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

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. Introduction. – The Kardar-Parisi-Zhang (KPZ) equation [1] describes the stochastic growth of a continuum interface driven by white noise. In one dimension it is at the center of the so-called KPZ class which contains a number of well-studied models sharing the same universal behavior at large scale. For all these models one can define a height field. For example, in particle transport models such as the asymmetric simple exclusion process (ASEP) on a lattice, the local density is a discrete analog to the height gradient [2, 3]. In the limit of weak asymmetry, ASEP converges [4], upon rescaling space and time to the KPZ equation. In the large scale limit, all models in the KPZ class (in particular ASEP and the KPZ equation) are expected to converge to a universal process called the KPZ fixed point [5, 6]. Note that the KPZ fixed point is universal with respect to the microscopic dynamics but still depends on the geometry of the space considered (full-line, half-line, circle, segment with boundary conditions). An important question is the nature of the steady state. While the global height grows linearly in time with non trivial t fluctuations, the height gradient, or the height differences between any two points, will reach a stationary distribution. It was noticed long ago [1,7,8] that the KPZ equation on the full line admits the Brownian motion as a stationary measure. It was proved rigorously in [4, 9], and in [10] for periodic boundary conditions. For ASEP, stationary measures were studied on the full and half-line [11, 12] and exact formulas were obtained on an interval using the matrix product ansatz [13]. The large scale limit of the stationary measures for ASEP on an interval was studied in [14,15]. The processes obtained there as a limit can be described in terms of textbook stochastic processes such as Brownian motions, excursions and meanders, and they should correspond to stationary measures of the KPZ fixed point on an interval. For the KPZ equation, while the stationary measures are simply Brownian in the full-line and circle case, the situation is more complicated (not translation invariant, not Gaussian) in the cases of the half-line and the interval. One typically imposes Neumann type boundary conditions (that is, we fix the derivative of the height field at the boundary) so that stationary measures depend on boundary parameters and involve more complicated stochastic processes (see below). For the KPZ equation on the halfline with Neumann type boundary condition, it can be shown [16] that a Brownian motion with an appropriate drift is stationary (the drift must be proportional to the boundary parameter). This specific stationary measure was studied in [17] for the equivalent directed polymer problem for which the boundary parameter measures the attractiveness of the wall. However, based on the analysis of stationary measures of ASEP on a half-line [11], it was expected that more complicated stationary measures for the KPZ equation also exist. The question of the stationary measure for the KPZ equation on the interval [0, L] has also remained open. In a recent breakthrough, Corwin and Knizel obtained [18] an explicit formula for the Laplace transform of the stationary height distribution (for L = 1, and for some range of parameters). This Laplace transform formula relates the stationary measure to an auxiliary stochastic process called continuous dual Hahn process. This construction