Polynomial mixing for white-forced Kuramoto-Sivashinsky equation on the whole line

TL;DR

This paper proves that the white-forced Kuramoto-Sivashinsky equation on the whole line exhibits polynomial mixing towards a unique invariant measure, using coupling methods and the Foiaș-Prodi estimate.

Contribution

It develops a new coupling criterion for polynomial mixing and applies it to establish ergodicity of the KSE on the whole line under stochastic forcing.

Findings

Proved polynomial mixing for the KSE with stochastic forcing.

Established a general criterion for polynomial mixing using coupling methods.

Combined coupling with Foiaș-Prodi estimates to demonstrate ergodicity.

Abstract

Our goal in this paper is to investigate ergodicity of white-forced Kuramoto-Sivashinsky equation (KSE) on the whole line. Under the assumption that sufficiently many directions of the phase space are stochastically forced, we can prove that the dynamics is attractive toward a unique invariant probability measure with polynomial rate of any power. In order to prove this, we further develop coupling method and establish a sufficiently general criterion for polynomial mixing. The proof of polynomial mixing for KSE is obtained by the combination of the coupling criterion and the Foia\c{s}-Prodi estimate of KSE on the whole line.