Averaging principle for slow-fast systems of stochastic PDEs with rough coefficients

TL;DR

This paper establishes an averaging principle for slow-fast stochastic PDEs with rough coefficients, proving existence, tightness, and convergence of solutions to an averaged equation in complex reaction-diffusion systems.

Contribution

It introduces conditions for existence of solutions and demonstrates the averaging principle for stochastic PDEs with unbounded, discontinuous nonlinearities.

Findings

Existence of martingale solutions under rough coefficients

Tightness of the laws of slow motions

Convergence to an averaged equation

Abstract

In this paper, we consider a class of slow-fast systems of stochastic partial differential equations where the nonlinearity in the slow equation is not continuous and unbounded. We first provide conditions that ensure the existence of a martingale solution. Then we prove that the laws of the slow motions are tight, and any of their limiting points is a martingale solution for a suitable averaged equation. Our results apply to systems of stochastic reaction-diffusion equations where the reaction term in the slow equation is only continuous and has polynomial growth.