Stationary solutions to the stochastic Burgers equation on the line

TL;DR

This paper establishes the existence and uniqueness of extremal invariant measures for the stochastic Burgers equation on the real line, and proves convergence of solutions to these measures over time.

Contribution

It proves the existence of unique indecomposable invariant measures for each mean and shows convergence of solutions to these measures, advancing understanding of long-term behavior.

Findings

Unique indecomposable invariant measures for each mean $a$.

Solutions from perturbations converge to stationary measures.

Characterization of invariant measures for stochastic Burgers equation.

Abstract

We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or extremal, if it cannot be represented as a convex combination of other invariant measures. We show that for each $a\in\mathbb{R}$, there is a unique indecomposable law of a spacetime-stationary solution with mean $a$, in a suitable function space. We also show that solutions starting from spatially-decaying perturbations of mean-$a$ periodic functions converge in law to the extremal space-time stationary solution with mean $a$ as time goes to infinity.