Stationary measures of the open KPZ equation: A short proof

In this note, we give a short probabilistic proof of the two-layer Brownian representation of the stationary measure of the open Kardar--Parisi--Zhang (KPZ) equation on a finite interval with arbitrary Neumann boundary parameters, conjectured by Barraquand and Le Doussal and recently established in full generality by Dunlap, Gu and Rosati. Our starting point is Bryc's two-layer Gibbs representation of the stationary measure of the open asymmetric simple exclusion process (ASEP) under Liggett's condition, together with our previously established probabilistic interpretation of the associated Bryc polynomials in terms of inhomogeneous birth--death Markov chains. Martingale estimates yield uniform asymptotics for these polynomials under weakly asymmetric scaling. Combined with known convergence results from open ASEP to open KPZ, these estimates recover the stationary height profile of the open KPZ equation as the first marginal of a reweighted pair of Brownian motions.