Ergodicity results for the open KPZ equation

We give a new proof of existence as well as two proofs of uniqueness of the invariant measure of the open-boundary KPZ equation on [0,1], for all possible choices of inhomogeneous Neumann boundary data. Both proofs yield an exponential convergence result in total variation when combined with the strong Feller property which was recently established in [Knizel-Matetski, arXiv:2211.04466]. An important ingredient in both proofs is the construction of a compact state space for the Markov operator to act on, which measurably contains all of the usual H\"older spaces modulo constants. Along the way, the strong Feller property is extended to this larger class of initial conditions. The arguments do not rely on exact descriptions of the invariant measures, and some of the results generalize to the case of a spatially colored noise and other boundary conditions.