Half-Space Stationary Kardar-Parisi-Zhang Equation.

We study the solution of the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth of an interface of height (, ) on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a wall at . The boundary condition corresponds to an attractive wall for , and leads to the binding of the polymer to the wall below the critical value . Here we choose the initial condition (, 0) to be a Brownian motion in with drift . When , the solution is stationary, i.e. remains at all times a Brownian motion with the same drift, up to a global height shift (0, ). We show that the distribution of this height shift is invariant under the exchange of parameters and . For any , we provide an exact formula characterizing the distribution of (0, ) at any time , using two methods: the replica Bethe ansatz and a discretization called the log-gamma polymer, for which moment formulae were obtained. We analyze its large time asymptotics for various ranges of parameters , . In particular, when , the critical stationary case, the fluctuations of the interface are governed by a universal distribution akin to the Baik-Rains distribution arising in stationary growth on the full-line. It can be expressed in terms of a simple Fredholm determinant, or equivalently in terms of the Painlevé II transcendent. This provides an analog for the KPZ equation, of some of the results recently obtained by Betea-Ferrari-Occelli in the context of stationary half-space last-passage-percolation. From universality, we expect that limiting distributions found in both models can be shown to coincide.