Invariant Measure And Its Numerical Approximation For Stochastic Schrodinger Lattice System

As an important infinite dimensional dynamical systems,stochastic lattice systems not only discrete spatial structure but also consider the interference of noise(such as random disturbance or uncertint erference),which are widely used in the fields of biology,chemical reaction,brain science,image processing,etc.For stochastic lattice dynamic systems,the long-time dynamic behaviors have received extensive attention,including random attractors and invariant measures.Considering the influence of random interference,uncertainty and time delay,this paper will study invariant measures and its numerical approximation for regime-switching stochastic Schrodinger lattice system and stochastic delay Schrodinger lattice system.The paper is divided into four chapters.Chapter 1 introduces the research background of stochastic lattice systems and the main contents of this paper.Chapter 2 introduces some basic concepts and assumptions involved in this paper and proves the well-posedness of solution of stochastic Schr(?)dinger lattice system with Markovian switching.Chapter 3 considers the invariant measures and numerical approximation of stochastic Schrodinger lattice system with Markovian switching.Based on the well-posedness of solution of the system,we first investigate the existence of invariant measure of autonomous system corresponding to the underlying system in l2×S by Krylov-Bogolyubov’s method and uniform tail-ends estimates of solutions,where the dissipative conditions are relaxed by the stationary property of the Markov chain.And we prove the uniqueness of invariant measure of corresponding autonomous system under further conditions.Second,based on the tightness of the invariant measures set of its finite dimensional truncated systems,the convergence of invariant measures in Wasserstein sense as well as the convergence rate is studied between the underlying systems and its finite-dimensional truncated systems.Finally,based on the numerical invariant measure for the finite dimensional truncated system by the backward Euler-Maruyama(BEM)method,we can approximate numerically the invariant measure for the underlying systems with the convergence rate.Chapter 4 studies the ergodicity and numerical approximation of invariant measure for stochastic delay Schrodinger lattice system.First,by proving the exponential convergence of the solution of the stochastic delay lattice system with regard to the initial value,we prove the uniqueness of invariant measure of the system,which along with the existence of invariant measure,we can obtain the ergodicity of invariant measure of the system.Second,by AscoliArzalà theorem and the techniques in Chapter 3,we can prove the weak convergence of invariant measure between the underlying system and its finite dimensional truncated system.Finally,we study the numerical invariant measure of the finite dimensional truncated systems by virtue of the truncated Euler-Maruyama(TEM)method,and hence the finite dimensional numerical approximation of invariant measure of the stochastic delay Schrodinger lattice system can be obtained.