Homogenization And Dynamical Behaviors For Nonlocal Stochastic Heterogeneous Systems

Multi-scale simulation is a typical interdisciplinary problem involving mathematics,chemistry,physics,engineering,computer science,environmental science and other disciplines,such as fluid flows in porous media,composites materials in advanced technologies,and data assimilations in geophysical modeling.The multi-scale problems are widespreading in many scientific and engineering problems.Traditional multi-scale methods,including multi-grid method and adaptive method,are difficult to solve many practical problems.Therefore,it is necessary to explore much more efficient methods,such as the homogenization method.The theory of Homogenization is a mathematical method introduced in the 1970 s to study non-homogeneous materials.This method is used to analyze systems with two or more scales.It can map the information at the micro scale to the macro scale,and then derive the homogenized equation at the macro scale.That is to say,we can solve the original problem at the macro scale.In this dissertation,we adopt the homogenization method to study the multi-scale systems,and obtain the effective reduced equations of multi-scale systems.The multi-scale systems here stand for the heterogeneous systems in this paper.Firstly,we study the homogenization of stochastic partial differential equations with nonlocal operators by martingale method.As the generators of L(?)vy processes,there are two different kinds of the nonlocal operators: integrable jump kernel and non-integrable jump kernel.Furthermore,we apply the homogenization results to the Zakai equations for a signal-observation system driven by non-Gaussian noise.Secondly,we deal with homogenization of a nonlocal model with L(?)vy-type operator of rapidly oscillating coefficients.Finally,we study the stochastic Schr(?)dinger equation with nonlocal Laplace operator by using the two-scale convergence method.We obtain a homogenized Schr(?)dinger equation.This thesis is organized as follows.In Chapter 1,we discuss the research background,significance,research status,and main results.In Chapter 2,we recall basic knowledge and related theorems in measure theory,stochastic analysis,differential equation and multi-scale method,and some inequalities needed in this thesis.In Chapter 3,we study a class of nonlocal heterogeneous systems described by stochastic partial differential equations.The nonlocal operator in the system is the generator of a L(?)vy process of an integrable jump kernel.We get an estimation result by using the Taylor expansion.Then,we deduce that the martingale solution of the nonlocal heterogeneous equation converges to the martingale solution of a local equation.Further,we apply this convergence result to the homogenization of the Zakai equation of a signal-observation system driven by non-Gaussian noise.And an effective approximation is also obtained.In Chapter 4,we study a class of nonlocal heterogeneous systems with the nonlocal operator as the generator of a L(?)vy process of non-integrable jump kernel.Here,the Taylor expansion formula is no longer applicable.We examine the properties of solutions in a new fractional Sobolev space by studying the nonlocal gradient operator and the nonlocal divergence operator.By martingale method,we prove the convergence of martingale solutions for nonlocal heterogeneous equations.But different from the third chapter,the homogenized equation here is still a nonlocal equation.Finally,we apply this convergence result to the homogenization of the Zakai equation of a signal-observation system driven by ?-stable L(?)vy process.In Chapter 5,we deal with homogenization of a nonlocal model with L(?)vy-type operator of rapidly oscillating coefficients.This nonlocal model describes mean residence time and other escape phenomena for stochastic dynamical systems with non-Gaussian L(?)vy noise.We derive the convergence rate of the effective model.This enables efficient analysis and simulation of escape phenomena under non-Gaussian fluctuations.In Chapter 6,we study the homogenization problem of the Schr(?)dinger equation with nonlocal Laplacian operator.By studying the properties of the nonlocal gradient operator and the nonlocal divergence operator,and combining with the two-scale convergence method,we derive an effective Schr(?)dinger equation.The Schr(?)dinger equations and the Zakai equations are widely used in physics,and we study stochastic homogenization of them.This makes it possible to study the dynamical behaviors of Schr(?)dinger equations and Zakai equations obtained under heterogeneous conditions.The wide existence of heterogeneous media in our life makes our research more practical.At the same time,we have also extended the form of nonlocal operators in the equation to cover more practical problems.Finally,in Chapter 7,we summarize our research work in this thesis,and outline some future research topics.