Stochastic averaging for two-time-scale stochastic partial differential equations with fractional Brownian motion

Abstract

Abstract In this paper, we are concerned with a class of stochastic partial differential equations that have a slow component driven by a fractional Brownian motion with Hurst parameter 0 H 1 ∕ 2 and a fast component driven by a fast-varying diffusion. We will establish an averaging principle in which the fast-varying diffusion process acts as a “noise” and is averaged out in the limit. The slow process is shown to have a limit in the L 2 sense, which is characterized by the solution to a stochastic partial differential equation driven by a fractional Brownian motion with Hurst parameter 0 H 1 ∕ 2 whose coefficients are averages of that of the original slow process with respect to the stationary measure of the fast-varying diffusion. In the end, one example is given to illustrate the feasibility and effectiveness of results obtained.