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Stochastic averaging for two-time-scale stochastic partial differential equations with fractional Brownian motion. (English) Zbl 1418.60083
Summary: 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.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
60H05 Stochastic integrals
Full Text: DOI
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