Averaging Principle for Nonautonomous Slow-Fast Systems of Stochastic Reaction-Diffusion Equations: The Almost Periodic Case

Abstract

We study the validity of an averaging principle for a slow-fast system of stochastic reaction diffusion equations. We assume here that the coefficients of the fast equation depend on time, so that the classical formulation of the averaging principle in terms of the invariant measure of the fast equation is not anymore available. As an alternative, we introduce the time depending evolution family of measures associated with the fast equation. Under the assumption that the coefficients in the fast equation are almost periodic, the evolution family of measures is almost periodic. This allows to identify the appropriate averaged equation and prove the validity of the averaging limit. *

Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients

“…Fu, Wan and Liu proved the strong averaging principle for stochastic hyperbolic-parabolic equations with slow and fast time-scales in [13]. Cerrai and Lunardi studied the averaging principle for nonautonomous slow-fast systems of stochastic reaction-diffusion equations in [7]. For some further results on this topic, we refer to [4,12,22,23] and the references therein.However, the references we mentioned above always assume that the coefficients satisfy Lipschitz conditions, and there are few results on the average principle for SPDEs with nonlinear terms.…”

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

“…In 2017, the averaging principle has been presented for nonautonomous slow-fast system of stochastic reactiondiffusion equations by Cerrai [25]. But, the system of this paper was driven by Gaussian noises, which is considered as an ideal noise source and only can simulate fluctuations near the mean value.…”

Averaging principle for a stochastic cable equation

“…Wang and Xu [34] investigated the stochastic av-eraging method for neutral stochastic delay equations driven by fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). Other interesting examples of studying of averaging for stochastic differential equations can be found in papers [1,12,15,18].…”