Luận án tiến sĩ: Stochastic delay equations and invariant measure for the wave equation with noise

Stochastic delay equations and invariant measure for the wave equation with noise by Xi Zhao Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Supervised by Professor Carl Mueller Department of Mathematics The College Arts and Sciences University of Rochester Rochester, New York 2006 UMI Number: 3245866 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion.

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Box 1346 Ann Arbor, MI 48106-1346 il Curriculum Vitae The author was born in Shijiazhuang, Hebei province, China on December 24th, 1979. She attended the Department of Mathematics at the Nankai University, Tianjin, China in 1998 and graduated with a bachelor of science degree in Math- ematics in 2002. In the same year, she was admitted to the Department of Math- ematics at the University of Rochester, Rochester NY. After earning a master of science degree in 2004, she chose probability as her research area and was su- pervised by Prof.

Mueller as a Ph. She was granted a Teaching Assistantship from the Department of Mathematics during the academic years 2002-2006. Hi Acknowledgments I would like to express my deepest gratitude to my advisor Prof.Carl Mueller for his guidance, encouragement, inspiration, supervision and patience in the past four and half years. It is him who introduced me to the area of probability and stochastic process and led me to stochastic differential equations and delay equations.

I am deeply impressed by his wisdom in thinking and solving problems in probability, differential equations, analysis and physics. I am fortunate to have him as my thesis advisor and greatly appreciate his time and efforts for each meeting, discussion, reading and correcting my manuscripts. I gratefully acknowledge those who served as my committee members in my oral defense: Prof. I thank them for very useful suggestions and discussions.

I thank all the professors who taught me one or two courses during my Ph. study, who pass their knowledge without reservation and teach us not only how to learn but also how to think: Prof. Greenleaf and Prof. Many thanks to Dr.

Kijung Lee, who led me through the reading course for two semesters. I appreciate his thorough discussion for every important concept and quick reply to every question. I am very grateful to the secretaries of the Department of Mathematics: Joan Robinson, Fran Crawford, Hazel McKnight. They cordially, faithfully and iv patiently helped me with all administrative paper work since the first.

day I came to the department. I am gratefully indebted to my family. My parents have always been sup- portive and encouraging. My dear husband Li’s love always strengthens, comforts and encourages me.

He takes most of the responsibilities to take care of our son when this thesis was being written. I have had the good chance to have many friends at University of Rochester who help to turn 4 years of study into 4 years of life. I look forward to having fun with you again. Abstract This thesis is divided into two major parts.

First we study the moment stability of the trivial solution of a linear differential delay equation in the presence of additive and multiplicative white noise. The stability of the first moment for the solutions of a linear differential delay equation under stochastic perturbation is identical to that of the unperturbed system. However, the stability of the second moment is altered by the perturbation. We obtain, using Laplace transform tech- niques, necessary and sufficient conditions for the second moment to be bounded.

Then we establish the stability criteria for stochastic differential equations with Markovian switching using the comparison principle. These criteria include sta- bility in probability, asymptotic stability in probability, stability in the pth mean, asymptotic stability in the pth mean and the pth moment exponential stability of such equations. Next, we study the uniqueness of the invariant measure for the wave equation with noise. We will use a coupling technique and others from the theory of Markov chains on general state spaces.

The application of these Markov chain results leads to straightforward proofs of ergodicity of SDEs. The key points which need to be verified are the existence of a Lyapunov function including returns to a compact set, a uniformly reachable point from within that set and some smoothness of the probability densities. vi Table of Contents Curriculum Vitae ii Acknowledgments iii Abstract 1 Stochastic differential delay equations 1.2 Preliminaries of functional differential equation .3 Moment stability: the system with perturbation 2 Stability criteria for SDDE with Markovian switching 29 2.2 | Basic comparison principle. Q Q HQ nu vn và va 40 3 Invariant measure for the wave equation with noise 3.1 Introduction: Coupling method .2 Ergodicity for the Markov chain through coupling .3 Application to the stochastic wave equation.

vii A Proof of Theorem 1.5 62 Bibliography 64 1 Stochastic differential delay equations Stochastic differential delay equations were first introduced by Ité [2] in the 1960s. Fundamental results including existence and uniqueness of solutions, stochastic stability, numerical approximation, etc. have only been developed in the last decade. See [3] for a recent survey of these results.

In spite of the efforts of many researchers, this field is still in its infancy. For example, conditions for the moment stability of some linear stochastic differential delay equations with constant coefficients are still not known. The Lyapunov function method is useful to study the stability of differential equation and has been developed for both differential delay equations and stochas- tic differential equations. In the 1990s, Mao extended this method to stochastic functional equations [7, section 5].

Because of the results of Mao, we have some results for the stability of stochastic differential delay equations (see [7 Sec 5. In this chapter, we study the moment stability of the trivial solution of a linear differential delay equation in the presence of additive and multiplicative white noise. The stability of the first moment for the solutions of a linear dif- ferential delay equation under stochastic perturbation is identical to that of the unperturbed system. However, the stability of the second moment is altered by the perturbation.

We obtain, using Laplace transform techniques, necessary and sufficient conditions for the second moment to be bounded. This chapter is organized as follows. We will first briefly present the mathe- matical preliminaries for the linear differential delay equations needed for the rest of the paper. Then we will examine the effect of stochastic perturbations on the behavior of the mean and variance of the stochastic differential delay equation.1 Introduction Consider a stochastic differential delay equation of the form dy = f(y, yr)dt + g(y, yr)dW (t) (1.1) where y,(t) = y(t —7),7 > 0 and W(t) is a standard Wiener process.

In the deterministic case dy = f(y, yr)dt (1.2) For any y*, such that f(y*, y*) = 0, y* is a steady state of (1. Now linearize equation (1.1) around y*, we get dz = (ax + bz;)dt + (ơạz + ơiz; + ơa)dW (t) (1.2 Preliminaries of functional differential equa- tion When o; = 0 in (1.4) , we have the linear differential delay equation z'(t) = ax(t) + br(t — 1) (1.5) This differential equation has been studied extensively in [1]. Now we will state some of the main results.1 The characteristic equation of a homogeneous linear differen- tial equation with constant coefficients is obtained from the equation by looking x for nontrivial solutions of form e”c, c is a constant.5) has a nontrivial solution e*‘c if and only if the characteristic equation h(A) = A-—a—be* =0 (1.2 The fundamental solution of (1.5) is a solution of (1.5), whose Laplace transform is h~'(X).3 (Existence and convolution of Laplace Transform) If f : [Ũ, oo) — R is measurable and satisfies \f(x)| sae" — t € [0, 00) for some constants a and b, then the Laplace transform C(f) defined by L(A) = [Pes nat exists and is an analytic function of A for ReA > 6. If the function f * g is defined by f « 9(t) = fo f(t — s)9(s)ds, then L(f *9) = L(F)L(g) The fundamental solution of (1.5) can be introduced in two equivalent ways.

It is the solution of (1.5) whose Laplace transform is h~'(X) and equivalently, a solution of (1.5) with initial condition 0_ i_-1<6<0 +(86) = 1 if@=0 In what follows, we will denote by X(t) the fundamental solution of (1. It is not hard to show that X(t) is bounded and |X(t)} < me” for some constants m and n.4 The solution X(t) of equation (1.5) with initial data given above is the fundamental solution; that is Also, for anyc >n X(t) =Ƒ e*h“(A)dA — t>0 c where n is the exponent associated with the bound on X(t). see the proof in ref [1]. Through the fundamental solution, the general solution of (1.5) with initial condition 7(0) = y(0), where —1 < 6 < 0, is given by ap(t) = X(t)e(0) + [ : X(t — 1 -s)y(s)ds (1.7), the asymptotic behavior of x,(t) is determined by the fundamental solution X(t).

Now, we have an important theorem for X(t).5 If ao = max{Re(X) : h(A) = 0} is negative, then for any ag < a < 0, there is a constant K = K(qa), such that the fundamental solution X(t) satisfies the inequality [X(t)| << Ke* (>0) See the proof in the appendix.5, the solution of (1.5) with y(@) approaches 0 as t — oo if and only if ao < 0. The region in the (a, b)-plane such that ag < 0 is given in [1].8) Now, we will give the estimation of œo and K(ø). Let 4 = a9 +77 be the solution of the characteristic equation with the largest real part.6), we have |a9 — al < |ble~°° , where ao is given by the maximum real solution of — a12 (a — a)? — b2e~22o + [areeos Ti =0 Lemma 1.6 For any @ > ao, the constant K = K(a) in Theorem 1.5 is given by (|œ — aole® + | bl) log2 K(a) <1+€(a)+ < IBịx (1. where (oie) 1 | palele"* a+ be~(S†1) — œạ €íœ) = 2m k, ble-e (@ — a9 + iz)h(at+ xt Proof: Using the inverse Laplace transform, we have X(t) = I, h-(A)e*da, where 1 /at7 (a) Tc 271 Ja—-iT Let g(A) = h71(A) — (A — a) 71, so X()=— fae Mt ar+ fr _a ay)“ y-1 te*dd At Therefore, [gretaAJe*dd| =| + 0 A-a ~ 0) ay A x IA et /„,Ia0) À)|dA + e9 Hence, we can take K(a) =1+ j_ lø()ldA Noting that a + be” — ao (A — ao)h(A) we have, when Re(XA) = a and |Im(A)| > 2| ble~* |a — ag| + | ble“? 9S À)|< TOQUE = Ble) Thus 1| r2lble~*.

1 — se = ee |a— ep ao] +' || ble~* Ị << I. (Aas 2m L b|e~% g(œ + /z)dz| + 7 Dyes z(z — | b|e~* )4z _ (la — aole* + | d]) log 2 and inequality (1.9) is defined when b # 0 and a > 0. When b = 0, we can simply take K(a) = 1 whenever a > do The inequality (1.9) gives an estimate for K(a) for all parameters. When a <0 and |b| < —a, we have the following compact estimation for all bounds on Z6).