Study On The Approximation Problem Of Random Evolution Equations In Hilbert Spac

This paper mainly studies the Trotter-Kato approximation system of mild solutions for Mc Kean-Vlasov type stochastic evolution equations and stochastic evolution equations with finite delays in a real separable Hilbert space.The paper is divided into the following four chapters:In Chapter 1,firstly,the background and research status of approximation problems of stochastic evolution equations and operator semigroup theory are introduced.Then the related concepts and known conclusions that need to be used in the later research are given.Finally,the main research contents of this paper are summarized.In Chapter 2,some known conclusions on the approximation problems of general stochastic evolution equations are generalized.The Trotter-Kato approximation system of Mc Kean-Vlasov type stochastic evolution equations on the mild solution is studied.The two function terms in this kind of equations depend on the state and probability distribution of the mild solution at a given time,which has not been considered before.In this chapter,firstly,the existence and uniqueness of the mild solution of such equations are proved.On this basis,the Trotter-Kato approximation system for the mild solution of this kind of equations is established,and the weak convergence of the corresponding induced probability measures and the error estimation of the approximation system are obtained.Then,as an application,the classical limit theorem on the parameter dependence of such equations under the approximation system is given;Finally,a Mc Kean-Vlasov type stochastic heat equation is used to illustrate the effectiveness of the conclusions.In Chapter 3,the application scope of approximation theory in stochastic evolution equations is further expanded.When stochastic evolution equations have finite delays,the Trotter-Kato approximation system of mild solution is studied.Since it is known that the stochastic evolution equations with finite delays has a unique mild solution,the Trotter-Kato approximation system of such equations can be established directly,and the approximation error can be estimated at the same time.Based on this,in this chapter,the classical limit theorem on parameter dependence of stochastic evolution equations with finite delays is given as an application of approximation system in this kind of equations.Finally,a stochastic heat equation with finite delays is given,which shows that the relevant conclusions are of practical significance.In Chapter 4,the previous research results of this paper are summarized,and three prospects for future research are put forward: firstly,the conclusions obtained in Chapter 3 can be further extended to consider the approximation problems of neutral stochastic partial differential equations;On this basis,the impulsive term can also be introduced into the equations,that is,the approximation problems of impulsive neutral stochastic partial differential equations can be studied;In addition,new approximation theories such as Euler-Maruyama approximation system can also be considered.