Invariant measures for stochastic Burgers equation on unbounded domains

TL;DR

This paper studies the stochastic damped Burgers equation on the real line, proving existence, uniqueness, and the existence of invariant measures for its solutions using probabilistic and analytical methods.

Contribution

It establishes the existence and uniqueness of solutions and invariant measures for the stochastic Burgers equation on unbounded domains, extending previous results to the entire real line.

Findings

Existence and uniqueness of mild solutions

Solutions are uniformly bounded in time

Existence of invariant measures via Krylov-Bogolioubov theorem

Abstract

In this paper, we investigate the stochastic damped Burgers equation with multiplicative noise defined on the entire real line. We demonstrate the existence and uniqueness of a mild solution to the stochastic damped Burgers equation and establish that the solution is uniformly bounded in time. Furthermore, by employing the uniform estimates on the tails of the solution, we obtain the tightness of a family of probability distributions of the solution. Subsequently, by applying the Krylov-Bogolioubov theorem, we establish the existence of invariant measures.