Optimal convergence rates in the averaging principle for slow-fast SPDEs driven by multiplicative noise

TL;DR

This paper establishes optimal convergence rates for the averaging principle in slow-fast SPDEs with multiplicative noise, demonstrating strong and weak convergence orders using Poisson equation techniques.

Contribution

It provides the first proof of optimal convergence orders in the averaging principle for a class of slow-fast SPDEs with multiplicative noise.

Findings

Strong convergence order of 1/2 achieved

Weak convergence order of 1 achieved

Method based on Poisson equation effectively applied

Abstract

In this paper, we study a class of slow-fast stochastic partial differential equations with multiplicative Wiener noise. Under some appropriate conditions, we prove the slow component converges to the solution of the corresponding averaged equation with optimal orders 1/2 and 1 in the strong and weak sense respectively. The main technique is based on the Poisson equation.