Invariant measures for the open KPZ equation: an analytic perspective

TL;DR

This paper provides a rigorous stochastic analytic proof for the invariant measures of the open KPZ equation, employing regularity structures and boundary layer analysis, advancing understanding of boundary effects in stochastic growth models.

Contribution

It introduces a novel analytic approach using regularity structures and boundary layer analysis to prove invariant measures for the open KPZ equation, independent of boundary parameters.

Findings

Established a central limit theorem for boundary nonlinearity.

Proved invariance of measures without boundary parameter restrictions.

Applied regularity structures to boundary layer analysis in KPZ.

Abstract

The ergodic theory of the open KPZ equation has seen significant progress in recent years, with explicit invariant measures described in a series of works by Corwin--Knizel, Barraquand--Le Doussal, and Bryc--Kuznetsov--Wang--Weso{\l}owski. In this paper, we provide a stochastic analytic proof of the formula for the invariant measures. Our approach starts from the Gaussian invariant measure for the case of homogeneous boundary conditions. We approximate the inhomogeneous problem by a homogeneous one with a singular boundary potential. Using tools including change of measure, time reversal for Markov processes, and It\^o's formula, we then reduce the problem to analyzing the KPZ nonlinearity in a thin boundary layer. Finally, using the theory of regularity structures, we establish a central limit theorem for the time-integrated nonlinearity near the boundary, which completes the proof of the invariance. Although it is known that different boundary parameters give rise to distinct physical regimes for the invariant measures, our method is robust and does not rely on any particular choice of boundary parameters.