Stationary solutions to the stochastic Burgers equation on the line

Alexander Dunlap, Cole Graham, Lenya Ryzhik

Published 2019-10-16Version 1

We consider the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a noise that is white in time and colored in space. We show that, in a suitable space, there is a unique ergodic spacetime-stationary solution with any given mean. We also prove that the solution of the initial value problem with an initial condition that is an $L^1(\mathbb{R})$-perturbation of a constant $a$ converges to the ergodic spacetime-stationary solution with mean $a$.