  This note, dedicated in Harold Widom's memory, is an expanded version of a
lecture I gave in fall 2021 at the MSRI program "Universality and Integrability
in Random Matrices and Interacting Particle Systems". I will focus on the
behavior of the stationary measure for the open KPZ equation, a paradigmatic
model for interface growth in contact with boundaries. Much of this will review
elements of my joint work with A. Knizel as well as with H. Shen, as well as
subsequent works of W. Bryc, A. Kuznetsov, Y. Wang, and J. Wesolowski and of G.
Barraquand and P. Le Doussal. The basis for this advance is fundamental work of
B. Derrida, M. Evans, V. Hakim and V. Pasquier from 1993, of T. Sasamoto, M.
Uchiyama and M. Wadati from 2003, and of W. Bryc and J. Wesolowski from 2010
and 2017. I will try to explain how all of this fits together, without laboring
details for the sake of exposition.
  Though this work does not directly follow from Harold Widom's own work, it
(and a great deal of my research) is very much inspired by his and Craig
Tracy's work on ASEP.
