Guillaume Barraquand: Stationary measures in the open KPZ universality class

Abstract:

A solution to the Kardar-Parisi-Zhang stochastic PDE, started from a two-sided Brownian motion on \(\mathbf{R}\) at time \(0\), will remain a two-sided Brownian motion for all later times, up to a global height shift. For the same stochastic PDE defined on an interval with boundary conditions, however, stationary measures are more complicated, and have been characterized only recently through the analysis of discrete models. They can be described in terms of two Brownian motions reweighted by exponential functionals. I will review these results, and explain why the appearance of such measures becomes very natural using its relation to Gibbsian line ensembles associated to the KPZ equation and its discrete analogues.