The Laplace transform of the stationary measure of the open KPZ equation in the shock region

The stationary increment measure of the open Kardar-Parisi-Zhang equation on a bounded interval can be represented as the independent sum of a Brownian motion and another process defined via an explicit Radon-Nikodym derivative with respect to the measure induced by a Brownian motion. In the fan region, its finite-dimensional Laplace transform was previously expressed in terms of the continuous dual Hahn process. The main contribution here consists of computing the finite-dimensional Laplace transform for a range of parameters in the shock region by an analytic continuation argument. As an application, we provide a direct proof of the convergence of the finite-dimensional distributions of the increment height function of open WASEP under the stationary measure in the shock region, by computing the convergence of finite-dimensional Laplace transform without relating to the open KPZ equation.