Markov limits of steady states of the KPZ equation on an interval

TL;DR

This paper proves the equivalence of two probabilistic descriptions of stationary measures for the KPZ equation on an interval and analyzes their large-scale behavior, connecting recent results to Markov process representations.

Contribution

It establishes the equivalence of two descriptions of KPZ stationary measures and rigorously connects recent limit results to Markov process representations.

Findings

Proved the equivalence of two probabilistic descriptions of stationary measures.

Provided rigorous proofs of results by Barraquand and Le Doussal.

Analyzed large-scale behavior of KPZ stationary measures and their Markov process representations.

Abstract

This paper builds upon the research of Corwin and Knizel who proved the existence of stationary measures for the KPZ equation on an interval and characterized them through a Laplace transform formula. Bryc, Kuznetsov, Wang and Wesolowski found a probabilistic description of the stationary measures in terms of a Doob transform of some Markov kernels; essentially at the same time, another description connecting the stationary measures to the exponential functionals of the Brownian motion appeared in work of Barraquand and Le Doussal. Our first main result clarifies and proves the equivalence of the two probabilistic description of these stationary measures. We then use the Markovian description to give rigorous proofs of some of the results claimed in Barraquand and Le Doussal. We analyze how the stationary measures of the KPZ equation on finite interval behave at large scale. We investigate which of the limits of the steady states of the KPZ equation obtained recently by G. Barraquand and P. Le Doussal can be represented by Markov processes in spatial variable under an additional restriction on the range of parameters.