Mild Stochastic Sewing Lemma, SPDE in Random Environment, and Fractional Averaging

TL;DR

This paper introduces a stochastic sewing lemma with quantitative bounds for mild processes, applies it to SPDEs driven by fractional Brownian motions in random environments, and establishes a fractional averaging principle for non-stationary fast environments.

Contribution

It presents a new stochastic sewing lemma with estimates, and extends fractional averaging principles to non-stationary environments for SPDEs.

Findings

Established uniform $L^p$ bounds for mild processes.

Proved a fractional averaging principle for SPDEs in non-stationary environments.

Applied the sewing lemma to study SPDEs driven by fractional Brownian motions.

Abstract

Our first result is a stochastic sewing lemma with quantitative estimates for mild incremental processes, with which we study SPDEs driven by fractional Brownian motions in a random environment. We obtain uniform $L^p$-bounds. Our second result is a fractional averaging principle admitting non-stationary fast environments. As an application, we prove a fractional averaging principle for SPDEs.