Weighted Almost Automorphic Behavior Of Solutions Of Two Kinds Of Stochastic Evolution Equations

This thesis mainly discusses some basic properties and theories of 52-weighted pseudo almost automorphic and Sγ2-weighted pseudo almost automorphic stochastic processes, by virtue of which we study the weighted almost automorphic behavior of two kinds of stochastic evolution equations. The whole contents is divided into four chapters.Chapter 1 is the part of introduction, which mainly introduces the research backgrounds and the main purpose of this thesis and also include the organization of this thesis.Chapter 2 is the preliminaries, which briefly explain the definitions, lemmas and nota-tions that we will use throughout the thesis. This chapter mainly introduces the definitions of almost automorphic stochastic process, weighted pseudo almost automorphic stochastic process,S2-weighted pseudo almost automorphic stochastic process and Sγ2-weighted pseudo almost automorphic stochastic process, and some proof of basic properties and lemmas are also included. Besides, we also introduce some results of infinite delay spaces.In chapter 3, we introduce the definition of S2-weighted pseudo almost automorphic stochastic process, then consider the following linear and nonlinear stochastic differential equations in the form dx(t)=Ax(t)dt+f(t)dt+η(t)dW(t), t ∈ R (1.1) and dx{t)= Ax(t)+f(t, x(t))dt+η(t, x(t))dW(t), t ∈ R (1.2) the existence and uniqueness of S2-weighted pseudo almost automorphic solutions. Then we study the asymptotic behavior of solutions to above equations under S^-weighted pseudo almost automorphic coefficients without global Lipschitz conditions.In chapter 4, we introduce the definition of Sγ2-weighted pseudo almost automorphic stochastic process, then consider the following non-autonomous stochastic neutral functional evolution equations with infinite delay in the abstract form d[u(t)+f(t, ut)]= [A(t)u(t)+g(t, ut)]dt+σ(t, ut)dW(t), t ∈ R (1.3) the existence and uniqueness of square-mean weighted pseudo almost automorphic solution-s. First, we introduce the definition of Sγ2-weighted pseudo almost automorphic stochastic process with infinite delay. In addition, we establish the completeness of the space and some useful composition theorems. Finally, we investigate the asymptotic behavior of solutions to above equation under Sγ2-weighted pseudo almost automorphic coefficients.