Averaging Principle For Two-Time-Scale Stochastic Functional Differential Equations

Two-time-scale stochastic differential equations are widely used in control,optimization,network systems and economic systems,etc.For example,plane disk motion,climate model and intracellular biochemical reactions can be modeled by two-time-scale stochastic differential equations.Unfortunately,because the difference between the two time scales is large and fast-slow variables are coupled with each other,which lead to system is difficult to deal with.Hence we want to use a simpler system to replace the original complex system.So the averaging principle provides an effective method for the qualitative analysis of multi-scale complex systems.In addition,delays are ubiquitous in everyday life.Sometimes delays may affect the dynamics of the system.The influence of delays on the system cannot be ignored in modeling,so it is very necessary to consider delays in two-time-scale stochastic differential equations.Hence this thesis studies the problem of the averaging principle for two-time-scale stochastic functional differential equations.This thesis is mainly divided into the following four chapters.Chapter 1 introduces the research background and progress of the averaging principle,and summarizes the main content of this thesis.Chapter 2 introduces the basic knowledge such as the definitions of non-anticipative functional derivability,continuity and local boundedness.Then we generalize the mixed functional Ito formula to the product mixed functional Ito formula which be useful for studying the averaging principle for two-time-scale stochastic functional differential equations.Chapter 3 mainly proves the existence,uniqueness and regularity of the solution of the functional Poisson equation,and gives the estimates of the solution and its derivatives.The parameter in the Poisson equation belongs to infinite-dimensional space,which can be used to study stochastic average and limit problems in probability theory,etc.Chapter 4 mainly studies the averaging principle for two-time-scale stochastic functional differential equations.Firstly,the existence and uniqueness of the solution are proved.Secondly,the existence,uniqueness and exponential ergodicity of the invariant measure of the solution of the frozen equation are investigated.Then the moment estimate is given.Finally,the strong fluctuation estimate of the integrated function of the solution is shown by using the functional Poisson equation.Strong convergence of the averaging principle is demonstrated based on strong fluctuation estimate.