| Stochastic differential equations are often applied to study various complex phenomena perturbed by random influences in many fields such as physics,biology and so on.A very important and difficult topic in the researches of stochastic differential equations is multi-scale stochastic differential equations.Multi-scale problem exists widely in almost all phenomena and applications of the nature and the human society.Due to the appearance of the multi-scale phenomena and random influences,it is difficult to investigate corresponding problems.For a long time,an important task of mathematicians is to develop valid methods to study the analytical solutions,the approximate solutions and the dynamical behaviors.Amplitude equations and the theory of random invariant manifolds are two kinds of important methods.The main idea of both is to construct reduced equations,by which we can obtain the approximate solutions of the original equations and study the corresponding dynamical behaviors.The main goal of this dissertation is to investigate the asymptotic behaviors of multi-scale stochastic differential equations.We apply amplitude equations and the theory of random invariant manifolds to the depth analysis of some classes of multiscale stochastic differential equations from different perspectives,and attempt to present some new research results.The dissertation consists of six chapters:Chapter 1 is the introduction,in which we systematically introduce the related research background and important results of multi-scale differential equations,then introduce amplitude equations and the theory of random invariant manifolds.Chapter 2 is the preliminary,in which we introduce basic concepts and important results of stochastic differential equations and random dynamical systems.In Chapter 3 and 4,we study the approximate solutions of stochastic partial differential equations with nonlinearities perturbed by additive and multiplicative noise at the same time.We obtain the approximate solutions via amplitude equations.Then,the error between the approximate solutions and the original ones is rigorously estimated.In Chapter 5,with the help of the theory of random invariant manifolds,we study the approximation of the dynamical behaviors of a class of stochastic partial differential equations perturbed by multiplicative noise,and prove that the original equations have the random invariant manifolds and foliations with exponential tracking property.Moreover,we also prove that when the small parameter tends to 0,the random invariant manifolds and foliations of the original equations can converge to those of the limiting equations.Finally,we present a brief conclusion of this dissertation,and prospect the next attractive researches in Chapter 6. |
