Averaging Principle For Multiscale Stochastic Differential Equations

In many fields,some complex process phenomena can be simulated and analyzed by stochastic differential equations In addition.It is found in practice that many phenomena or processes have multi time characteristics or multi-scale effects Therefore,it is very natural to describe and analyze these phenomena or processes with the theory of multi-scale stochastic dynamic system,and the qualitative analysis of this kind of stochastic system has also become a research hotspot.The average method is a powerful tool to simplify complex stochastic dynamic systems,which is widely used in the study of various models.This paper mainly studies the average principle of two kinds of stochastic differential equations.Firstly,the average results of stochastic evolution equations are discussed.It is proved that under appropriate conditions,the solution of the original equation is close to the solution of the average equation Secondly,the averaging principle of two-scale stochastic differential equations driven by Brownian motion under non-Lipschitz condition is studied.The structure of this paper is as follows:In the first part,some basic concepts and related definitions and lemmas in the theory of stochastic dynamic systems are given.In the second part,we study the averaging principle of a class of neutral stochastic evolution equations with impulses by transforming the time scale of the original equation and "averaging" the coefficient function of the original equation.The effective equation of the original equation is obtained.Then,by estimating the error in the appropriate space of the solution process of the original equation and the solution process of the effective equation after the time scale transformation,the random average principle in the sense of L2 convergence can be obtained.The third part mainly discusses the averaging principle of two-scale Hilfer fractional derivative stochastic differential equations.The Brownian motion noise in the system disturbs both fast motion process and slow motion process.Under appropriate conditions,when the slow variable is fixed,a“fixed”equation of the fast variable equation can be obtained,and the invariant measure of exponential mixing exists and is unique.Using khasminskii technique,the auxiliary process（（?）tε,（?）tε）is constructed for analysis.Next,we use Jensen inequality,B-D-G inequality,Gronwall inequality and other computational techniques to prove the solution estimation of the equation and estimate |Xtε-（?）tε|.Then,due to the exponential ergodicity of the fast variable equation,the auxiliary process（?）tε and average equation solution process deviation between Xt.Finally,it is proved that when the time scale parameter tends to zero,the solution process of the slow variable equation converges to the solution process of the averaging equation in the sense of L2.The last part,we summarize this paper and make a prospect for future work.