Asymptotic Behavior Of Stochastic Lattice Systems And Wong-Zakai Approximations

Stochastic lattice systems are known as the spatial discretization of stochastic partial differential equations,which have wide applications in many fields.The study on the long term dynamics of the solutions of stochastic lattice systems is helpful to understand or predict the future trends of the systems.At present,the asymptotic behavior of lattice systems with additive or multiplicative white noise has been studied in depth and fruitful results have been obtained.In this paper,we study the existence of pullback attractors of two types of stochastic lattice systems and the convergence of pullback attractors of their Wong-Zakai approximations.This dissertation is organized as follows:In Chapter 1,we briefly introduce the research background and significance of stochastic lattice systems,and then recommend the main research results and basic structure of this dissertation.In Chapter 2,we introduce the concepts of non-autonomous stochastic dynamical systems and three main methods of Wong-Zakai approximations.In Chapter 3 and Chapter 4,we study the asymptotic behavior of solutions of the pLaplacian lattice systems by using the method of colored noise approximating white noise.In Chapter 3,under weaker conditions,we obtain the existence of pullback absorbing set for the p-Laplacian lattice system with nonlinear colored noise,we derive uniform estimates on the tails of solutions,we also prove the system has a unique pullback attractor.In Chapter 4,we study the p-Laplacian lattice system with multiplicative white noise.we first transform the stochastic p-Laplacian lattice system into a pathwise deterministic system.Then,we prove the existence of the pullback attractor of system,and consider the uniform compactness of the pullback attractor of its approximations.Finally,we prove the convergence of the pullback attractor from colored noise to white noise.In Chapter 5 to Chapter 7,we study the well-posedness and long-term dynamical behavior of a class of second-order stochastic lattice systems and their Euler approximations of Brownian motion.In Chapter 5,we prove the Euler approximation of system driven by nonlinear white noise has a unique pullback attractor.In Chapter 6 and Chapter 7,we consider the second-order stochastic lattice systems driven by multiplicative or additive white noise,respectively.Firstly,we transform the second-order stochastic lattice systems into a pathwise deterministic systems,and prove the existence of pullback attractors of the secondorder lattice systems driven by multiplicative or additive white noise.At the same time,we prove the uniform compactness of the pullback attractors of the Euler approximations of the second-order stochastic lattice systems driven by multiplicative or additive white noise.Finally,we prove the pullback attractors of the respective Euler approximations converge to the pullback attractors of the original systems driven by multiplicative or additive white noise as the step-length of the Wiener shift approaches zero.