Some Results On Limit Theory For Stochastic Partial Differential Equations

The stochastic partial differential equation is the partial differential equation with random terms or random coefficients,and it is an important tool to analyze and simulate the real problem.Stochastic partial differential equations have wide applications in many fields,such as chemistry,theoretical physics,financial mathematics,life sciences,control problem and so on.Therefore,the study of the limit theory for stochastic partial differential equations has attracted the attention of many probabilistic scholars,which is of great significance.Experts and scholars found that the spatial integral of the solution to the stochastic heat equation in 1+1 dimension,driven by a space-time white noise with flat initial data,is similar to a sum of independent and identically distributed random variables,and they used the Malliavin calculus and Stein’s method to study the central limit theorem and functional central limit theorem for the spatial integral of the form∫[0,R]u(t,x)dx.After that,the mathematical community set off a boom in research on the limit theory for spatial integrals of solutions to stochastic partial differential equations.In this thesis,we consider some results on limit theory for spatial integrals of solutions to stochastic partial differential equations.The thesis is composed of four chapters.The first chapter is the introduction.We summarize the research background and current situation of stochastic partial differential equations and introduce the basic knowledge related to this thesis,including Malliavin calculus and some inequalities in probability theory.At the same time,the basic framework of this thesis is given.In Chapter 2,we mainly study the almost sure central limit theorems for the stochastic partial differential equations.We prove the almost sure central limit theorems for the spatial integrals of the solutions to the stochastic heat equation,parabolic Anderson model and stochastic wave equation,using central limit theorems,the Poincare type inequality and the moment estimates for the Malliavin derivatives of the solutions to these equations.These results enrich the limit theory for spatial integrals of solutions to stochastic partial differential equations.In Chapter 3,we study the law of the iterated logarithm for the stochastic partial differential equations.Inspired by the localization property of solutions to the 1+1 dimensional stochastic partial differential equations driven by space-time white noise and the skillful partition of the interval,we get the Hartman-Wintner type law of the iterated logarithm,the nonclassical law of the iterated logarithm and the functional law of the iterated logarithm for the spatial integrals of the solutions to the 1+1 dimensional stochastic heat equation and stochastic wave equation driven by space-time white noise,which improve the limit theory for spatial integrals of solutions to stochastic partial differential equations.In Chapter 4,we study the precise asymptotics for the stochastic partial differential equations.Using the Burkholder-Davis-Gundy inequality,we can obtain the moment estimates for the spatial integrals of the solutions to stochastic partial differential equations.Thus,we prove the precise asymptotics for the complete convergence and the complete moment convergence for spatial integrals of the solutions to stochastic heat equation and stochastic wave equation,which greatly enriches the theoretical results for spatial integrals of solutions to stochastic partial differential equations.At the end of the thesis,we summarize the main work and achievements of this thesis and put forward some questions and assumptions for the follow-up research work.