Markov limits of steady states of the KPZ equation on an interval

This paper builds upon the research of Corwin and Knizel (2021) who proved the exis-tence of stationary measures for the KPZ equation on an interval and characterized them through a Laplace transform formula. Bryc et al. (2022) found a probabilistic description of the stationary measures in terms of a Doob transform of some Markov kernels; essentially at the same time, an-other description connecting the stationary measures to the exponential functionals of the Brownian motion appeared in Barraquand and Le Doussal (2022).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 (2022). We analyze how the stationary measures of the KPZ equation on [0, T] behave at large scale, as T goes to infinity. We also analyze the behaviour of the stationary measures of the KPZ equation on [0, T] without rescaling, when T goes to infinity. Finally, we analyze the measures on [0, infinity) at large scale, which according to Barraquand and Le Doussal (2022) should correspond to stationary measures of a hypothetical KPZ fixed point on [0, infinity).