Stationary Measure of the Open KPZ Equation through the Enaud-Derrida Representation

TL;DR

This paper proves the weak convergence of measures describing the open KPZ stationary state using the Enaud-Derrida representation, providing a new probabilistic proof and constructing the measure for general interval lengths.

Contribution

It offers an alternative proof of the open KPZ stationary measure's probabilistic formulation and constructs the measure on intervals of any length for all parameters in the fan region.

Findings

Weak convergence of measures to the open KPZ stationary measure

First construction of the measure on arbitrary interval lengths

Probabilistic proof avoiding finite-dimensional distribution analysis

Abstract

Recent works of Barraquand and Le Doussal and Bryc, Kuznetsov, Wang, and Wesolowski gave a description of the open KPZ stationary measure as the sum of a Brownian motion and a Brownian motion reweighted by a Radon-Nikodym derivative. Subsequent work of Barraquand and Le Doussal used the Enaud-Derrida representation of the DEHP algebra to formulate the open ASEP stationary measure in terms of the sum of a random walk and a random walk reweighted by a Radon-Nikodym derivative. They show that this Radon-Nikodym derivative converges pointwise to the Radon-Nikodym derivative that characterizes the open KPZ stationary measure. This article proves that the corresponding sequence of measures converges weakly to the open KPZ stationary measure. This provides an alternative proof of the probabilistic formulation of the open KPZ stationary measure, which avoids dealing explicitly with finite dimensional distributions. We also provide the first construction of the measure on intervals of a general length and for the full range of parameters in the fan region $(u+v>0)$.