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

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.

Duke Scholars

Author Alexander J Dunlap Mathematics