Weighted Pseudo Almost Automorphic Solutions For Non-autonomous Stochastic Differential Equations

In this paper, we consider the existence and uniqueness of the solutions which are weighted pseudo almost automorphic in distribution for a class of non-autonomous stochastic differential equations in a Hilbert space. In Chapter1, we provide some relative definations, lemmas and propositions as preliminaries. In Chapter2, we prove our main result. In conclusion, we use the Banach contraction mapping principle and exponential dichotomy property to obtain our main results.Liu and Sun [12] introduced the concept of almost automorphy in distribution and studied the almost automorphy in distribution solutions of stochastic differential equations driven by Levy noise. Defination1An H-valued stochastic process X(t) is said to be almost automorphic in distribution if its law μ(t) is a P(H)-valued almost automorphic mapping, i.e., for every sequence of real numbers{s’n}, there exist a subsequence{sn} and a V(H)-valued mapping μ(t) such that hold for all t∈R.Chen and Lin [3] investigated the square-mean pseudo almost automorphic process and its applications. Defination2An L2-continuous process f(t):Râ†'L2(P, H) is said to be square-mean pseudo almost automorphic if it can be decomposed as J.Blot and his colleague [1] had researched the abstract differential equations, and proved some results for the Weighted pseudo almost automorphic functions. We ex-tended the abstract differential equations into C2(P, H)), defined the weighted pseudo almost automorphy on it. In this paper, we consider the existence and uniqueness of the solutions which are weighted pseudo almost automorphic in distribution for a class of non-autonomous stochastic differential equations of the form where A(t) is a family of closed linear operators satisfying the Acquistapace-Terrani conditions (see [10,11]),/(t, x), g(t, x) are weighted pseudo almost automophic in t E R for each x∈L2(P, H), and f, g are assumed to satisfy Lipschitz conditions with respect to x.In our proof, we use exponential dichotomy property. And befor provide the defi-nation of ED, we define the evolution family.Defination3A family of bounded linear operators{U(t, s):t≥s, t, s E R} on C2(P,H) associated with A(t) is said to be an evolution family of operators if the following conditions hold: is strongly continuous, for every t> s;Defination4An evolution family{U(t, s):t≥s, t, s E R} is said to exponen-tial dichotomy (ED), if there are projectors P(t), t E R, being uniformly bounded and strongly continuous in t and two constants K≥1and ω>0such that(1) P(t)U(t,s)=U(t,s)P(s);(2) the restriction U{t, s):UQ{t,s)L2(P, H)â†'Q(t)L2(P, H) of U(t, s) is invert-ible, and UQ(s,t)=(UQ(t,s))-1, for t> s;(3)‖U(t,s)P(s)‖≤Ke-ω(t-s),for t≥s, and‖U(t,s)Q(s)‖≤Keω(t-s),fort<s, where Q(s)=I-P(s).Defination5An L2-continuous stochastic process x(t)t∈R is called a mild solu-tion of the problem (0.0.1) if it satisfies the corresponding stochastic integral equation: for all t≥α0and for each α0∈R. Then, we transfer the solution of (0.0.1) into the form: And consider if it is continuous, being existence and uniqueness or pseudo almost automorphic.