Mixing for the Burgers equation driven by a localised two-dimensional stochastic forcing

TL;DR

This paper proves that the one-dimensional Burgers equation with localized stochastic forcing exhibits mixing behavior, using controllability and a general mixing criterion, advancing understanding of stochastic PDEs with low-dimensional noise.

Contribution

It introduces a novel approach combining controllability and mixing criteria to analyze the Burgers equation with localized stochastic forcing.

Findings

Established mixing property for the stochastic Burgers equation

Utilized controllability results for damped conservation laws

Provided a framework for analyzing low-dimensional stochastic PDEs

Abstract

We consider the one-dimensional Burgers equation perturbed by a stochastic forcing, which is assumed to be white in time and localised and low-dimensional in space. We establish a mixing property for the Markov process associated with the problem in question. The proof is based on a general criterion for mixing and a recent result on global approximate controllability to trajectories for damped conservation laws.