Stochastic averaging for two-time-scale stochastic partial differential equations with fractional Brownian motion. (February 2019)

• Record Type: Journal Article
• Authors: Li, Zhi; Yan, Litan
• Is Part Of: Nonlinear analysis. Volume 31(2019)
• Page Start: 317
• Page End: 333
• Publication Date: 2019-02
• Subjects: 60H15 -- 60G15 -- 60H05; Stochastic averaging -- Fast-slow SPDEs -- Fractional Brownian motion; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nahs.2018.10.002
• Languages: English
• ISSNs: 1751-570X

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.