Exponential mixing for the stochastic Kuramoto-Sivashinsky equation on the 1D torus

TL;DR

This paper proves that the stochastic Kuramoto-Sivashinsky equation on the 1D torus has a unique invariant measure that is exponentially attractive, improving understanding of its long-term behavior without requiring small anti-diffusion.

Contribution

It establishes exponential convergence to equilibrium for the stochastic KSE without the need for small anti-diffusion, using coupling and Lyapunov functions.

Findings

Unique invariant measure exists and is exponentially attractive.

Convergence rate towards equilibrium is exponential.

Smallness condition on anti-diffusion coefficient is removed.

Abstract

In this paper, we study the large-time behaviors of the Kuramoto-Sivashinsky equation (KSE) on the 1D torus while being subjected to random perturbation via additive Gaussian noise. It is well-known that under suitable assumptions on the stochastic forcing, the KSE admits a unique invariant probability measure. In this work, we make further progress on the topic of ergodicity by addressing the issue of convergence rate toward equilibrium. In comparison with the previous results, we can prove that the unique invariant probability measure is exponentially attractive and smallness condition of anti-diffusion coefficient is not necessary here. The proof relies on a coupling argument while making use of Lyapunov functions motivated by those of deterministic equations.