Stationary Solutions to the Stochastic Burgers Equation on the Line

We consider invariant measures for the stochastic Burgers equation on R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document}, 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∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in {\mathbb {R}}$$\end{document}, 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.