Effective Dynamical Behaviors Of Stochastic And Multiscale Evolutionary Equations

This thesis is devoted to the study of effective dynamical behaviors for stochastic or multiscale evolutionary equations.The first issue is the homogenization problems for stochastic Schrodinger equations and stochastic parabolic equations with large periodic potentials,where the initial data depends on the scale parameter.The second one is the effective approximation for the Schrodinger equation with a rapidly oscillating potential and the associated Bohmian trajectory.The third issue is the estimates of the Kantorovich-Rubinstein distance for solutions of non-local Fokker-Planck equations with weakly differentiable coefficients,and the smooth approximation for a class of non-local Fokker-Planck equations.And these non-local Fokker-Planck equations are associated to certain stochastic differential equations with non-Gaussian Levy noises.This thesis is organized as follows.In Chapter 1,the research background and relevant progress are expounded.Then,our results are stated.In Chapter 2,basic knowledge and theorems about Brownian motion and Levy process in stochastic analysis are reviewed.The definitions and relations of several types of solutions of stochastic partial differential equations are presented.Some inequalities used in this thesis are also recalled.In Chapter 3,the homogenization for a stochastic Schrodinger equation with a large periodic potential is studied.To obtain the homogenized stochastic Schrodinger equation,the main idea is to use Bloch wave theory to build adequate oscillating test functions and to pass to the limit using two-scale convergence.It is proved that the solution of the original equation can be approximately factorized as the product of a fast oscillating cell eigenfunction and a slowly varying solution of the homogenized equation.In Chapter 4,the effective approximation of a stochastic partial differential equation with a large potential in a periodic medium are investigated,where the initial value depends on the scale parameter and has oscillation and periodicity.Based on the Bloch wave theory and the two-scale convergence method,an efficient approximation stochastic partial differential equation is derived,and its coefficients depend on the initial value.In Chapter 5,the time-oscillating Schrodinger equation and associated quantum Bohmian trajectory are investigated.First,based on homogenization method,the convergence of the wave function governed by the time-oscillating Schrodinger equation is proved under certain assumptions.Then,the limit of the corresponding Bohmian measure is derived.Finally,according to the connection between the Bohmian measure and the Bohmian flow,it is proved that the corresponding Bohmian trajectory converges locally in a measure on the finite time interval,and the limit trajectory coincides with the effective system of the time-oscillating Schrodinger equation.This is beneficial for the efficient simulation of the Bohmian trajectories in oscillating potential fields.In Chapter 6,the Kantorovich-Rubinstein distance for solutions of the non-local FokkerPlanck equations,which correspond to stochastic differential equations with non-Gaussian Levy noises,is studied.Based on superposition principle,an upper bound estimate of the Kantorovich-Rubinstein distance for the solutions of two non-local Fokker-Planck equations with different coefficients is obtained,where the coefficients are all weakly differentiable.This leads to a smooth effective approximation for a class of non-local Fokker-Planck equations.Finally,in Chapter 7,our research work and innovation are summarized,and further questions are also discussed.