Steady state of the KPZ equation on an interval and Liouville quantum mechanics
Abstract: We obtain a simple formula for the stationary measure of the height ﬁeld evolving according to the Kardar-Parisi-Zhang equation on the interval [0, L] with general Neumann type boundary conditions and any interval size. This is achieved using the recent results of Corwin and Knizel (arXiv:2103.12253) together with Liouville quantum mechanics. Our formula allows to easily determine the stationary measure in various limits: KPZ ﬁxed point on an interval, half-line KPZ equation, KPZ ﬁxed point on a half-line, as well as the Edwards-Wilkinson equation on an interval.
Keywords: Liouville / quantum / KPZ equation / ﬁxed / easily / Zhang / Neumann / Kardar
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