Invariant measures for the open KPZ equation: the Gaussian case

TL;DR

This paper extends the proof of invariance of drifted Brownian motions to the open KPZ equation with boundary conditions, using Stein's equation and integration by parts.

Contribution

It provides a new proof that Brownian motion with constant drift is invariant for the open KPZ equation with boundary conditions, generalizing previous results.

Findings

Brownian motion with drift is invariant for the open KPZ equation with boundary conditions

The approach of Stein's equation and integration by parts is effective in this context

The invariance holds modulo height shifts

Abstract

In [arXiv:2409.08465], Quastel and Gu use Stein's equation and integration by parts to give a direct proof that drifted Brownian motions are stationary (modulo height shifts) for the full-line KPZ equation. In this article, we consider the open KPZ equation with boundary conditions $\partial_x h(t,0) = \partial_x h(t,1) = \alpha$ for a general real parameter $\alpha$, and emulate the approach of Quastel and Gu to provide a similar proof that Brownian motion with constant drift $\alpha$ is invariant (modulo height shifts) in this case.