Abstract: The KPZ equation has attracted great attention since its inception in the 80s, first in physics, then in the fields of random matrix theory, integrable probability, and stochastic PDEs. In this talk we will focus on the “open KPZ equation” which models stochastic interface growth over a spatial interval [0,1] with mixed Neumann boundary conditions at 0 and 1. For given boundary parameters, when started from any initial height profile the interface should converge to a unique stationary measure in terms of height-function differences. We will describe how to construct such stationary measures. Along the way, we will encounter orthogonal polynomials in the Askey-Wilson scheme, the asymmetric simple exclusion process, and precise asymptotics of q-Gamma functions. No prior knowledge of any of these subjects will be assumed, though.
This talk is based on my joint work with Alisa Knizel and Hao Shen, and also involves important contributions from Wlodzimierz Bryc, Jacek Wesolowski and Yizao Wang.