Stationary solutions to the stochastic Burgers equation on the line

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.

68 pages. This post-publication version removes a claim in the statement of Proposition 5.2 whose proof was incorrect. The applications of this statement in the remainder of the paper are replaced with slightly modified arguments that avoid the use of this claim, and so the main results of the paper remain unchanged. The authors are grateful to Yu Gu for pointing out the error