This paper builds upon the research of Corwin and Knizel who proved the
existence of stationary measures for the KPZ equation on an interval and
characterized them through a Laplace transform formula. Bryc, Kuznetsov, Wang
and Wesolowski found a probabilistic description of the stationary measures in
terms of a Doob transform of some Markov kernels; essentially at the same time,
another description connecting the stationary measures to the exponential
functionals of the Brownian motion appeared in work of Barraquand and Le
Doussal.
Our first main result clarifies and proves the equivalence of the two
probabilistic description of these stationary measures. We then use the
Markovian description to give rigorous proofs of some of the results claimed in
Barraquand and Le Doussal. We analyze how the stationary measures of the KPZ
equation on finite interval behave at large scale. We investigate which of the
limits of the steady states of the KPZ equation obtained recently by G.
Barraquand and P. Le Doussal can be represented by Markov processes in spatial
variable under an additional restriction on the range of parameters.
