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This note, dedicated in Harold Widom's memory, is an expanded version of alecture I gave in fall 2021 at the MSRI program "Universality and Integrabilityin Random Matrices and Interacting Particle Systems". I will focus on thebehavior of the stationary measure for the open KPZ equation, a paradigmaticmodel for interface growth in contact with boundaries. Much of this will reviewelements of my joint work with A. Knizel as well as with H. Shen, as well assubsequent 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 ofB. 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 2010and 2017. I will try to explain how all of this fits together, without laboringdetails 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 CraigTracy's work on ASEP.

Some recent progress on the stationary measure for the open KPZ equation