Existence of stationary stochastic Burgers evolutions on $\mathbf{R}^2$ and $\mathbf{R}^3$

TL;DR

This paper proves the existence of stationary solutions for the stochastic Burgers equation in dimensions less than four, driven by gradient noise, and links these solutions to the KPZ equation with stationary gradients.

Contribution

It establishes the existence of spacetime-stationary solutions for the stochastic Burgers equation on <4 with gradient noise, extending understanding of these stochastic PDEs.

Findings

Existence of stationary solutions for stochastic Burgers in <4

Solutions are gradients of KPZ solutions with stationary gradients

Method involves tightness of time-averaged laws in weighted spaces

Abstract

We prove that the stochastic Burgers equation on $\mathbf{R}^{d}$, $d<4$, forced by gradient noise that is white in time and smooth in space, admits spacetime-stationary solutions. These solutions are thus the gradients of solutions to the KPZ equation on $\mathbf{R}^{d}$ with stationary gradients. The proof works by proving tightness of the time-averaged laws of the solutions in an appropriate weighted space.