The Existence Of Periodic Solutions For Stochastic Functional Differential Equations With Infinite Delay In Distribution

In recent years,with the continuous development of stochastic differential equa-tion theory,time delay phenomenon has been widely concerned.At the same time,for differential dynamical systems,the study of periodic solution problem is always the central topic.In this paper,we will study the existence of periodic solutions for stochastic functional differential equations with infinite delay in distribution.dX(t)=f(t,Xt)dt+g(t,Xt)dW(t).Firstly,considering that the delay is infinite,we choose an appropriate norm and define an appropriate phase space accordingly.In this space,referring to the Halanay's criterion of deterministic differential equations,the law of large numbers for stochastic functional differential equations with infinite delay is constructed.Then the existence of periodic solutions in distribution is proved by linear growth condition and local Lipschitz condition.The paper is divided into five chapters.The first chapter mainly introduces the research background and present situation of stochastic differential equations,s-tochastic periodic solutions and delay functional differential equations.In the second chapter,the basic concepts needed in this paper are introduced,and the preliminary theorems for stochastic functional differential equations with infinite delay are given and proved.In the third chapter,we study the existence of periodic solutions for stochastic functional differential equations with infinite delay in distribution.The fourth chapter makes LaSalle type stationary principe as a corollary of this article.The fifth chapter summarizes this paper.