Stochastic averaging for a class of two-time-scale systems of stochastic partial differential equations

This paper focuses on systems of stochastic partial differential equations that have a slow component driven by a fractional Brownian motion and a fast component driven by a fast-varying diffusion. We 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 L₂ sense, which is characterized by the solution of a stochastic partial differential equation driven by a fractional Brownian motion whose coefficients are averages of that of the original slow process with respect to the stationary measure of the fast-varying diffusion. This averaging principle paves a way for reduction of computational complexity. The implication is that one can ignore the complex original systems and concentrate on the average systems instead.