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

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.