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.

Expanded version, typo in (6.6) corrected

References in corpus (10)

• Exponential functionals of Brownian motion, I: Probability laws at fixed time
• Exponential functionals of Brownian motion, II: Some related diffusion processes
• The KPZ Limit of ASEP with Boundary
• Askey--Wilson polynomials, quadratic harnesses and martingales
• Singular SPDEs in domains with boundaries
• Markov limits of steady states of the KPZ equation on an interval
• The spectral expansion approach to index transforms and connections with the theory of diffusion processes
• Dual representations of Laplace transforms of Brownian excursion and generalized meanders
• On the continuous dual Hahn process
• The Doob-McKean identity for stable Lévy processes

Cited by in corpus (11)

• From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval
• Stationary measures of the KPZ equation on an interval from Enaud-Derrida's matrix product ansatz representation
• Approximating the stationary distribution of the ASEP with open boundaries
• High temperature behaviors of the directed polymer on a cylinder
• KPZ fluctuations in finite volume
• Local KPZ behavior under arbitrary scaling limits
• Askey-Wilson signed measures and open ASEP in the shock region
• Fluctuations of random Motzkin paths II
• Localization length of the $1+1$ continuum directed random polymer
• Stationary measure for six-vertex model on a strip
• Limits of open ASEP stationary measures near a boundary