Stationary measures for the log-gamma polymer and KPZ equation in half-space

Abstract

We construct explicit one-parameter families of stationary measures for the Kardar–Parisi–Zhang equation in half-space with Neumann boundary conditions at the origin, as well as for the log-gamma polymer model in a half-space. The stationary measures are stochastic processes that depend on the boundary condition as well as a parameter related to the drift at infinity. They are expressed in terms of exponential functionals of Brownian motions and gamma random walks. We conjecture that these constitute all extremal stationary measures for these models. The log-gamma polymer result is proved through a symmetry argument related to half-space Whittaker processes which we expect may be applicable to other integrable models. The KPZ result comes as an intermediate disorder limit of the log-gamma polymer result and confirms the conjectural description of these stationary measures from Barraquand and Le Doussal (2021). To prove the intermediate disorder limit, we provide a general half-space polymer convergence framework that extends works of (J. Stat. Phys. 181 (2020) 2372–2403; Electron. J. Probab. 27 (2022) Paper No. 45; Ann. Probab. 42 (2014) 1212–1256).