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

Bin Pei; Yong Xu; Yuzhen Bai

doi:10.3934/dcdsb.2019213

Article Contents

2020, Volume 25, Issue 3: 1141-1158. Doi: 10.3934/dcdsb.2019213

This issue Previous Article Global asymptotic stability of nonconvex sweeping processes Next Article Global stability of the predator-prey model with a sigmoid functional response

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
^* Corresponding author: Yong Xu

Received: November 2018

Revised: February 2019

Early access: September 2019

Published: March 2020

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)

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Abstract

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.

Keywords:

Averaging principle,
mild solution,
fractional Brownian motion,
fast-slow,
stochastic partial differential equations.

Mathematics Subject Classification: Primary: 60G22; Secondary: 60H15.

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References

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