Averaging Principles for Nonautonomous Two-Time-Scale Stochastic Reaction-Diffusion Equations with Jump

Ruifang Wang,Yong Xu

doi:10.1155/2020/9864352

Abstract

In this paper, we aim to develop the averaging principle for a slow-fast system of stochastic reaction-diffusion equations driven by Poisson random measures. The coefficients of the equation are assumed to be functions of time, and some of them are periodic or almost periodic. Therefore, the Poisson term needs to be processed, and a new averaged equation needs to be given. For this reason, the existence of time-dependent evolution family of measures associated with the fast equation is studied and proved that it is almost periodic. Next, according to the characteristics of almost periodic functions, the averaged coefficient is defined by the evolution family of measures, and the averaged equation is given. Finally, the validity of the averaging principle is verified by using the Khasminskii method.

Highlights

Complexity where ε ≪ 1 is a positive parameter and α is a sufficiently large fixed operators
Where u is the solution of the averaged equation (2). e theory of the averaging principle, originated by Laplace and Lagrange, has been applied in celestial mechanics, oscillation theory, radiophysics, and other fields. e firstly rigorous results for the deterministic case were given by Bogolyubov and Mitropolskii [1]
Volosov [2] and Besjes [3] promoted the development of the averaging principle. en, great interest has appeared in its application to dynamical systems under random perturbations

Summary

A Priori Bounds for the Solution

With all notations introduced above, system (1) can be rewritten in the following abstract form: duε(t) 􏼂A1(t)uε(t) + B1 t, uε(t), vε(t)􏼁􏼃dt. According to the Burkholder–Davis–Gundy inequality, by proceeding as Proposition 4.2 in [12], we can get t cp(t). According to the Gronwall inequality, we get E sup uε(r) pH ≤ cp,T􏼐1 +‖x‖pH +‖y‖pH􏼑, t ∈ 􏼂0, t1􏼃. + 􏽚 􏽚 U1(t + h, r)G1 r, uε(r), z􏼁N􏽥 1(dr, dz) ≔ 􏽘 Iit. By proceeding as the proof of Proposition 4.4 in [12] and (60), fix θ ∈ [0, θ), for any p ≥ 1, and it is possible to show that. According to eorem 13.2 in [43] and the above lemma, by using Chebyshev’s inequality, we can get that the family of probability measures 􏼈L(uε)􏼉ε∈(0,1] is tight in

An Evolution Family of Measures for the Fast Equation

The Averaged Equation

Averaging Principles