Steady state of the KPZ equation on an interval and Liouville quantum mechanics

TL;DR

This paper derives a simple formula for the stationary measure of the KPZ equation on an interval with various boundary conditions, connecting recent results with Liouville quantum mechanics to analyze different limits.

Contribution

It provides a unified formula for the stationary measure of the KPZ equation on an interval, incorporating boundary conditions and interval size, using recent theoretical advances.

Findings

Explicit stationary measure formula for KPZ on an interval

Analysis of limits including KPZ fixed point and Edwards-Wilkinson

Connection established between KPZ stationary measure 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.