# American Institute of Mathematical Sciences

March  2020, 25(3): 1141-1158. doi: 10.3934/dcdsb.2019213

## Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion

 1 School of Mathematical Sciences, Fudan University, Shanghai, 200433, China 2 Graduate School of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan 3 Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710072, China 4 School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

* Corresponding author: Yong Xu

Received  November 2018 Revised  February 2019 Published  September 2019

Fund Project: The research of B. Pei was partially supported by the NSF of China (11802216), the China Postdoctoral Science Foundation (2019M651334) and JSPS KAKENHI Grant Number JP18F18314 (Grant-in-Aid for JSPS Fellows). The research of Y. Xu was partially supported by the NSF of China (11702216), Shaanxi Province Project for Distinguished Young Scholars and the Fundamental Research Funds for the Central Universities. B. Pei would like to thank JSPS for Postdoctoral Fellowships for Research in Japan (Standard).

In this paper, we focus on fast-slow stochastic partial differential equations in which the slow variable is driven by a fractional Brownian motion and the fast variable is driven by an additive Brownian motion. We establish an averaging principle in which the fast-varying diffusion process will be averaged out with respect to its stationary measure in the limit process. It is shown that the slow-varying process $L^{p}(p \geq 2)$ converges to the solution of the corresponding averaging equation. To reduce the complexity, one can concentrate on the limit process instead of studying the original full fast-slow system.

Citation: Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
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