On averaging and mixing for stochastic PDEs

TL;DR

This paper investigates the convergence behavior of averaging methods for nonlinear stochastic perturbations of linear PDEs with imaginary spectra, demonstrating uniform convergence under mixing conditions.

Contribution

It establishes that if the effective equation is mixing, then the Krylov--Bogolyubov averaging convergence is uniform in time for certain stochastic PDEs.

Findings

Convergence in averaging is uniform in time under mixing conditions.

Effective equations with mixing properties ensure robust convergence.

Provides theoretical insights into stochastic PDEs with imaginary spectra.

Abstract

We examine the convergence in the Krylov--Bogolyubov averaging for nonlinear stochastic perturbations of linear PDEs with pure imaginary spectrum and show that if the involved effective equation is mixing, then the convergence is uniform in time.