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We construct explicit one-parameter families of stationary measures for theKardar-Parisi-Zhang equation in half-space with Neumann boundary conditions atthe origin, as well as for the log-gamma polymer model in a half-space. Thestationary measures are stochastic processes that depend on the boundarycondition as well as a parameter related to the drift at infinity. They areexpressed in terms of exponential functions of Brownian motions and gammarandom walks. We conjecture that these constitute all extremal stationarymeasures for these models. The log-gamma polymer result is proved through asymmetry argument related to half-space Whittaker processes which we expect maybe applicable to other integrable models. The KPZ result comes as anintermediate disorder limit of the log-gamma polymer result and confirms theconjectural description of these stationary measures from arXiv:2105.15178. Toprove the intermediate disorder limit, we provide a general half-space polymerconvergence framework that extends works of arXiv:1804.09815, arXiv:1901.09449,arXiv:1202.4398.

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