Markov processes related to the stationary measure for the open KPZ equation

TL;DR

This paper characterizes the stationary measures of the open KPZ equation on [0,1] using a Markov process derived from Brownian motion and connects it to the continuous dual Hahn process through Laplace transform identities.

Contribution

It introduces a probabilistic Markov process description for the stationary measures of open KPZ and establishes a novel link with the continuous dual Hahn process via Laplace transform equivalences.

Findings

The Markov process $Y$ describes the stationary measure for open KPZ.

Laplace transforms of $Y$ and the continuous dual Hahn process are shown to be equal.

The approach connects probabilistic and integrable structures of open KPZ.

Abstract

We provide a probabilistic description of the stationary measures for the open KPZ on the spatial interval $[0,1]$ in terms of a Markov process $Y$, which is a Doob's $h$ transform of the Brownian motion killed at an exponential rate. Our work builds on a recent formula of Corwin and Knizel which expresses the multipoint Laplace transform of the stationary solution of the open KPZ in terms of another Markov process $\mathbb T$: the continuous dual Hahn process with Laplace variables taking on the role of time-points in the process. The core of our approach is to prove that the Laplace transforms of the finite dimensional distributions of $Y$ and $\mathbb T$ are equal when the time parameters of one process become the Laplace variables of the other process and vice versa.