Pseudo Almost Automorphic Type Solutions To Two Classes Of Stochastic Evolution Equations

In this article, we mainly discuss some basic properties of pseudo almost automorphic functions and the existence and uniqueness of pseudo almost automorphic mild solutions to some stochastic evolution equations. This thesis are divided into four sections.Firstly, Section1has been focused on mainly the research backgrounds and some main results of this thesis. Besides a series of recent theoretical achievements to the theory of pseudo almost automorphic functions, stepanov-like pseudo almost automorphic functions and square-mean almost automorphic stochastic processes in abstract spaces.Then as the preliminaries, in Section2, we will list certain knowledge that will be used throughout this thesis. This part mainly include some basic definitions, notations, theo-rems, lemmas, methods and properties about almost automorphic functions, pseudo almost automorphic functions, Stepanov-like pseudo almost automorphic functions, square-mean almost automorphic stochastic process, square-mean pseudo almost automorphic stochas-tic process and S2-almost automorphy for stochastic process. In addition, we introduce briefly the concepts, fundamental results and some relative notations of probability space, Brownian motion with operator theory.Next, in Section3we mainly discuss the existence and uniqueness of pseudo almost automorphic mild solutions to the stochastic functional differential equations in the form listed below where r≥0is a fixed constant, and A is the infinitesimal generator of a Co-Semigroup {R(t)}t≥0on LP(P,H) and Bi,i=1,2are bounded linear operators, W(t) is a two-sides standard one-dimensional Brownian motion defined on the filtered probability space (Ω,F, P,FT) with Ft=σ{W(u)-W(v):u,v≤t}. Here f and g are appropriate func-tions to be specified later. In this section, we establish some of our main results by use Schauder’s fixed point theorem, Ito isometry, Holder’s inequality combined with Banach contraction principle, basic properties and composition theorems of p-th mean pseudo al-most automorphic functions as well as some proper factors, and then we get the mild solutions of the stochastic functional differential equations in a separable Hilbert space. The obtained results can be seen as a generalization of pseudo almost automorphy cases.Finally, in Section4, we are focused on discussing the existence and uniqueness of S2-pseudo almost automorphic mild solutions for the following linear and nonlinear stochastic differential equations in the forms and where A is the infinitesimal generator of a Co-semigroup {T(t)}t≥0on L2(P, H), and W(t) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space (Ω,F,P,Ft), where Ft=σ{W(u)-W(v);u,v≤t}. Here f and g are appropriate functions specified later. First, we introduce a new concept of stochastic processes called S2-pseudo almost automorphy for stochastic processes, which generalizes the notation of square-mean pseudo almost automorphy or S2-almost automorphy for stochastic processes. And then we present some properties of such functions and apply this new concepts to investigate the existence and uniqueness of S2-almost automorphic mild solutions to the above stochastic differential equations in a real separable Hilbert space.