Invariant Structures And Related Properties For Stochastic Dynamical Systems

There have been increasing interests in stochastic dynamical systems and their applications. Research in this subject involves stochastic analysis, partial differential equations, dynamical systems, harmonic analysis and others. Stochastic partial or ordinary differential equations arise as mathematical models from modern physics, chemistry, biology, economics and other applied disciplines. This makes research in this area meaningful and relevant.This Ph.D. thesis is devoted to the study of invariant structures and related topics for stochastic dynamical systems. It is divided into five chapters.In Chapter1, we introduce basic theory of Stochastic process, stochastic dynam-ical and stochastic differential equations.In Chapter2, we study the existence of invariant foliations for a class of stochastic partial differential equations with dynamical boundary conditions. Then we consider the approximation of the invariant foliations by analytic expansion method.Specially, we consider the following system: where D (?) Rn, and W1, W2are two independent Wiener processes. Boundary con-ditions of this type are usually called dynamical boundary condition, because on the boundary it involves the Ito differential of the unknown function with respect to time. Under proper conditions, we prove the existence of invariant foliations for the above system, and then we let∈â†'0, from which we can get the approximation result of in-variant foliations between the stochastic differential equation with dynamical bound-ary conditions and the stochastic differential equation with deterministic boundary conditions.In Chapter3, we consider the geometric shape of smooth stable manifolds for the following class of stochastic partial differential equation: where w is a scalar Brownian motion, A is a linear unbounded operator which is the infinitesimal generator of a strongly continuous semigroup. The nonlinear term F is assumed to be Ck. Symbol o indicates the Stratanovich differential. In stochastic dy-namical systems, we often use a stochastic transform to convert a stochastic system into a random system, and then we will approximate the Lyapunov-Perron integral equation (whose solution provides the graph of a random stable manifold) of the ran-dom system. We will deduce the results for the original stochastic system by the corresponding inverse stochastic transformation.In Chapter4, we consider a class of stochastic partial differential equations under the small perturbation of the domain. Specifically, we consider the following system: and the following perturbation system: where e is a small parameter, w is a scalar Brownian motion, and nonlinear functions f(u), g[u) and f∈(u),g∈(u) satisfy Iipschtiz conditions. We assume that the resolvent operator of A has convergence property under the perturbation of the domain. From the relationship of resolvent operator and analytic semigroup, and in every compact subset of (0, T], we have the convergence result of the analytic semigroup. Through a series of analysis, we deduce the result of convergence of the solution for stochastic partial differential equations when the perturbation parameter∈tends to zero, in the sense of mean square.Finally, in Chapter5, we discuss some future research topics and some exten-sions of the present research.