The Asymptotic Properties Of Stochastic Functional Differential Equations

This dissertation mainly considers stochastic functional differential equations with finite and infinite delays.The main research includes two aspects:firstly,we discuss the tools that can be used to study the asymptotic properties or behavior of stochastic functional differential equations,namely LaSalle-type theorem,asymptotic log-Harnack inequality.Secondly,we study the asymptotic properties or behaviors of stochastic functional differential equations,that is,stability in distribution and ergodicity.This paper is divided into the following six chapters:The first chapter firstly introduces the research background of stochastic functional dif-ferential equations.Then we present the research status of LaSalle-type theorem,asymptotic log-Harnack inequality and ergodicity for stochastic functional differential equations,and the specific questions to be addressed below.Chapter 2 introduces some basic knowledge,main-ly including the basic concepts and mathematical tools required below,with emphasis on the Wasserstein distance and its properties.In the next three chapters,we discuss these two aspects mentioned above in detail.In chapter 3,we prove the LaSalle-type theorem for nonautonomous stochastic function-al differential equations with infinite delay,which includes the existence and uniqueness of global solution and asymptotic attraction of this solution.In addition,we give three important corollaries from which asymptotic attraction,almost sure stability,and asymptotic bounded-ness can be deduced.Finally,two examples are given to illustrate the results of this chapter.In chapter 4,we consider the stochastic functional differential equations with finite delay.We give the sufficient conditions of stability in distribution for stochastic functional differential equations with super linear delay coefficients.In addition,we prove that the mapping process is Feller process,and then it follows from stability in distribution that the solution mapping has a unique invariant probability measures.Finally,two examples are given to illustrate the results of this chapter.In chapter 5,we consider stochastic functional differential equations with infinite delay and non-Lipschitz coefficients.Under the local non-Lipschitz and linear growth conditions,we prove the existence and uniqueness of global solution,and prove that the solution mapping is a strong Markovian Process and a Feller process.Then,using the method of asymptotic cou-pling,exponential ergodicity of stochastic functional differential equations with infinite delay and Holder continuous coefficients is established.Similarly,we use the method of asymptotic coupling to prove the asymptotic log-Harnack inequality for stochastic functional differential equation with non-Lipstchitz coefficients,from which the solution mapping process is asymp-totically strong Feller.In chapter 6,we summarize the main results of this dissertation and propose some questions that can be further studied.