3.7.12) is true. Then it is easy to see that the validity of (3.7.1) is clear. Indeed, this is immediate by Proposition 2.1.3 since A is supposed to generate a C0 -semigroup T (t) satisfying T (t) ≤ eµt , t ≥ 0, for some µ ∈R1 . Since the null solution is exponentially stable in mean square, ∞ the term 0 EXs (x0 )H ds is well deﬁned and there exists a symmetric and nonnegative operator R ∈ L(H) (see, e.g. Da Prato and Zabczyk [1]) such that for any x ∈ H, ∞ EXs (x)2H ds = Rx, xH 0

and for any x ∈ D(A), (LR·, ·H )(x) = −x2H . Hence, Λ(x) = Rx, xH + αx2H for arbitrary x ∈ H. It is obvious that Λ(·) ∈ Cb2 (H) and αx2H ≤ Λ(x) ≤ (R + α)x2H for arbitrary x ∈ H. This proves (a) and (b). To prove (c), we note that A is the inﬁnitesimal generator of the C0 -semigroup T (t), t ≥ 0, satisfying T (t) ≤ eµt , which implies x, AxH ≤ µx2H , x ∈ D(A), by Proposition 2.1.4. Hence we have for any x ∈ D(A), (L · 2H )(x) = 2x, AxH + tr(BxQ(Bx)∗ ) ≤ (2µ + B2 trQ)x2H . Therefore, (LΛ)(x) = (LR·, ·H )(x) + α(L · 2H )(x) ≤ −x2H + α(2λ + B2 trQ)x2H

= − 1 + α(2λ + B2 trQ) x2H for any x ∈ D(A). Letting α be small enough and using (b) yield the condition (c). This proves the theorem. In a similar way to Theorem 3.7.1, we can also derive the following ultimate boundedness result which is quite useful in investigating invariant measures of mild solutions of the stochastic evolution equation (3.3.1). Theorem 3.7.2 Assume the condition (3.7.1) holds with µ = 0. Suppose Xt (x0 ), t ≥ t0 ≥ 0, is the solution of (3.3.1) with Xt0 = x0 . If there exists a function Λ(·, ·) ∈ C 1,2 (R+ × H; R1 ) satisfying the following:

142

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

(a) Λ(·, ·) satisﬁes all the conditions of using Itˆ o’s formula in Theorem 1.2.7; (b) |Λt (t, x)| + xH Λx (t, x)H + x2H Λxx (t, x) ≤ cx2H , x ∈ H, for some number c = c(T ) ≥ 0, t ∈ [t0 , T ], T ∈ R+ ; (c) c1 x2H − k1 ≤ Λ(t, x) ≤ c2 x2H + k2 , ∀x ∈ H, t ≥ 0; (d) (LΛ)(t, x) ≤ −c3 Λ(t, x) + k3 , ∀x ∈ D(A), t ≥ 0, where ci > 0, ki ≥ 0, i = 1, 2, 3, are some constants and L is the inﬁnitesimal generator deﬁned as in Theorem 3.7.1 of the Markov process Xt (x0 ), then Xt , t ≥ t0 , satisﬁes EXt (x0 )2H ≤ α1 x0 2H · e−β1 (t−t0 ) + α2 ,

t ≥ t0 ,

Xt0 = x0 ∈ H, (3.7.14)

for some α1 > 0, α2 ≥ 0 and β1 > 0. Conversely, suppose (3.7.14) holds and deﬁne T +t0 Λ(t0 , x0 ) = EXs (x0 )2H ds + αx0 2H for all x0 ∈ H,

t0 ≥ 0,

t0

where Xt0 = x0 and T , α are two proper constants, independent on t0 ∈ R+ , x0 ∈ H. Assume Λ(·, ·) lies in the domain of the inﬁnitesimal generator of Xt and satisﬁes all the conditions of Theorem 1.2.7 and (b), then there exist constants ci > 0, ki ≥ 0, i = 1, 2, 3, such that Conditions (c) and (d) above hold. In nonlinear situations, generally speaking, we have diﬃculty in showing Λ(t, x) ∈ C 1,2 (R+ × H; R1 ); it is therefore appropriate to use ﬁrst order approximation technique as we did in Section 3.5 to study exponential decay of the solutions of Equation (3.3.1). Proposition 3.7.1 Suppose the null solution of Equation (3.7.11) is exponentially stable in mean square and the relation (3.7.12) holds. Then the solution Xt (x0 ) of (3.3.1) with t0 = 0, X0 = x0 is exponentially decayable in mean square if 2xH F (t, x)H + tr G(t, x)QG(t, x)∗ − (Bx)Q(Bx)∗ < θx2H + γ · e−µt (3.7.15) for some positive constants γ > 0, µ > 0 and 0 < θ < β1 /α1 . Proof

Let Λ0 (x) = Rx, xH + αx2H

as deﬁned in the proofs of Corollary 3.7.1 for arbitrary x ∈ H and (L0 R·, ·H )(x) = −x2H for any x ∈ D(H) where L0 is the inﬁnitesimal generator corresponding to the equation (3.7.11). Hence, R ≤ α1 /β1 . On the other hand, since Λ0 (x) ∈

Stability of Nonlinear Stochastic Diﬀerential Equations

143

C 2 (H; R+ ) by using Corollary 3.7.1, it suﬃces for our purposes to show that Λ0 (x) satisﬁes the condition (d) in Theorem 3.7.1. Since (LΛ0 )(t, x) − (L0 Λ0 )(x) 1 = Λ0 (x), F (t, x)H + tr Λ0 (x) G(t, x)QG(t, x)∗ − (Bx)Q(Bx)∗ 2

= 2(R + αI)x, F (t, x)H + tr (R + αI) G(t, x)QG(t, x)∗ − (Bx)Q(Bx)∗

≤ 2(R + α)xH F (t, x)H +(R + α)tr G(t, x)QG(t, x)∗ − (Bx)Q(Bx)∗ = (R + α) 2xH F (t, x)H + tr G(t, x)QG(t, x)∗ − (Bx)Q(Bx)∗ for arbitrary x ∈ D(A). By (3.7.10) and the assumption (3.7.15), (LΛ0 )(t, x) satisﬁes the condition (d) in Theorem 3.7.1 if we choose α > 0 small enough and a suitable T > 0 in (3.7.10). This proves the proposition. Also, we use the ﬁrst order approximation to derive some results about exponential ultimate boundedness in mean square of nonlinear equations, based on the same property of the corresponding linear equations. In fact, in a similar way to Theorem 3.5.5, we have the following results. Proposition 3.7.2 Consider the equation (3.7.11) and let {Xt0 (x0 ), t ≥ 0} be its mild solution. Assume that this solution is exponentially ultimately bounded in mean square and satisﬁes the relation (3.7.14). Let {Xt (x0 ), t ≥ 0} be a solution of the equation (3.3.1) and suppose 2xH F (t, x)H +tr G(t, x)QG(t, x)∗ −(Bx)Q(Bx)∗ ≤ θx2H +M, ∀x ∈ H, (3.7.16) for any constant M > 0 and some 0<θ<

max s> 0∨

ln α1 β1

1 − α1 e−β1 s . α1 /β1 + α2 s

(3.7.17)

Then the solution of (3.3.1) is exponentially ultimately bounded in mean square. Proof The proofs are quite similar to those in Theorem 3.5.3 and are therefore omitted here (also, see Liu and Mandrekar [2]).

3.7.2

Ultimate Boundedness and Invariant Measures

In this section, we shall consider the existence and uniqueness of invariant measures of mild solutions by means of boundedness of moments and by using

144

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Lyapunov function characterization methods. Let Xt (x0 ), t ≥ 0, be the mild solution of (3.3.10) whose coeﬃcients F (·) and G(·) satisfy the usual Lipschitz and linear growth conditions. Lemma 3.7.1 Suppose the mild solution Xt (x0 ), t ≥ 0, of (3.3.10) is ultimately bounded in mean square satisfying, for instance, (3.7.14). Then for any invariant measure ν of the Markov process Xt (x0 ), y2H ν(dy) ≤ α2 < ∞, (3.7.18) H

where α2 is the constant in (3.7.14). Proof Let f (y) = y2H and fn (y) = χ[0,n] (f (y))f (y), y ∈ H, where χB (·) is the indicator function of B ⊂ R1 . We note that fn (·) ∈ L1 (H, B(H), ν; R1 ). From the assumption of ultimate boundedness (3.7.14), there is a constant α2 ≥ 0 such that lim sup Ex0 f (Xt ) ≤ α2 t→∞

for any x0 ∈ H,

(3.7.19)

where Ex0 means the conditional expectation under the condition X0 = x0 . Using the Ergodic theorem for Markov processes with invariant measures (cf. Yosida [1]), there exists the following limit N 1 Pk fn (x0 ) = fn∗ (x0 ), N →∞ N

lim

ν − a.s.

(3.7.20)

k=1

and

Eν fn∗ = Eν fn ,

(3.7.21)

where Pk fn (x0 ) = H fn (y)P (k, x0 , dy) and Eν fn = H fn (x)dν(x). ¿From the inequality fn (x) ≤ f (x) and the assumption of ultimate boundedness of the solution (3.3.10), we have for any x ∈ H N N 1 1 lim sup Pk fn (x) ≤ lim sup Pk f (x) ≤ α2 , N →∞ N N →∞ N k=1

k=1

which, together with (3.7.20), immediately yields fn∗ (x) ≤ α2 (ν-a.s.). Therefore, from this inequality, we have Eν fn∗ ≤ α2 . (3.7.22) The results (3.7.21), (3.7.22) and the fact fn (x) ↑ f (x) (n → ∞) imply that Eν f = lim Eν fn = lim Eν fn∗ ≤ α2 < ∞. n→∞

n→∞

Stability of Nonlinear Stochastic Diﬀerential Equations

145

We are now in a position to obtain the main results in this subsection. First of all, we establish the uniqueness of invariant measures of mild solutions. Theorem 3.7.3 Suppose the solution Xt (x0 ), t ≥ 0, of (3.3.10) is exponentially ultimately bounded in mean square. Suppose for each R > 0, δ > 0 and ε > 0, there exists T0 = T0 (R, δ, ε) > 0 such that P Xt (x1 )−Xt (x2 ) > δ < ε for any x1 , x2 ∈ HR whenever t ≥ T0 , H

(3.7.23) where HR = {x ∈ H : xH ≤ R}. Then there exists at most one invariant measure. Proof Let µi , i = 1, 2, be invariant measures of (3.3.10). Then by Lemma 3.7.1, for each ε > 0, there exists R > 0 such that µi (H\HR ) < ε. Let b ψ ∈ Cw (H), the space of all bounded weakly continuous functions on H (cf. Kantorovich and Akilov [1]). We ﬁrstly claim that there exists T = T (ε, R, ψ) > 0 such that (3.7.24) (Pt ψ)(x1 ) − (Pt ψ)(x2 ) ≤ ε for x1 , x2 ∈ HR if t ≥ T. H

Indeed, to this end, let L be a weakly compact set in H and recall that the weak topology on L is equivalent to the topology deﬁned by the metric d(x, y) =

∞

(1/2k )|ek , x − yH |,

x, y ∈ L,

k=1

for any orthonormal set {ek } of H. We shall ﬁrst prove that for each δ > 0 and ε > 0, there exists a number T1 = T1 (ε, R, δ, ψ) > 0 such that t ≥ T1 implies P ψ(Xt (x1 )) − ψ(Xt (x2 )) ≤ δ ≥ 1 − ε for all x1 , x2 ∈ HR . By the ultimate boundedness, there exists T2 = T2 (ε, R) > 0 such that t ≥ T2 implies P Xt (x) ∈ HR ≥ 1 − ε/3 for any x ∈ HR . Note that ψ(·) on HR is uniformly continuous with respect to the metric d. Hence, there exists a δ > 0 such that x, y ∈ HR and d(x, y) ≤ δ implies |ψ(x) − ψ(y)| ≤ δ. Note also that there exists an integer J > 0 such that ∞ k=J+1

(1/2k )|ek , x − yH | ≤ δ /2

for all x, y ∈ HR .

146

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Now choose T1 ≥ T2 such that t ≥ T1 implies J P ek , Xt (x1 ) − Xt (x2 )H ≤ δ /2 ≥ 1 − ε/3

for all x1 , x2 ∈ HR ,

k=1

a case which is possible by using the condition (3.7.23). Hence, for t ≥ T1 we have P ψ(Xt (x1 )) − ψ(Xt (x2 )) ≤ δ ≥ P Xt (x1 ), Xt (x2 ) ∈ HR , d(Xt (x1 ), Xt (x2 )) ≤ δ J ≥ P Xt (x1 ), Xt (x2 ) ∈ HR , (1/2k )ek , Xt (x1 ) − Xt (x2 )H ≤ δ /2 k=1 ≥ P Xt (x1 ), Xt (x2 ) ∈ HR , ek , Xt (x1 ) − Xt (x2 )H ≤ δ /2, k = 1, 2, · · · J

≥ 1 − ε/3 − ε/3 − ε/3 = 1 − ε. Now for the given ε > 0, choosing T > 0 such that t ≥ T implies ε P ψ(Xt (x1 )) − ψ(Xt (x2 )) ≤ ε/2 ≥ 1 − 4M0 where M0 = sup |ψ(x)| < ∞. Then E ψ(Xt (x1 )) − ψ(Xt (x2 )) ≤ ε/2 + 2M0 (ε/4M0 ) = ε as required. In order to conclude our proof, ﬁrst note that ψ(x)mi (dx) = [Pt ψ](x)mi (dx), i = 1, 2. H

H

On the other hand, we have for any t ≥ T , ψ(x1 )m1 (dx1 ) − ψ(x2 )m2 (dx2 ) H H = [ψ(x1 ) − ψ(x2 )]m1 (dx1 )m2 (dx2 ) H H = [Pt ψ](x1 ) − [Pt ψ](x2 ) m1 (dx1 )m2 (dx2 ) H H ≤ [Pt ψ](x1 ) − [Pt ψ](x2 )m1 (dx1 )m2 (dx2 ) H H = + + [Pt ψ](x1 ) − [Pt ψ](x2 )m1 (dx1 )m2 (dx2 ) HR

H\HR

HR

≤ ε + 2(2M0 )ε + 2M0 ε2 .

H\HR

Stability of Nonlinear Stochastic Diﬀerential Equations

147

Since ε > 0 is arbitrary, we deduce ψ(x1 )m1 (dx1 ) = ψ(x2 )m2 (dx2 ) H

H

which immediately implies m1 (·) = m2 (·). Corollary 3.7.2 Assume the solution Xt (x0 ), t ≥ 0, of (3.3.10) is exponentially ultimately bounded in mean square. Suppose the conditions of Propositions 3.3.1 and 3.3.2 hold, then (3.7.23) is satisﬁed. In other words, in this case there exists at most one invariant measure. To establish existence results similar to those in Section 3.5 for mild solutions, ﬁrst notice that in this situation the condition (3.5.13), for instance, does not make sense any more. A suitable version however is possible with the V -norm replaced by a H-norm plus some additional assumptions on the associated inﬁnitesimal generator A. For instance, we impose the following assumption: (H6) A is self-adjoint and has eigenvectors {ek }, k = 1, 2, · · ·, which form an orthonormal basis of H and eigenvalues {−λk } ↓ −∞ as k → ∞. Suppose the condition (H6) holds and the solution of

Theorem 3.7.4 (3.3.10) satisﬁes 1 T

T

EXs (x0 )2H ds ≤ M (1 + x0 2H )

(3.7.25)

0

for some M > 0 and any T ≥ 0. Then there exists an invariant measure for the equation (3.3.10). To obtain this result, we ﬁrst show the following lemma. Lemma 3.7.2

Suppose the conditions in Theorem 3.7.4 hold. Then 1 t mt (·) = P (s, x0 , ·)ds for t ≥ 0 t 0

is weakly compact. Here P (t, x0 , Γ), Γ ∈ B(H), is the transition probability of the solution process Xt (x0 ), t ≥ 0. Proof show

By a well-known result (cf. Gihman and Skorohod [1]), it suﬃces to 1 T

0

∞ T k=1

Ex2k (t)dt

is uniformly convergent in T ≥ 0,

148

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where xk (t) = Xt (x0 ), ek H and {ek } is the orthonormal basis given in (H6). Since the assumption (H6) holds, the semigroup T (t) has the representation T (t)h =

∞

e−λk t ek , hH ek , h ∈ H,

k=1

which, by the deﬁnition of mild solutions, immediately yields −λk t

xk (t) = e

xk +

t

e−λk (t−r) ek , F (Xr (x0 ))H dr

0

t

+

e−λk (t−r) ek , G(Xr (x0 ))dWr H

0

where xk = x0 , ek H , k = 1, 2, · · ·. Hence, Ex2k (t)

3e−2λk t x2k

=

t 2 + 3E e−λk (t−r) ek , F (Xr (x0 ))H dr 0

2 t +3E e−λk (t−r) ek , G(Xr (x0 ))dWr H . 0

Letting N be large enough so that λN > 0 and using H¨ older’s inequality, we can deduce that for any integer m > 0, N +m k=N

1 T

T

E

0

t

2 e−λk (t−r) ek , F (Xr (x0 ))H dr dt

0

T

≤

0

2 E F (Xr (x0 ))H dr C1 (1 + x0 2H ) ≤ 4δ(λN − δ)T δ(λN − δ)

for a small δ > 0 and some C1 > 0. In a similar way, it can be deduced that N +m k=N

1 T

0

T

E

t

2 e−λk (t−r) ek , G(Xr (x0 ))dWr H dt

0

≤

trQ

T 0

EG(Xt (x0 ))2 dt C2 trQ(1 + x0 2H ) ≤ 2λN T λN

for some C2 > 0. Thus, N +m k=N

1 T Exk (t)2 dt T 0 ≤ 3x0 2H /2λN + 3(C1 + C2 )(1 + x0 2H )[1/δ(λN − δ) + trQ/λN ]

which tends to zero uniformly in T ≥ 0 as N → ∞.

Stability of Nonlinear Stochastic Diﬀerential Equations Proof of Theorem 3.7.4 mn (Γ) =

1 n

149

For integer n ≥ 0, deﬁne n

P (s, x0 , Γ)ds,

Γ ∈ B(H).

0

Then mn (·) is a probability measure and y2H mn (dy) ≤ M (1 + x0 2H ). H

Hence, for each ε > 0, there exists R > 0 such that mn (HR ) > 1 − ε, HR = {x ∈ H : xH ≤ R}. By Lemma 3.7.2, mn (·), n ≥ 0, is weakly compact and there exists a subsequence, still denoted by mn (·), which is weakly convergent to some probability measure m(·). By a similar argument to the proof of Theorem 3.5.6, it can be deduced that m(·) is an invariant measure of P (t, x0 , ·). The proof is now complete. Remark Nowadays, there exists an extensive literature on the topic of invariant measures of the stochastic evolution equations (3.3.10). In particular, various extensions of Theorems 3.7.3 and 3.7.4 in one way or another have been made by some researchers. The reader is referred to the existing references such as Da Prato and Zabczyk [2] for more material in this respect. Example 3.7.1 Consider the following stochastic heat equation 

 dy(x, t) = ∂ 2 /∂x2 y(x, t)dt + cy 2 (x, t)/(1 + |y(x, t)|) dBt1 + g(x)dBt2 0 < x < 1, c > 0,  y(0, t) = y(1, t) = 0, y(x, 0) = y0 (x), y0 (·), g(·) ∈ L∞ (0, 1), (3.7.26) where Bti , i = 1, 2, are mutually independent real Brownian motions. In this situation, we take H = L2 (0, 1) and A = d2 /dx2 with D(A) = {y ∈ H : y , y ∈ H, y(0) = y(1) = 0}. Then it is easy to deduce Ay, yH ≤ −π 2 y2H√ . Also note that the condition (H6) is satisﬁed since A has eigenvectors { 2 sin nπx} and eigenvalues {−n2 π 2 }, n = 1, 2, · · ·. Now let Λ(y) = y2H , then (L · 2H )(y) ≤ −2[π 2 − c2 /2]y2H + g2L∞ (0,1) ,

y ∈ D(A).

If 2π 2 > c2 , then the solution is exponentially ultimately bounded in mean square. Moreover, by virtue of Theorem 3.7.4 there exists an invariant measure. We also have (Ld · 2H )(y) ≤ −[2π 2 − c2 ]y2H ,

y ∈ D(A),

where Ld is deﬁned exactly as that in Proposition 3.3.2. Hence, if 2π 2 > c2 , by virtue of Theorem 3.7.3, the invariant measure is unique. Furthermore, if g(·) = 0, the null solution is exponentially stable in mean square.

150

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

3.8

Decay Rates of Systems

The stochastic diﬀerential equation (3.2.1) or (3.3.1) could be commonly illustrated as the perturbed stochastic system by white noise sources of a deterministic diﬀerential equation. For instance, consider the system (3.3.1) which could be regarded as the perturbed stochastic system of the corresponding deterministic one t dXt = T (t)x0 + T (t − s)F (s, Xs )ds, X0 = x0 ∈ H. (3.8.1) 0

We have shown in Example 3.2.1 that when the solution of (3.8.1) is exponentially decayable (for instance, suppose F (t, x) = 0 and A generates a stable C0 -semigroup T (t), t ≥ 0, on H), it cannot be generally deduced that its perturbed stochastic system (3.3.1) remains exponentially decayable, despite the fact that the solution might remain decayable with a slower decay. To put this more precisely, suppose H = L2 (0, 1) and Wq (t, x); t ≥ 0, x ∈ [0, 1] is an H-valued Wiener process with associated covariance operator Q, trQ < ∞, determined by a positive deﬁnite kernel q(x, y) ∈ L2 [0, 1] × [0, 1]; R1 , q(x, x) ∈ L2 (0, 1; R1 ). Let A be a linear strictly elliptic diﬀerential operator of the second order, for instance, the Laplace operator on [0, 1], F (t, ·) = 0 and G(t, ·) in (3.3.1) is the operator deﬁned by G(t, ·)h(·) = (1 + t)−µ h(·) for some constant µ > 0 and arbitrary h(·) ∈ L2 (0, 1). Then we may formulate the following problem with null initial data ∂ 2 Xt (x) dt + (1 + t)−µ dWq (t, x), t ≥ 0, x ∈ [0, 1], ∂x2 X0 (x) = 0, x ∈ [0, 1]; Xt (0) = Xt (1) = 0, t ≥ 0.

dXt (x) =

It is easy to obtain the explicit mild solution: t Xt (x) = T (t − s)(1 + s)−µ dWq (s, x),

(3.8.2)

t ≥ 0,

0

where T (t), t ≥ 0, is the strongly continuous semigroup generated by the ∂2 Laplace operator ∂x It may be shown by a direct computation that for 2. arbitrarily given µ > 0, the Lyapunov exponents lim sup t→∞

log EXt (·)2H =0 t

and

lim sup t→∞

log Xt (·)H = 0 a.s. t

Stability of Nonlinear Stochastic Diﬀerential Equations

151

However, we want to point out that although the mild solution is not exponentially decayable, we show below that the solution is decayable with a slower, polynomial, decay rate.

3.8.1

Decay in the p-th Moment

Consider the semilinear stochastic diﬀerential equation (3.3.1) in the separable, real Hilbert space H. Assume Λ(t, x) : R+ × H → R+ is any function diﬀerentiable of the ﬁrst order in R+ and twice Fr´echet diﬀerentiable in H. Denote it by Λ(t, x) ∈ C 1,2 (R+ × H; R+ ). Assume that Λ(t, x) satisﬁes all the conditions of using Itˆ o’s formula in Theorem 1.2.7. For our purpose, we introduce the following operator L: (LΛ)(t, x) := Λt (t, x) + Λx (t, x), Ax + F (t, x)H 1 + · tr Λxx (t, x)G(t, x)QG(t, x)∗ , 2

t ≥ 0,

x ∈ D(A). (3.8.3)

Deﬁnition 3.8.1 Let p ≥ 2 and λ(t) ↑ ∞, as t → ∞, which is some nondecreasing, continuous function deﬁned for suﬃciently large t > 0. The mild solution of Equation (3.3.1) is said to be decayable with rate λ(t) in the p-th moment if there exist a positive constant γ > 0 and function O(λ(t)), t ≥ 0, such that log EXt (x0 )pH lim sup ≤ −γ, (3.8.4) log O(λ(t)) t→∞ holds for any X0 = x0 ∈ H, where O(λ(t)) is the big oh of λ(t). Theorem 3.8.1 Let Λ(t, x) ∈ C 1,2 (R+ × H; R+ ) and ψ1 (t), ψ2 (t) be two non-negative, continuous functions. Assume that there exist a positive constant m > 0 and real numbers ν, θ ∈ R1 such that for some p ≥ 2, m

(1). xpH λ(t) ≤ Λ(t, x), (t, x) ∈ R+ × H; (2). |Λt (t, x)| + |Λ(t, x)| + xH Λx (t, x)H + x2H Λxx (t, x) ≤ cxpH , x ∈ H, for some constant c = c(T ) > 0, t ∈ [0, T ], T ∈ R+ ; (3). (LΛ)(t, x) ≤ ψ1 (t) + ψ2 (t)Λ(t, x), (t, x) ∈ R+ × D(A); (4). t t log 0 ψ1 (s)ds ψ2 (s)ds lim sup ≤ ν, lim sup 0 ≤ θ. (3.8.5) log λ(t) log λ(t) t→∞ t→∞ Then, if γ = m − θ − ν > 0, we have lim sup t→∞

log EXt (x0 )pH ≤ −γ. log λ(t)

(3.8.6)

In other words, the solution of Equation (3.3.1) is decayable with rate λ(t) in the p-th moment sense.

152

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Proof In order to show (3.8.6), let us carry out an approximating solution procedure exactly as we did in Section 3.3. To this end, applying Itˆ o’s formula to Λ(t, x) and the strong solution Xtn of (3.3.6) yields

t

Λ(t, Xtn ) = Λ(0, X0n ) +

t

(Ln Λ)(s, Xsn )ds + 0

Λx (s, Xsn ), G(s, Xsn )dWs H ,

0

(3.8.7) where (Ln Λ)(t, x) = Λt (t, x) + Λx (t, x), Ax + R(n)F (t, x)H ∗ 1 + tr Λxx (t, x)R(n)G(t, x)Q R(n)G(t, x) , 2

x ∈ D(A),

and n ∈ ρ(A), the resolvent set of A. Now, for ﬁxed n ∈ R+ and arbitrary m ∈ R+ , deﬁne an increasing sequence of stopping times t inf t > 0 : 0 Λx (s, Xsn ), G(s, Xsn )dWs H > m , n τm = ∞ if the set is empty. n Clearly, for any ﬁxed n, τm ↑ ∞, as m → ∞, and since

t

Λx (s, Xsn ), G(s, Xsn )dWs H ,

t ∈ R+ ,

0

is a continuous localmartingale, it follows that for ﬁxed n ≥ 1, and any m ≥ 1, E

n t∧τm

Λx (s, Xsn ), G(s, Xsn )dWs H = 0,

t ∈ R+ .

0

Therefore, taking expectations in (3.8.7) and using Conditions (2) and (3) in Theorem 3.8.1 yield n n n EΛ(t ∧ τm , Xt∧τ n ) ≤ EΛ(0, X0 ) + m

+

n t∧τm

n t∧τm

ψ1 (s) + ψ2 (s)EΛ(s, Xsn ) ds

0

E Λx (s, Xsn ), (R(n) − I)F (s, Xsn )H

0

∗ 1 + tr R(n)G(s, Xsn )Q R(n)G(s, Xsn ) Λxx (s, Xsn ) 2 −G(s, Xsn )QG(s, Xsn )∗ Λxx (s, Xsn ) ds (3.8.8) where R(n) = nR(n, A), R(n, A) is the resolvent of A.

Stability of Nonlinear Stochastic Diﬀerential Equations

153

Firstly, letting m → ∞ and using the Fatou’s lemma in (3.8.8), then taking n → ∞, together with Condition (2) above and using Proposition 1.3.6, it follows that t EΛ(t, Xt ) ≤ EΛ(0, x0 ) + ψ1 (s) + ψ2 (s)EΛ(s, Xs ) ds. 0

Hence, by virtue of the Gronwall’s lemma we derive t t EΛ(t, Xt ) ≤ EΛ(0, x0 ) + ψ1 (s)ds exp ψ2 (s)ds . 0

0

Hence, for arbitrary ε > 0, in view of Condition (4) we have that for any t ∈ R+ suﬃciently large, log EΛ(t, Xt ) ≤ log EΛ(0, x0 ) + λ(t)ν+ε + log λ(t)θ+ε , which, letting ε → 0 and using Condition (1), immediately implies

log λ(t)−m EΛ(t, Xt ) log EXt (x0 )pH lim sup ≤ lim sup log λ(t) log λ(t) t→∞ t→∞

≤ − m − (ν + θ)

as required. Next, let us investigate a class of stochastic diﬀerential equations which includes as a special case (3.8.2) to illustrate the theory derived above. Example 3.8.1 Let µ > 1/2 be a positive constant. Consider the following stochastic partial diﬀerential equation ∂2 −µ dYt (x) = ∂x α(Yt (x))dWq (t, x), t > 0, x ∈ (0, 1), 2 Yt (x)dt + (1 + t) Yt (0) = Yt (1) = 0, t ≥ 0, Y0 (x) = y0 (x), (3.8.9) where Wq (t, x) is the Q-Wiener process deﬁned exactly as in the beginning of the section, and α(·) : R1 → R1 is a certain bounded, Lipschitz continuous function with |α(y)| ≤ C, C > 0, for each y ∈ R1 . Let H = L2 [0, 1]. Clearly, d2 on this occasion F (t,u) = 0, G(t, u) = (1 + t)−µ α(u) and Au = dx 2 u(x) in (3.8.3) with D(A) = u(x) ∈ H : u(x), u (x) are absolutely continuous with u (x), u (x) ∈ H, u(0) = u (0) = 0 . Introduce a Lyapunov function on H in the following way 1 Λ(t, u) = (1 + t)2(π∧µ) u(x)2 dx, ∀u ∈ H. 0

154

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

A simple computation gives that for any u ∈ D(A), t ≥ 0, (LΛ)(t, u) ≤ C 2

1

q(x, x)dx < ∞. 0

In view of Theorem 3.8.1, we obtain the result that the mild solution of (3.8.9) is polynomially decayable in mean square. Moreover, lim sup t→∞

3.8.2

log EXt (x0 )2H ≤ − 2(π ∧ µ) − 1) . log t

Almost Sure Pathwise Decay

When one tries to investigate almost sure decay rate, the following diﬃculty might be encountered. The usual method, i.e., consider the decay of solution processes in p-th moment and then (sometimes additional conditions are imposed) deduce their pathwise decay, may not work any more. This is because in the usual investigation of exponential decay such as Theorem 3.2.2, a Borel-Cantelli lemma type of technique is used. This method usually involves an argument of convergence of a certain series with terms in the form (3.2.12). For exponential decay, this series which consists of exponential decay terms is always convergent. But, for general decay rates, this series could be divergent. In this section, we shall present a diﬀerent approach to handle this diﬃculty. Deﬁnition 3.8.2 Suppose λ(t) ↑ ∞, as t → ∞, is some non-decreasing, continuous function deﬁned for suﬃciently large t > 0. The solution of Equation (3.3.1) is said to be almost surely decayable with rate λ(t) if there exist a positive constant γ > 0 and function O(λ(t)), t ≥ 0, such that lim sup t→∞

log Xt (x0 )H ≤ −γ log O(λ(t))

a.s.

holds for any x0 ∈ H, where O(λ(t)) is the big oh of λ(t). In order to obtain the main results, we need the following exponential martingale inequality. Recall that W 2 ([0, T ]; L) denotes the family of all L(K, H)valued predictable {Ft }-adapted processes Φ(t, ω), t ≥ 0, such that E

T

tr Φ(t, ω)QΦ(t, ω)∗ dt < ∞ for T ≥ 0.

0

Lemma 3.8.1 Assume J(t, ω) ∈ H is an arbitrary H-valued continuous T process with 0 EJ(t, ω)2H dt < ∞. Let Φ(t) ∈ W 2 ([0, T ]; L) and T , a, b be

Stability of Nonlinear Stochastic Diﬀerential Equations

155

any positive numbers. Then t P sup J(s), Φ(s)dWs H 0≤t≤T 0 a t − tr J(s) ⊗ J(s)(Φ(s)QΦ(s)∗ ) ds > b ≤ e−ab , 2 0 (3.8.10) where Wt is some Q-Wiener process and J(t) ⊗ J(t) is the linear operator deﬁned by (J(t) ⊗ J(t))h = J(t), hHJ(t) for any h ∈ H, t ≥ 0. For every integer n ≥ 1, deﬁne the stopping time  t  J(s), Φ(s)dW inf t ≥ 0 :  s H  0 t τn = + 0 tr J(s) ⊗ J(s)(Φ(s)QΦ(s)∗ ) ds > n ,    ∞ if the set is empty,

Proof

and the process

t

χ[[0,τn ]] (s)J(s), Φ(s)dWs H t a − χ[[0,τn ]] (s)tr J(s) ⊗ J(s)(Φ(s)QΦ(s)∗ ) ds 2 0

xn (t) = a

0 2

where [[·, ·]] is the stochastic interval (cf. M´etivier [1]). Clearly, xn (t) is bounded and o’s formula to

τn ↑ ∞ almost surely as n → ∞. Applying Itˆ exp xn (t) , we obtain t

exp xn (t) = 1 + exp xn (s) dxn (s) 0

a2 t + exp xn (s) χ[[0,τn ]] (s)tr J(s) ⊗ J(s)(Φ(s)QΦ(s)∗ ) ds 2 0 t

= 1+a exp xn (s) χ[[0,τn ]] (s)J(s), Φ(s)dWs H . 0

Therefore, one easilysee that exp xn (t) is a nonnegative martingale over can

R+ with E exp xn (t) = 1, t ≥ 0. Hence, by Doob’s maximal inequality, we get that

P sup exp xn (t) ≥ eab ≤ e−ab E exp xn (T ) = e−ab . 0≤t≤T

That is, t P sup χ[[0,τn ]] (s)J(s), Φ(s)dWs H 0≤t≤T

0

156

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations a t − χ[[0,τn ]] (s)tr J(s) ⊗ J(s)(Φ(s)QΦ(s)∗ ) ds > b ≤ e−ab . 2 0

Now the required result follows by letting n → ∞. For arbitrary Λ(t, x) ∈ C 1,2 (R+ × H; R+ ), consider the equation (3.3.1) and deﬁne the operator (QΛ)(t, x) := tr Λx ⊗ Λx (t, x)G(t, x)QG(t, x)∗ , t ≥ 0, x ∈ H. (3.8.11) Theorem 3.8.2 Let λ(t) ↑ ∞, as t → ∞, be some non-decreasing, continuous function satisfying the properties that there exists a T ≥ 0 such that (i). log λ(t) is uniformly continuous over [T, ∞); (ii). There exists a non-negative constant τ ≥ 0 such that log t lim supt→∞ log log λ(t) ≤ τ ; (iii). For all s, t ≥ T , λ(s)λ(t) ≥ λ(s + t). Let Λ(t, x) ∈ C 2,1 (R+ × H → R+ ) and assume ψ1 (t), ψ2 (t) are two nonnegative continuous functions on R+ . Suppose that there exist a positive constant m > 0, and real numbers ν, θ ∈ R1 such that for some p ≥ 2, (1). xpH λ(t)m ≤ Λ(t, x), (t, x) ∈ R+ × H; (2). |Λt (t, x)| + |Λ(t, x)| + xH Λx (t, x)H + x2H Λxx (t, x) ≤ cxpH , x ∈ H, for some constant c = c(T ) > 0, t ∈ [0, T ], T ∈ R+ ; (3). (LΛ)(t, x) + (QΛ)(t, x) ≤ ψ1 (t) + ψ2 (t)Λ(t, x), x ∈ D(A), t ∈ R+ ; (4). t t log 0 ψ1 (s)ds ψ2 (s)ds lim sup ≤ ν, lim sup 0 ≤ θ. log λ(t) log λ(t) t→∞ t→∞ Then, if γ = m − θ − τ − ν > 0, we have almost surely lim sup t→∞

γ log Xt (x0 )H ≤− . log λ(t) p

In other words, the solution of Equation (3.3.1) is almost surely decayable with rate λ(t). Proof This is similar to the proof of Theorem 3.8.1: applying Itˆ o’s formula to Λ(t, x) and the strong solution Xtn of (3.3.6), we can derive that for any t ≥ 0, Λ(t, Xtn )

= Λ(0, X0n ) +

t

(Ln Λ)(s, Xsn )ds + 0

0

t

Λx (s, Xsn ), R(n)G(s, Xsn )dWs H

Stability of Nonlinear Stochastic Diﬀerential Equations t t n n = Λ(0, X0 ) + (LΛ)(s, Xs )ds + Λx (s, Xs ), G(s, Xs )dWs H 0

157

0

+I1 (t, n) + I2 (t, n)

(3.8.12)

where

t

t

(Ln Λ)(s, Xsn )ds −

I1 (t, n) = 0

t

I2 (t, n) =

(LΛ)(s, Xsn )ds, 0

Λx (s, Xsn ), [R(n) − I]G(s, Xsn )dWs H ,

0

and (Ln Λ)(t, x) = Λt (t, x) + Λx (t, x), Ax + R(n)F (t, x)H ∗ 1 + tr Λxx (t, x)R(n)G(t, x)Q R(n)G(t, x) , 2

x ∈ D(A),

for any t ≥ 0, x ∈ D(A). By virtue of the uniform continuity of log λ(t), for any ε > 0 there exist two positive integers N = N (ε) and k1 = k1 (ε) such that if k−1 ≤ t ≤ 2kN , k ≥ k1 (ε), we have 2N k log λ N − log λ(t) ≤ M. 2 On the other hand, owing to Lemma 3.8.1 we have t t u P ω : sup Λx (s, Xs ), G(s, Xs )dWs H − (QΛ)(s, Xs )ds > v 0≤t≤w 0 0 2 ≤ e−uv for any positive constants u, v and w. In particular, taking u = 2,

v = log

k − 1 2N

,

w=

k , 2N

k = 2, 3, . . . ,

we then apply the well-known Borel-Cantelli lemma to obtain the existence of a random integer k0 (ε, ω) > 0 such that almost surely

t

Λx (s, Xs ), G(s, Xs )dWs H ≤ log

0

k − 1 2N

t

(QΛ)(s, Xs )ds

+ 0

for all 0 ≤ t ≤ 2kN , k(ω) ≥ k0 (ε, ω). Substituting this into (3.8.12) and using Condition (3), we see that for almost all ω ∈ Ω, Λ(t, Xtn ) ≤ log

k − 1 2N

+ Λ(0, X0n ) +

t

(LΛ)(s, Xsn )ds + 0

t

(QΛ)(s, Xsn )ds 0

158

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations + I1 (t, n) + I2 (t, n) + I3 (t, n) t k − 1 n ≤ log + Λ(0, X ψ1 (s) + ψ2 (s)Λ(s, Xsn ) ds ) + 0 N 2 0 +I1 (t, n) + I2 (t, n) + I3 (t, n), (3.8.13)

where

t I3 (t, n) =

(QΛ)(s, Xs ) − (QΛ)(s, Xsn ) ds

0

for all t ∈ [0, ], k ≥ k0 (ε, ω) ∨ k1 (ε). So by the well-known Gronwall’s lemma, we derive that almost surely k − 1 Λ(t, Xtn ) ≤ Λ(0, X0n ) + log + |I1 (t, n)| + |I2 (t, n)| t 2N t (3.8.14) +|I3 (t, n)| + ψ1 (s)ds · exp ψ2 (s)ds k 2N

0

0

for all 0 ≤ t ≤ 2kN , k ≥ k0 (ε, ω) ∨ k1 (ε). By virtue of Proposition 1.3.6, there exists a subsequence Xtnk of Xtn , denote it still by Xtn , such that Xtn → Xt almost surely, as n → ∞ uniformly with respect to t ∈ [0, 2kN ], i.e., there exists Ωk ⊂ Ω with P (Ωk ) = 0 such that for any ω ∈ Ω\Ωk , Xtn → Xt , as n → ∞, uniformly with respect to t ∈ [0, 2kN ]. Hence, we have, by using Condition ∞ (2), that for any ω ∈ Ω\ k=1 Ωk , Ii (t, n) → 0, i = 1, 2, 3, as n → ∞ for all 0 ≤ t ≤ 2kN , k ≥ k0 (ε, ω) ∨ k1 (ε). Hence, letting n → ∞ in (3.8.14), we may deduce that almost surely k − 1 t t Λ(t, Xt ) ≤ Λ(0, x0 ) + log + ψ (s)ds exp ψ2 (s)ds 1 N 2 0 0 for all 0 ≤ t ≤ 2kN , k ≥ k0 (ε, ω) ∨ k1 (ε). Using Condition (4) and the uniform continuity of log λ(t), for the preceding ε > 0 there exists a positive integer k2 (ε, ω) > 0 such that k − 1 log Λ(t, Xt ) ≤ log Λ(0, x0 ) + λ(t)(ν+@) + log log + (θ + M) log λ(t) 2N for all k−1 ≤ t ≤ 2kN , k ≥ k0 (ε, ω) ∨ k1 (ε) ∨ k2 (M, ω), which, letting ε → 0 and 2N using Condition (1), immediately implies that almost surely

log Xt (x0 )H 1 log λ(t)−m Λ(t, Xt ) m − [ν + τ + θ] lim sup ≤ lim sup ≤− log λ(t) log λ(t) p t→∞ t→∞ p as required. Clearly, replacing λ(t) in Theorem 3.8.2 by et , 1 + t or log t leads to exponential, polynomial or logarithmic decay, respectively. In particular, as an

Stability of Nonlinear Stochastic Diﬀerential Equations

159

immediate consequence of Theorem 3.8.2, we may apply it to the equation (3.8.9) to obtain its almost sure decay rate. Indeed, to this end we may introduce the following Lyapunov function on H, Λ(t, u) = (1 + t)2(π∧µ−δ)

1

u(x)2 dx,

∀u ∈ H,

t ≥ 0,

0

where 2µ > δ > 1 is some constant to be determined later on. A direct computation easily yields that for any u ∈ D(A), t ≥ 0, (LΛ)(t, u) ≤

2(π ∧ µ − δ)(1 + t)2(π∧µ)−1−δ − 2π(1 + t)2π∧ν−δ

+(1 + t)−δ C 2 ≤ (1 + t)−δ C 2

1

u(x)2 dx

0

1

q(x, x)dx 0 1

q(x, x)dx < ∞, 0

and

(QΛ)(t, u) ≤ 4(1 + t)−δ

1

q(x, x)dx Λ(t, u).

0

Using Theorem 3.8.2, we can obtain that if 2(π ∧ µ) − δ − (1 − δ) > 0, i.e., µ > 1/2 (letting δ → 1), the mild solution of (3.8.9) is polynomially decayable in the almost sure sense. Moreover, lim sup t→∞

3.9

log Xt (x0 )H ≤ − (π ∧ µ) − 1/2 log t

a.s.

Stabilization of Systems by Noise

In Example 3.2.1, we have shown that although the deterministic system dXt = AXt dt, where A generates an exponentially stable C0 -semigroup, is exponentially stable, its stochastically perturbed system might fail to be exponentially stable if the perturbation does not “decay fast”. However, in some situations we ﬁnd that even though the deterministic system is not exponentially stable, it is still possible to choose some proper noise sources so as to make its perturbed system exponentially stable. To put this clearly, let us go back to Example 2.4.1:  ∂X (x) 2 Xt (x) t  ∂t = ∂ ∂x + rXt (x), t > 0, 0 < x < π, 2 (3.9.1) Xt (0) = Xt (π) = 0, t ≥ 0,  X0 (x) = x0 (x), 0 ≤ x ≤ π.

160

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Recall that if we assume r = r0 , a constant, then this equation has an explicit solution ∞ 2 Xt (x) = an e−(n −r0 )t sin nx n=1

∞

with x0 (x) = n=1 an sin nx. Therefore, it is immediate to obtain exponential stability if r0 < 1 despite the fact that for r0 ≥ 1, the null solution is generally not stable. Now suppose that r is random, and it is modelled as r0 + r1 B˙ t so that the equation (3.9.1) becomes dYt (x) =

∂2 + r0 Yt (x) dt + r1 Yt (x)dBt , 2 ∂x t > 0, 0 < x < π,

Yt (0) = Yt (π) = 0,

t ≥ 0; Y0 (x) = y0 (x),

(3.9.2) 0 ≤ x ≤ π,

where Bt is a one-dimensional standard Brownian motion. In Example 2.4.1, it was shown that when r0 < 1, i.e., the unperturbed system (3.9.1) is stable, the perturbed system (3.9.2) remains pathwise exponentially stable if r12 < 2(1 − r0 ). On the other hand, since a new noise term is included in the perturbed system (3.9.2), a natural problem now arises: as r0 ≥ 1, is it possible to obtain any stability results for the perturbed system (3.9.2)? In other words, is it possible to stabilize the unperturbed (3.9.1) by a suitable noise source? In this chapter, we shall provide suﬃcient conditions for exponential stabilization of a class of (stochastic) diﬀerential equations in inﬁnite dimensions.

3.9.1

Nonlinear Deterministic Equations

First of all, recall that V ;→ H ≡ H ∗ ;→ V ∗ , and β is a positive constant such that uH ≤ βuV , ∀u ∈ V. Consider the following deterministic nonlinear equation dXt = A(t, Xt )dt, t ≥ 0, (3.9.3) X0 = x0 ∈ H, where A(t, ·) : V → V ∗ , A(t, 0) = 0, t ∈ R+ , is a measurable family of nonlinear operators satisfying the following hypothesis: (H7). There exist a continuous function ν(t), t ≥ 0, and a real number ν0 ∈ R1 such that 2u, A(t, u)V,V ∗ ≤ ν(t)u2H where lim sup t→∞

1 t

for all t ∈ R+ ,

t

ν(s) ds ≤ ν0 . 0

u ∈ V,

Stability of Nonlinear Stochastic Diﬀerential Equations

161

Now we may present the following question. If the system (3.9.3) is not exponentially stable, is it possible to stabilize it by using a stochastic perturbation, for simplicity of the type B(t, Xt )B˙ t ? Here Bt , t ≥ 0, is some standard real Brownian motion, and B(t, ·) : H → H satisﬁes B(t, 0) = 0 and the following condition: (H8). B(t, u) − B(t, v)2H ≤ λ(t)u − v2H ,

t ∈ R+ ,

u, v ∈ V,

where λ(t), t ≥ 0, is a nonnegative continuous function such that 1 t lim sup λ(s)ds ≤ λ0 t→∞ t 0 for some number λ0 ≥ 0. The answer to this question will be aﬃrmative if we choose a suitable B(·, ·). Indeed, consider the following stochastic perturbed diﬀerential equation on H dYt = A(t, Yt )dt + B(t, Yt )dBt , t ≥ 0, (3.9.4) Y0 = y0 ∈ H. The following result can be used to handle this problem. Theorem 3.9.1 Assume the strong solution Yt (y0 ), t ≥ 0, of (3.9.4) satisﬁes the condition that Yt (y0 )H = 0 almost surely for all t ≥ 0 provided y0 H = 0. In addition to hypotheses (H7) and (H8), assume the following: B(t, u), u2H ≥ ρ(t)u4H

for all t ∈ R+ ,

u ∈ H,

where ρ(t), t ≥ 0, is a nonnegative continuous function such that 1 t lim inf ρ(s) ds ≥ ρ0 for some ρ0 ∈ R+ . t→∞ t 0 Then, the following relation lim sup t→∞

1 ν0 + λ 0 log Yt (y0 )H ≤ − ρ0 − t 2

holds almost surely for any y0 ∈ H with y0 H = 0. In particular, if 2ρ0 > ν0 + λ0 the null solution of Equation (3.9.4) is almost surely exponentially stable. Fix y0 H = 0 and then it is easy to deduce by Itˆ o’s formula that t 2 log Yt 2H ≤ log y0 2H + 2 Ys , B(s, Ys )H dBs 0 Ys H t 2Ys , A(s, Ys )V,V ∗ + B(s, Ys )2H 2Ys , B(s, Ys )2H ds. + − Ys 2H Ys 4H 0 (3.9.5) Proof

162

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Due to the exponential martingale inequality, i.e., Lemma 3.8.1, we have t 2 P ω : sup 2 Ys , B(s, Ys )H dBs 0≤t≤w 0 Ys H t 2u − Ys , B(s, Ys )2H ds > v ≤ e−uv 4 0 Ys H for any positive constants u, v and w. Assigning ε > 0 arbitrarily and taking u = α,

v = 2α−1 log k,

w = kε,

k = 1, 2, 3, . . . ,

where 0 < α < 1, we then apply the well-known Borel-Cantelli lemma to show that there exists an integer k0 (ε, ω) > 0 for almost all ω ∈ Ω such that

t

0

2 Ys , B(s, Ys )H dBs ≤ 2α−1 log k + α Ys 2H

0

t

2 Ys , B(s, Ys )2H ds Ys 4H

for all 0 ≤ t ≤ kε, k ≥ k0 (ε, ω). Substituting this into (3.9.5) and using the conditions in Theorem 3.9.1, we see that for the preceding ε > 0, there exists a positive integer k1 (ε) > 0 such that almost surely log Yt H t t t 1 ≤ (ν(s) + λ(s))ds − (1 − α) ρ(s)ds log y0 2H + 2α−1 log k + 2t 0 0 1 2 −1 ≤ log y0 H + 2α log k + ν0 + λ0 + ε t − (1 − α) 2ρ0 + ε t . 2t for all (k − 1)ε ≤ t ≤ kε, k ≥ k0 (ε, ω) ∨ k1 (ε), which immediately implies lim sup t→∞

log Yt H 1 ≤ ν0 + λ0 + ε − (1 − α) 2ρ0 + ε t 2

Therefore, letting α → 0 and ε → 0 yields log Yt (y0 )H ν0 + λ0 lim sup ≤ − ρ0 − t 2 t→∞

a.s.

a.s.

as required. As an immediate consequence, let us apply Theorem 3.9.1 to the simple linear case described at the beginning of this section. Indeed, we may formulate 2 this problem by setting V = H01 ([0, π]), H = L2 ([0, π]), A(t, u) = ∂∂xu2 + r0 u, u ∈ V , B(t, u) = r1 u, u ∈ H. It is obvious that this equation has a unique trivial solution if y0 H = 0 by the uniqueness of solutions. Since A(t, 0) = 0, B(t, 0) = 0, it is easy to conclude, for instance, by an argument of backward stochastic diﬀerential equations, that Yt (y0 )H = 0 for all t ≥ 0 almost surely

Stability of Nonlinear Stochastic Diﬀerential Equations

163

if y0 H = 0. Now it is easy to check that Theorem 3.9.1 can be applied to this problem with ρ0 = λ0 = r12 and ν0 = 2(r0 − 1). Thus, it can be deduced that the strong solution of (3.9.4) satisﬁes r2 log Yt (y0 )H lim sup a.s. ≤ − 1 − (r0 − 1) t 2 t→∞ for any y0 H = 0. That is, we get pathwise exponential stability with probability one if r12 > 2(r0 − 1), a case which actually improves the corresponding results in Example 2.4.1. In particular, for any r0 ∈ R1 , the above result shows that it is always possible to choose a suitable multiplicative noise source to stabilize the unperturbed system (3.9.1) when r = r0 . To close this subsection, let us investigate a nonlinear example. Example 3.9.1 Let O be an open, bounded subset in Rn with regular boundary and 2 ≤ p < ∞. Consider the Sobolev spaces V = W01,p (O), H = L2 (O) with its usual inner product, and the monotone operator A: V → V ∗ deﬁned as n ∂u(x) p−2 ∂u(x) ∂v(x) v, AuV,V ∗ = − dx ∂xi ∂xi ∂xi O i=1 + au(x)v(x)dx, ∀u, v ∈ V, O

where a ∈ R . We also let B(t, u) = bu, u ∈ H, where b ∈ R1 and Bt be a standard real Brownian motion. Then, the condition (H5) in Section 3.2 turns out to be 1

2u, A(t, u)V,V ∗ + B(t, u)2 = −2upV + 2au2H + b2 u2H , u ∈ V. Condition (i) of Theorem 3.2.1 requires 2a + b2 < 0, i.e., a < 0 and b2 < −2a. Alternatively, (ii) will hold whenever (2a + b2 )β 2 − 2 < 0, i.e., b2 < 2β −2 − 2a. Therefore, Theorem 3.2.1 guarantees almost surely exponential stability of the null solution only for these values of a and b for which the deterministic system dXt = A(t, Xt )dt is exponentially stable and the random perturbation is small enough. However, it is easy to prove by an argument similar to the above that Theorem 3.9.1 can be applied in this situation to ensure exponential stability of the null solution for suﬃciently large perturbations although the deterministic system is unstable. Note that, in this case, it is not diﬃcult to see that 2au2H if p > 2, p 2 ∗ 2u, A(t, u)V,V = −2uV + 2auH ≤ (2a − 2β −2 )u2H if p = 2. Therefore, by virtue of Theorem 3.9.1 it immediately follows that the strong solution of (3.9.4) satisﬁes 2 1 −2 b /2 − a −2 if p > 2, lim sup log Yt (y0 )H ≤ − b /2 − a + β if p = 2, t→∞ t

164

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

for any y0 H = 0. Consequently, we get almost surely exponential stability if 2a if p > 2, 2 b > 2a − 2β −2 if p = 2.

3.9.2

Nonlinear Stochastic Equations

It is interesting that a noise source can also be used to stabilize stochastic diﬀerential equations. For instance, let us go back to Equation (3.9.2) once again. When r0 > 1, r1 ∈ R1 with r12 ≤ 2(r0 − 1), we do not know whether or not the null solution is exponentially stable. Indeed, we may show below that if the system is perturbed by another multiplicative noise source of the ¯t is another one-dimensional Brownian motion ¯˙ t , where B same type, say r2 Yt B independent of Bt , it can be speciﬁcally deduced that the system dYt (x) =

∂2 ¯t , + r0 Yt (x) dt + r1 Yt (x)dBt + r2 Yt (x)dB 2 ∂x t ≥ 0, 0 ≤ x ≤ π,

Yt (0) = Yt (π) = 0,

t ≥ 0;

Y0 (x) = y0 (x),

(3.9.6)

x ∈ [0, π],

becomes pathwise exponentially stable once again if r2 ∈ R1 is chosen large enough. Consider the following stochastic system ¯t , t > 0, dYt = A(t, Yt )dt + B(t, Yt )dBt + H(t, Yt )dB (3.9.7) Y0 = y0 ∈ H, ¯t are two independent standard real Brownian motions on where Bt and B the same probability space (Ω, F, P ). A(t, ·) : V → V ∗ , A(t, 0) = 0, is a family of nonlinear measurable operators, and B(t, ·), H(t, ·) : H → H are both Lipschitz continuous, B(t, 0) = H(t, 0) = 0, and satisfy the following assumptions: (H9). There exists a continuous function ν˜(t), t ≥ 0, and ν˜0 ∈ R1 such that 2u, A(t, u)V,V ∗ + B(t, u)2H ≤ ν˜(t)u2H , and lim sup t→∞

1 t

t ≥ 0,

u ∈ V,

t

ν˜(s) ds ≤ ν˜0 . 0

˜ ˜0, There exist nonnegative continuous functions λ(t), ρ˜(t), t ≥ 0, and λ ρ˜0 ∈ R+ such that 2 ˜ H(t, u)2H ≤ λ(t)u H,

u, H(t, u)2H ≥ ρ˜(t)u4H ,

t ≥ 0, t ≥ 0,

u ∈ H, u ∈ H,

Stability of Nonlinear Stochastic Diﬀerential Equations where lim sup t→∞

1 t

t

˜ ˜0, λ(s)ds ≤λ 0

lim inf t→∞

1 t

165

t

ρ˜(s) ds ≥ ρ˜0 . 0

Theorem 3.9.2 Assume the strong solution Yt (y0 ), t ≥ 0, of (3.9.7) satisﬁes that Yt (y0 )H = 0 for all t ≥ 0 almost surely provided y0 H = 0. Suppose the hypothesis (H9) holds. Then the strong solution of Equation (3.9.7) satisﬁes ˜0 1 ν˜0 + λ lim sup log Yt (y0 )H ≤ − ρ˜0 − a.s. (3.9.8) 2 t→∞ t ˜ 0 , the null for any y0 ∈ H with y0 H = 0. In particular, if 2˜ ρ0 > ν˜0 + λ solution of (3.9.7) is exponentially stable in the almost sure sense. Proof Fix arbitrarily y0 ∈ H such that y0 H = 0. Then, by virtue of Itˆ o’s formula, it follows that log Yt 2H

t

2 Ys , A(s, Ys )V,V ∗ ds Y 2H s 0 1 t 2B(s, Ys )2H 4Ys , B(s, Ys )2H + ds − 2 2 0 Ys H Ys 4H t 1 2H(s, Ys )2H 4Ys , H(s, Xs )2H + ds − 2 0 Ys 2H Xs 4H t t 2Ys , B(s, Ys )H 2Ys , H(s, Ys )H ¯ + dB + dBs . s 2 Ys H Ys 2H 0 0 t 2 1 t 2B(s, Ys )2H ∗ ≤ log y0 2H + Y , A(s, Y ) ds + ds s s V,V 2 2 0 Ys 2H 0 Ys H 1 t 2H(s, Ys )2H 4Ys , H(s, Xs )2H + ds − 2 0 Ys 2H Xs 4H t t 2 2 ¯ + Y , B(s, Y ) dB + s s H s 2 2 Ys , H(s, Ys )H dBs . 0 Ys H 0 Ys H (3.9.9) = log y0 2H +

Taking into account the hypotheses of the present theorem, applying Lemma 3.8.1 to the last two terms of the right hand side in (3.9.9) and carrying out a similar argument to that in the proof of Theorem 3.9.1, it is easy to deduce that ˜0 1 ν˜0 + λ lim sup log Yt (y0 )H ≤ − ρ˜0 − a.s. 2 t→∞ t

166

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

As an immediate consequence, we can apply this result to stabilize (3.9.2). Recall that we had no idea whether (3.9.2) is exponentially stable or not when r12 ≤ 2(r0 − 1), r0 > 1, r1 ∈ R1 . Indeed, we can handle this problem in the form of (3.9.6) by introducing another multiplicative noise source. More precisely, for the example modelled by (3.9.6), one can easily check that ν˜0 = 2(r0 − 1) + r12 ,

˜ 0 = ρ˜0 = r2 . λ 2

Therefore, the relation (3.9.8) becomes lim sup t→∞

r2 − r2 1 1 log Yt (y0 )H ≤ − 2 − (r0 − 1) t 2

a.s.

In other words, when r22 > 2(r0 − 1) + r12 , the null solution is exponentially stable in the almost sure sense.

3.10

Lyapunov Exponents and Stabilization

In the history of the study of asymptotic properties (mainly in ﬁnite dimensional spaces), one of the most important methods for handling stochastic stability of ﬁnite dimensional stochastic systems was the development of the so-called Lyapunov exponent approach. This is certainly the stochastic counterpart in some sense of the notion of characteristic exponents introduced in Lyapunov’s classic work on asymptotic (exponential) stability of deterministic systems. Under some circumstances, this approach provides necessary and suﬃcient conditions for stability, but signiﬁcant computational problems must be solved. In inﬁnite dimensional cases, the Lyapunov exponent method, especially for nonlinear stochastic systems, needs to use sophisticated mathematics and is far from being a mature subject area. In this section, we would like to inaugurate the study of Lyapunov exponent methods by presenting some elementary results, and then applying them to the stabilization problem for a class of linear stochastic evolution equations. Consider the following deterministic equation in H, dXt = AXt dt, t ≥ 0, (3.10.1) X0 = x0 ∈ H. We shall prove that if A generates a C0 -semigroup, then this equation can be stabilized, in terms of Lyapunov exponents, by suitable noise sources. To this end, we assume that A generates a strongly continuous semigroup of bounded operators T (t), t ≥ 0, on H. It is then well known that Xt (x0 ) = T (t)x0 , t ≥ 0, is the unique mild solution of (3.10.1). For arbitrary x0 ∈ H

Stability of Nonlinear Stochastic Diﬀerential Equations

167

with x0 H = 0, we deﬁne the Lyapunov exponent of the equation (3.10.1) as follows: log Xt (x0 )H λX (x0 ) = lim sup . t t→∞ Lemma 3.10.1 If the operator A generates a C0 -semigroup T (t), t ≥ 0, then for any initial value x0 ∈ H, λX (x0 ) ≤ λ < ∞, where λ ∈ R1 is independent of x0 ∈ H. Proof This may be deduced from the fact that since T (t), t ≥ 0, is a C0 -semigroup, there exist constants λ ∈ R1 and M ≥ 1 such that T (t) ≤ M · exp(λt),

∀t ≥ 0.

Let us next consider the following stochastic system dYt = AYt dt + GYt dBt , t ≥ 0, Y0 = y0 ∈ H,

(3.10.2)

where A : D(A) ⊂ H → H, generates a C0 -semigroup T (t), t ≥ 0, G ∈ L(H, H). Assume that T (t) and G commute for any t ≥ 0, and Bt , t ≥ 0, is a real standard Brownian motion. It is easy to deduce that there exists a unique mild solution to the problem (3.10.2) and the solution is given by the following formula 1 Yt (y0 ) = exp Bt G − tG2 T (t)y0 . (3.10.3) 2 For arbitrary y0 ∈ H with y0 H = 0, the Lyapunov exponent of the system (3.10.2) is deﬁned as: λY (y0 , ω) = lim sup t→∞

log Yt (y0 , ω)H . t

From Lemma 3.10.1 and the next theorem, it will follow that λY (y0 , ω) < ∞ almost surely. Theorem 3.10.1 Let y0 ∈ H with y0 H = 0. Suppose Yt (y0 ), t ≥ 0, is the mild solution of (3.10.2) given by (3.10.3); then the following inequality holds: λY (y0 , ω) ≤ λX (y0 ) − α a.s.

168

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where λX (y0 ) is the Lyapunov exponent of (3.10.1) with initial value X0 = y0 and 1 α = − lim sup log exp[−(1/2)tG2 ] > 0. t→∞ t Proof

We have by virtue of (3.10.3) that

λY (y0 , ω) 1 1 = lim sup log exp Bt G − tG2 T (t)y0 2 H t→∞ t 1 1 2 ≤ lim sup log exp[(−1/2)tG ] exp(Bt G) + lim sup log T (t)y0 H t→∞ t t→∞ t 1 1 ≤ λX (y0 ) + lim sup log exp[(−1/2)tG2 ] + lim sup log exp(Bt G). t→∞ t t→∞ t (3.10.4) Note that lim sup t→∞

1 1 log exp(Bt G) ≤ lim sup log exp(G|Bt |) = 0, t t→∞ t

since by the strong law of large numbers lim supt→∞ |Bt |/t = 0 with probability one. Therefore, substituting this into (3.10.4) immediately yields the desired result. A similar argument to Theorem 3.10.1 can be applied to the following stochastic system driven by multiplicative white noise sources, n dZt = AZt dt + k=1 bk Zt dBtk , t ≥ 0, (3.10.5) Z0 = z0 ∈ H, where bk ∈ R1 , k = 1, · · · , n, and Bt1 , · · · , Btn are mutually independent, real Brownian motions. Then we have: Theorem 3.10.2 Let z0 ∈ H with z0 H = 0. Suppose Zt (z0 ), t ≥ 0, is the mild solution of (3.10.5) and λZ (z0 , ω) is the corresponding Lyapunov exponent, then the following inequality holds: 1 2 λZ (z0 , ω) ≤ λX (z0 ) − bk 2 n

a.s.

k=1

Corollary 3.10.1 Suppose that the operator A generates a C0 -semigroup; then the deterministic system (3.10.1) can be stabilized by noise.

Stability of Nonlinear Stochastic Diﬀerential Equations

169

Proof Let us ﬁx z0 ∈ H with z0 H = 0. Lemma 3.10.1 ensures the existence of a constant λ > 0 such that λX (z0 ) ≤ λ. Choosing bk ∈ R1 , k = 1, · · · , n, properly such that 1 2 bk > λ, 2 n

k=1

then we have λZ (z0 , ω) < 0 almost surely. The proof is complete by means of Theorem 3.10.2. Next, we shall apply the theory derived above to some partial diﬀerential equations of parabolic type. Let O be a bounded domain in Rn with smooth boundary ∂O. Consider the diﬀerential operator of the form n ∂ ∂ A= aij (x) + a(x), ∂xi ∂xj i,j

where the coeﬃcients satisfy the following assumptions: ¯ the closure (1). aij (·), a(·) are suﬃciently smooth real-valued functions on O, of O; (2). aij (·) = aji (·) for any i, j ∈ {1, · · · , n}; (3). µ1 ξ2Rn ≤ i,j aij ξi ξj ≤ µ2 ξ2Rn , ξ2Rn = ξ12 + · · · + ξn2 , ξi ∈ R1 , i = 1, · · · , n; ∂aij (·) (4). ∂x ≤ µ3 for any 1 ≤ i, j, k ≤ n; k (5). |a(·)| ≤ µ4 , where µi > 0, i = 1, · · · , 4. It is well known (Pazy [1]) that there exists an orthonormal basis of L2 (O), {ej }, j = 1, 2, · · ·, consisting of eigenvectors of the operator A such that Aej = λj ej ,

where

λj ↓ − ∞ as

j → ∞.

Moreover, the operator A generates an analytic semigroup of bounded operators T (t), t ≥ 0, on the space L2 (O). We will study the following parabolic equation with Dirichlet boundary conditions: ∂X (x) t = AXt (x), ∀t ≥ 0, x ∈ O; ∂t X0 (x) = x0 (x) ∈ L2 (O), ∀x ∈ O; Xt (x) = 0, ∀t ≥ 0, x ∈ ∂O. (3.10.6) It is easy to deduce that the unique mild solution of (3.10.6) is given by the following formula Xt (x) = T (t)x0 (x) =

∞ j=1

exp(tλj )xj0 ej (x)

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where x0 (x) =

∞

xj0 ej (x),

xj0 = x0 , ej L2 (O) .

(3.10.7)

j=1

Proposition 3.10.1 Let x0 (·)L2 (O) = 0 and j0 be the smallest integer j ≥ 1 in the expansion (3.10.7) of x0 such that xj00 = 0. Then the Lyapunov exponent of (3.10.6) exists and is given by λX (x0 ) = λj0 . Proof

It is immediate to deduce ∞ ∞ 2 1/2 1 1 exp(tλj0 )xj0 log exp(tλj )xj0 ej ≤ log t t L2 (O) j=0 j=j 0

1 = λj0 + log x0 L2 (O) t and ∞ 1 1 1 exp(tλj )xj0 ej ≥ log exp(tλj0 )xj00 = λj0 + log |xj00 |. log 2 t t t L (O) j=0

The desired result follows. Now let us consider the following stochastic perturbation of the deterministic problem (3.10.6) ∂Yt (x) = AYt (x) + bk Yt (x)dBtk , ∂t n

k=1 2

Y0 (x) = y0 (x) ∈ L (O),

∀t ≥ 0, x ∈ O;

∀x ∈ O, and Yt (x) = 0, ∀t ≥ 0, x ∈ ∂O, (3.10.8)

where bk = 0, k = 1, · · · , n, and Bt1 , · · · , Btn are mutually independent, real Brownian motions. It is easy to show that the mild solution of the equation (3.10.8) is equal to Yt (x) = exp

n k=1

n 1 bk Btk exp − b2k t Xt (x), 2 k=1

where Xt (x) is the mild solution of the equation (3.10.6). It is now easy to prove the following theorem using the above formula and the fact that limt→∞ Btk /t = 0 almost surely for k = 1, · · · , n.

Stability of Nonlinear Stochastic Diﬀerential Equations

171

Proposition 3.10.2 The Lyapunov exponent of the system (3.10.8) almost surely exists as a limit which is non-random and the following formula holds: λY (y0 , ω) = λX (y0 ) −

1 2 1 2 bk = λj0 − bk 2 2 n

n

k=1

k=1

a.s.

In particular, we have the following: if y01 = y0 , e1 L2 (O) = 0, then the stochastic system (3.10.8) is exponentially stable in the almost sure sense if and only if its top Lyapunov exponent satisﬁes 1 2 bk < 0. 2 n

λ1 −

k=1

In conjunction with Proposition 3.10.2, the stabilization strategy by noise above can be stated in a slightly diﬀerent way. We may say that the Lyapunov exponents of the linear equation (3.10.1) are essentially diﬀerent from those of its perturbed Itˆ o equation (3.10.2) because of the inclusion of multiplicative white noise sources. However, we may show that this is not the case if we consider the Stratonovich version of (3.10.2). Precisely, we will show that the Lyapunov exponents of (3.10.2) in the Stratonovich interpretation are as same as those of its unperturbed deterministic model for a wide range of stochastic perturbations. Let us consider that a linear noise is added to the problem (3.10.1) in a Stratonovich sense dYt = AYt dt + GYt ◦ dBt , t ≥ 0, (3.10.9) Y0 = y0 ∈ H, where A generates a C0 -semigroup T (t), t ≥ 0, and G : D(G) ⊂ H → H is assumed to be the generator of a C0 -group, denoted by S(t), t ∈ R1 , satisfying D(A) ⊂ D(G). We assume that for each y0 ∈ H, there exists a unique mild solution of (3.10.9) (see Da Prato and Zabczyk [1] or Kunita [2] for suitable conditions). Theorem 3.10.3 Assume that A and S(t), t ≥ 0, commute. Then the null solution of Equation (3.10.1) is exponentially stable if and only if the null solution of (3.10.9) is exponentially stable in the almost sure sense. Proof Let y0 ∈ D(A) and Yt (y0 ) be a solution of (3.10.9). Deﬁne the transformation Zt by the following Zt = Zt (y0 ) = S −1 (Bt (ω))Yt (y0 ). Now, it is not diﬃcult to check that dZt = S −1 (Bt (ω))dYt − S −1 (Bt (ω))GYt ◦ dBt = S −1 (Bt (ω))AYt dt = AZt dt,

(3.10.10)

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

and Z0 = y0 . Consequently, the process Zt = Zt (y0 ) = Xt (y0 ) = T (t)y0 is the unique mild solution of Equation (3.10.1). Since D(A) is dense in H and Zt , T (t) both are linear, Zt (y0 ) = Xt (y0 ) for any y0 ∈ H. Assume that the null solution of (3.10.1) is exponentially stable. This means that there exist constants M ≥ 1, µ > 0 such that T (t) ≤ M e−µt for all t ≥ 0. On the other hand, as the operator G is the generator of the C0 -group S(t), it is well known (see Pazy [1]) that there exist C ≥ 1, γ ∈ R1 such that S(t) ≤ Ceγ|t| for all t ∈ R1 . Then, since lim

t→∞

|Bt | =0 t

a.s.

there exists Ω0 ⊂ Ω, P (Ω0 ) = 0 such that if ω ∈ Ω\Ω0 , lim

t→∞

µ−γ

|Bt | = µ, t

and there exists T (ω) ≥ 0 such that for all t ≥ T (ω), µ−γ|Bt |/t ≥ µ/2. Thus, for arbitrarily given y0 ∈ D(A), ω ∈ Ω\Ω0 , and taking into account the fact that Xt (y0 ) = T (t)y0 , Yt (y0 )H = S(Bt (ω))Xt (y0 )H ≤ M Ceγ|Bt (ω)| e−µt y0 H ≤ M Cy0 H e−(µ−

γ|Bt (ω)| )t t

≤ M Cy0 H e−µ0 t ,

∀t ≥ T (ω),

where µ0 = µ/2. Therefore, the null solution of (3.10.9) is exponentially stable in the almost sure sense. Conversely, if G is an operator such that the null solution of (3.10.9) is almost surely exponentially stable, there exist Ω0 ⊂ Ω, P (Ω0 ) = 0 and constants α > 0, θ > 0 such that if ω ∈ Ω\Ω0 , then Yt (y0 )H ≤ αy0 H e−θt ,

∀t ≥ T0 (ω)

for some random variable T0 (ω) ≥ 0. Now, for any ﬁxed ω ∈ Ω\Ω0 , the equation (3.10.10) implies T (t)y0 H ≤ S(−Bt (ω))Yt (y0 )H ≤ αCy0 H e−(θ−

γ|Bt (ω)| )t t

,

∀t ≥ T0 (ω).

On the other hand, we can assure the existence of Ω0 ⊂ Ω, P (Ω0 ) = 0 satisfying that for all ω ∈ Ω\Ω0 , there exists T1 (ω) ≥ 0 such that for all t ≥ T1 (ω), we have γ|Bt | θ θ− ≥ . t 2 ¯ 0 , it is easy to deduce that ¯ 0 = Ω ∪ Ω and take any ﬁxed ω ∈ Ω\Ω Denote Ω 0 0 T (t)y0 H ≤ M e−µt y0 H ,

∀t ≥ T¯(ω),

where T¯(ω) = max{T0 (ω), T1 (ω)}. The proof is now complete.

Stability of Nonlinear Stochastic Diﬀerential Equations

173

Note that if we consider the particular case Gy = σy for some σ ∈ R1 , the C0 -group S(t) is given by S(t) = eσt I, and the hypotheses in Theorem 3.10.3 are fulﬁlled. Therefore, the null solution of (3.10.1) is exponentially stable if and only if the null solution of Equation (3.10.9) is exponentially stable in the almost sure sense.

3.11

Notes and Comments

Important progress was made on the stability of nonlinear stochastic diﬀerential equations in inﬁnite dimensional spaces over the last two decades. The main results in Subsection 3.1.1 are taken from Ichikawa [6]. Absolute stability in the class Lk of ﬁnite dimensional diﬀerential equations in Subsection 3.1.2 has been extensively studied since it appears in some feedback control systems (Hahn [1]). The systems (3.1.22), (3.1.23) are often called stochastic diﬀerential equations of Lue’s type. In ﬁnite dimensional cases, Morozan [1] showed that the system (3.1.23) is absolutely stable in the class Lk deﬁned in Subsection 3.1.2 and that the null solution is exponentially stable in mean square for each g ∈ Lk if and only if the linear equation (3.1.22) is stable. The material in Subsection 3.1.2 is mainly due to Ichikawa [4]. Following the classic work of Pardoux [1] who established the fundamental results on existence and uniqueness of solutions of stochastic nonlinear partial diﬀerential equations of monotone type in which a coercive condition plays an important role, Chow [2] pointed out that under some circumstances the coercive condition actually takes the role of a stability criterion. The main results in Section 3.2 which are due to Caraballo and Liu [1] improved and generalized those in Chow [2] to non-autonomous cases. Stochastic stability of partial diﬀerential equations using ﬁnite dimensional approximations can be found in Yavin [1], [2]. In the history of the study of asymptotic stability of stochastic systems, the Lyapunov function method is probably the most inﬂuential tool, for instance, see Khas’minskii [1]. The basic technique involved is to construct, ﬁrstly, a proper Lyapunov function and then deal with the stability of the nonlinear case by means of the ﬁrst order approximation procedure. In inﬁnite dimensional cases, the ﬁrst investigation in this respect was due to Khas’minskii and Mandrekar [1]. The general non-autonomous version presented in Theorem 3.4.1 is taken from Liu [5]. The relation between ultimate boundedness in the mean square sense and invariant measures of stochastic diﬀerential equations has been pointed out by Miyahara [1], [2] in ﬁnite dimensional cases. But the basic ideas go back at least to Wonham [2] and Zakai [1], [2]. The corresponding investigation in inﬁnite dimensional cases was carried out in Liu and Mandrekar [1] for strong solutions and Ichikawa [5], Liu and Mandrekar [2]

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

for mild solutions. The inﬁnite dimensional generalizations, Theorem 3.5.6, Corollary 3.5.2 and Theorem 3.6.1, of ﬁnite dimensional results are presented in Chow and Khas’minskii [1]. The reader can also ﬁnd one approach in Ethier and Kurtz [1] to deal with the existence of invariant measures by constructing a Lyapunov function and solving the relevant martingale problem. The research about decay rates of ﬁnite dimensional Itˆ o’s stochastic diﬀerential equations goes back at least to Mao [1] in which polynomial decay was investigated. The material in Section 3.8.2 is closely related to Liu [1]. One of the most interesting topics in stability theory is the so-called stabilization by white noise sources of deterministic (stochastic) systems. In this respect there is a systematic statement in ﬁnite dimensions in Khas’minskii [1]. There exists some related work, for instance, in Arnold, Crauel and Wihstutz [1], Mao [2] and Scheutzow [1] among others. The presentation in Section 3.9 is mainly taken from Caraballo, Liu and Mao [1]. Some material about Lyapunov exponents such as Theorem 3.10.1 in Section 3.10 is taken from Kwieci´ nska [2]. But the theorem 3.10.3 is obtained in Caraballo and Langa [1] in spite of the fact that in ﬁnite dimensional cases, Arnold [4] has proved that the deterministic system (3.10.1) can be stabilized by a suitable Stratonovich linear noise if and only if trA < 0.

Chapter 4 Stability of Stochastic Functional Diﬀerential Equations

In this chapter, we shall investigate stability properties of stochastic functional diﬀerential equations in inﬁnite dimensions. We begin with an argument of reducing the stability problem of retarded functional linear deterministic equations to a class of C0 -semigroups of bounded linear operators so as to ﬁnd exact regions of stability. The characteristic conditions of mean square exponential stability established in Chapter 2 for linear equations are extended to a class of stochastic linear functional equations with time lag. A kind of coercive condition is formulated to secure desired decay behavior of strong solutions for nonlinear stochastic functional diﬀerential equations. The methods of Lyapunov and Razumikhin functionals are emphasized and contrasted to describe stability properties of mild solutions for semilinear stochastic evolution equations with memory.

4.1

Linear Deterministic Equations

Recall that S denotes a separable Banach space with norm ·S over the real ﬁeld R1 and suppose r ≥ 0 is a given constant. Let Cr = C([−r, 0]; S) denote the Banach space of all continuous S-valued functions on [−r, 0] with the usual supremum norm · Cr , which is deﬁned by φCr = max−r≤θ≤0 φ(θ)S for any φ ∈ Cr . For arbitrary real numbers a ≤ b, t ∈ [a, b] and any continuous function u(·) : [a − r, b] → S, ut denotes the element of Cr given by ut (θ) = u(t + θ) for θ ∈ [−r, 0]. In this section, we wish to consider the following abstract integral equation on S t u(t) = T (t)u(0) + 0 T (t − s)F us ds, t ≥ 0, (4.1.1) u0 = φ ∈ Cr , where F : Cr → S is a bounded linear operator with norm F and {T (t)}, t ≥ 0, is a strongly continuous semigroup of bounded linear operators over S with its inﬁnitesimal generator AT . It can be proved by the standard Picard iteration procedure that for arbitrarily given φ ∈ Cr , there exists a unique

175

176

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

continuous function u(·)(φ) : [−r, ∞) → S, which solves the equation (4.1.1). Clearly, all the stability deﬁnitions in Section 1.4 can be applied to Equation (4.1.1) in an obvious manner. For instance, we say that the null solution of (4.1.1) is stable if for each ε > 0, there exists δ > 0 such that the solution u(t)(φ) satisﬁes that for any φCr < δ, u(t)(φ)S < ε for any t ≥ 0.

4.1.1

Stable Semigroups (Finite Delays)

In order to investigate the stability of Equation (4.1.1), we deﬁne a family of operators U (t) : Cr → Cr given by U (t)φ = ut (φ),

t ≥ 0,

where ut (φ) = u(t + ·)(φ), t ≥ 0, denotes the solution of (4.1.1). As an immediate consequence, all the stability deﬁnitions of Equation (4.1.1) can be restated by means of U (t), t ≥ 0, in a straightforward way. For instance, we say that the null solution of (4.1.1) is asymptotically stable if it is stable and there exists δ > 0 such that the relation φCr < δ implies lim U (t)φCr = 0.

t→∞

By a standard argument, we may deduce immediately from the solution of (4.1.1) that: Proposition 4.1.1 U (t), t ≥ 0, is a strongly continuous semigroup of bounded linear operators on Cr satisfying the condition that for any φ ∈ Cr , t ≥ 0, U (t)φCr ≤ M φCr · e(µ+M F )t , −µr U (t)φCr ≤ M e−µr φCr · e(µ+M F e )t ,

if if

µ ≥ 0, µ < 0,

(4.1.2)

where T (t) ≤ M · eµt , M ≥ 1, µ ∈ R1 , for all t ≥ 0. Proof The linearity of U (t) is immediate and its strong continuity follows from the fact that solutions of (4.1.1) are continuous. The semigroup property follows from that for arbitrary t, t˜ ≥ 0, φ ∈ Cr , u(t + t˜)(φ)

t

T (t + t˜ − s)F us (φ)ds +

= T (t + t˜)φ(0) + 0

t+t˜

T (t + t˜ − s)F us (φ)ds t

t t˜ ˜ = T (t) T (t)φ(0) + T (t − s)F us (φ)ds + T (t˜ − s)F ut+s (φ)ds

0 t˜

T (t˜ − s)F ut+s (φ)ds.

= T (t˜)u(t)(φ) + 0

0

Stability of Stochastic Functional Diﬀerential Equations

177

By the uniqueness of solutions of Equation (4.1.1), this implies that ut+t˜(φ) = ut˜(ut (φ)). On the other hand, from (4.1.1) we have that for arbitrary t ≥ −r, t u(t)(φ)S ≤ M eµt φ(0)S + M F eµ(t−s) us (φ)Cr ds. 0

If µ ≥ 0, then for t ≥ 0, ut (φ)Cr ≤ M e φCr + M F

t

eµ(t−s) us (φ)Cr ds,

µt

0

and if µ < 0, then for t ≥ 0, ut (φ)Cr ≤ M e−µr eµt φCr + M F e−µr

t

eµ(t−s) us (φ)Cr ds. 0

By virtue of the well-known Gronwall lemma, we have the desired (4.1.2). Deﬁne the following operator AU : D(AU ) ⊂ Cr → Cr by D(AU ) = φ ∈ Cr : φ ∈ Cr , φ(0) ∈ D(AT ), φ− (0) = AT φ(0) + F φ , AU φ(θ) = φ (θ), −r ≤ θ ≤ 0,

φ ∈ D(AU ).

It can be shown (cf. see Travis and Webb [1]) that AU is the inﬁnitesimal generator of the C0 -semigroup {U (t)}t≥0 . In addition, Propositon 4.1.1 actually provides a suﬃcient condition for the null solution of (4.1.1) to be stable or asymptotically stable in terms of the growth rate of the semigroup {T (t)}t≥0 and the norm F . Corollary 4.1.1 (4.1.1). Assume

Suppose u(t)(φ), t ≥ 0, is the solution of the equation µ + M F e−µr ≤ 0;

(4.1.3)

then the null solution is stable. Moreover, assume µ + M F e−µr < 0;

(4.1.4)

then the null solution is globally asymptotically stable in the sense that it is stable and for any φ ∈ Cr , the relation limt→∞ ut (φ)Cr = 0 holds. However, we would also like to point out that the stability condition (4.1.3) or (4.1.4) is not easy to apply to many practical situations. For instance, in a variety of situations we ﬁnd that the stability of (4.1.1) still remains true for some time retarded parameters r > 0, which do not satisfy (4.1.3) or (4.1.4). In the sequel, we shall carry out a detailed investigation of this problem so as

178

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

to establish some more eﬀective stability criteria. In particular, for the sake of simplicity we shall always assume in the remainder of this subsection that {T (t)}t≥0 is a strongly continuous semigroup of bounded linear operators over S satisfying T (t) ≤ eµt for all t ≥ 0, where µ ∈ R1 is some real number. Remark It is obvious that Proposition 4.1.1 remains true even for certain nonlinear operators, for instance, if the term F is supposed to be a class of nonlinear operators F (·) : Cr → S with global Lipschitz constants, denoted still by F . That is, the term F satisﬁes ˜ S ≤ F · φ − φ ˜ C F (φ) − F (φ) r

for any φ, φ˜ ∈ Cr .

Proposition 4.1.2 Let F denote the operator norm of linear operator F . If µ ≥ −F and Re λ > F + µ where Re λ denotes the real part of the complex number λ, then (AU − λI)−1 exists and has domain all of Cr . Given ψ ∈ Cr , we must solve

Proof

(AU − λI)φ = ψ. That is, φ − λφ = ψ,

φ (0) = λφ(0) + ψ(0) = AT φ(0) + F φ.

(4.1.5)

This means that θ φ(θ) = eλθ φ(0) + 0 eλ(θ−s) ψ(s)ds, φ(0) = (AT − λI)−1 (ψ(0) − F φ).

θ ∈ [−r, 0],

(4.1.6)

The mapping · x → (AT − λI)−1 ψ(0) − F eλ· x + eλ(·−s) ψ(s)ds 0

is a strict contraction from S to S since by the well-known Hille-Yosida theorem (AT − λI)−1 F (eλ· x) ≤ F xS S Re λ − µ for all x ∈ S. Then (4.1.6) and hence (4.1.5) has a unique solution. But this means that (AU − λI) is onto and injective and the proof is complete. Proposition 4.1.3 If µ ≥ −F and Re λ > F +µ, then (AU −λI)−1 ≤ 1/(Re λ − F − µ).

Stability of Stochastic Functional Diﬀerential Equations

179

Proof Let φ = (AU − λI)−1 ψ for ψ ∈ Cr . Suppose ε > 0 and θ ∈ [−r, 0] has the property that φ(θ)S ≥ φCr − ε. Using (4.1.6) and Hille-Yosida theorem, we have θ φ(θ)S ≤ eλθ (AT − λI)−1 [ψ(0) − F φ] + eλ(θ−s) ψ(s)ds S

S

0

eRe λθ 1 − eRe λθ ≤ (ψCr + F φCr ) + ψCr Re λ − µ Re λ 1 − ( Reµ λ )(1 − eRe λθ ) F eRe λθ = φCr + ψCr . Re λ − µ Re λ − µ But this implies 1 − ( Reµ λ )(1 − eRe λθ ) Re λ − µ − F eRe λθ φCr ≤ ε + ψCr . Re λ − µ Re λ − µ Since

1 − ( Reµ λ )(1 − eRe λθ ) 1 ≤ , Re λ − µ − eRe λθ F Re λ − µ − F

the assertion follows. As an immediate by-product of Propositions 4.1.2 and 4.1.3, we may deduce the following results which will help us to formulate more reﬁned stability conditions than those in Corollary 4.1.1. If −F = µ, then for arbitrary φ ∈ Cr , t ≥ 0,

Corollary 4.1.2

U (t)φCr ≤ φCr . Proof

(4.1.7)

By virtue of Propositions 4.1.2 and 4.1.3 and the fact that (I − λAU )−1 = (AU − (1/λ)I)−1 (−1/λ),

it is easy to deduce (choosing µ ≥ −F if necessary) that (I − λAU )−1 exists with domain Cr and for all real λ with 0 < λ < 1/(F + µ), (I − λAU )−1 ≤

1 . 1 − λ(F + µ)

Then it is easy to get the desired result by using the following result of Crandall and Liggett [1], lim (I − (t/n)AU )−n φ = U (t)φ for all φ ∈ Cr .

n→∞

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Corollary 4.1.3 positive integer n,

If −F > µ, then for arbitrary φ ∈ Cr , t ≥ 0, and each

U (t)φCr n−1 ≤ (−F /µ)n + F k (1 − (−F /µ)n−k )eµ(t−(k+1)r) tk /k! φCr . k=0

(4.1.8) Furthermore, there exists a unique φ0 ∈ Cr such that U (t)φ0 = φ0 for arbitrary t > r and limt→∞ U (t)φ = φ0 for all φ ∈ Cr . Proof Since −F > µ, it is easy to show that (4.1.7) remains true by virtue of Corollary 4.1.2 (µ can always be chosen larger than any given µ). Then, for t ≥ 0 t u(t)(φ)S ≤ eµt φ(0)S + F 0 eµ(t−s) us (φ) Cr ds µt ≤ (−F /µ) + (1 − (−F /µ))e φCr .

(4.1.9)

Then, for arbitrary t ≥ r, U (t)φCr ≤ (−F /µ) + (1 − (−F /µ))eµ(t−r) φCr .

(4.1.10)

However, since eµ(t−r) ≥ 1 for 0 ≤ t ≤ r, (4.1.10) holds for all t ≥ 0. In a similar manner, we can substitute the inequality (4.1.10) into (4.1.9) and integrate to obtain U (t)φCr ≤ (−F /µ)2 + (1 − (−F /µ)2 )eµ(t−r) +F (1 − (−F /µ))eµ(t−2r) t φCr

for all t ≥ 0. An induction argument yields (4.1.8). By virtue of (4.1.10), U (t), t > r, is a commutative family of strict contractions on Cr and therefore has a unique ﬁxed point. That is, for arbitrary t > r, there exists φt ∈ Cr such that U (t)φt = φt . Thus, U (s)φt = U (s)U (t)φt = U (t)U (s)φt for any s > r which implies φt = U (s)φt , i.e., φs = φt . The last statement now follows from (4.1.8). As a direct consequence of Corollaries 4.1.2 and 4.1.3, we can immediately deduce that the null solution of (4.1.1) is stable if −F = µ and asymptotically stable if −F > µ.

Stability of Stochastic Functional Diﬀerential Equations

181

Example 4.1.1 Let S = C0 [0, π], the space of all continuous real-valued functions over [0, π] which are zero at the ends 0 and π, and has the usual supremum norm. Let AT : S → S be deﬁned by AT y(x) = ∂ 2 y(x)/∂x2 ,

D(AT ) = {y(·) ∈ S : ∂ 2 y(x)/∂x2 ∈ S}.

Then AT is the inﬁnitesimal generator of a strongly continuous semigroup of bounded linear operators T (t) satisfying T (t) ≤ e−t , t ≥ 0, and for an arbitrarily given real number θ, there exists a unique mild solution of the following equation   ut (x, t) = uxx (x, t) + θ · u(x, t − r), 0 ≤ x ≤ π, t ≥ 0, u(0, t) = u(π, t) = 0, t ≥ 0, (4.1.11)  u(·, t) = φ(t)(·) ∈ Cr , −r ≤ t ≤ 0. Moreover, the arguments above apply and the null solution of (4.1.11) is stable if θ = 1 and asymptotically stable if θ < 1. Under some circumstances, some more reﬁned results of stability than Corollaries 4.1.2 and 4.1.3 can be derived. In fact, it is possible to determine the exact regions of stability for Equation (4.1.1) by a diﬀerent argument. To illustrate this, let us ﬁrst explore some compact properties of solutions of (4.1.1). Lemma 4.1.1 Let {T (t)}t≥0 be a strongly continuous semigroup of bounded linear operators on S. Assume also that T (t) : S → S is compact for each t ≥ 0. Let B be a bounded subset of S and {fγ ; γ ∈ Γ} a family of continuous functions from some ﬁnite interval [c, d] ⊂ (0, ∞) to B. Then d W = { c T (s)fγ (s)ds; γ ∈ Γ} is a precompact subset of S. Proof Let L = {T (t)x; t ∈ [c, d], x ∈ B}. We will show that L is totally bounded. For any ε > 0, by virtue of the uniform continuity of the mapping T (·) : [c, d] → L(S, S), there exist c = t1 < t2 < · · · < tn = d such that T (ti ) − T (t) ≤

ε 2C

for

t ∈ [ti−1 , ti ],

i = 2, · · · , n,

(4.1.12)

where C > 0 is some bound of B. Since for each ti , T (ti )B is totally bounded, there exist {xi1 , xi2 , · · · , xik(i) } ⊂ B such that if x ∈ B, then T (ti )xij − T (ti )xS ≤ ε/2

for some

xij .

(4.1.13)

The total boundedness of L now follows from (4.1.12) and (4.1.13). Since L is precompact, so is the closed convex hull of L. The lemma follows since W is contained in the closed convex hull of (d − c)L.

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Proposition 4.1.4 Suppose that T (t), t ≥ 0, deﬁned in (4.1.1) is compact for each t > 0. Then the mapping (t, φ) → ut = ut (φ) deﬁned by the solution of (4.1.1) is compact in φ for each ﬁxed t > r. Proof Let {φγ : γ ∈ Γ} be a bounded subset of Cr and t > r. For each γ ∈ Γ, deﬁne fγ ∈ Cr by fγ = ut (φγ ). Then, for θ ∈ [−r, 0] we have t + θ > 0, and so fγ (θ) = ut (φγ )(θ) = u(φγ )(t + θ) = T (t + θ)φγ (0) t+θ + T (t + θ − s)F us (φγ )ds. 0

We ﬁrst show that this family is equicontinuous. By virtue of (4.1.2), one may show that {F us (φγ ); s ∈ [0, t], γ ∈ Γ} is bounded by a constant, say M > 0. Let γ ∈ Γ, 0 < c < t − r, −r ≤ θ˜ < θ ≤ 0, and observe that ˜ S fγ (θ) −fγ (θ) ˜ γ (0)S ≤ T (t + θ)φγ (0) − T (t + θ)φ t+θ˜ t+θ + T (t + θ − s)F us (φγ )ds − T (t + θ − s)F us (φγ )ds S 0 0 t+θ˜ T (t + θ − s) − T (t + θ˜ − s) F us (φγ )ds + S ˜ t+θ−c ˜ t+θ−c T (t + θ − s) − T (t + θ˜ − s) F us (φγ )ds + S

0

µt ˜ ˜ ≤ T (t + θ) − T (t + θ)φ + 2cM eµt γ (0)S + |θ − θ|M e +M t sup T (t + θ − s) − T (t + θ˜ − s). ˜ s∈[0, t+θ−c]

One can now use the uniform continuity of T (s), s ∈ [c, t], in L(S, S) to demonstrate the claimed equicontinuity. Next, we show that for each ﬁxed θ ∈ [−r, 0], {fγ (θ) : γ ∈ Γ} is precompact in S. Indeed, {T (t + θ)φγ (0) : γ ∈ Γ} is precompact since t + θ > 0 and φγ (0)S is bounded (independent of γ). We will show that t+θ L= T (t + θ − s)F us (φγ )ds : γ ∈ Γ 0

is totally bounded. Observe that for 0 < c < t + θ, we have t+θ T (t + θ − s)F us (φγ )ds ≤ cM eµt t+θ−c

S

for all γ ∈ Γ. By Lemma 4.1.1, if 0 < c < t + θ, then t+θ−c Lc = T (t + θ − s)F us (φγ )ds : γ ∈ Γ 0

(4.1.14)

Stability of Stochastic Functional Diﬀerential Equations

183

is precompact in S. This fact, together with (4.1.14), yields the precompactness of L. Now applying the well-known Arzela-Ascoli theorem to {fγ ; γ ∈ Γ} concludes the proof. For each scalar λ, deﬁne a linear operator ∆(λ) : D(AT ) → S by ∆(λ)x = AT x − λx + F (eλ· x),

x ∈ D(AT ),

where eλ· x ∈ Cr is deﬁned by (eλ· x)(θ) = eλθ x,

θ ∈ [−r, 0].

Note that we still use Cr here to denote its complexiﬁcation. In particular, we will say that λ satisﬁes the “characteristic equation” of (4.1.1) provided ∆(λ)x = 0 for some x = 0. Suppose D is an arbitrary linear operator and let σ(D), P σ(D) denote the spectrum and point spectrum sets of D, respectively. Recall that U (t), t ≥ 0, is the strongly continuous semigroup of bounded linear operators on Cr deﬁned by the solutions of (4.1.1). Lemma 4.1.2 For t > r, σ(U (t)) is a countable set and is compact with only one possible accumulation point, 0, and if µ = 0 ∈ σ(U (t)), then µ ∈ P σ(U (t)). Proof The lemma follows immediately from Proposition 2.4 and Theorem 6.26 in Kato [1], p. 185. Lemma 4.1.3 For t > r, P σ(U (t)) = etP σ(AU ) plus possibly {0}. More speciﬁcally, if µ = µ(t) ∈ P σ(U (t)) for some t > r and µ = 0, then there exists λ ∈ P σ(AU ) such that eλt = µ. Furthermore, if {λn } consists of all distinct points in P σ(AU ) such that eλn t = µ, then for arbitrary k, the kernel Ker(U (t) − µI)k is the closed linear extension of the linearly independent sets Ker(AU − λn I)k , where n ranges over eλn t = µ. Proof

See Lemma 22.1 and the exercise after it in Hale [1].

Lemma 4.1.4 Let S(t), t ≥ 0, be an arbitrary strongly continuous semigroup of bounded linear operators on S and suppose that for some s > 0 the spectral radius ρ of S(s) is not zero and τ = (1/s) ln ρ. Then for all γ > 0 there exists a constant M (γ) ≥ 1 such that S(t)xS ≤ M (γ)e(τ +γ)t xS Proof

for all t ≥ 0, x ∈ S.

See Lemma 22.2 in Hale [1], p. 112.

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Lemma 4.1.5 There exists a real number ν such that Re λ ≤ ν for all λ ∈ σ(AU ) and if γ is any real number there exist only a ﬁnite number of λ ∈ P σ(AU ) such that γ ≤ Re λ. Proof The existence of the constant ν follows immediately from Propositions 4.1.2 and 4.1.3. Indeed, one can choose ν = max{0, F + µ}. Assume that {λk } is an inﬁnite sequence of distinct points in P σ(AU ) such that Re λk > γ for all k, where γ is a given real number. By Lemma 4.1.3, eλk t ∈ P σ(U (t)) for a ﬁxed t > r. If {eλk t } is inﬁnite, then P σ(U (t)) has an accumulation point diﬀerent from zero, a fact that contradicts Lemma 4.1.2. If {eλk t } is ﬁnite, then eλnk t = µ = constant for some inﬁnite subsequence {λnk }. Then Ker(U (t) − µI) is inﬁnite dimensional, since it contains the linearly independent sets Ker(AU − λnk I) by Lemma 4.1.3. But this contradicts Theorem 5.7.3 in Hille and Phillips [1], which claims that the set Ker(U (t) − µI) is ﬁnite dimensional. Thus the assumption is false and the proof is complete.

Now we are in a position to state the desired stability results. Proposition 4.1.5 Suppose β is some real number such that if λ satisﬁes the characteristic equation of (4.1.1), Re λ ≤ β. Then for each γ > 0, there exists a constant M (γ) ≥ 1 such that for all t ≥ 0, U (t)φCr ≤ M (γ)e(β+γ)t φCr .

(4.1.15)

Proof Suppose ν = 0 ∈ σ(U (t)) where t > r is some ﬁxed number. By Lemma 4.1.2, ν ∈ P σ(U (t)). Also by virtue of Lemma 4.1.3, ν = eλt where λ ∈ P σ(AU ). Then there exists φ = 0 ∈ D(AU ),

φ − λφ = 0.

(4.1.16)

But this is equivalent to φ(β) = eλθ φ(0),

φ(0) = 0,

φ− (0) = AT φ(0) + F φ.

(4.1.17)

Then ∆(λ)φ(0) = 0, and Re λ ≤ β by assumption. Thus the spectral radius of U (t) is less than or equal to etβ and (4.1.15) follows immediately by applying Lemma 4.1.4. Corollary 4.1.4 Let β be the smallest real number such that if λ satisﬁes the characteristic equation of (4.1.1), then Re λ ≤ β. If β < 0, then for all

Stability of Stochastic Functional Diﬀerential Equations

185

φ ∈ Cr , U (t)φCr → 0, as t → ∞. If β = 0, then there exists φ = 0 ∈ Cr such that U (t)φCr = φCr for all t ≥ 0. If β > 0, then there exists φ ∈ Cr such that U (t)φCr → ∞ as t → ∞. Proof The existence of β is a consequence of Lemma 4.1.5. The claim for β < 0 is immediate from (4.1.15). If β = 0, let x = 0 ∈ D(AT ) such that ∆(λ)x = 0 where Re λ = 0 (such a λ exists by Lemma 4.1.5). As in (4.1.16) and (4.1.17), φ(t) = eλt φ(0), φ(0) = x solves (AU − λI)φ = 0, φ = 0. Thus U (t)φ = eλt φ and U (t)φCr = |eit·Im λ |φCr = φCr . If β > 0, let x = 0 ∈ D(AT ) such that ∆(λ)x = 0 and Re λ > 0. Again φ(t) = eλt φ(0), φ(0) = x solves (AU − λI)φ = 0, φ = 0. Thus U (t)φ = eλt φ and U (t)φCr = eRe λt φCr → ∞ as t → ∞. Finally, let us apply Corollary 4.1.4 to an example whose exact stability region can be speciﬁed. Example 4.1.2 We wish to determine the exact region of stability of the linear partial diﬀerential equation  ∂u(x,t) 2 u(x,t)  ∂t = ∂ ∂x − au(x, t) − bu(x, t − r), 0 ≤ x ≤ π, t ≥ 0, 2 (4.1.18) u(0, t) = u(π, t) = 0, t ≥ 0,  u(x, t) = φ(t)(x), 0 ≤ x ≤ π, −r ≤ t ≤ 0, as a function of a, b and r, where the solutions are in the mild sense of (4.1.1) for S = L2 (0, π), and AT : S → S is deﬁned by d2 y (x), dx2 dy d2 y (x) ∈ S, D(AT ) = y(·) ∈ S : y(x), (x) are absolutely continuous, dx dx2

AT y(x) =

and y(0) = y(π) = 0 .

Then, it is well known that AT is closable on S and the closure of AT generates an analytic compact semigroup {T (t)}t≥0 on S with T (t) ≤ e−t , t ≥ 0. Let F : Cr → S be given by F φ = −aφ(0) − bφ(−r). The characteristic values of (4.1.18) are speciﬁed by the equation ∆(λ)f = [AT − (λ + a + be−λr )I]f = 0 for f ∈ D(AT )\{0}. Since the eigenvalues of AT are −n2 , n = 1, 2, · · ·, we have from Corollary 4.1.4 that the null solution of (4.1.18) is asymptotically stable if and only if all the roots of the equations λ + a + be−λr = −n2 ,

n = 1, 2, · · · ,

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

have negative real parts. The exact region of stability of (4.1.18) is obtained as an immediate consequence of the following useful result which is due to Hayes [1] and whose proof is referred to, for instance, Hale and Lunel [1], p. 416. Proposition 4.1.6 All the roots of the equation (z + µ)ez + ν = 0, where µ and ν are real numbers, have negative real parts if and only if µ > −1,

µ + ν > 0,

ν < ρ sin ρ − µ cos ρ,

where ρ = π/2 if µ = 0, or ρ is the root of ρ = −µ tan ρ in (0, π) if µ = 0.

4.1.2

Stable Semigroups (Inﬁnite Delays)

In all attempts to obtain an extension of stability results in the last section to functional equations with inﬁnite delays, one meets some diﬃculties immediately. For instance, one will ﬁnd the usual phase space Cr , (r = ∞ at the moment) does not work appropriately for the stability analysis. Some essential results for stability such as Proposition 4.1.4 do not remain true any more due to the invalidation of the compactness of the solution semigroups. Therefore, all the investigation becomes more delicate. Consider the following linear functional diﬀerential equation with inﬁnite retarder on S du(t) = AT u(t) + F ut , dt

t ≥ 0,

or its integrated form t u(t) = T (t)φ(0) + 0 T (t − s)F us ds, u0 (θ) = φ(θ), θ ≤ 0,

t ≥ 0,

(4.1.19)

where φ is an element in some phase space Cg to be speciﬁed below, {T (t)}t≥0 is a compact analytic semigroup on the Banach space S with the generator AT . F : Cg → S is a bounded linear operator and for each u(·) : (−∞, ∞) → S and t ≥ 0, ut is a mapping from (−∞, 0] to S deﬁned by ut (θ) = u(t + θ) for θ ≤ 0. For our stability purposes, we assume g(s) = e−γs ,

s ≤ 0,

for some constant γ > 0. Deﬁne the following Banach space Cg = φ : (−∞, 0] → S; φ is continuous and lim eγs φ(s)S = 0 , s→−∞

equipped with the norm φCg =

sup

−∞≤s≤0

eγs φ(s)S .

Stability of Stochastic Functional Diﬀerential Equations

187

It may be proved by the standard “method of steps” that for any φ ∈ Cg , there exists one and only one u(·) = u(·)(φ) : (−∞, ∞) → S such that u0 (·) = φ(·) and (4.1.19) is satisﬁed for all t ≥ 0. Moreover, let U (t) : Cg → Cg be given by U (t)φ = ut (φ), t ≥ 0. Then it can be proved that {U (t)}t≥0 is a C0 -semigroup on Cg with some inﬁnitesimal generator AU . In contrast to the ﬁnite delay case, in general, U (t) : Cg → Cg is no longer compact. This fact simply implies that the spectrum of U (t) : Cg → Cg is much more complicated than its counterpart in the ﬁnite delay case. In order to handle its stability, we need the notions of the essential spectrum and its radius. Let G : D(G) ⊂ S be a closed operator with a closed domain. We denote by ess(G) the set of essential spectrum of G (cf. Browder [1]), and the radius of ess(G) is denoted by re (G). It was proved in Nussbaum [1] that re (G) = inf k ∈ R1 ; α(G(N )) ≤ kα(N ) for every bounded subset N of S , where α(·) is the Kuratowski measure of noncompactness. Moreover, it is known that if λ0 belongs to the spectrum of G but not to the essential spectrum, then λ0 is in the point spectrum of G. Theorem 4.1.1 that

Assume that there exist positive constants M and µ such T (t) ≤ M · e−µt ,

t ≥ 0.

Then the radius of the essential spectrum of the solution semigroup {U (t)}t≥0 of (4.1.19) satisﬁes re (U (t)) ≤ M · e− min{µ,γ}t ,

t ≥ 0.

(4.1.20)

Proof Deﬁne ﬁrst of all an operator S(t) : Cg → Cg as follows: for each φ ∈ Cg , S(t)φ is the solution of the initial value problem u0 = φ and u(t) = T (t)φ(0) for t ≥ 0. Therefore, for each t ≥ 0 and θ ∈ (−∞, −t] we have [U (t) − S(t)](φ)(θ) = 0. By using the Arzela-Ascoli theorem and carrying out a similar argument to Proposition 4.1.4, one can show that the mapping τ τ ∈ [0, t] → 0 T (τ − s)F (U (s)φ)ds ∈ S is compact. Therefore, we have the conclusion that U (t)−S(t) : Cg → Cg is compact for t > 0, and from (4.1.20), we get re (U (t)) = re (S(t)) ≤ S(t). On the other hand, for any φ ∈ Cg we have

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

S(t)φCg = sup eγs (S(t)φ)(s)S s≤0

= max ≤ max

sup eγs T (t + s)φ(0)S , sup eγs φ(t + s)S

−t≤s≤0

s≤−t γs −µ(t+s)

sup M · e e

−t≤s≤0

− min{µ,γ}t

≤M ·e

φ(0)S , e−γt sup eγ(t+s) φ(t + s)S

s≤−t

φCg .

This proof is complete. From the results above, it is obvious that if the semigroup {T (t)}t≥0 is stable, the point spectrum of {U (t)}t≥0 will determine its asymptotic behavior as t → ∞. Before proceeding to derive our main stability results, let us ﬁrst study some properties of the operator AU . Theorem 4.1.2 Let AU denote the inﬁnitesimal generator of the semigroup {U (t)}t≥0 on Cg . (i). If AU φ = λφ with φ = 0, then Re λ ≥ −γ and U (t)φ = eλt φ. Moreover, φ(θ) = eλθ φ(0) with φ(0) ∈ D(AT ) and φ(0) satisﬁes the characteristic equation (λI − AT )x − F (eλ· x) = 0; (4.1.21) (ii). If Re λ ≥ −γ and (4.1.21) has a nontrivial solution, then λ ∈ P σ(AU ); (iii). If µ ∈ P σ(U (t)) and µ = 0, then there exists a ﬁnite number λ ∈ P σ(AU ) such that eλt = µ. Proof

(i). Let z(t) = U (t)φ. As φ ∈ D(AU ) and AU φ = λφ, we know that dz(t) = AU z(t) = U (t)AU φ = U (t)λφ = λz(t). dt

Therefore, z(t) = eλt φ for t ≥ 0. On the other hand, we know that z(t)(θ) = (U (t)φ)(θ) = (U (t + θ)φ)(0) for θ < 0. This implies that eλt φ(θ) = eλ(t+θ) φ(0) for θ ≤ 0 from which it follows that φ(θ) = eλθ φ(0) for θ ≤ 0. Since F (U (t)φ) = F (eλt eλ· φ(0)) = eλt F (eλ· φ(0)), is, as a function of t ≥ 0, locally H¨ older continuous, we know by the standing theory of partial diﬀerential equations that (U (t)φ)(0) is indeed a solution of

Stability of Stochastic Functional Diﬀerential Equations

189

du(t)/dt = AT u(t) + F (ut ). This implies that φ(0) ∈ D(AT ) and λeλt φ(0) = AT eλt φ(0) + eλt F (eλ· φ(0)), completing the proof of (i). The veriﬁcation of (ii) is straightforward, where λ ∈ P σ(AU ) with the corresponding eigenvector eλ· φ(0). (iii). The existence of λ is an immediate consequence of Lemma 4.1.3. We need only to show the ﬁniteness of such λ ∈ P σ(AT ). Clearly, all solutions to eλt = µ have the form λn =

ln µ i2πn + , t t

n = 0, ±1, ±2, · · · .

As AT generates an analytic semigroup, there exists an integer n0 such that for |n| ≥ n0 , λn belongs to the resolvent set of AT . This means that the characteristic equation (4.1.21) with λ = λn , |n| ≥ n0 , is equivalent to x = (λI − AT )−1 F (eλ· x).

(4.1.22)

If λn ∈ P σ(AU ), then (4.1.22) has a solution xn such that xn S = sup eλn θ φ(0)S = 1. θ≤0

However, xn S = (λn I − AT )−1 F (eλn · xn )S M M F eλn · xn S = F → 0, ≤ |λn | |λn | as n → ∞, a contradiction. This completes the proof. The following theorem and its corollary will give useful information to ensure stability behavior on the semigroup {U (t)}t≥0 . Theorem 4.1.3

For any ε > 0, the set L := λ ∈ C; Re λ > − min(µ, γ) + ε

contains only a ﬁnite number of points of P σ(AU ). Proof The set L is a subset of the resolvent set of AT and thus if λ ∈ L, then the characteristic equation (4.1.21) is equivalent to (4.1.22). Therefore, λ0 ∈ P σ(AU ) ∩ L if and only if 1 ∈ P σ(F (λ)), where F (λ)x := (λI − A)−1 F (eλ· x) for each x ∈ S. As F (λ0 ) : S → S is compact, 1 is an isolated point of the spectrum of F (λ0 ). Clearly, the function λ → F (λ) is analytic in L. So either L ⊂ P σ(AU ) or P σ(AU ) is isolated in L. But the ﬁrst claim is impossible by

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

virtue of Theorem 4.1.2. So P σ(AU ) is isolated in L. A similar argument to that of (iii) in Theorem 4.1.2 also shows that P σ(AU ) ∩ L is ﬁnite. An immediate consequence of the above theorem is the following asymptotic stability criterion: Corollary 4.1.5 Suppose the conditions in Theorem 4.1.1 hold. If Re λ < 0 for any solution λ such that the characteristic equation (4.1.21) has a nontrivial solution, then there exist constants C > 0 and µ > 0 such that U (t) ≤ C · e−µt ,

t ≥ 0.

Proof By virtue of Theorem 4.1.3, ∆ := sup Re (P σ(AU )) < 0. Theorem 4.1.2 implies sup |λ| : λ ∈ P σ(U (t)) = e∆t . By Theorem 4.1.1, we have µ1 > 0 and C1 > 0 such that r(U (t)) ≤ C1 · e−µ1 t . This yields the conclusion by using Lemma 4.1.4. As an illustration of the preceding results, let us investigate the following linear partial Lotka-Volterra integrodiﬀerential equation. Consider the equation t ∂ ∂2 k(t − s)u(x, s)ds, u(x, t) = ν 2 u(x, t) + ∂t ∂x −∞ t ≥ 0, x ∈ (0, π), ∂ u(x, t) = 0, x = 0, π, ∂x

Example 4.1.3

(4.1.23)

where ν is a positive constant. As usual, we let S = C([0, π]; R1 ) and deﬁne d2 y d2 y 2 1 AT y(x) = ν dx 2 (x) for y(·) ∈ D(AT ) = {y(·) ∈ C ([0, π]; R ); dx2 (0) = d2 y dx2 (π)

= 0}. Then the closure of AT generates an analytic compact semigroup in S. We also assume that k : [0, ∞) → R1 is continuous and there exists γ > 0 such that ∞

eγt |k(t)|dt < ∞.

0

Deﬁne F : Cg → S by

0

F (φ)(x) =

x ∈ [0, π].

k(−θ)φ(x, θ)dθ, −∞

Then F (φ)S = sup |F (φ)(x)| ≤ φCg x∈[0,π]

0

∞

eγt |k(t)|dt

Stability of Stochastic Functional Diﬀerential Equations

191

for arbitrary φ ∈ Cg . Consequently, the null solution of (4.1.23) is asymptotically stable if Re λ < 0 for all λ ∈ C such that the equation 0 d2 v λ− k(−u)eλu du v(x) = ν · 2 (x), v(0) = v(π) = 0, dx −∞ has a nontrivial solution v ∈ C 2 ([0, π]; R1 ). As the eigenvalues of AT are given by −νj 2 , j = 0, 1, 2, · · ·, we conclude that the null solution of (4.1.23) is asymptotically stable if Re λ < 0 for all solutions λ of the equations 0 λ− k(−θ)eλθ dθ = −νj 2 , j = 0, 1, · · · . (4.1.24) −∞

Note that the latter is equivalent to requiring that the null solution of each of the following scalar ordinary diﬀerential equations t dy 2 (t) = −νj y(t) + k(t − s)y(s)ds dt −∞ is asymptotically stable for j = 0, 1, · · ·. What remains is to ﬁnd suﬃcient conditions to ensure that Re λ < 0 for each solution of (4.1.24). To this end, let us introduce the following results whose proofs are referred to Lenhart and Travis [1]. Consider the equation ∞ L du bi u(t − s)dmi (s), (t) = AT u(t) + au(t) + dt 0 i=1 where AT generates a C0 -semigroup on some Banach space, bi , 1 ≤ i ≤ L, are real constants, and mi : [0, ∞) → R1 are given measures such that ∞ ∞ |dmi (s)| = 1, dmi (s) ≥ 0, 1 ≤ i ≤ L. 0

0

We assume that the spectrum of AT contains only real numbers {θn } with θ1 < 0 being the largest. The above discussion has demonstrated the importance of studying the roots of the associated characteristic equation λ = a + θn +

L i=1

bi

∞

e−λs dmi (s),

n = 1, 2, · · · .

(4.1.25)

0

Theorem 4.1.4 All the roots of the characteristic equations (4.1.25) have negative real parts for all mi , 1 ≤ i ≤ L, if and only if L i=1

|bi | ≤ −(a + θ1 )

and

L i=1

bi < −(a + θ1 ).

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

4.2

Stability Equivalence and Reduction of Neutral Equations

Bearing the deterministic results of the last section in mind, we are now ready to study stochastic stability of a class of neutral linear stochastic functional diﬀerential equations (4.2.6) below. Before moving forward, we want to mention that it turns somewhat inconvenient to use some of the results derived in Section 4.1 such as Theorem 4.1.3 to ensure stability of equations with inﬁnite retarder when AT as deﬁned there does not generate a compact analytic semigroup. This fact suggests that a diﬀerent scheme should be carried out to deal with more general (stochastic) evolution equations. In this section, we shall ﬁrst establish some equivalent relations among L2 -stability, uniformly asymptotic and exponential stabilities for a class of stochastic differential equations (4.2.3) with inﬁnite delays. As a consequence, we can apply these relations by means of proper transformations to the investigation of neutral stochastic evolution equations (4.2.6) in which we are especially interested.

4.2.1

Stability of Retarded Stochastic Systems

We assume {hj }, 0 < h1 < · · · < hj · · ·, is a sequence of real numbers which satisﬁes lim hn = ∞, n→∞

and lim

n→∞

ln n = 0. hn

(4.2.1)

Let A be a linear operator, generally unbounded, and the inﬁnitesimal generator of a C0 -semigroup T (t), t ≥ 0, of bounded linear operators on the separate Hilbert space H. Suppose {Fj } ∈ L(H) and {Gj } ∈ L(H, L(K, H)) are two families of bounded linear operators, respectively. To introduce a proper phase space for future equations, let us assume that there is a sequence of nonnegative numbers {dj } such that Fj ∨ Gj ≤ dj where a ∨ b = max{a, b} for any a, b ∈ R1 , and ∞

dj e2δhj < ∞

(4.2.2)

j=1

for some positive number δ. Let (Ω, F, P ) be a probability space equipped with a ﬁltration {Ft }t≥0 which satisﬁes the usual conditions. For our stability

Stability of Stochastic Functional Diﬀerential Equations

193

purpose, we intend to introduce the following space X: X := L2 (Ω, H) ⊕ L2 ((−∞, 0) × Ω; H), where L2 ((−∞, 0) × Ω; H) = ψ : (−∞, 0] × Ω → H is F − measurable and 0 ∞ d2j e2δhj Eψ(s)2H ds < ∞ −hj

j=1

with norm ψ2L2 =

∞

d2j e2δhj

0

−hj

j=1

Eψ(s)2H ds.

It is straightforward to check that X is a Hilbert space with the inner product ψ˜1 , ψ˜2 X = Eψ1 (0), ψ2 (0)H +

∞

d2j e2δhj

j=1

0

−hj

Eψ1 (s), ψ2 (s)H ds

where ψ˜1 = (ψ1 (0), ψ1 ) and ψ˜2 = (ψ2 (0), ψ2 ) ∈ X, and the norm ˜ X= ψ

˜ ψ ˜ X, ψ,

where ψ˜ = (ψ(0), ψ) ∈ X. Our major concern is to investigate stability of a wide class of neutral stochastic evolution equations (4.2.6) in Subsection 4.2.4. To this end, let us ﬁrst study the following stochastic diﬀerence diﬀerential equation over H: for any t ≥ 0, ψ˜ ∈ X,  ˜ = T (t)ψ(0) + t T (t − s) ∞ Fj v(s − hj , ψ)ds ˜   v(t, ψ) j=1 0 t ∞ ˜ + 0 T (t − s) j=1 Gj v(s − hj , ψ)dW t ≥ 0, s,   2 2 ˜ ˜ v(0, ψ) = ψ(0) ∈ L (Ω, H), v(t, ψ) = ψ(t) ∈ L ((−∞, 0) × Ω; H), t < 0, (4.2.3) where Wt , t ≥ 0, is some given K-valued Q-Wiener process with ﬁnite trace class covariance operator Q with respect to {Ft }t≥0 . It can be shown that under the condition (4.2.2), the equation (4.2.3) does make sense in X. In particular, by a standard argument we may establish the following existence and uniqueness of (4.2.3) whose proof is referred to Liu [7]. Theorem 4.2.1 Let Fj , Gj , j = 1, 2, · · ·, be two families of bounded linear operators such that (4.2.2) holds. Then the equation (4.2.3) has a

194

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

unique solution satisfying the property that there exist constants C > 0 and µ > 0 such that for any real number T ≥ 0, ψ˜ = (ψ(0), ψ) ∈ X, ˜ 2 ≤ C · eµT ψ ˜ 2. sup Ev(t, ψ) H X t∈[0,T ]

˜ Deﬁnition 4.2.1 For t ≥ 0, ψ˜ ∈ X and the corresponding solution v(t, ψ) 2 t ˜ t ˜ of (4.2.3), deﬁne v (ψ) ∈ L ((−∞, 0) × Ω; H) and v˜ (ψ) ∈ X, t ≥ 0, as ˜ = {v(t + s, ψ) ˜ : s ∈ (−∞, 0)}, v t (ψ)

˜ = (v(t, ψ), ˜ v t (ψ)), ˜ v˜t (ψ)

˜ (t) : X → X, t ≥ 0, as and map U ˜ ˜ (t)ψ˜ = v˜t (ψ), U

ψ˜ ∈ X.

˜ (t) is a linear mapping. The following corollary Clearly, for any t ≥ 0, U ˜ (t)}, t ≥ 0, which may be easily deduced from Theorem 4.2.1 shows that {U is actually a family of continuous linear operators from X into itself. Corollary 4.2.1

For t ≥ 0 and ψ˜ ∈ X, ˜ X ≤ c · eνt ψ ˜ X ˜ (t)ψ U

for some constants c ≥ 1 and ν > 0. In addition to the properties shown in Corollary 4.2.1, we can actually prove ˜ (t), t ≥ 0, is a strongly continuous that the bounded linear operator family U semigroup (see Liu [7] for more details). ˜ (t) : X → X, t ≥ 0, deﬁned in Deﬁnition Theorem 4.2.2 For the family U ˜ 4.2.1, we have that for arbitrary ψ ∈ X, ˜ (t)U ˜ (s)ψ˜ = U ˜ (t + s)ψ˜ for any s, t ∈ [0, ∞); (i) U (ii) there exist constants C ≥ 1 and µ > 0 such that ˜ (t) ≤ C · eµt , U

t ≥ 0;

˜ (t) is strongly continuous in the mean square sense, i.e., for any ψ˜ ∈ X, (iii) U ˜ 2 = 0. ˜ (t)ψ˜ − ψ lim U X

t→0+

To establish stability of solutions for the equation (4.2.3) with initial data ψ˜ = (ψ(0), ψ) ∈ X, we ﬁrst derive the following lemma which is a generalization of the usual uniform boundedness principle for continuous linear operators. Let S be a real separable Banach space. A map ρ(·) : S → R1 is a seminorm if it satisﬁes the following conditions:

Stability of Stochastic Functional Diﬀerential Equations

195

(i) |ρ(x + y)| ≤ |ρ(x)| + |ρ(y)| for any x, y ∈ S; (ii) |αρ(x)| = |ρ(αx)| for any α ≥ 0, x ∈ S. Lemma 4.2.1 Let ρn (·), n ≥ 1, be a family of continuous semi-norms on the Banach space S such that for each x ∈ S, supn ρn (x) < ∞. Then there exists a positive constant C < ∞ such that for all x ∈ S, sup ρn (x) ≤ CxS . n

Proof The proof is a straightforward variant of the corresponding arguments of Lemma 13, p. 53 in Dunford and Schwartz [1]. Deﬁnition 4.2.2 The null solution of (4.2.3) is said to be L2 -stable in mean if for every ψ˜ ∈ X, we have ∞ ˜ 2 dt < ∞. Ev(t, ψ) H 0

A direct application of Lemma 4.2.1 to Deﬁnition 4.2.2 produces the following statement which is equivalent to the above relation. Lemma 4.2.2 If the null solution of (4.2.3) is L2 -stable in mean, then there exists a constant C > 0 such that for each ψ˜ ∈ X, ∞ ˜ 2 dt ≤ Cψ ˜ 2 < ∞. Ev(t, ψ) H X 0

Proof

For each t ≥ 0, deﬁne on the Hilbert space X the seminorm n 1/2 ˜ ˜ 2 dt ρn (ψ) = Ev(t, ψ) H 0

for each n ≥ 1. It is easy to see that for each ﬁxed n, ρn (·) is a continuous seminorm by linear property of the solution (4.2.3). Thus, by virtue of Lemma 4.2.1, ˜ ≤ C 1/2 · ψ ˜ X sup ρn (ψ) n

for some C > 0. Let 0 < r < ∞. For the purpose of our stability analysis, we restrict the initial space of (4.2.3) to a closed Hilbert subspace of X Xr := ψ˜ : ψ˜ ∈ X and ψ(t)H = 0 a.s. on (−∞, −r) .

196

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

˜ t ∈ R+ , Theorem 4.2.3 For arbitrary initial data ψ˜ ∈ Xr , let v(t, ψ), be the solution of the equation (4.2.3). Then the following three notions of stability are equivalent: (i). The null solution of (4.2.3) is L2 -stable in mean; (ii). The null solution of (4.2.3) is uniformly asymptotically stable in mean square in the sense of Deﬁnition 2.3.1; (iii). The null solution of (4.2.3) is exponentially stable in mean square. The proofs of the theorem are divided into several lemmas and propositions below. Proposition 4.2.1 For arbitrary ψ˜ ∈ Xr , assume that the null solution of (4.2.3) is L2 -stable in mean. Then it is also stable in mean square and the following relation ˜ 2 =0 lim Ev(t, ψ) H

t→∞

holds. In order to prove this, we ﬁrst show the following lemma. Lemma 4.2.3 Let δ be an arbitrarily given real number, then any solution of (4.2.3) satisﬁes the equation

˜ = eδ t T (t)ψ(0) − δ v(t, ψ)

t

˜ eδ (t−s) T (t − s)v(s, ψ)ds 0

t

∞

j=1 ∞

eδ (t−s) T (t − s)

+ 0

t

eδ (t−s) T (t − s)

+ 0

˜ Fj v(s − hj , ψ)ds ˜ Gj v(s − hj , ψ)dW s

j=1

for any t ≥ 0 and ψ˜ ∈ Xr . Proof

Indeed, substituting

t

˜ = T (t)ψ(0) + v(t, ψ)

T (t − s) 0

t

T (t − s)

+ 0

∞ j=1

∞

˜ Fj v(s − hj , ψ)ds

j=1

˜ Gj v(s − hj , ψ)dW s

(4.2.4)

Stability of Stochastic Functional Diﬀerential Equations

197

into the right-hand side of (4.2.4) and using Fubini-type theorem for stochastic integrals, i.e., Proposition 1.3.4, we obtain

eδ t T (t)ψ(0) − δ

t

eδ (t−s) T (t − s) T (s)ψ(0) +

0

s

T (s − u) 0

∞

˜ Fj v(u − hj , ψ)du +

t

∞

∞

eδ (t−s) T (t − s)

+ 0

T (s − u) 0

j=1

s

∞

˜ Gj v(u − hj , ψ)dW u ds

j=1

˜ Fj v(s − hj , ψ)ds

j=1 t

eδ (t−s) T (t − s)

+ 0

˜ Gj v(s − hj , ψ)dW s

j=1

δ t

δ t

T (t)ψ(0) + (1 − e )T (t)ψ(0) t s ∞ ˜ −δ eδ (t−s) T (t − u) Fj v(u − hj , ψ)duds

=e

0

−δ

0 t

s

T (t − u)

eδ (t−s) 0

t

0

∞

0

eδ (t−s) T (t − s) 0

t

T (t − u)

∞

0 t

−

0

T (t − u) 0

∞

˜ Gj v(u − hj , ψ)dW u

t

eδ (t−u) T (t − u) 0

∞

j=1 ∞

eδ (t−s) T (t − s) 0 t

eδ (t−s) T (t − s) 0

∞

˜ Gj v(u − hj , ψ)dW u

j=1 t

+

˜ Fj v(u − hj , ψ)du

j=1

−

+

∞ j=1

t

+

˜ Fj v(u − hj , ψ)du

j=1

eδ (t−u) T (t − u)

˜ Gj v(s − hj , ψ)dW s

j=1

= T (t)ψ(0) +

˜ Fj v(s − hj , ψ)ds

j=1 t

+

˜ Gj v(u − hj , ψ)dW u ds

j=1 ∞

eδ (t−s) T (t − s)

+

j=1 ∞

j=1

˜ Fj v(s − hj , ψ)ds ˜ Gj v(s − hj , ψ)dW s

198

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations t ∞ ˜ = T (t)ψ(0) + T (t − u) Fj v(u − hj , ψ)du 0

j=1

t

T (t − u)

+ 0

∞

˜ ˜ Gj v(u − hj , ψ)dW u = v(t, ψ).

j=1

This shows that any solution of (4.2.3) satisﬁes (4.2.4). 2 Proof of Proposition 4.2.1. First assume that the system ∞ (4.2.3) is L 2 ˜ stable in mean, i.e., there exists a constant M1 > 0 such that 0 Ev(t, ψ) dt ≤ H 2 µt ˜ M1 ψX by Lemma 4.2.2. Suppose T (t) ≤ M e , t ≥ 0, for some constants M ≥ 1, µ > 0, and let δ = −2µ in Lemma 4.2.3, then we have from (4.2.4) that

˜ 2 Ev(t, ψ) H t 2 ˜ H ds ≤ 16 M 2 e−4µt Eψ(0)2H + 4µ2 E M e−µ(t−s) v(s, ψ) +E

0 t

M e−µ(t−s)

0

∞

˜ H ds Fj · v(s − hj , ψ)

2

j=1

∞ 2 t ˜ +E e−2µ(t−s) T (t − s) Gj v(s − hj , ψ)dW s 0

˜ 2 + ≤ C1 ψ X

H

j=1 ∞

˜ 2 ds Ev(s, ψ) H

0

0 ∞ ∞ 1 2 2 ∞ 2 2 ˜ + j d Ev(s, ψ) ds + Eψ(s) ds j H H j 2 j=1 0 −hj j=1 which, by using the assumptions of Proposition 4.2.1, immediately implies ˜ 2 ≤ C2 ψ ˜ 2 + M1 ψ ˜ 2 Ev(t, ψ) H X X 0 ∞ 2 ˜ 2 + ˜ 2, d2j e2δhj ψ + Eψ(s) ds ≤ C3 ψ X H X −hj

j=1

where C1 , C2 and C3 are some proper positive constants. Therefore, the null solution of (4.2.3) is stable in mean square. Now let 0 ≤ t0 ≤ t, we have 2 ˜ 2 ≤ 16 E ˜ Ev(t, ψ) e−2µ(t−t0 ) T (t − t0 )v(t0 , ψ) H H t 2 ˜ +E 2µ e−2µ(t−s) T (t − s)v(s, ψ)ds t0

H

Stability of Stochastic Functional Diﬀerential Equations ∞ t 2 ˜ +E e−2µ(t−s) T (t − s) Fj v(s − hj , ψ)ds t0

199

H

j=1

∞ 2 t ˜ +E e−2µ(t−s) T (t − s) Gj v(s − hj , ψ)dW s t0

j=1

˜ 2 + 2µM 2 ≤ 16 C3 M 2 e−µ(t−t0 ) ψ X

H

∞

˜ 2 ds Ev(s, ψ) H

t0

∞ ∞ 1 2 2 ∞ ˜ 2 ds +M 2 e−2µs ds j d Ev(s, ψ) j H 2 j t0 t0 −hj j=1 j=1 ∞ ∞ 1 2 2 ∞ ˜ 2 ds +M 2 trQ j d Ev(s, ψ) j H j 2 j=1 t0 −hj j=1 ∞ ˜ 2 + C4 ˜ 2 ds ≤ C4 e−µ(t−t0 ) ψ Ev(s, ψ) X H +C4

+C4

J

∞

j 2 d2j

j=1 ∞ j=J+1

j 2 d2j

t0 ∞

t0 −hj

˜ 2 ds Ev(s, ψ) H

∞

t0 −hj

˜ 2 ds Ev(s, ψ) H

where C4 is some positive constant. Let ε > 0 be an arbitrarily given constant. By virtue of Theorem 4.2.1 and L2 -stability in mean of (4.2.3), the last term of the right hand side of the above inequality can be made less than ε/3 if J is suﬃciently large. The L2 -stability in mean implies that the second and the third terms can be made less than ε/3 if t0 is suﬃciently large. Finally, the ﬁrst term can be made less than ε/3 if t − t0 is suﬃciently large. Therefore, ˜ 2 < ε. This concludes the proof. if t is large then Ev(t, ψ) H Proposition 4.2.2 Let 0 < r < ∞ and the null solution of (4.2.3) be L2 -stable in mean. Then for arbitrarily given ε > 0, there exists T (ε) > 0 such that for all ψ˜ ∈ Xr , ˜ 2 ≤ εψ ˜ 2 if t ≥ T (ε). ˜ (t)ψ U X X In other words, the null solution of (4.2.3) is uniformly asymptotically stable in mean square. Proof Suppose (4.2.3) is L2 -stable in mean. For any ψ˜ ∈ Xr , we have by using Fubini’s theorem that

200

∞

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

˜ 22 dt = v (ψ) L t

0

∞

=

d2j e2δhj

j=1

=

∞

0

−hj

hj d2j e2δhj

∞ j=1

−r

˜ 2 dtdu Ev(t, ψ) H

∞

˜ 2 dt + Ev(t, ψ) H

0

0

−r

Eψ(t)2H dt .

hj d2j e2δhj < ∞ due to (4.2.2). We also have

0

−r

˜ 2 dudt Ev(t + u, ψ) H

∞

j=1

It is easy to see that

0

−hj

0

j=1 ∞

∞

d2j e2δhj

Eψ(t)2H dt ≤

1 ˜ 2 · e−2δhJ ψ X d2J

where hJ is the ﬁrst number of {hj } such that hJ ≥ r. Hence, there exists constant M > 0 such that for all ψ˜ ∈ Xr , ∞ ∞ ∞ 2 2 ˜ ˜ ˜ 22 dt ≤ M ψ ˜ 2 . (4.2.5) ˜ U (t)ψXdt = Ev(t, ψ)H dt+ v t (ψ) X L 0

0

0

˜ 2 is contin˜ (t)ψ Without loss of generality, we suppose 0 < ε < 1. Since U X ˜ uous, there exists, using (4.2.5), a ﬁrst time t0 (ε, ψ) > 0 such that for any ˜ X = 1, ψ˜ ∈ Xr with ψ ˜ ψ ˜ 2 = ε, ˜ (t0 (ε, ψ)) U X ˜ and for t ∈ [0, t0 (ε, ψ)),

˜ 2 > ε. ˜ (t)ψ U X

By virtue of (4.2.5), we obtain the inequality ˜ ≤ εt0 (ε, ψ)

˜ t0 (ε,ψ)

˜ 2 dt ≤ M, ˜ (t)ψ U X

0

˜ X = 1, which immediately yields for any ψ˜ ∈ Xr with ψ ˜ ≤ t0 (ε, ψ)

M =: T (ε). ε

Hence, as a result of Proposition 4.2.1, we have if t ≥ T (ε), then for any ˜ X = 1, ψ˜ ∈ Xr with ψ ˜ 2 = U ˜ U ˜ ψ ˜ 2 ≤ M U ˜ ψ ˜ 2 = M ε, ˜ (t)ψ ˜ (t − t0 (ε, ψ)) ˜ (t0 (ε, ψ)) ˜ (t0 (ε, ψ)) U X X X which proves the proposition.

Stability of Stochastic Functional Diﬀerential Equations

201

Observe that the implication (iii) =⇒ (i) in Theorem 4.2.3 is straightforward. Therefore, to conclude our proofs, it suﬃces to show the null solution of (4.2.3) is exponentially stable in mean square if it is uniformly asymptotically stable in mean square. But this is also immediate by carrying out a similar argument to Theorem 2.3.1. Therefore, the proof of Theorem 4.2.3 is complete. The following result gives a very useful criterion for the system (4.2.3) to be L2 -stable in mean by using a Lyapunov function method. Theorem 4.2.4 The system (4.2.3) is L2 -stable in mean if there exists a positive self-adjoint operator Q ∈ L(H ⊕ H) such that for any ψ˜ ∈ Xr , d ˜ v˜t (ψ) ˜ X ≤ −Ev(t, ψ) ˜ 2. Q˜ v t (ψ), H dt Proof Assume that there exists a positive self-adjoint operator Q ∈ L(H ⊕ H) satisfying the above inequality. Then ∞ ˜ 2 dt ≤ Qψ, ˜ ψ ˜ X ≤ Qψ ˜ 2, Ev(t, ψ) H X 0

˜ = ψ˜ ∈ X . where v˜0 (ψ) r

4.2.2

Stability of Neutral Stochastic Systems

A remarkable consequence of studying Equation (4.2.3) is that we can apply these results derived in Subsection 4.2.1, for instance, Theorem 4.2.3, to a wide class of linear neutral stochastic evolution equations of retarded type to establish their stability properties. Deﬁne Cr = φ ∈ C([−r, 0]; L2 (Ω, H)) : φ ∈ F, max Eφ(s)2H < ∞ s∈[−r,0]

with norm φ2Cr = max Eφ(s)2H s∈[−r,0]

for any φ ∈ Cr .

Consider the following abstract linear stochastic neutral functional diﬀerential equation: u(t, φ) −

m

m Dj u(t − rj , φ) = T (t) φ(0) − Dj φ(−rj )

j=1

t

j=1 m

T (t − s)

+ 0

t

T (t − s)

+ 0

j=1 m j=1

Aj u(s − rj , φ)ds Bj u(s − rj , φ)dWs ,

t ≥ 0,

202

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations u0 (·) = φ(·) ∈ Cr , (4.2.6)

where {T (t)}t≥0 is some given C0 -semigroup deﬁned on the Hilbert space H with inner product ·, ·H and 0 < r1 < r2 < · · · < rm = r are constants. Further Aj and Dj are bounded linear operators from H into H, the Bj are bounded linear operators from H into L(K, H), j = 1, · · · , m, and Wt , t ≥ 0, is a given standard Q-Wiener process on the Hilbert space K with ﬁnite trace class covariance operator Q. By employing a sequence of steps as in the proof of Theorem 4.2.1, it is possible to establish that for each φ ∈ Cr , there exists a unique u(·, φ) ∈ C([−r, ∞); L2 (Ω; H)) satisfying (4.2.6) and {U (t)}t≥0 deﬁned by U (t)φ = ut (φ) is a strongly continuous semigroup from Cr into Cr in the sense of Theorem 4.2.2. To apply the stability results obtained in Subsection 4.2.1 to Equation (4.2.6), we ﬁrst study some properties of initial data between the equations (4.2.3) and(4.2.6). For arbitrary φ ∈ Cr , r > 0, deﬁne the transform ψ(s) = φ(s) −

m

Dj φ(s − rj ),

s ∈ [−r, 0],

(4.2.7)

j=1

s < −r.

ψ(s) = 0,

Then, we have the following property. Lemma 4.2.4 Let R be the transform deﬁned by (4.2.7), and denote it by Rφ = ψ˜ = (ψ(0), ψ). Then R is a bounded linear operator from Cr into Xr , r > 0. Proof The linearity is clear. To prove boundedness, note that ψ(s)H = 0 almost surely if s < −r and m 2 2 2 ˜ sup ψ(s) 1 + . ≤ 2φ D j X Cr

−r≤s≤0

j=1

Hence, ˜ 2 = Eψ(0)2 + ψ X H

∞

d2j e2δhj

j=1

≤

Eψ(0)2H

+

≤ 2φ2Cr 1 +

∞

j=1 m

Dj 2

j=1

−hj

d2j e2δhj

0

Eψ(s)2H ds

0

−r

Eψ(s)2H ds

1+r

∞ j=1

d2j e2δhj

< ∞.

Stability of Stochastic Functional Diﬀerential Equations

203

Thus, there exists a constant M > 0 such that ˜ X = RφX ≤ M φC . ψ r

By virtue of Lemma 4.2.4, we may deﬁne a Hilbert subspace Mr = RCr ⊂ Xr of X which is the closure of RCr in the Hilbert space X. Then Mr will be taken as the initial data space for the equation (4.2.6) in the remainder of this section. For each solution u(·, φ) of (4.2.6), we may extend its domain to (−∞, ∞) by letting u(t, φ) = 0, t ≤ −r. Making a change of variables v(t) = u(t) −

m

Dj u(t − rj ),

t ∈ R1 ,

(4.2.8)

j=1

we get u(t) = v(t) +

∞

˜ k v(t − r˜k ), D

k=1

˜ k : H → H is a bounded linear operator, 0 < where for each k ≥ 1, D r˜1 < r˜2 < · · · and each r˜k is of the form n1 r1 + n2 r2 + · · · + nm rm for some nonnegative integers n1 , n2 , · · · , nm such that the above equality makes sense. Under the above change of variables, v(t) satisﬁes the following initial value problem of retarded stochastic evolution equation with inﬁnite delay ˜ = T (t)ψ(0) v(t, ψ) t m ∞ ˜ + ˜ ds ˜ k v(s − r˜k − rj , ψ) + T (t − s) Aj v(s − rj , ψ) D 0

j=1 t

T (t − s)

+ 0

m

˜ + Bj v(s − rj , ψ)

j=1

k=1 ∞

˜ dWs , ˜ k v(s − r˜k − rj , ψ) D

k=1

˜ = ψ(t), t ≤ 0, where ψ˜ = (ψ(0), ψ) ∈ M . v(t, ψ) r (4.2.9) To make our arguments compatible with the framework established in the previous subsection, let’s assume that there exists a sequence of nonnegative numbers {d˜j }, j = 1, 2, · · ·, such that ˜ j ≤ d˜j , D

∞ j=1

d˜j e2δ˜rj < ∞,

(4.2.10)

204

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

for some number δ > 0. Suppose {Fj } : H → H and {Gj } : H → L(K, H) are two given sequences of bounded linear operators and {hj } is a given increasing sequence of positive numbers such that m

∞ ∞ ˜ + ˜ = ˜ ˜ k v(s − r˜k − rj , ψ) D Aj v(s − rj , ψ) Fj v(s − hj , ψ),

j=1 m

˜ + Bj v(s − rj , ψ)

j=1

k=1 ∞

j=1 ∞ ˜ ˜ ˜ Dk v(s − r˜k − rj , ψ) = Gj v(s − hj , ψ). j=1

k=1

Then (4.2.9) can be rewritten in the form of (4.2.3) with initial value space Mr : t ∞ ˜ = T (t)ψ(0) + ˜ v(t, ψ) T (t − s) Fj v(s − hj , ψ)ds 0

t

T (t − s)

+ 0

j=1

∞

˜ Gj v(s − hj , ψ)dW s,

(4.2.11)

j=1

˜ = ψ(t), t ≤ 0, where ψ˜ = (ψ(0), ψ) ∈ M . v(t, ψ) r The following proposition shows that the formulation of {Fj }, {Gj } and {hj }, j = 1, 2 · · ·, is justiﬁed in deﬁning the equation (4.2.11). Proposition 4.2.3 Suppose the relation (4.2.10) holds and {hj } is deﬁned as in (4.2.11). Let dj = j ∨ Gj , j = 1, 2, · · · , where a ∨ b = max{a, b} F ∞ for any a, b ≥ 0, then j=1 dj e2δhj < ∞ for some positive constant δ. Proof

Note that

˜ α + · · · + Aj D ˜ α , 1 ≤ αj < ∞, 1 ≤ ji ≤ m, 1 ≤ i ≤ n ≤ m, Fj = Aj1 D j1 n jn i ˜ α + · · · + Bj D ˜ α , 1 ≤ αj < ∞, 1 ≤ ji ≤ m, 1 ≤ i ≤ n ≤ m, Gj = Bj1 D j1 n jn i r˜j1 + rj1 = r˜j2 + rj2 = · · · = r˜jn + rjn = hj . Therefore, Fj ∨ Gj ≤ max (Ai ∨ Bi ) d˜αj1 + · · · + d˜αjn , 1≤i≤m

and ∞ j=1

(Fj ∨ Gj ) · e

2δhj

≤ max (Ai ∨ Bi ) 1≤i≤m

∞

d˜αj1 + · · · + d˜αjn e2δhj

j=1

≤ (me2δr ) max (Ai ∨ Bi ) 1≤i≤m

∞ j=1

d˜j e2δ˜rj < ∞.

Stability of Stochastic Functional Diﬀerential Equations

205

This completes the proof. To apply the stability results in Subsection 4.2.1 such as Theorem 4.2.3 to (4.2.11), we need to prove the validity of (4.2.1) for the sequence {hj } deﬁned in (4.2.11). In fact, we have the following result whose proof is referred to the appendix. Proposition 4.2.4 then

Let {hn } be the sequence deﬁned in the equation (4.2.11); lim

n→∞

ln n = 0. hn

Next, let us study an example to illustrate the procedure from (4.2.6) to (4.2.11). Example 4.2.1 ential equation

Consider the following stochastic neutral diﬀerence diﬀer-

∂2 d u(x, t) − Du(x, t − r) = u(x, t) − Du(x, t − r) dt 2 ∂x m + Aj u(x, t − rj )dt (4.2.12)

j=1

+

m

Bj u(x, t − rj )dBt ,

t ≥ 0,

j=1

u(0, t) = u(π, t) = 0, t ≥ 0, where x ∈ [0, π], t ≥ 0, 0 < r1 < r2 < · · · < rm = r, D, Aj and Bj , j = 1, 2, · · · , m, are real numbers with |D| < 1, and Bt , t ≥ 0, is some one-dimensional standard Brownian motion. Let v(x, t) = u(x, t) − Du(x, t − r), Then u(x, t) = v(x, t) +

∞

x ∈ [0, π],

t ∈ R+ .

Dj v(x, t − jr)

j=1

and the ﬁrst equation of (4.2.12) can be rewritten as m ∞ ∂2 k dv(x, t) = v(x, t − r v(x, t)dt + A ) + D v(x, t − kr − r ) dt j j j ∂x2 j=1 k=1

+

m j=1

Bj v(x, t − rj ) +

∞ k=1

Dk v(x, t − kr − rj ) dBt .

206

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

In that case we have ˜ j = |Dj |, D

r˜j = jr,

so ∞

d˜j e2δ˜rj =

j=1

∞

|Dj |(e2δr )j =

j=1

∞

|D|e2δr

j

<∞

j=1

if δ > 0 is suﬃciently small so that |D|e2δr < 1. Now we are in a position to obtain the main stability results of (4.2.6) in the section. We ﬁrst establish an equivalent result of stability between (4.2.6) and (4.2.11). Theorem 4.2.5 The system (4.2.6) is L2 -stable in mean if and only if the system (4.2.11) is L2 -stable in mean. Proof Firstly, assume the system (4.2.11) is L2 -stable in mean. Let φ ∈ Cr and for ψ˜ = Rφ ∈ Mr , ˜ + u(t, φ) = v(t, ψ)

∞

˜ ˜ j v(t − r˜j , ψ) D

j=1

where ∞

˜ j = d˜j , D

d˜j e2δ˜rj < ∞,

δ > 0.

j=1

Therefore, for any t ≥ 0, ˜ 2 +2 Eu(t, φ)2H = Ev(t, ψ) H

∞

˜ D ˜ H ˜ j v(t − r˜j , ψ) Ev(t, ψ),

j=1

+2

+

∞

˜ D ˜ H ˜ j v(t − r˜j , ψ), ˜ k v(t − r˜k , ψ) ED

(4.2.13)

1=j
˜ 2. ˜ j v(t − r˜j , ψ) ED H

j=1

Integrating both sides of (4.2.13), using Lemma 4.2.4 and the assumptions of

Stability of Stochastic Functional Diﬀerential Equations

207

Theorem 4.2.5, we obtain that for some positive constants M , C1 , C2 , ∞ Eu(t, φ)2H dt 0 ∞ ∞ 1 ˜ 2 ED ˜ 2 2 dt ˜ ≤2 Ev(t, ψ) v(t − r ˜ , ψ) j j H H j=1 0 ∞ ∞

+2

j
˜ j v(t − r˜j )2 ED ˜ k v(t − r˜k )2 ED H H

0

˜ 2 + +C1 ψ X

∞ j=1

˜ 2 +2 ≤ C1 ψ X +2

∞

∞

d˜2j

∞

−˜ rj

dt

˜ 2 dt Ev(t, ψ) H

˜ 2 + ˜ X M ψ d˜j M 1/2 ψ X

j=1

˜ 2 + d˜j d˜k M ψ X

˜ 2 + · M ψ X

12

1=j

0

−˜ rj

0

−˜ rj

Eψ(t)2H dt

Eψ(t)2H dt

12 (4.2.13)

12

12 Eψ(t)2H dt −˜ rk 0 ∞ 2 2 ˜ ˜ dj M ψX + + Eψ(t)2H dt 0

j=1

∞ ˜ 2 + ˜j ≤ C2 ψ d X

−˜ rj 0

−˜ rj

j=1

Eψ(t)2H dt

12 2

< ∞,

which is precisely the deﬁnition of L2 -stability in mean of the solution of (4.2.6). We now assume that the system (4.2.6) is L2 -stable in mean. For arbitrarily given ψ˜ ∈ RCr , let ψ˜ = Rφ for some φ ∈ Cr . Then, from (4.2.8) we have for any t ≥ 0, ˜ 2 = Eu(t, φ)2 − 2 Ev(t, ψ) H H

m

Eu(t, φ), Dj u(t − rj , φ)H

j=1

+2

+

m

EDj u(t − rj , φ), Dk u(t − rk , φ)H

(4.2.14)

1=j
EDj u(t − rj , φ)2H .

j=1

Integrating both sides of (4.2.14) over [0, ∞), we obtain that for some constant C > 0,

208

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

∞

˜ 2 dt ≤ Ev(t, ψ) H

0

·

m

Eu(t, φ)2H dt +

Dj · Dk

1=j
Dj 2

∞

∞

−rj

Eφ(t)2H dt

Eu(t, φ)2H dt +

Eu(t, φ)2H dt +

0

j=1

+ 2C

1/2

φCr

m

Dj

Eu(t, φ)2H dt

0

Eφ(t)2H dt

0

−rj

Eφ(t)2H dt

Cφ2Cr

j=1

−r

Eφ(t)2H dt

m m + 2 Dj · Dk + Dj 2 Cφ2Cr +

Let ˜ = ρn (ψ)

n

˜ 2 dt Ev(t, ψ) H

0

−r

j=1

1=j
1/2

< ∞,

0

+

12

12

0

−r

0

∞

0

0

+2

≤

∞

Dj

Cφ2Cr

Eu(t, φ)2H dt + 2

0

j=1 m

+

∞

1/2

Eφ(t)2H dt < ∞.

n ≥ 0,

ψ˜ ∈ RCr .

0

Since RCr is dense in Mr , it is easy to see that ρn (·) may be extended to a family of continuous semi-norms on Mr . Then by Lemma 4.2.1, there exists a constant C > 0 such that for any ψ˜ ∈ Mr , ∞ 1/2 ˜ = ˜ 2 dt ˜ X. ρ(ψ) Ev(t, ψ) ≤ C 1/2 ψ H 0

That is, the null solution of (4.2.11) is L2 -stable in mean. Lastly, we shall show the following counterpart of Theorem 4.2.3, which states stability equivalent relations for the equation (4.2.6). Theorem 4.2.6 Under the relation (4.2.10), the following three notions of stability are equivalent: (i). The null solution of (4.2.6) is L2 -stable in mean; (ii). The null solution of (4.2.6) is uniformly asymptotically stable in mean square; (iii). The null solution of (4.2.6) is exponentially stable in mean square. Proof All the proofs are quite similar to those in Theorem 4.2.3 except for the implication (i) =⇒ (ii). Assume the null solution of (4.2.6) is L2 -stable

Stability of Stochastic Functional Diﬀerential Equations

209

in mean, then by Theorem 4.2.5, the null solution of (4.2.11) is L2 -stable in mean. We can conclude by using Theorem 4.2.3 that for arbitrarily given ε > 0, there exists number T (ε) ≥ 0 such that ˜ 2 ≤ εψ ˜ 2 ˜ (t)ψ U X X

(4.2.15)

for any ψ˜ ∈ Mr if t ≥ T (ε). However, for any given φ ∈ Cr , we have ψ˜ = Rφ where ψ˜ ∈ Mr . Moreover, by Lemma 4.2.4 we know that there exists a constant C > 0 such that ˜ 2 ≤ Cφ2 . ψ (4.2.16) X Cr Also for ψ˜ = Rφ, ˜ + Eu(t, φ)2H = Ev(t, ψ)

∞

˜ 2 ˜ j v(t − r˜j , ψ) D H

j=1

˜ 2 + 2E ≤ 2Ev(t, ψ) H

∞

˜ H d˜j v(t − r˜j , ψ)

2

j=1 ∞ J ˜ 2 +2 ˜ 2 ˜j ≤ 2Ev(t, ψ) d d˜j Ev(t − r˜j , ψ) H H j=1 ∞

+

j=1

˜ 2 . d˜j Ev(t − r˜j , ψ) H

j=J+1

∞ ˜ (t), t ≥ 0, is Let j=1 d˜j = M < ∞. Since, in view of Theorem 4.2.3, U uniformly asymptotically stable in mean square, it follows that for arbitrarily given ε > 0, we can ﬁnd J such that ∞ j=J+1

˜ 2 ≤ d˜j Ev(t − r˜j , ψ) H

ε φ2Cr . 2M C

Then, by (4.2.15) and (4.2.16) we can ﬁnd T (ε) ≥ 0 such that ˜ 2 ≤ ε φ2 , Ev(t, ψ) H Cr 4 and J

˜ 2 ≤ ε φ2 d˜j Ev(t − r˜j , ψ) H Cr 4M j=1

if t ≥ T (ε). Hence, we see that for the given ε > 0 above, there exists T (ε) ≥ 0 such that for σ ∈ [−r, 0], Eu(t + σ, φ)2H ≤ εφ2C whenever t ≥ T (ε), i.e., r

U (t), t ≥ 0, is uniformly asymptotically stable in mean square.

210

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Example 4.2.2 We now consider the following stochastic neutral functional diﬀerential equation as an illustrative example to close this section. X(x, t − 1) X(x, t − 1) ∂2 X(x, t − 1) d X(x, t) − X(x, t) − = dt + dBt , 2 ∂x2 2 2 0 ≤ x ≤ 1, t ≥ 0, (4.2.17) subject to the boundary condition X(0, t) = X(1, t) = 0,

t ≥ 0,

and the initial condition 0 ≤ x ≤ 1,

X(x, t) = φ(x, t),

−1 ≤ t ≤ 0,

where φ ∈ C([0, 1] × [−1, 0]; R1 ) and Bt , t ≥ 0, is a standard one dimensional Brownian motion. Let 1 y(x, t) = X(x, t) − X(x, t − 1), 2 then 1 1 X(x, t) = y(x, t) + y(x, t − 1) + 2 y(x, t − 2) + · · · . 2 2 So, (4.2.17) can be reduced to a retarded equation with inﬁnite delay ∞ ∂2 1 y(x, t)dt + y(x, t − j)dBt , 2 ∂x 2j j=1

dy(x, t) =

(4.2.18)

0 ≤ x ≤ 1, t ≥ 0, subject to the boundary condition 0 ≤ x ≤ 1,

y(0, t) = y(1, t) = 0,

t ≥ 0,

and the initial condition y(x, t) = ψ(x, t),

0 ≤ x ≤ 1,

t ≤ 0.

Assume ψ : [0, 1] × (−∞, 0] → R1 has the following Fourier series expansion ψ(x, t) =

∞

ψn (t) sin(nπx);

n=1

we seek solutions of (4.2.17) of the form y(x, t) =

∞ n=1

yn (t) sin(nπx).

Stability of Stochastic Functional Diﬀerential Equations Then dyn (t) = −n2 π 2 yn (t) +

211

∞ 1 y (t − j)dBt . j n 2 j=1

We can regard the totality of (4.2.17) as a stochastic diﬀerence diﬀerential ∞ equation in the sequence space H = {(ln ) : j=1 ln2 < ∞} of the form ˜ = Av(t, ψ) ˜ + dv(t, ψ)

∞

˜ Dj v(t − j, ψ)dB t,

j=1

where A is the unbounded operator which takes the j-th coordinate of H onto itself multiplied by −(jπ)2 , and Dj is the bounded operator which maps each element of H into its multiple of 21j . In this case, hj = j, and if we let δ = ln 2/4, then 1 ∞ ∞ ∞ 1 √ j 1 j 2δhj Dj e = ( 2) = < ∞. 2j 2 j=1 j=1 j=1 Let

∞ ∞ ˜ 2 1 t 1 2 v(t, ψ) H ˜ ˜ Λ(˜ v (ψ)) = − v (s − j, ψ)ds, 2 2 n=1 j=1 0 22j n t

where ˜ = (v1 (t, ψ), ˜ v2 (t, ψ), ˜ · · · , vn (t, ψ), ˜ · · ·) ∈ H. v(t, ψ) ˜ we have Then along a dense set of trajectories v˜t (ψ), ∞ ∞ ∞ ˜ ˜ d(EΛ(˜ v t (ψ))) Evn2 (t − j, ψ) ˜ +1 = − n2 π 2 Evn2 (t, ψ) 2j dt 2 n=1 j=1 2 n=1 − ≤ −

∞ ∞ ˜ 1 Evn2 (t − j, ψ) 2j 2 n=1 j=1 2 ∞

˜ = −Ev(t, ψ) ˜ 2. Evn2 (t, ψ) H

n=1

Thus, for a dense set of points {ψ} in H,

˜ ≤ EΛ(ψ) ˜ − 0 ≤ EΛ(˜ v (ψ)) t

∞

˜ 2 dt. Ev(t, ψ) H

0

However,

ψ(0)2H 2 ˜ ≤ 1/2 · ψ ˜ X for all ψ˜ ∈ X. is a continuous function on X and, in fact, Λ(ψ) Therefore, ∞ ˜ 2 dt ≤ Λ(ψ) ˜ ≤ 1/2 · ψ ˜ 2 Ev(t, ψ) ˜ = Λ(ψ)

H

X

0

for all ψ˜ ∈ X1 . This proves that the system (4.2.17) is L2 -stable in mean.

212

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

4.3

Decay Criteria of Stochastic Delay Diﬀerential Equations

In this section, we shall investigate decay of strong solutions for a class of nonlinear stochastic delay diﬀerential equations. Typically, we are concerned with the following system on V ∗ :  t A(s, u(s, φ), u(s − τ (s), φ))ds  u(t, φ) = φ(0) + t 0 + 0 B(s, u(s, φ), u(s − τ (s), φ))dWs ,  u(t, φ) = φ(t) ∈ V, t ∈ [−r, 0],

∀t ≥ 0,

(4.3.1)

where φ(t, ω) : [−r, 0] × Ω → V , r ≥ 0, is some given initial datum such that φ(t) is F0 -measurable for any t ∈ [−r, 0] and sup−r≤s≤0 Eφ(s)2V < ∞. τ : [0, ∞) → [0, r] is a certain diﬀerentiable function satisfying dτ (t)/dt ≤ 0 for all t ≥ 0. A(t, ·, ·) : V × V → V ∗ and B(t, ·, ·) : V × V → L(K, H) are two families of measurable nonlinear operators satisfying that t ∈ [0, ∞) → A(t, x, y) ∈ V ∗ , t ∈ [0, ∞) → B(t, x, y) ∈ L(K, H) are Lebesgue measurable for any x, y ∈ V . For the purpose of existence and uniqueness of strong solutions, the following assumptions similar to those in Section 1.3 are imposed on (4.3.1): for any T ≥ 0, there exist constants α > 0, p > 1 and θ, λ, γ ∈ R1 such that (a) (Coercivity). 2x, A(t, x, y)V,V ∗ + B(t, x, y)2L0 2

≤

−αxpV

+

λx2H

+

θy2H

+ γ,

∀x, y ∈ V,

(4.3.2) 0 ≤ t ≤ T,

where · L02 denotes the Hilbert-Schmidt norm B(t, x, y)2L0 = tr(B(t, x, y)QB(t, x, y)∗ ); 2

(b) (Growth). There exists a constant C > 0 such that A(t, x, y)V ∗ ≤ C(1 + xp−1 + yp−1 V V ),

∀x, y ∈ V,

0 ≤ t ≤ T; (4.3.3) (c) (Monotonicity). For arbitrary x1 , x2 , y1 , y2 ∈ V, and 0 ≤ t ≤ T , 2x1 − x2 , A(t, x1 , y1 ) − A(t, x2 , y2 )V,V ∗ + B(t, x1 , y1 ) − B(t, x2 , y2 )2L0 2 2 2 ≤ λ x1 − x2 H + y1 − y2 H . (d) (Continuity). The map (ξ, η) ∈ R1 → w, A(t, x + ξu, y + ηv)V,V ∗ ∈ R1 is continuous for all x, y, u, v, w ∈ V and 0 ≤ t ≤ T ;

Stability of Stochastic Functional Diﬀerential Equations (e) (Lipschitz). There exists a constant L > 0 such that B(t, u, v) − B(t, u ˜, v˜)L02 ≤ L u − u ˜V + v − v˜V , ∀u, u ˜, v, v˜ ∈ V, 0 ≤ t ≤ T.

213

(4.3.4)

Let M p (a, b; V ) denote the space of all V -valued processes (Xt )t∈[a,b] , −∞ < a ≤ b < ∞, which are measurable from [a, b] × Ω into V and satisfy b EXt pV dt < ∞. a

It may be proved (see Appendix) that for each given initial datum φ ∈ M p (−r, 0; V ) ∩ L2 (Ω; C(−r, 0; H)), there exists a unique process u(t, φ) ∈ M p (−r, T ; V ) ∩ L2 (Ω × [−r, T ]; H) (strong solution) satisfying Equation (4.3.1) and u(·, φ) ∈ C(0, T ; H) almost surely where C(a, b; H) denotes the space of all continuous functions from [a, b] into H. If T is replaced by ∞, u(t, φ), −r ≤ t < ∞, is called a global strong solution of (4.3.1). Unless otherwise speciﬁed, we always suppose that there exists a unique global strong solution of (4.3.1), which satisﬁes the above conditions (a)–(e). In particular, we will show below that a proper version of the coercive condition (a) plays a role of decay criterion.

4.3.1

Nonlinear Coercive Conditions for Decay

For the purpose of decay, we formulate the following coercive condition on the equation (4.3.1): (H10) There exist constants α > 0, θ ∈ R+ , λ ∈ R1 , and a nonnegative continuous function γ(t), t ∈ R+ , and µ > 0 such that 2x, A(t, x, y)V,V ∗ + B(t, x, y)2L0 2

≤ −αxpV + λx2H + θy2H + γ(t),

x, y ∈ V,

where p ≥ 2 and γ(t)eµt is integrable on [0, ∞). Also recall that there exists a positive constant β > 0 such that xH ≤ βxV ,

∀x ∈ V.

(4.3.5)

Then we are in a position to state our stability results in the following form. Theorem 4.3.1 Suppose the condition (H10) holds and assume u(t, φ), t ≥ 0, is a global strong solution of the equation (4.3.1) with initial datum

214

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

φ ∈ M 2 (−r, 0; V ) ∩ L2 (Ω; C(−r, 0; H)), then there exist constants τ > 0, C = C(φ) > 0 such that Eu(t, φ)2H ≤ C(φ) · e−τ t ,

∀t ≥ 0,

(4.3.6)

if either one of the following hypotheses holds (i) λ < 0, −λ > θ, (∀p ≥ 2). (ii) ν > θ where ν = α/β 2 − λ, (in the particular case p = 2). Proof We only show the case (ii). Case (i) can be similarly proved. Firstly, from (4.3.5) and (H10) it is easy to deduce that for any t ≥ 0, 2x, A(t, x, y)V,V ∗ + B(t, x, y)2L0 2 ≤ −νx2H + θy2H + γ(t),

x, y ∈ V.

(4.3.7)

Since ν > θ, it is possible to ﬁnd a suitable number ε ∈ (0, ν) such that θeεr ≤ ν − ε.

(4.3.8)

Applying Itˆ o’s formula to the strong solution u(t, φ), t ≥ 0, yields that eεt u(t)2H −φ(0)2H t t = ε eεs u(s)2H ds + 2 eεs u(s), A(s, u(s), u(s − τ (s)))V,V ∗ ds 0 0 t εs +2 e u(s), B(s, u(s), u(s − τ (s)))dWs H t0 + eεs tr(B(s, u(s), u(s − τ (s)))QB(s, u(s), u(s − τ (s)))∗ )ds. 0

Now, deﬁne an increasing sequence of stopping times t inf t > 0 : 0 eεs u(s), B(s, u(s), u(s − τ (s)))dWs H > n , σn = ∞ if the set is empty. Clearly, σn ↑ ∞, as n → ∞, and

t

eεs u(s), B(s, u(s), u(s − τ (s)))dWs H ,

t ∈ R+ ,

0

is a continuous localmartingale, so it follows that for any n ≥ 1, E 0

t∧σn

eεs u(s), B(s, u(s), u(s − τ (s)))dWs H = 0,

t ∈ R+ .

Stability of Stochastic Functional Diﬀerential Equations

215

Therefore, it deduces from (4.3.7) that for all t ≥ 0, Eeε(t∧σn ) u(t ∧ σn )2H ≤

Eφ(0)2H

t∧σn

− (ν − ε)E

εs

e

u(s)2H ds

+ θE

0

t∧σn

eεs u(s − τ (s))2H ds

0

t∧σn

γ(s)e[ε+µ−(µ∧ε)]s ds t∧σn 2 ≤ Eφ(0)H − (ν − ε)E eεs u(s)2H ds 0 t∧σn εr +θe E eε(s−τ (s)) u(s − τ (s))2H ds + C1 · e[ε−(µ∧ε)]t +E

0

0

∞

where C1 = 0 γ(s)eµs ds < ∞. Using (4.3.8) and the fact that τ (t) ≤ 0, t ≥ 0, we deduce that there exists a constant C2 > 0 such that t∧σn Eeε(t∧σn ) u(t)2H ≤ C2 − (ν − ε)E eεs u(s)2H ds 0 t∧σn εr +θe E eεs u(s)2H ds + C1 e[ε−(µ∧ε)]t 0

≤ C2 + C1 e[ε−(µ∧ε)]t . Letting n tend to inﬁnity in the above shows that there exists a constant C3 = C3 (r, φ) > 0 such that Eu(t, φ)2H ≤ C3 · e−(µ∧ε)t . In other words, the strong solution (4.3.1) is exponentially decayable in mean square and the proof is now complete. If we consider the non-delay formulation, i.e., θ = 0, Theorem 3.2.1 is an immediate deduction from Theorem 4.3.1. In particular, we may formulate the following fractional power type of coercivity condition to get a similar decay criterion to Corollary 3.2.1: (H11) There exist constants α > 0, λ ∈ R1 , 0 ≤ σ ≤ 1 and nonnegative continuous functions γ(t), ζ(t), t ∈ R+ , and µ > 0 such that 2x, A (t, x, y)V,V ∗ + B(t, x, y)2L0 2

≤

−αxpV

+

λx2H

+ ζ(t)y2σ H + γ(t),

x, y ∈ V,

(4.3.9)

where p ≥ 2 and ζ(t)eµt , γ(t)eµt , t ≥ 0, are both integrable on [0, ∞). Corollary 4.3.1 Suppose that (H11) holds and u(t, φ), t ≥ 0, is a global strong solution to the equation (4.3.1) with initial datum φ ∈ M 2 (−r, 0; V ) ∩

216

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

L2 (Ω; C(−r, 0; H)); then there exist constants τ > 0, C = C(φ) > 0 such that Eu(t, φ)2H ≤ C(φ) · e−τ t ,

∀t ≥ 0,

(4.3.10)

if either one of the following hypotheses holds: (i). λ < 0, (∀p ≥ 2). (ii). ν > 0 where ν = α/β 2 − λ, (particularly, for p = 2). By carrying out a similar argument to that of Theorem 3.2.2, we can actually obtain almost sure pathwise decay under the conditions (H10) and (H11) above. Precisely, we have the following: Theorem 4.3.2 Suppose that the condition (H10) or (H11) holds. Then there exist a subset Ω0 ⊂ Ω with P (Ω0 ) = 0 and a random variable T (ω) ≥ 0 such that for each ω ∈ Ω\Ω0 , the strong solution u(t, φ) of (4.3.1) satisﬁes u(t, φ)H ≤ M (φ) · e−λt ,

∀t ≥ T (ω),

for some positive constants M = M (φ) > 0 and λ > 0. Example 4.3.1 ential equation:

Consider the following semilinear stochastic partial diﬀer-

1 ∂2 y(t, x) √ dBt , y(t, x)dt + e−t/2 y(t − r, x) 3 dt + µ ∂x2 1 + |y(t − r, x)| t ≥ 0, x ∈ (0, 1), y(t, x) = φ(t, x), 0 ≤ x ≤ 1, t ∈ [−r, 0]; y(t, 0) = y(t, 1) = 0, t ≥ 0, (4.3.11)

dy(t, x) =

where φ ∈ C 2 ([0, 1] × [−r, 0]; R1 ) and µ > 0, r > 0 are two positive numbers. Bt , t ≥ 0, is a real standard Brownian motion. We can set this problem in our formulation by taking H = L2 [0, 1], V = H01 ([0, 1]), K = R1 , A(t, u, v) = 1 √ d2 −t/2 v(x) 3 and B(t, u, v) = µ · u(x)/(1 + |v(x)|), u, v ∈ V . dx2 u(x) + e It is easy to deduce that for suﬃciently small δ > 0 and u, v ∈ V , 2u, A(t, u, v)V,V ∗ +B(t, u, v)2L0 2

≤ −2π 2 u2H + (δ + µ)u2H + 1/δ · e−t vH . 2/3

(4.3.12)

Therefore, whenever 2π 2 > δ + µ > 0, or equivalently, 2π 2 > µ > 0 (note that δ > 0 is an arbitrary positive number), we easily deduce from Corollary 4.3.1 and Theorem 4.3.2 that for an arbitrary delay interval [−r, 0], r > 0, the strong solution is exponentially decayable in the mean square and also in the almost sure senses.

Stability of Stochastic Functional Diﬀerential Equations

4.3.2

217

Linear Stability Conditions

The criterion (H10) in Subsection 4.3.1 is quite useful to ensure exponential stability of the nonlinear system (4.3.1). If one is concerned with linear stochastic delay diﬀerential equations, it is possible to derive less restrictive conditions than (H10). Similarly to those in Subsection 2.2.2, a generalization of Theorem 2.2.5 with time delays can be formulated. Consider the following linear stochastic delay diﬀerential equation on V ∗ t t u(t, φ) = φ(0) + 0 Au(s, φ)ds + 0 Bu(s − τ (s), φ)dWs , t ≥ 0, u(t, φ) = φ(t) ∈ V, t ∈ [−r, 0], (4.3.13) where φ(t) : [−r, 0] × Ω → V , r ≥ 0, is some given initial datum such that φ(t) is F0 -measurable for any t ∈ [−r, 0] and sup−r≤s≤0 Eφ(s)2V < ∞. τ : [0, ∞) → [0, r] is a continuously diﬀerentiable function (of delay) satisfying 0 ≤ τ (t) ≤ t + r, dτ (t)/dt ≤ 0 for all t ≥ 0. A ∈ L(V, V ∗ ) and B ∈ L(V, L(K, H)) satisfy the condition that for some constants α > 0, λ ∈ R1 , 2x, AxV,V ∗ + x, ∆(I)xV,V ∗ ≤ −αx2V + λx2H ,

∀x ∈ V,

(4.3.14)

where for arbitrary P ∈ L(H, H), ∆(P ) ∈ L(V, V ∗ ) is deﬁned by y, ∆(P )xV,V ∗ = tr[B(x)∗ P B(y)Q],

x, y ∈ V.

It is known that under the condition (4.3.14), there exists a unique strong solution of (4.3.13). Moreover, A generates a strongly continuous semigroup T (t), t ≥ 0, and the strong solution is also a mild solution. On the other hand, note that under (4.3.14), T (t) maps H into V and there is a constant C > 0 such that ∞ e−2λt T (t)x2H dt ≤ Cx2V , ∀x ∈ V 0

(see Lions [1], Chap. IV, Theorem 1.1). Then the operators BT (t) and T (t)B deﬁned by (BT (t))(x)(y) = B(T (t)x)(y), (T (t)B)(x)(y) = T (t)(B(x)(y)),

∀x ∈ V,

y ∈ K,

∀x ∈ V,

y ∈ K,

belong to L(V, L(K, H)). We say that B and T (t) commute if (BT (t))(x) = (T (t)B)(x), Theorem 4.3.3 (H12)

∀x ∈ V.

Suppose that the following relations hold:

∃C ≥ 1, γ > 0 : T (t) ≤ C · e−γt ,

t ≥ 0,

218

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations ∞ ∗ T (t)∆(I)T (t)dt < 1.

(H13)

0

Assume also that B commutes with T (t). Then there exist positive constants µ, M such that the strong solution of (4.3.13) satisﬁes Eu(t, φ)2H ≤ M · φ21 e−µt , 0 where φ21 = max{Eφ(0)2H , −r Eφ(t)2H dt}.

t ≥ 0,

Proof Since u(t, φ) is the strong solution of (4.3.13), it is also the mild solution and satisﬁes t u(t, φ) = T (t)φ(0) + T (t − s)Bu(s − τ (s), φ)dWs , t ≥ 0, 0

which immediately implies u(t, φ)2H

2 t = + T (t − s)Bu(s − τ (s), φ)dWs 0 H t +2 T (t)φ(0), T (t − s)Bu(s − τ (s), φ)dWs , T (t)φ(0)2H

0

t ≥ 0.

H

Hence, it is easy to deduce that for any t ≥ 0, 2 t 2 2 Eu(t, φ)H = T (t)φ(0)H + E T (t − s)Bu(s − τ (s), φ)dWs , 0

H

since T (t)φ(0) is F0 -measurable, and consequently t E T (t)φ(0), T (t − s)Bu(s − τ (s), φ)dWs 0

= 0.

H

By standard properties of stochastic integrals and since B commutes with T (t), we have t 2 E T (t − s)Bu(s − τ (s), φ)dWs H 0 t = E tr (T (t − s)Bu(s − τ (s), φ))∗ T (t − s)Bu(s − τ (s), φ)Q ds 0 t = ET ∗ (t − s)∆(I)T (t − s)u(s − τ (s), φ), u(s − τ (s), φ)H ds. 0

Let λ > 0 be a constant to be determined later. From the last equation, ∞ ∞ eλt Eu(t, φ)2H dt = eλt ET (t)φ(0)2H dt 0 0 t ∞ eλt E T ∗ (t − s)∆(I)T (t − s)u(s − τ (s), φ), u(s − τ (s), φ)H dsdt. + 0

0

(4.3.15)

Stability of Stochastic Functional Diﬀerential Equations

219

Evaluating the ﬁrst term on the right hand side of (4.3.15), we obtain by (H12) ∞ C2 φ21 , eλt ET (t)φ(0)2H dt ≤ (4.3.16) 2γ − λ 0 if λ is such a number 0 < λ < 2γ. Also, Fubini’s theorem and the change of variables u = s − τ (s) yield

∞

t

E T ∗ (t − s)∆(I)T (t − s)u(s − τ (s), φ), u(s − τ (s), φ)H dsdt ∞ = eλs eλt E T ∗ (t)∆(I)T (t)u(s − τ (s), φ), u(s − τ (s), φ)H dtds 0 0 ∞ ∞ λt ∗ ≤ e T (t)∆(I)T (t)dt eλs Eu(s − τ (s), φ)2H ds 0 0 ∞ λr λs 2 ≤ f (λ)e e Eu(s, φ)H ds −r ∞ ≤ f (λ)eλr φ21 + f (λ)eλr eλs Eu(s, φ)2H ds, λt

e 0

0 ∞

0

(4.3.17) f (λ) =

where

∞

0

e T (t)∆(I)T (t)dt . λt

∗

Using (4.3.15)–(4.3.17), we deduce that

∞

t

E T ∗ (t − s)∆(I)T (t − s)u(s − τ (s), φ), u(s − τ (s), φ)H dsdt

C2 ≤ f (λ)eλr 1 + φ21 + f (λ)eλr 2γ − λ t ∞ · eλt E T ∗ (t − s)∆(I)T (t − s)u(s − τ (s), φ), u(s − τ (s), φ)H dsdt. eλt

0

0

0

0

The continuity of functions deﬁned by integrals depending on parameters and (H13) yield lim f (λ)eλr < 1. + λ→0

Then, we can choose λ properly such that 0 < λ < 2γ, f (λ)eλr < 1. Consequently, there exists a constant C1 > 0, depending only on λ, such that 0

∞

eλt

t

ET ∗ (t − s)∆(I)T (t − s)u(s − τ (s), φ), u(s − τ (s), φ)H dsdt

0

≤ C1 φ21 . (4.3.18)

220

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

From (4.3.16) and (4.3.18) it follows that for each λ > 0 small enough, there exists a positive constant M1 = M1 (λ) such that ∞ eλt Eu(t, φ)2H dt ≤ M1 φ21 . (4.3.19) 0

Then, applying Itˆ o’s formula to the process eλt u(t, φ)2H , t ≥ 0, and carrying out a similar argument to Theorem 3.2.1, yields the required results. Remark Under the same conditions as in Theorem 4.3.3, we may also obtain almost sure pathwise exponential decay of (4.3.13). If B and T (t) do not commute, it is possible to formulate less general conditions than those in Theorem 4.3.3 to ensure the desired exponential decay. The reader is referred to Caraballo [1] for more details in this respect.

4.4

Razumikhin Type Stability Theorems

In this section, we will investigate stability of mild solutions for semilinear stochastic functional diﬀerential equations, based on the ideas of constructing Lyapunov functions rather than functionals, in the spirit of Razumikhin in ﬁnite dimensions. Consider the following stochastic retarded evolution equation in the Hilbert space H:  t t  u(t) = T (t)φ(0) + 0 T (t − s)F (us )ds + 0 T (t − s)G(us )dWs , (4.4.1) t ≥ 0,  u0 (·) = φ(·) ∈ Cr = C([−r, 0]; H), where T (t), t ≥ 0, is some C0 -semigroup of bounded linear operators on H with unbounded inﬁnitesimal generator A, and F : C([−r, 0]; H) → H and G : C([−r, 0]; H) → L(K, H) are two measurable nonlinear mappings satisfying F (0) = 0 and G(0) = 0. To ensure the existence and uniqueness of solutions for (4.4.1), we impose the following Lipschitz condition on the coeﬃcients F and G: for some k > 0, F (x) − F (y)H + G(x) − G(y) ≤ kx − yCr ,

(4.4.2)

where x, y ∈ Cr . Since Itˆ o’s formula is only applicable to strong solutions, we introduce the following approximating system of (4.4.1) as we did in Section 3.3 for non delay stochastic systems: du(t) = Au(t)dt + R(n)F (ut )dt + R(n)G(ut )dWt , t ≥ 0, (4.4.3) u(t) = R(n)φ(t) ∈ D(A), t ∈ [−r, 0],

Stability of Stochastic Functional Diﬀerential Equations

221

where n0 ≤ n ∈ ρ(A) for some natural number n0 , the resolvent set of A and R(n) = nR(n, A). It is possible to derive the following approximation result by a similar argument to Proposition 1.3.6. Proposition 4.4.1 Under the hypothesis (4.4.2), the equation (4.4.3) has for each n ≥ n0 a unique strong solution un (t) ∈ D(A) lying in Lp (Ω; C(0, ∞; H)), p ≥ 2. Moreover, for any T ≥ 0, un (t) → u(t) of (4.4.1) almost surely as n → ∞ uniformly with respect to [0, T ]. In inﬁnite dimensional spaces, for the purpose of deriving stability results, the construction of appropriate Lyapunov functionals rather than functions is a natural generalization of the Lyapunov direct method in ﬁnite dimensional spaces. We next derive a stability result of (4.4.1) to show that the situation in treating time delay stochastic systems by using this approach could become complicated. Let Λ(·) : C([−r, 0]; H) → R+ be an arbitrary continuous nonlinear functional and u(t, φ) be the solution of (4.4.1) with initial datum φ ∈ C([−r, 0]; H). Proposition 4.4.2 Suppose v(·), l(·) : R+ → R+ are two continuous nondecreasing functions, v(t) and l(t) are positive for t > 0, v(t) is convex with v(0) = l(0) = 0. If there is a continuous nonlinear functional Λ(·) : C([−r, 0]; H) → R+ such that v(φ(0)2H ) ≤ Λ(φ) ≤ l(φ2Cr ), ∀φ ∈ C([−r, 0]; H), (4.4.4) EΛ(ut (φ)) ≤ EΛ(us (φ)), ∀t ≥ s ≥ 0. Then the null solution of (4.4.1) is stable in mean square. Proof For any ε > 0, there is a 0 < δ = δ(ε) < ε such that l(δ) < v(ε). If φ2Cr ≤ δ, then the inequalities on Λ(·) above imply v(Eu(t, φ)2H ) ≤ Ev(u(t, φ)2H ) ≤ EΛ(ut (φ)) ≤ Λ(φ) ≤ l(φ2Cr ) ≤ l(δ) < v(ε),

t ≥ 0,

by the properties of the mean. Therefore, Eu(t, φ)2H < ε for any t ≥ 0. In spite of the formal simplicity of the above result, it is hard to apply Proposition 4.4.2 directly to practical problems even though H is ﬁnite dimensional. The reason is twofold. On the one hand, instead of the usual Lyapunov functions in ﬁnite dimensional spaces, a Lyapunov functional as above must be constructed properly, a case which is usually not easy to handle. On the other hand, the condition (4.4.4) is actually diﬃcult to check because of the inclusion of the solution itself which is not known explicitly in

222

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

most situations. One of the most eﬀective ways to deal with these problems is to study stability of retarded systems of the type (4.4.1) by using a method introduced by Razumikhin. Let C 2 (H; R+ ) denote the family of all nonnegative functions Λ(x) on H which are twice continuously Fr´echet diﬀerentiable. If Λ ∈ C 2 (H; R+ ), deﬁne an operator LΛ for any φ ∈ Cr with φ(0) ∈ D(A), by 1 (LΛ)(φ) = Λx (φ(0)), Aφ(0)+F (φ)H + tr Λxx (φ(0))G(φ)QG∗ (φ) , (4.4.5) 2 where Q is the covariance operator of the Wiener process W . Then we can assert the following results. Theorem 4.4.1 Let p ≥ 2. Assume that there exists a continuous nonlinear function Λ ∈ C 2 (H; R+ ) and constants ci > 0, i = 1, · · · , 4, λ > 0 such that for any x ∈ H, Λx (x)H ≤ c3 xp−1 H ,

(4.4.6)

E(LΛ)(ψ) ≤ −λEΛ(ψ(0)),

(4.4.7)

c1 xpH ≤ Λ(x) ≤ c2 xpH , Λxx (x) ≤ c4 xp−2 H ,

provided ψ ∈ C(Ω × [−r, 0]; H) with ψ(0) ∈ D(A) satisﬁes EΛ(ψ(θ)) < qEΛ(ψ(0)) for all − r ≤ θ ≤ 0 for some q > 1. Then the null solution of (4.4.1) is exponentially stable in p-th moment. Moreover, for arbitrary φ ∈ C(Ω × [−r, 0]; H), Eu(t, φ)pH ≤

c2 EφpCr e−γt , c1

∀t ≥ 0,

where γ = min λ, log(q) . r Proof Fix the initial data φ ∈ C(Ω × [−r, 0]; H) and write u(t, φ) = u(t) simply. Let ε ∈ (0, γ) be one arbitrary number and set γ¯ = γ − ε. Deﬁne U (t) = max eγ¯ (t+θ) EΛ(u(t + θ)) , t ≥ 0. −r≤θ≤0

Obviously, U (t) is well deﬁned and continuous. We claim that D+ U (t) = lim sup h→0+

U (t + h) − U (t) ≤0 h

for t ≥ 0.

To show this, for each ﬁxed t0 ≥ 0, deﬁne θ¯ = max θ ∈ [−r, 0] : U (t0 ) = eγ¯ (t0 +θ) EΛ(u(t0 + θ)) .

(4.4.8)

Stability of Stochastic Functional Diﬀerential Equations

223

Obviously, θ¯ is well deﬁned, θ¯ ∈ [−r, 0] and ¯ ¯ U (t0 ) = eγ¯ (t0 +θ) EΛ(u(t0 + θ)).

If θ¯ < 0, one has ¯ ¯ eγ¯ (t0 +θ) EΛ(u(t0 + θ)) < eγ¯ (t0 +θ) EΛ(u(t0 + θ)) for all θ¯ < θ ≤ 0.

It is therefore easy to observe that for any h > 0 small enough, ¯

¯ eγ¯ (t0 +h) EΛ(u(t0 + h)) ≤ eγ¯ (t0 +θ) EΛ(u(t0 + θ)). Hence U (t0 + h) ≤ U (t0 ) and D+ U (t0 ) ≤ 0. If θ¯ = 0, then eγ¯ (t0 +θ) EΛ(u(t0 + θ)) ≤ eγ¯ t0 EΛ(u(t0 )) for any θ ∈ [−r, 0]. Therefore, EΛ(u(t0 + θ)) ≤ e−¯γ θ EΛ(u(t0 )) ≤ eγ¯ r EΛ(u(t0 )) for any θ ∈ [−r, 0].

(4.4.9)

In the case of EΛ(u(t0 )) = 0, (4.4.6) and (4.4.9) imply that u(t0 + θ) = 0 for all θ ∈ [−r, 0] almost surely. Recall that F (0) = 0 and G(0) = 0, it then follows that u(t0 + h) = 0 almost surely for all h > 0, hence U (t0 + h) = 0 and D+ U (t0 ) = 0. On the other hand, in the case of EΛ(u(t0 )) > 0, (4.4.9) implies EΛ(u(t0 + θ)) ≤ eγ¯ r EΛ(u(t0 )) < qEΛ(u(t0 )) for any θ ∈ [−r, 0] as eγ¯ r < q. Let ν = q−eγ¯ r > 0, it then follows from the continuity of EΛ(u(t)) and (4.4.6) that for some h > 0 small enough, ν EΛ(u(t0 + θ)) ≤ eγ¯ r + EΛ(u(t0 )) for any θ ∈ [0, h]. 2 Now we need to introduce the strong solution sequence {un (t)} of (4.4.3) such that by Proposition 4.4.1, un (t) → u(t) in C(0, T ; H), ∀T ≥ 0, uniformly with respect to t as n → ∞ almost surely. Consequently, for some constant ν δ ∈ 0, 4+2ν EΛ(u(t0 )) , there are a suﬃciently small constant h > 0 and number N > 0 large enough such that for n ≥ N , it follows that for any s ∈ [t0 , t0 + h], EΛ(u(s)) > EΛ(u(t0 )) − δ > 0, EΛ(u(s + θ)) < EΛ(u(t0 + θ)) + δ, θ ∈ [−r, 0],

224

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations eγ¯ r EΛ(u(t0 )) < eγ¯ r EΛ(u(s)) + δ, EΛ(un (s)) > EΛ(u(s)) − δ > 0, γ ¯r

e EΛ(u(s)) < eγ¯ r EΛ(un (s)) + δ, EΛ(un (s + θ)) < EΛ(u(s + θ)) + δ, θ ∈ [−r, 0]. These immediately imply EΛ(un (s + θ)) < eγ¯ r EΛ(un (s)) + 4δ < eγ¯ r EΛ(un (s)) + ν(EΛ(u(s)) − δ) < eγ¯ r EΛ(un (s)) + νEΛ(un (s)) = qEΛ(un (s)) for any θ ∈ [−r, 0].

(4.4.10)

In addition to (4.4.7), (4.4.10) implies that E(LΛ)(uns ) ≤ −λEΛ(un (s)), ∀s ∈ [t0 , t0 + h].

(4.4.11)

Applying Itˆ o’s formula to the function eγ¯ t Λ(u) along the strong solutions un (t) ¯ ∈ [0, h], of (4.4.2), we can derive, by using (4.4.11), that for any h ¯ ¯ eγ¯ (t0 +h) EΛ(un (t0 + h))

t0 +h¯ ≤ eγ¯ t0 EΛ(un (t0 )) + (¯ γ − λ) eγ¯ s EΛ(un (s))ds t0 t0 +h¯ γ ¯s n + e EΛx (u (s)), (R(n) − I)F (uns )H ds t0 ¯ 1 t0 +h γ¯ s + Ee tr[Λxx (un (s))R(n)G(uns )Q(R(n)G(uns ))∗ ]ds 2 t0 ¯ 1 t0 +h γ¯ s − Ee tr[Λxx (un (s))G(uns )QG(uns )∗ ]ds, 2 t0

which, letting n → ∞, immediately yields ¯ ¯ ≤ eγ¯ t0 EΛ(u(t0 )) + eγ¯ (t0 +h) EΛ(u(t0 + h))

¯ t0 +h

(¯ γ − λ)eγ¯ s EΛ(u(s))ds

t0

≤ eγ¯ t0 EΛ(u(t0 )). (4.4.12) Then it must be the case that eγ¯ s EΛ(u(s)) ≤ eγ¯ t0 EΛ(u(t0 )),

∀s ∈ [t0 , t0 + h].

So it must hold that EU (t0 + h) = EU (t0 ) for any h > 0 suﬃciently small, and hence D+ U (t0 ) = 0. Since t0 is arbitrary, the inequality (4.4.8) is shown to hold for any t ≥ 0. It now follows immediately from (4.4.8) that U (t) ≤ U (0),

∀t ≥ 0.

Stability of Stochastic Functional Diﬀerential Equations

225

Also, (4.4.6) implies eγ¯ t EΛ(u(t)) ≤ U (t) ≤ U (0) ≤ c2 EφpCr ,

∀t ≥ 0.

Note that ε is arbitrary, it thus follows that EΛ(u(t)) ≤ c2 EφpCr e−γt ,

∀t ≥ 0,

which, by virtue of (4.4.6), immediately yields Eu(t)pH ≤

c2 EφpCr e−γt , c1

∀t ≥ 0.

Under the same conditions as in Theorem 4.4.1, one can deduce the almost sure exponential stability of (4.4.1) by carrying out a similar argument to Theorem 3.2.2. Next we will apply Theorem 4.4.1 to a class of stochastic delay diﬀerential equations of the following form: t u(t) = T (t)φ(0) + T (t − s)f (u(s − τ1 (s)))ds 0 t + T (t − s)g(u(s − τ2 (s)))dWs , t ≥ 0, (4.4.13) 0

u0 (·) = φ(·) ∈ C(Ω × [−r, 0]; H), where τi (·) : R+ → [0, r], i = 1, 2, are both continuous, and f (·) : H → H

and g(·) : H → L(K, H),

f (0) = 0, g(0) = 0, are two nonlinear measurable mappings satisfying the usual Lipschitz continuous conditions. Corollary 4.4.1 Let p ≥ 2 and assume that there exists a nonlinear function Λ ∈ C 2 (H; R+ ), constants ci > 0, i = 1, · · · , 4, λ > 0, λ1 > 0 and λ2 > 0 such that for any x ∈ H, c1 xpH ≤ Λ(x) ≤ c2 xpH , Λxx (x)

Λx (x)H ≤ c3 xp−1 H , ≤

c4 xp−2 H ,

(4.4.14) (4.4.15)

and 1 Λx (φ(0)), Aφ(0) + f (φ(−τ1 (0)))H + tr Λxx g(φ(−τ2 (0)))Qg ∗ (φ(−τ2 (0))) 2 ≤ −λΛ(φ(0)) + λ1 Λ(φ(−τ1 (0))) + λ2 Λ(φ(−τ2 (0))), (4.4.16)

226

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

for any φ ∈ Cr with φ(0) ∈ D(A). If λ > λ1 + λ2 , then the null solution of (4.4.13) is exponentially stable in p-th moment. Its p-th moment Lyapunov 2 2 exponent should not be greater than (q i=1 λi − λ), where q ∈ (1, λ/ i=1 λi ) 2 is the unique root of λ − q i=1 λi = log(q)/r. Furthermore, the null solution is also exponentially stable in the almost sure sense. Proof

Deﬁne for φ ∈ C([−r, 0]; H), F (φ) = f (φ(−τ1 (0)))

and G(φ) = g(φ(−τ2 (0))).

Then (4.4.13) can be handled in the framework of (4.4.1). Moreover, in that case the operator LΛ deﬁned in (4.4.5) becomes (LΛ)(φ) = Λx (φ(0)), Aφ(0) + f (φ(−τ1 (0)))H 1 + tr Λxx (φ(0))g(φ(−τ2 (0)))Qg ∗ (φ(−τ2 (0))) 2 for any φ ∈ Cr with φ(0) ∈ D(A). If ψ ∈ C(Ω × [−r, 0]; H) with ψ(0) ∈ D(A) satisﬁes EΛ(ψ(θ)) < qEΛ(ψ(0)) for all θ ∈ [−r, 0] for some q > 1, then by Condition (4.4.16), E(LΛ)(ψ) ≤ −λEΛ(ψ(0)) + λ1 EΛ(ψ(−τ1 (0))) + λ2 EΛ(ψ(−τ2 (0))) 2 ≤ q λi − λ EΛ(ψ(0)). i=1

So, by virtue of Theorem 4.4.1, the null solution of (4.4.13) is exponentially stable in the p-th moment and also in the almost sure 2senses. Its p-th moment Lyapunov exponent should not be greater than (q i=1 λi − λ). The proof is now complete. Last, let us discuss an example to close this section. Example 4.4.1 Consider the semilinear stochastic partial diﬀerential equation with ﬁnite time lags r1 , r2 (r > r1 , r2 > 0), 0 ∂2 Z(t, x)dt + ν Z(t + θ, x)h(θ)dθ dt ∂x2 −r1 +α(Z(t, x))Z(t − r2 , x)dBt , (µ > 0, ν > 0), Z(t, 0) = Z(t, π) = 0, t ≥ 0, Z(s, x) = φ(s, x), s ∈ [−r, 0], x ∈ [0, π], φ(s, ·) ∈ H = L2 (0, π), φ(·, x) ∈ C([−r, 0]; R1 ),

dZ(t, x) = µ

(4.4.17)

Stability of Stochastic Functional Diﬀerential Equations

227

where Bt is a standard one dimensional Brownian motion. Here α(·) : R1 → R1 , h(·) : [−r1 , 0] → R1 are two bounded Lipschitz continuous functions such that |α(x)| ≤ L, |h(θ)| ≤ M , L > 0, M > 0, for any x ∈ R1 , θ ∈ [−r1 , 0]. Let A = µ∂ 2 /∂x2 with the domain ∂u ∂ 2 u 2 D(A) = u ∈ L2 (0, π) : ∈ L (0, π), u(0) = u(π) = 0 , , ∂x ∂x2 so it is easy to deduce Au, uH ≤ −µu2H , u ∈ D(A). On the other hand, suppose Eψ(θ)2H ≤ qEψ(0)2H ,

q > 1, ∀θ ∈ [−r, 0],

for ψ ∈ C(Ω × [−r, 0]; H) with ψ(0) ∈ D(A). It is clear that

'

(

0

E ψ(0), ν

ψ(θ)h(θ)dθ −r1

H

≤ q 1/2 (νr1 M )Eψ(0)2H

and Eα(ψ(0))ψ(−r2 )2H ≤ qL2 Eψ(0)2H . Then letting Λ(u) = u2H , we have by a direct computation that E(LΛ)(ψ) ≤ − 2µ + 2q 1/2 (νr1 M ) + qL2 Eψ(0)2H . By virtue of Theorem 4.4.1 and letting q → 1, one gets that if 2µ > 2(νr1 M )+ L2 , then for arbitrary 0 ≤ r2 ≤ r, the null solution of (4.4.17) is exponentially stable in the mean square and almost sure senses.

4.5

Notes and Comments

All the main results in Section 4.1 are due to Travis and Webb [1] and Milota [1]. The discussion of characteristic equations with inﬁnite delay is based on the work of Lenhart and Travis [1]. In deterministic cases, the reduction of a neutral equation to a retarded equation with inﬁinite delay and the notion of the L2 -stability was developed by Datko [3], [4]. The stochastic version presented in Section 4.2 is due to Liu [7] which is in the spirit of Datko’s work. The established result that L2 -stability implies asymptotic stability allows one to apply Lyapunov functional method to obtain stability criteria. Staﬀans [1] has shown that an ordinary neutral functional diﬀerential equation with a linear stable diﬀerence operator can be reduced to a retarded equation

228

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

with inﬁnite delay. It is hoped that this reduction can be extended to a class of stochastic partial neutral functional diﬀerential equations and applied to the study of their stability. The main results in Section 4.3 are taken from Caraballo, Liu and Truman [2] and Caraballo [1]. There exists some work on stability for mild solutions of nonlinear stochastic evolution equations with time delays. For instance, Jahanipur [1] considered stability of a class of stochastic delay evolution equations with monotone nonlinearity. In ﬁnite dimensional spaces, the Razumikhin type arguments take advantages of the structure of Rn while the Lyapunov functional type ones need handle diﬃculties caused by the inﬁnite dimensional nature of Cr . This is clearly illustrated by an example in Levin and Nohel [1]. The Razumikhin ideas were ﬁrstly extended to obtain a version for stochastic systems by Taniguchi [2] and subsequently by Mao [3], [4] in ﬁnite dimensional spaces. Theorem 4.4.1 and Corollary 4.4.1 are discussed and established in Liu and Shi [1] in the spirit of Mao [3] to deduce stability (in the sense of mild solutions) for more general stochastic functional diﬀerential equations in inﬁnite dimensions.

Chapter 5 Some Related Topics of Stability and Applications

In this chapter, we shall present some selected topics in connection with stability properties of stochastic diﬀerential equations in inﬁnite dimensions. The choice of the material reﬂects our own personal reference and the treatment here is somewhat sketchy. Some chosen material touches upon speciﬁc stochastic systems which could be regarded as a potential starting point for research in this area. Some other material reveals interesting and important relationships between the main topic, stability, of this book and topics from other branches of science or technology such as stochastic control. In particular, in Section 5.1 we shall begin with an exposition of stability for a class of stochastic parabolic equations with singular noise, e.g., distributed one. The problem of optimal feedback control and its relation with stablity of stochastic linear systems are studied in Section 5.2. Section 5.3 is wholly devoted to the investigation of feedback stabilization for a class of nonlinear stochastic models. In Sections 5.4 and 5.5, the stability results of the previous chapters are applied to some important models in chemical dynamics, ﬂuid dynamics and mathematical population biology.

5.1

Parabolic Equations with Boundary and Pointwise Noise

In order to motivate our theory, suppose O is a bounded open domain in Rn with smooth boundary ∂O. Let −A(x, ∂) = aα (x)∂ α , x ∈ O, |α|≤2

be a uniformly strongly elliptic operator of order two with smooth coeﬃcients aα (x). Consider a class of nonhomogeneous partial diﬀerential equations of mixed type with boundary noise:

229

230

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

∂y(t, x) = −A(x, ∂)y(t, x), x ∈ O, t ≥ 0, ∂t y(0, x) = y0 (x), x ∈ O, ∂y(t, x) + a(x)y(t, x) = g(x)f (y(t, x))B˙ t , x ∈ ∂O, ∂n

(5.1.1) t ≥ 0,

where y0 (x) ∈ L2 (O), g(x) ∈ L2 (∂O), ∂/∂n is the outward normal derivative, a(x) is a real positive function deﬁned on ∂O, f (·) is a certain real function deﬁned on R1 and B˙ t is some given white noise. We take H = L2 (O) and deﬁne D(A) as the closure in H 2 (O) of the sub¯ which consists of functions φ satisfying the boundary conspace of C 2 (O) dition ∂φ/∂n + aφ = 0. Let −A be the restriction of −A(x, ∂) on D(A). Then it is possible to show that −A generates an analytic semigroup T (t) of bounded operators (see, for instance, Yosida [1]). Let M be the mapping: L2 (∂O) → L2 (O) deﬁned by y(·) = M g(·), where y(x) is the solution of A(x, ∂)y(x) = 0, x ∈ O, ∂y(x) + a(x)y(x) = g(x), ∂n

x ∈ ∂O.

It can be deduced that (5.1.1) is described by the abstract stochastic evolution equations on H t Yt = T (t)y0 + AT (t − s)M gf (Ys )dBs , 0 t (5.1.2) θ = T (t)y0 + A T (t − s)bf (Ys )dBs , t ≥ 0, 0

where θ = 1/4 + ε, ε > 0 and b = A1−θ M g ∈ H. The new feature of this formulation is that the generator A appears in the integral term, and it will turn out that we can apply equations of (5.1.2) type to systems with not only boundary but point noise. In this section, we are mainly concerned about the stability of a class of stochastic evolution equations. The results of this discussion can be applied to some models such as (5.1.1). Precisely, let T (t) be a strongly continuous analytic semigroup of negative type on a real separable Hilbert space H and −A be its inﬁnitesimal generator. Then fractional power Aγ , 0 < γ < 1, is well deﬁned and that for each 0 < T < ∞ there exists a constant M = M (T ) > 0 such that Aγ T (t) ≤ M/tγ

for any 0 < γ < 1 and t ∈ (0, T ].

Consider the following stochastic evolution equation t Yt = T (t)y0 + Aθ T (t − s)bf (Ys )dBs , Y0 = y0 ∈ H,

0

t ≥ 0,

(5.1.3)

(5.1.4)

Some Related Topics of Stability and Applications

231

where 0 ≤ θ < 1/2, b ∈ H, f (y) is a real Lipschitz continuous function on D(Aη ) for some 0 ≤ η < 1, f (0) = 0, and Bt is a real standard Brownian motion deﬁned on some probability space (Ω, F, {Ft }t≥0 , P ). The existence, uniqueness and regularity of (5.1.4) have been established in Ichikawa [7]. To study stability of (5.1.4), following the ideas in Section 2.2 we consider the operator integral equation T P (t)y, yH = F T (T − t)y, T (T − t)yH + M T (r − t)y, T (r − t)yH t +f 2 (T (r − t)y)(A∗ )θ P (r)Aθ b, bH dr, P (T ) = F, y ∈ H, 0 ≤ t ≤ T, T ≥ 0, (5.1.5) where A∗ is the dual operator of A, and the associated equation Q(t)z, zH

= F T (T − t)Aθ z, T (T − t)Aθ zH + +Q(r)b, bH f 2 (T (r − t)Aθ z) dr,

T

M T (r − t)Aθ z, T (r − t)Aθ zH

t

Q(T ) = (A∗ )θ F Aθ , z ∈ H, 0 ≤ t ≤ T, T ≥ 0, (5.1.6) In the remaining of this section, we will always suppose 0 ≤ β = θ + η < 1/2. In particular, we have the following existence and uniqueness results of (5.1.5) and (5.1.6). Proposition 5.1.1 There exists a unique solution to the equation (5.1.5) P (t) ∈ L(D(A−θ ), D((A∗ )θ )) or (5.1.6) Q(t) ∈ L(H) which is nonnegative and strongly continuous on [0, T ) with P (t)L(D(A−θ ),D((A∗ )θ )) ≤

C , (T − t)2θ

Q(t) ≤

C (T − t)2θ

for some C = C(T ) > 0, where · L(D(A−θ ),D((A∗ )θ )) is the operator norm in L(D(A−θ ), D((A∗ )θ )). Moreover, if F = 0, both P (t) and Q(t) are strongly continuous on [0, T ]. The relation between P (t) and Q(t) is given by P (t) = (A∗ )−θ Q(t)A−θ , Q(t) = (A∗ )θ P (t)Aθ ,

t ∈ [0, T ], t ∈ [0, T ].

Proof It suﬃces to show, say, the uniqueness and existence of Q(t). The other claims could be similarly proved. To show uniqueness of Q(t), we consider T

R(t)z, zH =

R(r)b, bH f 2 (T (r − t)Aθ z)dr. t

232

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

This yields

T

R(t) ≤ C t

R(r) dr (r − t)2β

for some

C > 0.

Thus by using Gronwall’s inequality, we obtain R(t) = 0 almost surely with respect to t. To establish a solution of (5.1.6), we deﬁne Q0 (t)z, zH = F T (T − t)Aθ z, T (T − t)Aθ zH T + M T (r − t)Aθ z, T (r − t)Aθ zH dr t

and iterate a sequence of nonnegative operators T Qn (t)z, zH = Q0 (t)z, zH + Qn−1 (r)b, bH f 2 (T (r − t)Aθ z)dr. t

Then Qn (t) is well deﬁned and continuous on [0, T ). Moreover for each t ∈ [0, T ), Qn (t) is monotonically increasing in n. But we have an estimate Qn (t) ≤

T Qn−1 (r) C¯ ¯ + C dr 2θ (T − t) (r − t)2β t

for some number C¯ > 0. Thus we obtain Qn (t) ≤

C (T − t)2θ

for some number C = C(T ) > 0 independent of n. Hence, for each t there exists a strong limit Q(t) of Qn (t) and Q(t) has the desired properties. To examine the stability of the null solution of (5.1.4), it is useful to consider more general initial value y0 , e.g., y0 ∈ D(A−θ ). In fact, we have the following: Proposition 5.1.2

For each 0 ≤ t ≤ T and y0 ∈ Y ,

EP (t)Yt (y0 ), Yt (y0 )H = EF YT (y0 ), YT (y0 )H T + EM Yr (y0 ), Yr (y0 )H dr. t

Moreover, for each 0 ≤ t ≤ T and y0 ∈ D(A−θ ), 0 ≤ θ + η < 1/2, (y0 , P (0)y0 )D((A∗ )θ ) = EF YT (y0 ), YT (y0 )H +

T

EM Yr (y0 ), Yr (y0 )H dr, 0

where (·, ·)D((A∗ )θ ) denotes the duality between D(A−θ ) and D((A∗ )θ ).

Some Related Topics of Stability and Applications Proof

233

The proof is similar to that of Proposition 2.2.2.

Now we can derive our stability results for mild solutions of (5.1.4). The three statements below are equivalent:

Theorem 5.1.1

(i) The null solution of (5.1.4) is L2 -stable in mean, i.e., ∞ EYt (y0 )2H dt < ∞ for any y0 ∈ D(A−θ ).

(5.1.7)

0

(ii) There exists a solution 0 ≤ P ∈ L(D(A−θ ), D((A∗ )θ )) ∩ L(H) to the Lyapunov equation 2P y, −AyH + (A∗ )θ P Aθ b, bH f 2 (y) = −y2H , y ∈ D(A). (5.1.8) (iii) For each 0 < T < ∞, −2θ Ct y0 2−θ , 0 < t ≤ T, EYt (y0 )2H ≤ Ce−αt y0 2−θ , t > T, for some α > 0, C = C(T ) > 0, (5.1.9) where · −θ is the norm of D(A−θ ). Proof

We ﬁrst assume (i) holds, then ∞ EYt (Aθ z)2H dt < ∞ for any z ∈ H. 0

Thus from Proposition 5.1.2 with F = 0, M = I, we have T (A∗ )θ PT (0)Aθ z, zH = EYt (Aθ z)2H dt, 0

where PT (t) is the solution of (5.1.5). Thus (A∗ )θ PT (0)Aθ ↑ Q ≥ 0, as T → ∞, in L(H). Let P = (A∗ )−θ QA−θ , then PT (0) → P in a strong sense, as T → ∞, in L(H) ∩ L(D(A−θ ), D((A∗ )θ )). Since PT (t) = PT −t (0), we conclude that P satisﬁes (5.1.8). Hence, (i) implies (ii). Conversely, suppose P is a solution of (5.1.8). Then by Proposition 5.1.2, we have T (y0 , P y0 )D((A∗ )θ ) = EP YT (y0 ), YT (y0 )H + EYt (y0 )2H dt. 0

Thus (i) is true. Now we assume (iii) holds, then (i) follows immediately. Finally, we assume (i) holds and show (iii). To this end, using Theorem 3.1.1 we obtain EYt (y0 )2H ≤ C1 · e−αt y0 2H , y0 ∈ H,

234

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

for some α > 0, C1 > 0. Then for each 0 < T < ∞, we have EYt (y0 )2H ≤ C1 · e−α(t−T ) EYT (y0 )2H for any t ≥ T. On the other hand, we have from Proposition 5.1.2 that EYt (y0 )2H ≤ C2 t−2θ y0 2−θ ,

t ∈ (0, T ]

for some C2 > 0. Combining these two estimates, we obtain (5.1.9). Remark

The condition (5.1.7) may be replaced by ∞ EYt (y0 )2H dt ≤ Cy0 2−θ for some C > 0. 0

If y0 ∈ H, then (i) or (ii) implies EYt (y0 )2H ≤ M · e−αt y0 2H for some M ≥ 1 and α > 0. Next we give suﬃcient conditions for stability. Note that, in the present situation, the Lyapunov equation (5.1.8) is equivalent to ∞ (y0 , P y0 )D((A∗ )θ ) = T (t)y0 2H + (A∗ )θ P Aθ b, bH f 2 (T (t)y0 ) dt, 0

(5.1.10) where y0 ∈ D(A−θ ). We may equally consider as before ∞ Qy, yH = T (t)Aθ y2H + Qb, bH f 2 (T (t)Aθ y) dt, y ∈ H. (5.1.11) 0

As an immediate consequence, we obtain: If the inequality ∞ b2H |f (T (t)Aθ y)|2 dt ≤ ρy2H ,

Proposition 5.1.3

y ∈ H,

(5.1.12)

0

for some 0 ≤ ρ < 1 holds, then there exists a unique nonnegative solution to (5.1.10) and hence to (5.1.11). ∞ −λn t Corollary 5.1.1 Suppose T (t)y = ψn ψn , yH for some orn=1 e thonormal basis {ψn } of H and λn > 0. If f (y) = f, yH for some f ∈ H, then (5.1.12) is written as b2H

∞ n=1

fn2 /2λ1−2θ ≤ ρ < 1, fn = f, ψn H . n

(5.1.13)

Some Related Topics of Stability and Applications

235

Finally, we examine two examples to illustrate our theory derived above. Example 5.1.1 Firstly, consider the example (5.1.1) at the beginning of this section. It is known that (5.1.1) may be formulated in terms of the semigroup model (5.1.2). We may take β = 1/2 − ε for any small ε > 0. If we assume f (y) = f, yH for some f ∈ H, then there exists a unique {Ft }t≥0 -adapted solution of (5.1.2) in the space L2 ((0, T )×Ω; D(A1/4−ε ))∩C((0, T ]; L2 (Ω; D(A1/4−ε )))∩C([0, T ]; L4−ε (Ω; H)), more suggestively, L2 ((0, T ) × Ω; D(A1/4− )) ∩ C((0, T ]; L2 (Ω; D(A1/4− ))) ∩ C([0, T ]; L4− (Ω; H)). If we take O = (0, 1) and A = −d2 /dx2 + 1, D(A) = {y ∈ H 2 (0, 1) : y (0) = y (1) = 0}, then (5.1.1) may represent   ∂y(t, x)/∂t = ∂ 2 y(t, x)/∂x2 − y(t, x), x ∈ (0, 1), t ≥ 0, y(0, x) = y0 (x), x ∈ (0, 1), (5.1.14)  ∂y(t, 0)/∂x = −kf, y(t, 0)H B˙ t , ∂y(t, 1)/∂x = 0, t ≥ 0. It is easy to see that the solution of y (x) − y(x) = 0, y (0) = a, y (1) = 0, is given by m(x) = a cosh(1 − x)/ sinh 1. Note that T (t)y = e−t y, 1H + 2

∞

e−(1+n

2

π 2 )t

cos nπxcos nπx, yH

n=1

and

1

m0 =

m(x)dx = a 0

mn =

1

√

2 cos nπx · m(x)dx =

√

2a/(1 + n2 π 2 ).

0

If we assume f H = 1, then (5.1.13) with θ = 1/2 yields k2

1 2

This is satisﬁed if k

2

+

∞ n=1

1

nπ 2 < 1. 1 + n2 π 2

∞ 1 1 < 1. + 2 π 2 n=1 n2

236

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Thus if k 2 < 3/2, the mild solution of (5.1.14) is exponentially stable in mean square. Example 5.1.2 Consider the stochastic parabolic equation dy(t, x) = [∂ 2 y(t, x)/∂x2 ]dt + kδ(x − ξ)f, yH dBt , 0 < x, ξ < 1, y(t, 0) = y(t, 1) = 0, y(0, x) = y0 (x), (5.1.15) where δ(·) is the usual delta function at zero. In this case we take H = L2 (0, 1) and A = −d2 /dx2 , D(A) = H01 (0, 1) ∩ H 2 (0, 1). The semigroup model for (5.1.15) is t Yt = T (t)y0 + AT (t − s)gkf, Ys H dBs , (5.1.16) 0

where g is the Green’s function g(x, ξ) = Since

1

√

(1 − ξ)x, 0 ≤ x < ξ, (1 − x)ξ, ξ ≤ x ≤ 1.

2 sin nπx · g(x, ξ)dx =

√

2 sin nπξ/(nπ)2 ,

0

we have g ∈ D(A3/4− ). Hence, (5.1.16) is written as

t

Aθ T (t − s)bkf, Ys H dBs ,

Yt = T (t)y0 + 0

where θ = 1/4 + ε, ε > 0 and b = A1−θ g. Thus, all the conditions in Ichikawa [7] to ensure the existence and uniqueness of mild solutions are satisﬁed. To obtain a suﬃcient condition for stability we assume f H = 1 and apply (5.1.13) with θ = 1/2 to get ∞ k 2 1 < 1. π n=1 n2

Hence, if k 2 < 6, the system is exponentially stable in mean square.

5.2

Stochastic Stability and Quadratic Control

Let Xtu , t ≥ 0, be a family of properly deﬁned processes in H with X0u = x0 . The parameter u associated with each member is termed a control. Generally, each u may be identiﬁed with a speciﬁc member of a given family of functions

Some Related Topics of Stability and Applications

237

of Xtu which takes values of the form u = K(t, Xtu ) depending on t and Xtu . The object of control theory is to select the control so that the corresponding process possesses some desired properties. For instance, one object may be to transfer an initial state to some target set of H, for instance, {0}, that is, choose a value of u so that Xtu → 0 as t approaches some time, possibly inﬁnity. This may be developed further by associating to each control a cost J(u; x0 ). A problem of optimal control is to select u so that Xtu → 0, and meanwhile minimizes the cost J(u; x0 ) with respect to other controls of a speciﬁed class of the so-called “admissible comparison controls”. For many years, much eﬀort has been devoted to the study of optimal control and system theory for deterministic partial diﬀerential equations and functional diﬀerential equations, by Butkovskii [1], Curtain and Pritchard [1], Lions [2], Manitius [1] and Wang [2] among others. These equations are also known as distributed parameter systems and can be described by several mathematical models. In this section, we shall focus our attention on systems as above but subject to some random environmental eﬀects. More precisely, we shall formulate stochastic distributed parameter systems as inﬁnite dimensional stochastic diﬀerential equations. This formulation is helpful not only in considering the major concepts of controllability, observability and stability in system theory, but in posing various problems in optimal control and ﬁltering. We take this approach based on the theory of semigroups and stochastic evolution equations. In particular, we shall employ dynamic programming methods which enable us to determine the optimal feedback control and study its relationship with the stability property of linear stochastic systems. The design of controls to ensure exponential stability of mild solutions of a class of semilinear stochastic evolution equations is discussed in Section 5.3.

5.2.1

Optimal Control on a Finite Interval

Let (Ω, F, {Ft }t≥0 , P ) be a complete probability space and H, Ki , i = 1, 2, 3, real separable Hilbert spaces. Consider the controlled stochastic diﬀerential equation on H   dXt = (AXt + Bu(t))dt + G(Xt )dWt1 + F dWt2 +C(u(t))dWt3 , 0 ≤ t ≤ T, (5.2.1)  X0 = x0 ∈ H, where 0 ≤ T < ∞, A is the inﬁnitesimal generator of a strongly continuous semigroup T (t), t ≥ 0, on H, u(t) is a control with values in a real separable Hilbert space U , B ∈ L(U, H), G ∈ L(H, L(K1 , H)), F ∈ L(K2 , H), C ∈ L(U, L(K3 , H)), Wti are independent Ki -valued Wiener processes with covariance operators Qi , trQi < ∞, i = 1, 2, 3. For each u(t), it is easy to show as in Theorem 1.3.4 that there exists a unique mild solution of (5.2.1) in C(0, T ; L2 (Ω, F, P ; H)).

238

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

We can deﬁne the quadratic cost functional as follows: T J(u; x0 ) = ELXT , XT H + E M Xt , Xt H + N u(t), u(t)U dt (5.2.2) 0

where 0 ≤ L, M ∈ L(H) and 0 < N ∈ L(U ) with inverse N −1 ∈ L(U ). Our control problem is then to minimize (5.2.2) over L2 ([0, T ] × Ω; U ). In ﬁnite dimensions optimal control of this type is well known and is given in terms of the solution of a Riccati equation. This is one of the best examples for which dynamic programming gives a complete solution. We shall show that this is the case in inﬁnite dimensions as well. A feedback control is deﬁned as a map K(t, x) : [0, T ] × H → U such that the integral equation given by (5.2.1) with u(t) = K(t, Xt ) has a unique continuous solution Xt , 0 ≤ t ≤ T . If u(t) of this form is adapted to {FtW }, the σ-ﬁeld generated by Wti − Wsi , 0 ≤ s ≤ t ≤ T , T i = 1, 2, 3 with 0 Eu(t)2U dt < ∞, we say that u(t) is admissible. Note that the feedback control u = K(t, x) is admissible if K(t, x) : [0, T ] × H → U is measurable in t and satisﬁes K(t, x)U ≤ c(1 + xH ), K(t, x) − K(t, y)U ≤ cx − yH ,

c > 0, c > 0,

∀x ∈ H, ∀x, y ∈ H.

(5.2.3)

Indeed, the equation (5.2.1) with u(t) = K(t, Xt ) has a unique mild solution Xt , 0 ≤ t ≤ T , and u(t) = K(t, Xt ) is admissible. Now deﬁne Γ(S)u, vU = tr(C ∗ (v)SC(u)Q3 ), ∆(S)u, vU = tr(G∗ (v)SG(u)Q1 ),

S ∈ L(U ), S ∈ L(U ),

u, v ∈ U, u, v ∈ U,

(5.2.4)

and 1 (Lu Λ)(x) = Λ (x), Ax + BuH + Γ(Λ (x))u, uU + ∆(Λ (x))x, xH 2 +tr(F ∗ Λ (x)F Q2 ) , x ∈ D(A), u ∈ U, for any twice Fr´echet diﬀerentiable function Λ(x) on H. The following theorem gives suﬃcient conditions for optimality. ¯ x) and a Theorem 5.2.1 Suppose there exists a feedback control u ¯ = K(t, real function Λ(t, x) on [0, T ] × H with the properties: (a) Λ(t, x) is continuous on [0, T ] × H, (b) Λ(t, x) is twice Fr´echet diﬀerentiable in x, moreover, Λx (t, x)x1 and Λxx (t, x)x1 , x2 H are continuous for all x1 , x2 ∈ H, (c) Λ(t, x) is diﬀerentiable in t for any x ∈ D(A) and Λt (t, x) is continuous on [0, T ] × D(A), (d) |Λ(t, x)| + xH Λx (t, x)H + xH Λxx (t, x) ≤ c(1 + x2H ), c > 0, x ∈ H,

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(e) Λ(T, x) = Lx, xH , x ∈ H, (f ) 0 = Λt (t, x) + (Lu¯ Λ)(t, x) + M x, xH + N u ¯, u ¯U ≤ Λt (t, x) + (Lu Λ)(t, x) + M x, xH + N u, uU for any x ∈ D(A) and u ∈ U , ¯ x) is continuous and satisﬁes the condition (5.2.3). (g) K(t, ¯ x) is optimal and the minimal cost is Then the feedback control u ¯ = K(t, J(¯ u; x0 ) = Λ(0, x0 ). ¯ t , 0 ≤ t ≤ T , be the mild solution of (5.2.1) with u ¯ x). Proof Let X ¯ = K(t, Introducing an approximation of the form (2.2.7) to (5.2.1) and applying Itˆ o’s formula to Λ(t, x), we can show as in Proposition 2.2.2

T

¯ T ) − Λ(0, x0 ) = − EΛ(T, X

¯t, X ¯ t H + N u E M X ¯(t), u ¯(t)U dt.

0

Hence, ¯T , X ¯ T H + Λ(0, x0 ) = ELX

T

¯t, X ¯ t H +N u u; x0 ). E M X ¯(t), u ¯(t)U dt = J(¯

0

Repeating the same procedure for the solution Xt of (5.2.1) with arbitrary admissible control u(t), we obtain Λ(0, x0 ) ≤ ELXT , XT H +

T

E M Xt , Xt H +N u(t), u(t)U dt = J(u; x0 ).

0

Here the inequality follows from (f). To solve the quadratic cost problem (5.2.1) and (5.2.2), we need to ﬁnd a ¯ x) satisfying the conditions of Theorem function Λ(t, x) and a control u ¯ = K(t, 5.2.1. We seek a function Λ(t, x) of the form Λ(t, x) = Q(t)x, xH + q(t),

Q(t) ∈ L(H).

Then (f) yields the following Riccati equation: d Q(t)x, xH + 2Ax, Q(t)xH + M x, xH + ∆(Q(t))x, xH dt −Q(t)B[N + Γ(Q(t))]−1 B ∗ Q(t)x, xH = 0, x ∈ D(A), Q(T ) = L, q(t) = t

T

tr(F ∗ Q(r)F Q2 )dr

(5.2.5)

(5.2.6)

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

and the feedback control: u ¯ = −[N + Γ(Q(t))]−1 B ∗ Q(t)x.

(5.2.7)

So if we can establish a solution of the Riccati equation (5.2.5) shown below, the proof is complete. Theorem 5.2.2 The Riccati equation (5.2.5) has a unique solution in the class of self-adjoint nonnegative strongly continuous L(H)-valued functions. The control law (5.2.7) is optimal and the minimum cost is

T

J(¯ u; x0 ) = Q(0)x0 , x0 H +

tr(F ∗ Q(t)F Q2 )dt.

0

Proof In a manner parallel to ﬁnite dimensional cases, take the sequence of linear diﬀerential equations: d Q0 (t)x, xH + 2Ax, Q0 (t)xH + [M + ∆(Q0 (t))]x, xH = 0, x ∈ D(A), dt Q0 (T ) = L; d Qn (t)x, xH + 2[A − BKn−1 (t)]x, Qn (t)xH + [M + ∆(Qn (t))]x, xH dt ∗ +Kn−1 (t)[N + Γ(Qn (t))]Kn−1 (t)x, xH = 0, x ∈ D(A), Qn (T ) = L; (5.2.8) Kn (t) = [N + Γ(Qn (t))]−1 B ∗ Qn (t). By a similar argument to Proposition 2.2.1, we can show that these equations are equivalent to the following integral equations

T

T (r − t)[M + ∆(Q0 (r))]T (r − t)xdr

Q0 (t)x = 0

∗

+T (T − t)LT (T − t)x, x ∈ H, T Qn (t)x = Un∗ (r, t){M + ∆(Qn (r))

(5.2.9)

0

∗ +Kn−1 (r)[N + Γ(Qn−1 (r))]Kn−1 (r)}Un (r, t)xdr +Un∗ (T, t)LUn (T, t)x, x ∈ H,

where Un (t, s) is the perturbation of T (t) by −BKn−1 (t). Moreover, they have a unique solution. As in Proposition 2.2.2 we can show J(un ; x0 ) = Qn (0)x0 , x0 H + 0

T

tr(F ∗ Qn (t)F Q2 )dt

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where un is the control law un = −Kn (t)x,

n = 1, 2, · · · ,

u0 = 0.

Next we shall show Qn−1 (t) ≥ Qn (t) ≥ 0, n = 1, 2, · · ·. Set Rn (t) = Qn−1 (t) − Qn (t), then it satisﬁes d Rn (t)x, xH + 2[A − BKn−1 (t)]x, Rn (t)xH + ∆(Rn (t))x, xH dt ∗ +Kn−1 (t)[N + Γ(Qn−1 (t))]Kn−1 (t)x, xH = 0, x ∈ D(A), (5.2.10) Rn (T ) = 0; Rn (t)x =

T

Un∗ (r, t){∆(Rn (r))

0

∗ +Kn−1 (r)[N + Γ(Qn−1 (r))]Kn−1 (r)}Un (r − t)xdr,

(5.2.11)

x ∈ D(A). But (5.2.11) has a unique solution Rn (t) ≥ 0, thus necessarily Qn−1 (T ) ≥ Qn (t). Since Qn (t), n = 1, 2, · · ·, is the sequence of monotonically decreasing nonnegative operators, there exists a limit Q(t). Passing to the limit n → ∞ in (5.2.9) and then diﬀerentiating it, we can show that Q(t) satisﬁes (5.2.5). Letting n → ∞ in (5.2.9) yields T Q(t)x = U ∗ (r, t){M + ∆(Q(r)) + K ∗ (r)[N + Γ(Q(r))]K(r)}U (r, t)xdr 0

+U ∗ (T, t)LU (T, t)x, K(t) = [N + Γ(Q(t))]−1 B ∗ Q(t), (5.2.12) where U (t, s) is the perturbation of T (t) by −BK(t) and we have used the strong convergence of Un (t, s). The uniqueness of a solution of (5.2.5) (and hence (5.2.12)) and the rest of the theorem follows from Theorem 5.2.1.

5.2.2

Optimal Control on an Inﬁnite Interval

In this subsection, we ﬁrst take F = 0 in (5.2.1) and consider the resulting equation dXt = (AXt + Bu(t))dt + G(Xt )dWt1 + C(u(t))dWt3 , 0 ≤ t < ∞, X0 = x0 ∈ H, (5.2.13) and the quadratic cost functional ∞ J(u; x0 ) = (5.2.14) E M Xt , Xt H + u(t), u(t)U dt. 0

For admissible controls we take the class of feedback controls u = K(t, x) such that

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

(i) K(t, x) : [0, ∞) × H → U is measurable and for some c > 0, K(t, x)U ≤ c(1 + xH ),

K(t, x) − K(t, y)U ≤ cx − yH ,

for any x, y ∈ H, and (ii) EXt 2H → 0 as t → ∞, where Xt , t ≥ 0, is the mild solution of (5.2.13) with u = K(t, x). Deﬁnition 5.2.1 The system (5.2.13) (or (A, B, C, G)) is stabilizable if there exists K ∈ L(H, U ) such that the feedback law u = −Kx yields a L2 -stable mild solution Xt in mean, i.e., ∞ EXt (x0 )2H dt < ∞. 0

In this case, we also say that (A − BK, C, G) is stable. If (A, B, C, G) is stabilizable, then the control problem (5.2.13), (5.2.14) is meaningful one. In particular, similarly to Theorem 2.2.1, we may obtain the following results: Lemma 5.2.1 (A, B, C, G) is stabilizable if and only if there exists K ∈ L(H, U ) and 0 ≤ P ∈ L(H) such that 2(A − BK)x, P xH + [K ∗ Γ(P )K + ∆(P )]x, xH = −x, xH ,

x ∈ D(A),

where Γ(P ) and ∆(P ) are deﬁned by (5.2.4). Lemma 5.2.2 such that

If (A − BK, C, G) is stable, then there exists 0 ≤ Q ∈ L(H)

2(A − BK)x, QxH + (M + K ∗ N K)x, xH +[∆(Q) + K ∗ Γ(Q)K]x, xH = 0, Proof

x ∈ D(A).

(5.2.15)

Let QT (t), 0 ≤ t ≤ T , be the unique solution of

d dt Q(t)x,

xH + (A − BK)x, Q(t)xH + (M + K ∗ N K)x, xH +[∆(Q(t)) + K ∗ Γ(Q(t))K]x, xH = 0, x ∈ D(A),

(5.2.16)

Q(T ) = 0. Similarly to Proposition 2.2.2, we then have T QT (0)x0 , x0 H = E M Xt , Xt H + N KXt , KXt H dt 0

where Xt is the solution of (5.2.13) with u = −Kx. Since (A − BK, C, G) is stable, QT (0) is uniformly bounded in T . But QT (0) is monotonically

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243

increasing and non-negative. So there exists a limit Q ≥ 0 and it satisﬁes (5.2.15). ¯ Theorem 5.2.3 Suppose that there exists an admissible control u ¯ = −K(x) and a real-valued function Λ(x) on H such that (a) Λ(x) is twice Fr´echet diﬀerentiable and Λ(x), Λ (x), Λ (x) are continuous, (b) |Λ(x)| + xH Λ (x)H + x2H Λ (x) ≤ cx2H , x ∈ H, c > 0, (c) 0 = (Lu¯ Λ)(x) + M x, xH + N u ¯, u ¯U ≤ (Lu Λ)(x) + M x, xH + N u, uU for any x ∈ D(A) and u ∈ U , ¯ ¯ (d) K(x) − K(y) U ≤ cx − yH , x, y ∈ H, c > 0. ¯ Then u ¯ = −K(x) is optimal and J(¯ u; x0 ) = Λ(x0 ). Proof

As in Theorem 5.2.1 one can show t ¯ t ) − Λ(x0 ) = − ¯r , X ¯ r H + N u EΛ(X E M X ¯(r), u ¯(r)U dr 0

¯ t is the mild solution of (5.2.13) with u = u where X ¯. Note that |Λ(x)| ≤ cx2H 2 ¯ t → 0 as t → ∞. So and EX H

∞

Λ(x0 ) =

¯t, X ¯ t H + N u u; x0 ). E M X ¯(t), u ¯(t)U dt = J(¯

0

In a similar way, for any admissible control u we obtain ∞ Λ(x0 ) ≤ E M Xt , Xt H + N Xt , Xt H dt = J(u; x0 ). 0

Now we seek a function Λ(x) of the form Λ(x) = Qx, xH ,

0 ≤ Q ∈ L(H).

Then (c) above yields an algebraic Riccati equation 2Ax, QxH + {M + ∆(Q) − QB[N + Γ(Q)]−1 B ∗ Q}x, xH = 0,

x ∈ D(A), (5.2.17)

and the control law u = −Kx, K = [N + Γ(Q)]−1 B ∗ Q.

(5.2.18)

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Hence, to conclude our proof, we need only establish conditions which guarantee the existence of a solution to (5.2.17) and the admissibility of u ¯ = −Kx. Claim: If there exist operators K ∈ L(H, U ), 0 ≤ Q ∈ L(H) satisfying (5.2.15), then the Riccati equation (5.2.17) has a solution. Indeed, let QT (t), QT (t) be the solutions of (5.2.14) and of (5.2.5) with L = 0, respectively. Then 0 ≤ QT (t) ≤ QT (t) ≤ Q. Since QT (t) is monotonically increasing in T , there exists a limit of QT (t) as T → ∞ which is independent of t and satisﬁes (5.2.17). Now we are in a position to summarize one of the main results in this subsection as follows. Theorem 5.2.4 Suppose that there exists some K ∈ L(H, U ) such that (A − BK, C, G) is stable. Then there is a unique solution Q ≥ 0 for the Riccati equation (5.2.17). Moreover, the optimal control for (5.2.13) is the feedback law (5.2.18) and J(¯ u; x0 ) = Qx0 , x0 H . Proof The existence of a solution to (5.2.17) follows from Theorem 5.2.3. Since Qx0 , x0 H is the minimum cost, Q is unique. If we keep the term F in (5.2.13), we may consider optimal stationary controls in terms of invariant measures. Precisely, we consider the stochastic control system (5.2.1) together with the class of feedback controls K(x) : H → U with K(x) − K(y)U ≤ cx − yH ,

c > 0,

x, y ∈ H.

(5.2.19)

We say that the feedback control u = −K(x) is admissible if it satisﬁes (5.2.19) and the Markov process given by (5.2.1) with u = −K(x) has an invariant measure µK such that x2H µK (dx) < ∞.

(5.2.20)

H

Our control problem is to minimize J(u; x0 ) = M x, xH + N K(x), K(x)U µK (dx) H

over all admissible controls. Clearly, the cost is independent of the initial value x0 , thus it is also denoted by J(u). ¯ Theorem 5.2.5 Suppose that there exist an admissible control u ¯ = −K(x), a number γ and a real-valued function Λ(x) on H such that

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245

(a) Λ(x) is twice Fr´echet diﬀerentiable and Λ(x), Λ (x), Λ (x) are continuous, (b) |Λ(x)| + xH Λ (x)H + x2H Λ (x) ≤ c(1 + x2H ), x ∈ H, c > 0, (c) γ = (Lu¯ Λ)(x) + M x, xH + N u ¯, u ¯U ≤ (Lu Λ)(x) + M x, xH + N u, uU for any x ∈ D(A) and u ∈ U . ¯ Then u ¯ = −K(x) is optimal and J(¯ u) = γ. ¯ t (ξ) be the mild solution of (5.2.1) with u = −K(x) ¯ Proof Let X and x0 = ξ. As in Theorem 5.2.1 we can derive ¯ (ξ)) − Λ(ξ) EΛ (X tt

¯ r (ξ), X ¯ r (ξ)H + N K( ¯ X ¯ r (ξ)), K( ¯ X ¯ r (ξ))U dr. = γ − E M X 0

(5.2.21) ¯ t and P (·, t, ·) the transition function. Now let µ ¯ be an invariant measure of X Then ¯ EΛ(Xt (ξ))¯ µ(dξ) = Λ(η)P (ξ, t, dη) µ ¯(dξ) = Λ(ξ)¯ µ(ξ). H

H

H

H

Taking expectations of (5.2.21) with respect to µ ¯ yields that 0=

t

¯ ¯ γ− M ξ, ξH + N K(ξ), ¯(dξ) dr. K(ξ) U µ 0

Hence,

H

γ=

¯ ¯ M x, xH + N K(x), ¯(dx) = J(¯ u). K(x) U µ

H

Similarly, for any admissible control u = −K(x) we have γ ≤ J(u).

It is natural as we did in Theorem 5.2.3 to seek a function Λ(x) of the form Λ(x) = Qx, xH . Then (c) above yields the Riccati equation (5.2.17), the control law (5.2.18) and γ = tr(F ∗ QF Q2 ).

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

In particular, by a similar argument to those in Theorems 5.3.3 and 5.3.4 we can prove: Theorem 5.2.6 Suppose that there exists some K ∈ L(H, U ) such that (A−BK, C, G) is stable. Then the optimal control is the feedback law (5.2.18) and the minimum cost is J(¯ u) = tr(F ∗ QF Q2 ) where Q ≥ 0 is the unique solution of the Riccati equation (5.2.17). Example 5.2.1 Consider the scalar stochastic diﬀerence diﬀerential equation dy(t) = [a0 y(t) + a1 y(t − b) + u(t)]dt + σy(t)dBt , y(θ) = y0 (θ), −b ≤ θ ≤ 0, a0 > 0, a1 > 0, b > 0, and the cost functional J(u) =

∞

[y 2 (t) + N u2 (t)]dt, N > 0,

0

where Bt , t ≥ 0, is a standard one dimensional Brownian motion. Let us take the feedback control u = −ky. It is possible to show that all the conditions in Theorem 5.2.4 are satisﬁed 0 for k suﬃciently large. Then the feedback control u ¯(·) = −N −1 [Q00 y(t) + −b Q01 (θ)y(t + θ)dθ] is optimal where

Q=

Q00 Q01 Q10 Q11

and kernels Q01 (θ), Q11 (θ) of Q01 , Q11 and Q00 satisfy the following: 2Q00 a0 + Q10 (0) + Q01 (0) + 1 − N −1 Q200 + σ 2 Q00 = 0, dQ01 (θ) = (a0 − N −1 Q00 )Q01 (θ) + a0 Q11 (θ, 0), Q01 (−b) = a1 Q00 , dθ ∂ ∂ + Q11 (θ, η) − N −1 Q10 (θ)Q01 (η) = 0, ∂θ ∂η Q11 (−b, η) = a1 Q01 (η), Q11 (θ, −b) = Q10 (θ)a1 . The reader is also referred to Delfour, McCalla and Mitter [1] for some more details of this formulation.

5.3

Feedback Stabilization of Stochastic Diﬀerential Equations

If a control problem is posed as a well-formulated mathematical optimization problem, as in the introduction to Section 5.2, then it is natural at least

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247

to attempt to compute the optimizing control. Owing to the diﬃculty of the computational problem, this is not always possible. In addition, practical control problems are not usually posed as well-formulated mathematical optimization problems. The goal which the control is designed to accomplish may be phrased somewhat loosely. One may desire a control which will guarantee that a given target set is attained with probability one at some random time. It may also be desired that the control, which accomplishes a given task, does not take “large” values with a high probability. In ﬁnite dimensional cases, it is a long story in the use of Lyapunov function methods to design controls which will satisfy such qualitative requirements. See, for example, Geiss [1] for the deterministic and Kushner [1], [2] for the stochastic cases. In this section, based on the perturbation theory of semigroups and some stability results established in Section 3.3 we shall present a theory to design a proper state feedback control law u(t), t ≥ 0, so that the resulting process has exponential stability uniformly with respect to perturbations. Let H, U and K be three real separable Hilbert spaces. The question of stability of the semilinear stochastic evolution equation dXt = (AXt + F (Xt ))dt + G(Xt )dWt ,

X0 = x0 ,

t ≥ 0,

has been considered in Section 3.3, where A generates a strongly continuous semigroup T (t), t ≥ 0, on H, F (·) and G(·) are nonlinear and satisfy Lipschitz conditions with F (0) = 0 and G(0) = 0. That is, F and G are bounded perturbations. But in many practical situations, F and G might be unbounded, even uncertain. The problem of stabilization is to design a state feedback control law that assures exponential stability of the zero state uniformly with respect to the perturbations F and G. Precisely, we wish to consider the following controlled stochastic system dXt = (AXt + F (Xt ))dt + G(Xt )dWt + Bu(t)dt,

X0 = x0 ,

t ≥ 0, (5.3.1)

on H. Here B is a bounded linear operator from U to H, and G : H → L(K, H) is nonlinear and F : H → H is nonlinear unbounded operators. As usual, Wt , t ≥ 0, is some K-valued process with covariance operator Q such that trQ < ∞. Deﬁnition 5.3.1 A linear operator L from a subset D(L) ⊂ H, domain of L, to H is said to be dissipative if x, LxH ≤ 0

for all

x ∈ D(L).

The following perturbation result for linear operators is standard but essential in the arguments which follow. The reader is referred to, for instance, Pazy [1], for its proof.

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Lemma 5.3.1 Let L and T be linear operators in H such that D(L) ⊂ D(T ). Assume that L is the inﬁnitesimal generator of a C0 -semigroup of contractions. Suppose T is dissipative and satisﬁes T xH ≤ aLxH + bxH

for x ∈ D(L)

where 0 ≤ a < 1 and b ≥ 0. Then L + T is the inﬁnitesimal generator of a C0 -semigroup of contractions. We shall impose the following assumptions on the operators A, F (·) and G(·) of (5.3.1). A1. The pair {A, B} is exponentially stabilizable, i.e., there exists a D1 ∈ L(H, U ) such that A¯ := A + BD1 generates an exponentially stable contraction semigroup S(t), t ≥ 0, in H satisfying S(t) ≤ e−µt , t ≥ 0, for some µ > 0. A2. F (·) = F1 (·) + F2 (·), where F1 (·) ∈ {F : F is a linear operator in H and D(A) ⊂ D(F ) ⊂ H, x, F xH ≤ kx, AxH , k ≥ 0, and there exist constants 0 ≤ a < 1, b ≥ 0 such that F (x)H ≤ aAxH + bxH for all x ∈ D(A)}, and F2 (·) ∈ {F : F maps H into H and satisﬁes Lipschitz conditions with F (0) = 0, i.e., there exists a contant l1 > 0 such that F (x) − F (y)H ≤ l1 x − yH , x, y ∈ H}. A3. The linear operator BB ∗ is coercive, i.e., x, BB ∗ xH ≥ θx2H for some θ > 0 and x ∈ H, where B ∗ is the adjoint operator of B. A4. The operator G : H → L(K, H) is nonlinear and satisﬁes Lipschitz conditions with G(0) = 0, i.e., G(x) − G(y) ≤ l2 x − yH , x, y ∈ H for some constant l2 > 0. Theorem 5.3.1 Suppose the assumptions A1, A2, A3 and A4 hold. Then the mild solution of the system (5.3.1) with a linear feedback control law given by u(t) = D2 Xt , t ≥ 0, where D2 := (1 + k)D1 −

1 (2l1 + l22 · trQ)B ∗ ∈ L(H, U ), 2θ

(5.3.2)

is exponentially stable in mean square uniformly with respect to the perturbations F (·) and G(·). That is, EXt (x0 )2H ≤ M x0 2H e−λt

for some M ≥ 1 and λ > 0.

Moreover, the null solution is also pathwise exponentially stable in the almost sure sense. That is, there exists a random variable 0 < T (ω) < ∞ and constants M1 ≥ 1, λ1 > 0 such that for all t ≥ T (ω), Xt (x0 )H ≤ M1 x0 H e−λ1 t

a.s.

Some Related Topics of Stability and Applications Proof

249

Using the feedback control law (5.3.2), deﬁne Ar = (A + BD1 ) + (F1 + kBD1 ) = A¯ + Ap , F¯ = F2 − [(2l1 + l22 · trQ)/2θ]BB ∗ ,

then the system (5.3.1) deduces to dXt = [Ar Xt + F¯ (Xt )]dt + G(Xt )dWt ,

X0 = x0 ∈ H,

t ≥ 0,

(5.3.3)

with F¯ (0) = 0 and G(0) = 0. First of all, we show that Ar generates a C0 -semigroup. By the assumption A1, the semigroup S(t), t ≥ 0, generated by A¯ satisﬁes S(t) ≤ e−µt ≤ 1 for t ≥ 0. Therefore, if one can show that Ar = A¯ + Ap is dissipative and Ap is relatively bounded with respect to A¯ with relative bound a < 1, it is easy to deduce by Lemma 5.3.1 that Ar generates a C0 -semigroup of contractions in H. Indeed, by means of A2 and Proposition 2.1.4 we have x, Ar xH ≤ x, F1 xH + kx, BD1 xH ≤ kx, (A + BD1 )xH ≤ −kµx2H ≤ 0 for x ∈ D(A) ⊂ D(Ar ). Hence, Ar is dissipative. On the other hand, we have Ap xH ≤ F1 xH + kBD1 xH ≤ aAxH + b1 xH ¯ H + b2 xH ≤ aAx

¯ ⊂ D(Ap ), for x ∈ D(A) = D(A)

where b1 = b + kBD1 , b2 = b1 + aBD1 . Hence, Ar = A¯ + Ap generates a C0 -semigroup of contractions in H. In order to conclude the proof, using the assumptions A1-A4, we obtain 2x, (A + BD2 )x + F (x)H + tr(G(x)QG∗ (x))

l1 ≤ 2x, (A + BD1 )xH + 2x, (F1 + kBD1 )(x)H + 2 x, F2 (x) − BB ∗ x θ H l22 · trQ x, BB ∗ xH θ = −dx2H , for x ∈ D(A),

+trQG(x)G∗ (x) − ≤ −2(1 + k)µx2H

where d = 2(1 + k)µ > 0. By means of Theorems 3.3.1 and 3.3.2, it is easy to complete the proof. We shall present a stochastic version analogous to Burgers’ mathematical models of turbulence to illustrate the theory derived above.

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Example 5.3.1 Consider a controlled semilinear stochastic diﬀusion equation   dzt (x) − c∆zt (x)dt + f (zt (x))dt − u(t, x)dt − g(zt (x))dBt = 0, t ≥ 0, x ∈ O = (0, 1),  zt (x)|∂O = 0, t ≥ 0; z0 (x) = h(x) for x ∈ (0, 1), which describes a large class of physical problems such as stochastic heat transfer, chemical diﬀusions and turbulences. Here Bt , t ≥ 0, is a one dimensional standard Brownian motion. In particular, let c = 1/R, R > 0,

1 1/2

6x ∂z(x) 6 2 2 2 f (z) = − √ · z (x)dx , z(x)−R − x +√ R ∂x R 0

h(x) = sin x,

and g(ξ) = sin ξ. On this occasion, one may associate with this equation in an obvious way a semilinear stochastic evolution equation of parabolic type dXt = (AXt + F (Xt ))dt + Bu(t)dt + G(Xt )dBt , X0 (x) = sin x, √ on the real Hilbert space H = L2 (0, 1) where A = (1/R)(d2 /dx2 )+x2 +(6/ R) with D(A) = {z ∈ H : z , z ∈ H, z(0) = z(1) = 0} and B = I. It is also clear that for v, w ∈ H, G(v) − G(w)H ≤ v − wH .

(5.3.4)

If we assume that R = 2 and D1 = −1, it is easy to deduce by Proposition 2.1.4 that A¯ = A + BD1 generates an exponentially stable contraction semigroup in H. Indeed,

π2 6 1 2 ¯ z, AzH ≤ − + x + √ − 1 z2H ≤ − z2H 2 2 2 ¯ = D(A). The operator F (z) is given by for z ∈ D(A) 6x dz(x) F (z) = F1 (z) + F2 (z) = √ · + 4zH . dx 2 Since z, F1 (z)H

1 6 dz(x) = √ xz(x) dx dx 2 0 1 1 3 3 = √ xz 2 (x) − √ z 2 (x)dx, 0 2 2 0

therefore, 3 z, F1 (z)H = − √ 2

1

z 2 (x)dx ≤ 0, 0

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z(x) ∈ D(F1 ) = {z ∈ H : z (x) ∈ H, z(0) = z(1) = 0}. Thus F1 is dissipative. Furthermore, F1 (z)2H

1 2

= 18

x 0

dz(x) dx

2 dx.

Hence, it is easy to verify that for any ε > 0, F1 (z)H ≤ εAzH + ν(ε)zH

for z ∈ D(A),

where ν(ε) → ∞ as ε → 0, i.e., F1 is bounded with A-bound zero in H. For F2 (z), F2 (v) − F2 (w)H = 4|vH − wH | ≤ 4v − wH

for v, w ∈ H.

By choosing L1 = 4 and k = 0 in D2 , it is easy to see, using (5.3.4), that all the conditions of Theorem 5.3.1 are satisﬁed. Hence, the feedback system is exponentially stable in the mean square and almost sure senses.

5.4

Stochastic Models in Mathematical Physics

This section is devoted to the problem of existence and uniqueness of invariant measures for two classes of important stochastic models, stochastic reaction-diﬀusion equations and Navier-Stokes equations, under various conditions on the coeﬃcients of equations, motivated by applications. To this end, we use the results derived in Section 3.5 in the spirit of Lyapunov functions.

5.4.1

Stochastic Reaction-Diﬀusion Equations

Assume O ⊂ Rn is a bounded domain with smooth boundary ∂O. Let A(x) denote a second-order elliptic operator in O deﬁned as A(x)ψ(x) =

n ∂ ∂ψ(x) aij (x, ∇ψ) ∂xi ∂xj i,j=1

for any ψ ∈ C 2 (O),

∂ψ ∂ψ where ∇ψ = ( ∂x , · · · , ∂x ) and aij ’s are arbitrarily given functions on O×Rn 1 n such that the matrix [aij ]n×n is positive-deﬁnite. Let f (x, r, y) and σi (x, r, y), i = 1, 2, · · · , m, be real-valued continuous functions in (x, r, y) ∈ O ×R1 ×Rn and Wti (x), for i = 1, · · · , m, be independent Wiener processes in H = L2 (O) with norm · H so that Wt = (Wt1 , · · · , Wtm ) is a Q-Wiener process in K = H m = H × · · · × H. The kernel matrix of the covariance operator Q of Wt is denoted by q(x, y) = [δij qi (x, y)]m×m , where δij is the Kronecker

252

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

delta function and qi is the covariance kernel of Wti . We assume that qi ’s are continuous in O × O such that qi (x) = qi (x, x) satisﬁes qi (x) ≤ q0 .

sup x∈O i=1,···,m

Now consider the parabolic Itˆ o equation of the form: du(t, x) = A(x)u(t, x)dt + f (x, u(t, x), ∇u)dt +

m

σi (x, u(t, x), ∇u)dWti (x),

i=1

u(t, ·)∂O = 0, t ≥ 0, u(0, x) = ξ(x), x ∈ O,

(5.4.1) where ξ(·) is supposed to satisfy ξ2H =

O

|ξ(x)|2 dx < ∞.

In Equation (5.4.1), the operator A(x) on u is to be interpreted in a variational sense as follows: n ∂u ∂v Au, vH = dx, ∀v ∈ Cb1 (O). aij (x, ∇u) (5.4.2) ∂x ∂x i j O i,j=1 For our purpose, let us impose the following conditions: (P.1). The functions aij and f are continuous and there exist constants α > 0, p ≥ 2 and q ≥ 2 such that n

aij (x, y)yi yj + |rf (x, r, y)| ≤ α(1 + ypRn + |r|q ),

i,j=1

r ∈ R1 , x ∈ O, y ∈ Rn . Then the operator ˜ A(v) = A(·)v + f (·, ·, ∇v)

(5.4.3)

is well deﬁned as a continuous operator from V = W01,p (O) ∩ Lq (O) into its dual V ∗ . For w ∈ K = H m , we set B(v)w =

m

σi (·, ·, ∇v)wi .

i=1

Then B(·) : V → L02 (K, H) if we assume that:

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253

(P.2). σi (x, r, y)’s are continuous in x and locally Lipschitz continuous in r and y such that σ(x, r, y)2Rm ≤ c(1 + y2Rn + |r|2 )

for some

c > 0.

In addition, suppose that the following two conditions hold: (P.3). There exist constants α > 0 and θ, γ ∈ R1 such that n

aij (x, y)yi yj − rf (x, r, y) −

i,j=1

q0 σ(x, r, y)2Rm ≥ αypRn + θ|r|2 + γ 2

for any r ∈ R1 , x ∈ O and y ∈ Rn ; (P.4). The operator A˜ : V → V ∗ deﬁned by (5.4.3) is monotone in the sense that 2 2 ˜ − A(v) ˜ 2u − v, A(u) V,V ∗ + q0 σ(·, ·, ∇u) − σ(·, ·, ∇v)K ≤ −δu − vV ,

for some δ ≥ 0 and arbitrary u, v ∈ V . Under the conditions (P.1)–(P.4), it is easy to check that all the assumptions of Corollary 3.6.1 are satisﬁed. Hence, there exists an invariant measure for Equation (5.4.1). If, in Condition (P.4), the constant δ is strictly positive, then the condition (3.6.12) in Corollary 3.6.1 is true so that the invariant measure is unique. Precisely, the following result holds. Proposition 5.4.1 If the conditions (P.1)–(P.4) hold, then the parabolic Itˆ o equation (5.4.1) has an invariant measure µ with support in W01,p (O)∩Lq (O). Moreover, if δ > 0 in (P.4), the invariant measure µ is unique. Let A = A0 be a linear strongly elliptic operator in O ⊂ R3 with aij (x, y) = a ˜ij (x) ∈ Cb (O) and there exists α0 > 0 such that 3

a ˜ij (x)yi yj ≥ α0 y2R3 ,

∀x ∈ O, y ∈ R3 .

(5.4.4)

j=1

Consider the following stochastic reaction-diﬀusion equation:  3 ∂u du(t, x) = A0 (x)u(t, x)dt + f (u)dt + j=1 σj (x) ∂x dBtj (x)   j  +σ4 (x)dBt4 (x), t ≥ 0, x ∈ O,  u(t, ·) ∂O = 0, t ≥ 0,   u(0, x) = ξ(x) ∈ L2 (O),

(5.4.5)

where f (u) = (µu − νu3 ), µ and ν are positive constants, σi (x)’s ∈ Cb (O) and Bti (x)’s are independent Wiener random ﬁelds with covariance functions qi ’s, i = 1, · · · , 4.

254

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Let H = L2 (O), V = H01 (O) ∩ L4 (O) and K = H 4 . It is easy to check that the conditions (P.1)–(P.3) hold for p = 2 and q = 4. To verify (P.4), we note that, with A˜ = A0 + f and by (5.4.4) and (5.4.5), 2 ˜ ˜ 2u − v, A(u) − A(v) V,V ∗ + q0 σ(u) − σ(v)K 2 2 ≤ −2α0 u − vV + 2νu − vH + 3q0 σ02 u − v2V ≤ −(2α0 − 2ν/λ0 − 3q0 σ02 )u − v2V , ∀u, v ∈ V,

where λ0 =

inf 1

v∈H0 vH =0

v2V > 0, v2H

v2V =

O

∇v(x)2R3 dx,

and σ0 = supx∈O σi (x) for i = 1, 2, 3, 4. Hence, (P.4) holds for ν and q0 so small that δ = 2α0 − 2ν/λ0 − 3q0 σ02 > 0. In this case, Equation (5.4.5) has a unique invariant measure supported in H01 (O) ∩ L4 (O) by Proposition 5.4.1. Now we may consider a generalization of Equation (5.4.1) where the elliptic operator A0 (x) is replaced by the nonlinear operator A(x)v(x) = A0 (x)v(x) + A1 v(x), where A1 v(x) =

3 ∂ ∂v(x) p−2 ∂v(x) , ∂xj ∂xj ∂xj j=1

(5.4.6)

p > 2.

Then the corresponding stochastic reaction-diﬀusion equation is 3 ∂u(t, x) p−2 ∂u(t, x) ∂u(t, x) ∂ a ˜ij (x) + δij + f (u(t, x)) = ∂t ∂x ∂x ∂xj i j i,j=1 (5.4.7) 3 ∂u(t, x) ˙ j 4 + Bt (x) + σ4 B˙ t (x). σi (x) ∂xj i=1

It is well known that the nonlinear elliptic operator is coercive and monotone in W01,p (O) (cf. see Lions [1]). Let V = W01,p ∩ L4 (O). Under the same conditions as in (5.4.4) and (5.4.5), we can apply Proposition 5.4.1 to conclude the existence of a unique invariant measure for Equation (5.4.5) with support in W01,p ∩ L4 (O).

5.4.2

Stochastic Navier-Stokes Equations

The theory of Navier-Stokes equations occupies a central position in the study of nonlinear partial diﬀerential equations, dynamical systems, and modern scientiﬁc computation, as well as classical ﬂuid dynamics. The mathematical theory of stochastic Navier-Stokes equations is rather technical and

Some Related Topics of Stability and Applications

255

already very large despite being still incomplete. In spite of their intuitive physical background from ﬂuids, these stochastic equations have also been extensively studied without direct reference to the conventional theory of turbulence. Some pioneer work in this respect goes back at least to Bensoussan and Teman [1] and Viot [1] among others. Some recent developments of this expanding topic can be found in the existing literature such as Da Prato and Zabczyk [1], [2] and references therein. As an application of Lyapunov function approaches, we investigate in this subsection invariant measures of a randomly perturbed Navier-Stokes equation describing a two-dimensional turbulent ﬂow. Let O ⊂ R2 be a bounded domain with a smooth boundary ∂O. Let u(t, x) = (u1 , u2 )(t, x) be the velocity ﬁeld and p(t, x) be the pressure ﬁeld in an incompressible ﬂuid. Then, under a random perturbation by Gaussian white noises, a ﬂuid ﬂow is governed by the stochastic Navier-Stokes equation: dui (t, x) +

2 j=1

uj

2 ∂ui (t, x) ∂ 2 ui (t, x) 1 ∂p(t, x) dt = − dt + ν dt ∂xj ρ ∂xi ∂x2j j=1

+σi dBti (x), 2 ∂uj (t, x) j=1

∂xj

= 0,

x ∈ O,

i = 1, 2,

t ≥ 0,

where ρ > 0 is the constant ﬂuid density, ν > 0 the kinematic viscosity, σi ’s the variance parameters, and Bt (x) = (Bt1 (x), Bt2 (x)) is a random force with associated covariance operators Qi , trQi < ∞, given by a positive deﬁnite kernel qi (x, y) ∈ L2 (O × O), qi (x, x) ∈ L2 (O), i = 1, 2. In a vectorial notation, the above equation takes a simpler form: du(t, x) + (u · ∇)udt = − ρ1 ∇pdt + ν∆udt + σdBt (x), t > 0, ∇ · u = 0,

x ∈ O,

(5.4.8) where ∇, ∇· and ∆ are the conventional notations for the gradient, divergence and Laplacian operators, respectively, and

σ1 0 σ= . 0 σ2 The equation is subject to the initial-boundary conditions: u(0, x) = ξ(x), u|∂O = 0, where ξ(x) is some proper initial velocity ﬁeld. Let C0∞ = {v ∈ [C0∞ (O)]2 : ∇·v = 0} and H the closure of C0∞ in [L2 (O)]2 , V = {v ∈ [H01 (O)]2 : ∇·v = 0}.

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Let V ∗ denote the dual of V , then it is known that V ;→ H ;→ V ∗ and the embeddings are compact. Let Π be the orthogonal projector from [L2 (O)]2 unto H, and deﬁne for any v ∈ C0∞ , A(v) = νΠ∆v − Π[(v · ∇)v]. Then A(·) can be extended as a continuous operator from V to V ∗ . The above equation can be recast as a stochastic diﬀerential equation in the form: du(t) = A(u(t))dt + σdWt , (5.4.9) u(0) = ξ, ξ ∈ V, where Wt is a Q-Wiener process in H instead of [L2 (O)]2 . Indeed, let u ∈ V be a solution of (5.4.8). By the well-known embedding theorem for Sobolev spaces, we obtain ∞ 2 ∂u 2 E uj j ≤ C sup Eu(t)2[L1 (O)]2 Eu(t)2[L2 (O)]2 dt ∂x [L2 (R+ ×O)]2 t≥0 0 j=1 for some C > 0. This estimate implies 2 ∂u 2 E uj j ≤ M Eu21,2 , ∂x [L2 (R+ ×O)]2 j=1

where M > 0 does not depend on u. Hence, all the terms of the system (5.4.8) containing u(t, ·) for any t ≥ 0 belong to L2 (Ω × O), and ∇p ∈ [L2 (Ω × R+ × O)]2 . Therefore, the operator Π can be applied to both sides of (5.4.8). Since ∇p · wdx = − p div wdx = 0, O

O

for any w ∈ V , then Π∇p(t, ·) = 0 for almost all t ≥ 0. Note that by the deﬁnition of V , we have Π∂u/∂t = ∂u/∂t for almost all t ≥ 0. Let −νΠ∆H 2 = {−νΠ∆v : v ∈ H 2 (O)} and then we intend to show −νΠ∆H 2 = H. But this is immediate. Indeed, it is known (cf. Solonnikov [1]) that for any f ∈ H, there exists v ∈ H 2 (O) such that −νΠ∆v = f and vH 2 ≤ Cf H , where C > 0 does not depend on f ∈ H. Hence, −νΠ∆H 2 ⊃ H.

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Clearly, −Π∆H 2 ⊂ H. Applying Π to (5.4.8), we see easily from the above that u is a solution of (5.4.8). On the other hand, by employing a Galerkin approximation technique it can be deduced (cf. Vishik and Fursikov [1], p. 347) that the problem (5.4.9) above has a unique strong solution {uξ (t); t ≥ 0} satisfying: Euξ (T )2H + νE 0

T

2 T ∂uξ (t) 2 dt ≤ Eξ2H + trQ. ∂xi H 2 i=1

ξ 2 1/2 ∂u (t) 2 Hence, using the fact that uξ (t)V is equivalent to , we i=1 ∂xi H get 1 T C lim inf Euξ (t)2V dt ≤ trQ T →∞ T 0 2ν where C > 0 is some constant. Using Markov’s inequality, we obtain 1 T ξ lim lim inf P u (t)V > R dt = 0 R→∞ T →∞ T 0 which, together with Corollary 3.5.2, immediately yields the desired result, namely the existence of an invariant measure for the stochastic Navier-Stokes equation. The existence of an invariant measure for the Stochastic Navier-Stokes problem was also studied by sophisticated asymptotic analysis in Da Prato and Zabczyk [2] and, for the periodic boundary condition in Albeverio and Cruzerio [1] by the Galerkin approximation and the method of averaging. However, it is worth mentioning that the uniqueness and ergodicity questions cannot be answered by the method shown here and have to be dealt with by diﬀerent approaches, for instance, that one in Flandoli and Maslowski [1].

5.5

Stochastic Systems Related to Multi-Species Population Dynamics

We shall study a system of two interacting populations in which each population density follows a stochastic partial diﬀerential equation. Precisely, consider two populations both living in a bounded domain O ⊂ Rn . The evolution of their densities ui , i = 1, 2, deﬁned as number of individuals per unit volume or area, is a result of three competing factors. Firstly, both populations can migrate in O according to a macroscopic diﬀusion described by the Laplacian ∆. As the second factor, the interaction between these populations is modelled by the functions fi (u1 , u2 ) = ui − ai u2i ± bi u1 u2 ,

(5.5.1)

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Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

where ui ∈ R1 , ai , bi ≥ 0, i = 1, 2. This just implies that in the absence of the other population, the considered one will grow in accordance with the logistic law. The term ±bi u1 u2 describes two typical interactions, namely the predator-prey model when the signs in front of bi are diﬀerent and the competition one when the sign is negative in both terms. As the third factor, it is assumed that each population is randomly distributed by Gaussian noise B˙ ti , t ≥ 0, deﬁned on a complete probability space (Ω, F, P ) with a statedependent noise intensity σi (ui ), i = 1, 2. Thus, the system in which we are interested can be modelled as: ∂ ˙1 ∂t u1 (t, x) = ν1 ∆u1 (t, x) + f1 (u1 (t, x), u2 (t, x)) + σ1 (u1 (t, x))Bt (x), ∂ ˙2 ∂t u2 (t, x) = ν2 ∆u2 (t, x) + f2 (u1 (t, x), u2 (t, x)) + σ2 (u2 (t, x))Bt (x), (5.5.2) ν1 , ν2 > 0, t ∈ (0, T ], x ∈ O, where T > 0 is an arbitrary number. In addition, ui , i = 1, 2, satisfy the initial condition ui (0, x) = θi (x) ≥ 0,

¯ x ∈ O,

¯ → R+ and here O ¯ = O ∪ ∂O is the closure of O in Rn . where θi : Ω × O Moreover, ui , i = 1, 2, satisfy homogeneous Dirichlet boundary conditions, i.e., ui (t, x) = 0, x ∈ ∂O, t ∈ [0, T ], or homogeneous Neumann conditions, i.e., ∂ ui (t, x) = 0, x ∈ ∂O, t ∈ [0, T ], ∂n where n denotes the normal vector. We require that the boundary ∂O is ∂ smooth enough to guarantee that a Green function to ∂t − ∆ with the corresponding boundary conditions does exist. The formal problem (5.5.2) is a stochastic counterpart of a logistic population growth model with migration (cf. Murray [1]). Let B be a Borel subset of some separable Banach space B and H be a separable Hilbert space. Consider a strong Markov process (V (t), U (t))t≥0 in the product space B × H such that its second component U is governed by the stochastic evolution equation in H t t U (t) = T (t)u0 + T (t−s)F (V (s), U (s))ds+ T (t−s)G(U (s))dWs (5.5.3) 0

0

where u0 ∈ H, T (t), t ≥ 0, is some strongly continuous semigroup on H with its inﬁnitesimal generator A (generally unbounded) and W stands for a Wiener process with a trace class incremental covariance operator Q in a separable Hilbert space K. Both F : B × H → H and G : H → L(K, H) are Lipschitz continuous. Furthermore, the process V (t), t ≥ 0, is assumed to be pathwise continuous in B such that T EV (t)pB dt < ∞ 0

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259

holds for some p > 2 and each T ≥ 0. Deﬁne the operator L : C 2 (H) → C(B × D(A)) by 1 (LΛ)(g, h) = Ah, Λ (h)H + F (g, h), Λ (h)H + T r[Λ (h)G(h)QG∗ (h)] 2 for any Λ ∈ C 2 (H), (g, h) ∈ B × D(A). Furthermore, let K(r) be a centered open ball in H with radius r > 0. In addition to employing a strong solution approximation type of argument, the proofs of the following theorem can be similarly worked out to Theorem 3.3.3. Theorem 5.5.1 Assume F (g, ·) = 0 and G(0) = 0 for each g ∈ B. Suppose there exists a mapping Λ ∈ C 2 (H\{0}) satisfying that for each r > 0 and h ∈ H\K(r) |Λ(h)| + Λ (h)H + Λ (h) ≤ c(r)(1 + hqH ),

(5.5.4)

for some q > 0, c(r) > 0 and (LΛ)(g, h) ≤ −b(gB )Λ(h),

(g, h) ∈ B × (D(A)\{0}),

(5.5.5)

where b : R+ → R+ is continuous such that b(u) ≤ k(1 + u),

u ∈ R+

for some k > 0 and lim inf b(V (t)B ) > 0 t→∞

a.s.

(5.5.6)

Moreover, let Λ satisfy lr :=

inf

zH >r

Λ(z) > 0

(5.5.7)

for each r > 0, Λ(0) = 0 and Λ is continuous at zero. Then the solution U of (5.5.3) has the property limt→∞ U (t)H = 0 almost surely for every initial point u0 ∈ H. Next, we shall use Theorem 5.5.1 to derive some suﬃcient Lyapunov type conditions for stability of the null solution to the system (5.5.2). In particular, we shall consider the predator-prey case, i.e., f1 (u1 , u2 ) = u1 − a1 u21 + b1 u1 u2 , f2 (u1 , u2 ) = u2 − a2 u22 − b2 u1 u2 , where ui ∈ R+ , ai , bi ≥ 0, i = 1, 2. The corresponding results for the competition can be similarly obtained.

260

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Let H = L2 (O), Ai = νi ∆ with domains equal either to H 2 (O) ∩ H01 (O) or ∂ φ(x) = 0, x ∈ ∂O} in the case of Dirichlet or Neumann to {φ ∈ H 2 (O) : ∂n boundary conditions, respectively. Furthermore, let (Gi (φ)ψ)(x) = σi (φ(x))ψ(x),

¯ φ, ψ ∈ H, x ∈ O,

be the multiplication operators deﬁned by the real-valued mappings σi , i = 1, 2. Denote by Qi the covariance operators of W i , i = 1, 2. The basic idea to obtain stability is a comparison of the solution to (5.5.2) with an essentially simpler system which is given by dU1 (t) = (A1 U1 (t) + U1 (t) + b1 U1 (t)U2 (t))dt + G1 (U1 (t))dWt1 , (5.5.8) dU2 (t) = (A2 U2 (t) + U2 (t))dt + G2 (U2 (t))dWt2 , in the space H × H with initial conditions U1 (0) = θ1 and U2 (0) = θ2 . For our purpose, we impose the following conditions: ¯ → R1 are measurable, pathwise continuous and there exists (i) θi : Ω × O a deterministic constant kθ > 0 such that 0 ≤ θi (ω, x) ≤ kθ ¯ i = 1, 2. for any ω ∈ Ω and x ∈ O, (ii) The mappings σi : R1 → R1 , i = 1, 2, are globally Lipschitz continuous. (iii) σi (0) = 0, i = 1, 2. The strongly continuous semigroups deﬁned by their inﬁnitesimal generators Ai have the form (Ti (t)φ)(x) = Gi (t, x, y)φ(y)dy, φ ∈ H, x ∈ O, t ≥ 0, O

∂ where Gi (t, x, y) = g(νi t, x, y), i = 1, 2, and g is the Green function to ∂t − ∆. It can be shown that under the conditions (i)–(iii), the equation (5.5.2) has a pathwise unique nonnegative solution. So does (5.5.8) since the system (5.5.8) represents a particular case of (5.5.2). In what follows we need a comparison theorem for a single stochastic partial diﬀerential equation with progressively measurable random coeﬃcients. This is certainly a stochastic version of the standard ﬁnite dimensional stochastic comparison principle. The reader is referred to, for instance, Manthey and Zausinger [1] for its proofs. Consider the following two stochastic equations (k = 1, 2)

∂ (k) v (t, x) = (Av (k) )(t, x) + F (k) (ω, t, x, v (k) (t, x)) ∂t ˙ t (x), +G(ω, t, x, v (k) (t, x))W ¯ → R1 and homogeneous where t ≥ 0, x ∈ O with initial data θ(k) : Ω × O Dirichlet or Neumann boundary conditions and A = ν · ∆, ν > 0.

Some Related Topics of Stability and Applications

261

Theorem 5.5.2 Assume that θ(k) are uniformly bounded and let F (k) and G be uniformly Lipschitz continuous and uniformly bounded, k = 1, 2. If θ(1) ≥ θ(2)

a.s.

and F (1) (ω, t, x, v) ≥ F (2) (ω, t, x, v) ¯ v ∈ R , then v where (t, x) ∈ [0, ∞) × O, 1

(1)

≥v

(2)

a.s. holds almost surely.

By using Theorem 5.5.2, we can carry out a truncation procedure for solutions to show that for the equations (5.5.2) and (5.5.8), 0 ≤ u1 (t, x) ≤ U1 (t, x), 0 ≤ u2 (t, x) ≤ U2 (t, x),

(5.5.9)

¯ almost surely for any (t, x) ∈ R+ × O. Next we show a stability assertion that requires stability of the null solution for the second component of (5.5.3) in a stronger sense. This result is then used to derive stability for both components of (5.5.2) in H. Theorem 5.5.3 ¯ we have C(O),

Suppose that for some nonnegative initial function θ2 ∈ lim U2 (t)C(O) ¯ =0

(5.5.10)

t→∞

almost surely, and let one of the following conditions (a) and (b) be satisﬁed (a) 1 −β1 + 1 + kσ2 1 T rQ1 < 0, (5.5.11) 2 where −β1 stands for the ﬁrst eigenvalue of the operator A1 and kσ1 for the Lipschitz constant of σ1 ; (b) W 1 is a standard scalar Wiener process, σ1 (u)H ≥ κu, for some κ ≥ 0 and

u ≥ 0,

1 −β1 + kσ2 1 − κ2 < 0. 2

(5.5.12)

Then every solution (u1 , u2 ) of (5.5.2) such that u1 (0, x) = θ1 (x) with an ¯ and u2 (0, x) = θ2 satisﬁes arbitrary nonnegative θ1 ∈ C(O) lim u1 (t, ·)H = 0

t→∞

a.s.

and lim u2 (t, ·)C(O) ¯ =0

t→∞

a.s.

262

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

Proof Obviously, according to (5.5.9) it is enough to show U1 (t)H → 0 as t → ∞ almost surely. To this end, we use Theorem 5.5.1 with the function Λ(h) = h2γ H where γ > 0 will be speciﬁed later on. The conditions on Λ required in Theorem 5.5.1 are satisﬁed trivially except the condition (5.5.5), ¯ In the case (a), letting which is to be checked now with B = B = C(O). γ = 1, we get (LΛ)(g, h) = 2(A1 h, hH + h2H + b1 g · h, hH ) 1 + T r[G1 (h)Q1 G∗1 (h)], (g, h) ∈ B × D(A1 ), 2 where g · h means the usual multiplication of functions g and h. It follows that 1 2 (LΛ)(g, h) ≤ 2h2H −β1 +1+b1 gC(O) (g, h) ∈ B ×D(A1 ). ¯ + kσ1 T rQ1 , 2 Thus letting 1 b(u) := 2 β1 − 1 − kσ2 1 T rQ1 − b1 u , 2

u ∈ R+ ,

we obtain (5.5.5). Conditions (5.5.10) and (5.5.11) trivially yield (5.5.5) in the present case, so the assumptions of Theorem 5.5.1 are veriﬁed and we obtain U1 (t) → 0, t → ∞ almost surely which proves the case (a). In the case (b), we have 1 2 2 (LΛ)(g, h) ≤ 2γh2γ + 1 + b g + + (γ − 1)κ k − β , ¯ 1 1 C( O) H 2 σ1 for any (g, h) ∈ B × (D(A1 )\{0}), so we obtain (5.5.5) with 1 b(u) := 2γ β1 − 1 − kσ2 1 + (1 − γ)κ2 − b1 u , 2

u ∈ R+ .

Hence, choosing γ > 0 small enough and taking into account (5.5.10) and (5.5.12), we see that (5.5.6) in Theorem 5.5.1 is veriﬁed and, consequently, U1 (t)H → 0, t → ∞ almost surely which proves the case (b). By virtue of Theorem 5.5.3, in order to show stability of the zero solution of (5.5.2) it suﬃces to prove stability for a single equation for U2 in (5.5.8) ¯ where the “drift” is linear. However, the convergence of U2 in the C(O)-norm is needed, which is more diﬃcult to prove than the analogous convergence in ¯ for H. In the following theorem, we present some results on stability in C(O) the second component in (5.5.8). Together with Theorem 5.5.3, this will give suﬃcient conditions for the stability of the original system (5.5.2). Theorem 5.5.4

Suppose that W 2 is a standard scalar Wiener process.

Some Related Topics of Stability and Applications

263

(a) Consider the second equation in (5.5.8) with Neumann boundary conditions and assume κu ≤ σ2 (u)H , u ≥ 0, for some 0 ≤ κ ≤ kσ2 , and 1 1 + kσ2 1 − κ2 < 0. 2 Then lim U2 (t)C(O) ¯ =0

t→∞

a.s.

(b) If σ2 (u) = kσ2 u and 1 1 − kσ2 2 < 0, 2 then lim U2 (t)C(O) ¯ =0

t→∞

a.s.

Proof The second equation of (5.5.8) with Neumann boundary conditions and one-dimensional Wiener process is spatially homogeneous. So if we start from a spatially constant initial function the solution evolves in the space of constant functions. Therefore, taking a constant θ¯2 > 0 such that θ¯2 ≥ θ2 (x), ¯ a comparison theorem argument yields x ∈ O, U2 (t, x) ≤ v2 (t),

¯ x ∈ O,

where v2 (t) solves the one-dimensional stochastic diﬀerential equation dv2 (t) = v2 (t)dt + σ2 (v2 (t))dWt2 with v2 (0) = θ¯2 . A simple one-dimensional version of the proof of Thorem 5.5.3 yields v2 (t) → 0 as t → ∞ almost surely. In (b) the Neumann boundary condition is no longer demanded. To prove this assertion note that the Itˆ o formula implies U2 (t, x) = (T˜2 (t)θ2 )(x) · exp (1 − kσ2 2 /2)t + kσ2 Wt2 ¯ where T˜2 (t) is the semigroup generated by A2 in the space C(O). ¯ for x ∈ O, ˜ Obviously, T2 (t) is a contraction semigroup and exp (1 − kσ2 2 /2) + kσ2 Wt2 t−1 t → 0, as t → ∞ almost surely since Wt2 /t → 0 almost surely. Hence, U2 (t)C(O) ¯ → 0 as t → ∞, which completes this proof.

264

Stability of Inﬁnite Dimensional Stochastic Diﬀerential Equations

The interpretation of the above stability results in terms of a predator-prey system is very natural: in case of the prey is extinction, the predator that would have died out if the prey did not exist, dies out in presence of the prey as well. In the competition case the situation is much easier. We can choose here b1 = 0 in the ﬁrst equation of (5.5.8) and obtain a system of two decoupled equations. Hence, all results for the second equation summarized in Theorem 5.5.4 are true for both populations.

5.6

Notes and Comments

Semigroup models have been adopted by many researchers to deal with controlled partial diﬀerential equations for a long time. For instance, such a formulation for parabolic equations with boundary control has been extensively studied in Balakrishnan [1] and Lasiecka [1] among others. Some important problems such as quadratic control and stabilizability have been investigated by Lasiecka and Triggiani [1], [2]. Following their approaches, a semigroup model (5.1.2) for parabolic equations with boundary and pointwise noise is proposed in Ichikawa [7]. Semigroup models for boundary noise can be also found in Curtain [3] and Zabczyk [3]. Curtain [3] deals with the model (5.1.2) in which f (y) is independent of y and Zabczyk [3] gives a model in which boundary values satisfy stochastic diﬀerential equations. The existence and uniqueness of invariant measures for a class of stochastic evolution equations with boundary and pointwise noise has been investigated by Maslowski [2]. The control problem of (5.1.2) can be also considered as in Da Prato and Ichikawa [1]. The existence of an optimal control law is closely connected with the behavior of the Riccati operator equations as shown in Section 5.2.1. The quadratic problem (5.2.13) is discussed without using Theorem 5.2.3 in Ichikawa [1], and the optimal control problem with average cost is also investigated. For discrete-time quadratic control problems, Zabczyk [4] studies the existence of an optimal stationary strategy for a general system described by a linear diﬀerence equation. The uniqueness of the stationary measure related to this strategy is also investigated. There is an important problem which is not discussed in this book, that is, the quadratic cost problem for partially observable systems. In this case admissible controls are those dependent only on observations. It is known that the problem can be decomposed into two parts, ﬁltering and control, and this fact is known as the separation principle; see Curtain and Ichikawa [1] and Curtain [2]. After reducing the problems to those with complete observation one can use the results in Section 5.2 and obtain an optimal feedback control law on ﬁlters. The ﬁltering part can be solved

Some Related Topics of Stability and Applications

265

as a dual problem to the deterministic regulator problem. Hence, quadratic problems with incomplete observation may be solved using the approach in Section 5.2. A diﬀerent formulation is also presented by Bensoussan and Viot [1] in which they give necessary and suﬃcient conditions of optimality for linear stochastic distributed parameter systems, with convex diﬀerentiable payoﬀs and partial observation. In Theorem 5.3.1, the diﬀusion coeﬃcient G is deﬁned for all x ∈ H, which has the limitation that it does not allow unbounded operators. However, under some circumstances these stability results can be extended to the case as was shown in Li and Ahmed [1], [2]. The existence of an invariant distribution and the stationary solution for the 2-D stochastic Navier-Stokes problem was studied in Vishik and Fursikov [1] and, for periodic boundary conditions, in Albeverio and Cruzerio [1] by using the Galerkin approximation and method of averaging. Flandoli and Gatarek [1] considered rather general white noise and looked for solutions which are martingales or stationary for a class of stochastic Navier-Stokes equations. By the same techniques as in Section 3.9, the stabilization problem for a class of speciﬁc models, 2-D stochastic NavierStokes equations, was investigated in Caraballo, Langa and Taniguchi [1]. The material in Section 5.5 is mainly borrowed from Manthey and Maslowski [1].

Appendix

A

The Proof of Proposition 4.2.4

Proof

For any positive integer q, deﬁne O(q) = {hj : hj = n1 r1 + · · · + nm rm , n1 + · · · + nm = q}.

An elementary counting argument leads to

q+m−1 (q + m − 1)! #O(q) = = . m−1 q!(m − 1)! Let

q

j+m−1

(q) =

j=1

m−1

.

Since

we have

j+m−1 m−1

=

(q) ≤

j m+1 (1 + 1j ) · · · (1 +

m−1 j )

(m − 1)!

mm (m − 1)!

q+1

xm−1 dx < 1

<

mm j m−1 , (m − 1)!

mm (q + 1)m . m!

We now claim that if hα = qr1 for a positive integer q, then α ≤ (q). Suppose the contrary: there exist at least (q) terms of {hj } which are less than qr1 . This implies the existence of an hj such that qr1 > hj = n1 r1 + · · · + nm rm ,

n1 + · · · + nm ≥ q.

This is impossible since qr1 > hj ≥ r1 (n1 + · · · + nm ) ≥ qr1 . Hence, α≤

(q) ≤

mm (q + 1)m ≤ mm (q + 1)m . m! 267

268

Appendix

Observe (q) < (q + 1) for all q. We claim that for any n ∈ that [ (q), (q + 1)], hn ≥ qr1 . This is true. Indeed, note that if hn < qr1 , then hn < qr1 = hα , where n < α ≤ (q) which is impossible. Thus, for this n, ln n m ln m(q + 1) ≤ ln (q + 1) ≤ . hn r1 q On the other hand, for arbitrarily given ε > 0, there exists q0 such that (m ln m(q + 2))/(r1 q) is decreasing for q ≥ q0 and less then ε. Hence for that q0 and n ≥ (q0 ), lnhnn < ε. This completes the proof.

B

Existence and Uniqueness of Strong Solutions of Stochastic Delay Diﬀerential Equations

Theorem In addition to (a)–(e) of (4.3.2) to (4.3.4), assume that φ and τ satisfy the hypotheses at the beginning of Section 4.3. Then, there exists a unique strong solution to (4.3.1) on [0, T ] for all T ≥ 0. Proof Uniqueness. Suppose that u(t) and v(t) are two strong solutions of (4.3.1) on [0, T ]. Then, writing ρ(t) = t − τ (t) , t ≥ 0 , it follows that u(t) − v(t) =

t 0

A(s, u(s), u(ρ(s))) − A(s, v(s), v(ρ(s))) ds

t

+

B(s, u(s), u(ρ(s))) − B(s, v(s), v(ρ(s))) dWs , ∀t ∈ [0, T ].

0

Now, Itˆ o’s formula, the condition (c) in Section 4.3 and the fact u(ρ(t)) = v(ρ(t)) as ρ(t) ≤ 0 yield Eu(t) − v(t)2H t =2 Eu(s) − v(s), A(s, u(s), u(ρ(s))) − A(s, v(s), v(ρ(s)))V,V ∗ ds 0 t + EB(s, u(s), u(ρ(s))) − B(s, v(s), v(ρ(s)))2L0 ds 2 0 t t ≤λ Eu(s) − v(s)2H ds + Eu(ρ(s)) − v(ρ(s))2H ds 0

≤λ

0

0

t

ρ(t)

Eu(s) − v(s)2H ds +

Eu(s) − v(s)2H ds ρ(0)

≤λ

Appendix 269 t t Eu(s) − v(s)2H ds + Eu(s) − v(s)2H ds , ∀t ∈ [0, T ],

0

0

and then a Gronwall’s lemma type argument yields the required uniqueness. Existence. First of all, notice that since τ (t) ≤ 0 and τ (t) ∈ [0, r] for all t ≥ 0, there exist only three possible situations: Case i): limt→∞ τ (t) = δ > 0. Case ii): limt→∞ τ (t) = 0 but τ (t) > 0 for all t ≥ 0. Case iii): There exists T ∗ > 0 such that τ (t) > 0 for t ∈ [0, T ∗ ) and τ (t) = 0 for t ≥ T ∗ . Let us analyze each of them separately: Case i): As τ (t) ≥ δ for all t ≥ 0, we get that ρ(t) ≤ t − δ for all t ≥ 0. So, ρ(t) ≤ t − δ ≤ 0 for t ∈ [0, δ] and therefore the problem on [0, δ] can be rewritten as t t u(t) = φ(0) + A(s, u(s), φ(ρ(s)))ds + B(s, u(s), φ(ρ(s)))dWs , t ∈ [0, δ], 0

u(t) = φ(t),

0

t ∈ [−r, 0],

which is a nondelay problem. Now, observe that in the case without delays considered by Pardoux [1], the existence of strong solutions is proved under the following similar assumptions to (a)–(e). In fact, consider A(t, ·) : V → V ∗ , a family of nonlinear operators, and B(t, ·) : V → L(K, H), satisfying (a) (Coercivity). There exist α > 0, p > 1 and λ, γ ∈ R1 such that: 2x, A(t, x)V,V ∗ + B(t, x)2L0 ≤ −αxpV + λx2H + γ, 2

∀x ∈ V.

(b) (Growth). There exists c > 0 such that A(t, x)V ∗ ≤ c(1 + xp−1 V ),

∀x ∈ V.

(c) (Measurability). t ∈ (0, T ) → A(t, x) ∈ V ∗ is Lebesgue-measurable ∀x ∈ V, ∀T > 0. (d) (Continuity). ξ ∈ R1 → v, A(t, x + ξy)V,V ∗ ∈ R1 is continuous for all x, y, v ∈ V . (e) (Monotonicity). For all x, y ∈ V, 2x − y, A(t, x) − A(t, y)V,V ∗ + B(t, x) − B(t, y)2L0 ≤ λx − y2H . 2

(f) There exists k > 0 such that B(t, x) − B(t, y)2L0 ≤ kx − y2V , 2

∀x, y ∈ V.

(g) t ∈ (0, T ) → B(t, x) ∈ L(K, H) is Lebesgue-measurable ∀x ∈ V , ∀T > 0.

270

Appendix

However, it is not diﬃcult to check that the proofs in Pardoux [1] are also valid if one assumes some integral versions of the hypotheses (a) , (b) , (e) and (f) . In fact, it is suﬃcient to make the following assumptions instead of (a) , (b) , (e) and (f) : (A) There exist α > 0, p > 1 and λ, γ ∈ R1 such that for all w ∈ Lp (Ω × (0, T ); V ) ∩ L2 (Ω × (0, T ); H) and all t ∈ [0, T ],

t

2

t

EB(s, ws )2L0 ds 2 t t ≤ −α Ews pV ds + λ Ews 2H ds + γt;

Ews , A(s, ws )V,V ∗ ds + 0

0

0

0

(B) There exist positive constants c1 , c2 > 0 such that for all w ∈ Lp (Ω × (0, T ); V ) ∩ L2 (Ω × (0, T ); H),

T

p/(p−1) EA(t, wt )V ∗

0

T

(c1 Ewt pV + c2 ) dt;

dt ≤ 0

(E) For all w1 , w2 ∈ Lp (Ω × (0, T ); V ) ∩ L2 (Ω × (0, T ); H) and all t ∈ [0, T ],

t

t

Ews1 − ws2 , A(s, ws1 ) − A(s, ws2 )V,V ∗ ds +

2 0

0

EB(s, ws1 ) − B(s, ws2 )2L0 2

t

≤λ

Ews1 − ws2 2H ds. 0

(F) There exists k > 0 such that for all w1 , w2 ∈ Lp (Ω × (0, T ); V ) ∩ L2 (Ω × (0, T ); H) and all t ∈ [0, T ],

t

EB(s, ws1 ) 0

−

B(s, ws2 )2L0 2

ds ≤ k

t

Ews1 − ws2 2V ds. 0

Let A1 (t, wt ) = A(t, w(t), φ(ρ(t))) and B1 (t, wt ) = B(t, w(t), φ(ρ(t))) for w ∈ Lp (Ω × (0, T ); V ) ∩ L2 (Ω × (0, T ); H) and t ∈ [0, δ]. Our existence result will hold if we can prove that A1 and B1 satisfy (A), (B), (E) and (F). But this follows immediately from the assumptions (a), (b), (c) and (e) in Section 4.3. Let us prove (A), for instance. Indeed, for w· ∈ Lp (Ω × (0, δ); V ) ∩ L2 (Ω × (0, δ); H) and t ∈ [0, δ], we obtain 2

t

t

Ews , A1 (s, ws )V,V ∗ + EB1 (s, ws )2L0 ds 2 0 0 t t t Ews pV ds + λ Ews 2H ds + θ Eφ(ρ(s))2H ds + γt ≤ −α 0 0 0 t t ≤ −α Ews pV ds + λ Ews 2H ds + β 2 |θ| sup Eφ(s)2H + γ t. 0

0

−r≤s≤0

Appendix

271

Now, since (B), (E) and (F) can be similarly proved, we then obtain the existence of a strong solution on [0, δ]. By induction, the problem can be solved on [nδ, (n + 1)δ] for all natural numbers n ≥ 0 and therefore on [0, ∞). Case ii): In this case, we can choose a strictly increasing sequence {δn } such that δn → ∞, δ0 = 0, ρ(δn+1 ) = δn and ρ(t) ∈ [δn−1 , δn ] for all t ∈ [δn , δn+1 ] and for all n ≥ 0, where we write δ−1 = −r. Consequently, our equation can be solved on each [δn , δn+1 ] exactly as in Case i), and further on [0, ∞). Case iii): Firstly, we can prove the existence of a strong solution on [0, T ∗ ) in the same way as Case ii). Then, it is not diﬃcult to show that the solution u(t) tends to certain u(T ∗ ) ∈ L2 (Ω, FT ∗ , P ; H), as t → T ∗ . Now, on [T ∗ , ∞) the problem becomes t t u(t) = u(T ∗ ) + A(s, u(s), u(s))ds + B(s, u(s), u(s))dWs , T∗

T∗

which obviously has a unique strong solution and our proof is complete. In the deterministic framework, there exists a large literature on the existence of diﬀerent kinds of solutions to functional diﬀerential equations; see, for instance, Ruess [1] for a comprehensive description of recent results. For the stochastic case in a variational setting, Real [1] investigated the existence and uniqueness for a class of linear systems. In a similar spirit to Pardoux [1], Caraballo, Liu and Truman [1] and Caraballo, Garrido-Atienza and Real [1] established conditions to ensure existence and uniqueness of solutions of general stochastic functional diﬀerential equations.

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K. Yosida. [1] Functional Analysis. (6th edition), Springer-Verlag, New York, (1980). J. Zabczyk. [1] A note on C0 semigroups, Bull. Polish Acad. Sci. (Math.) 162, (1975), 895-898. [2] On stability of inﬁnite dimensional stochastic systems. Probab. Theory, Z. Ciesislski (ED), Banach Center Publications, 5, Warswa, (1979), 273281. [3] On decomposition of generators. SIAM J. Control Optim. 16, (1978), 523-534. [4] On optimal stochastic control of discrete-time systems in Hilbert space. SIAM J. Control. 13, (1975), 1217-1234. M. Zakai. [1] On the ultimate boundedness of moments associated with solutions of stochastic diﬀerential equations. SIAM J. Control. 5, (1967), 588-593. [2] A Lyapunov criterion for the existence of stationary probability distributions for systems perturbed by noise. SIAM J. Control. 7, (1969), 390-397.

Index

admissible control 238, 243, 244 algebraic Riccati equation 239, 243 asymptotic stability 30, 40 in probability 32 in p-th moment 32 almost sure 32

feedback control 238, 248 Feller semigroup, property 8, 127 ﬁltration 2 normal 2 satisfying the usual conditions 2 right-continuous 2

Bochner’s integral 3 strong 4 boundary noise 229 Burkholder-Davis-Gundy’s inequality 15, 97, 105

Gaussian measure 11 Green function 236

Chapman-Kolmogorov equation 7 characteristic equation 183 compact operator 41, 182 covariance operator 10 cross quadratic variation 10 decay, decayable 33 with rate λ(t) in the p-th moment 151 with rate λ(t) in the almost sure sense 154 diﬀerentiable (of a semigroup) 43 dissipative (of an operator) 247 distribution (of a random variable) 2 Doob’s inequality 8, 14 equivalent process 5 essential spectrum (of an operator) 187 exponential stability 40 in p-th moment 32 almost sure 32

hitting time 6 indistinguishable process 5 invariant measure 127 Itˆ o’s formula 16, 21 linear growth condition 102 Lipschitz continuous condition 23, 102 Lyapunov exponent 36, 167 exponent method 36 function 35 second (direct) method 34 spectrum 36 Markov process 7 martingale 8 continuous square integrable 9 mean square stability in average 61 measurable mapping 1 space 1 stochastic process 4 strongly 3

297

298 modiﬁcation of a stochastic process 5 multiplicity (of the exponent) 36 optimal feedback control 238, 243, 244 pointwise noise 236 predictable process 6 σ-ﬁeld 6 sets 6 probability measure 1 space 1 space, complete 2 process, adapted 4 continuous 4 increasing 9 integrable 8 progressively measurable 5 quadratic cost functional 238 quadratic variation 10 Q-Wiener process 10 random variable (simple) 1 Razumikhin method 220 semigroup (of a Markov process) 7 seminorm 194 solution, mild 23 solution, global strong 19 stability (of an operator) 39, 40 stability

Index almost sure 31, 63 in the Large 32 in probability 31 in mean square 33 in p-th moment 31 stabilizable (of a controllable system) 242 stabilization by noise 159 stochastic convolution 15 stochastic integral 13, 14 stochastically continuous 4 stochastic diﬀerential equation 18, 22 (neutral) functional diﬀerential equation 201 Navier-Stokes equation 254 reaction-diﬀusion equation 251 stopping time 2, 6 Stratonovich integral 21, 171 strictly dissipative 51 strong solution 19, 23 submartingale 8 supermartingale 8 tensor quadratic variation 9 trajectories of stochastic process 4 transition probability function, 7 homogeneous 7 ultimate boundedness in mean square (exponential) 121 uniform stability (in a class) 91 uniformly asymptotic stability in mean square 71, 196 Young’s inequality 100