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

    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.

    Source link


    Leave A Reply