Pathwise unique solutions and stochastic averaging for mixed stochastic partial differential equations driven by fractional Brownian motion and Brownian motion

TL;DR

This paper establishes existence, uniqueness, and an averaging principle for mixed stochastic partial differential equations driven by fractional Brownian motion and Brownian motion, combining pathwise and Itô calculus techniques.

Contribution

It introduces a new approach to prove existence and uniqueness for mixed SPDEs and establishes an averaging principle for systems with fractional Brownian motion.

Findings

Proved existence and uniqueness of solutions for mixed SPDEs.

Established an averaging principle in the mean square sense.

Derived the limit process relying on the invariant measure of the fast component.

Abstract

This paper is devoted to a system of stochastic partial differential equations (SPDEs) that have a slow component driven by fractional Brownian motion (fBm) with the Hurst parameter $H >1/2$ and a fast component driven by fast-varying diffusion. It improves previous work in two aspects: Firstly, using a stopping time technique and an approximation of the fBm, we prove an existence and uniqueness theorem for a class of mixed SPDEs driven by both fBm and Brownian motion; Secondly, an averaging principle in the mean square sense for SPDEs driven by fBm subject to an additional fast-varying diffusion process is established. To carry out these improvements, we combine the pathwise approach based on the generalized Stieltjes integration theory with the It\^o stochastic calculus. Then, we obtain a desired limit process of the slow component which strongly relies on an invariant measure of the fast-varying diffusion process.