  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 functions 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 arXiv:2105.15178. To
prove the intermediate disorder limit, we provide a general half-space polymer
convergence framework that extends works of arXiv:1804.09815, arXiv:1901.09449,
arXiv:1202.4398.
