SCALING LIMITS OF WEAKLY ASYMMETRIC INTERFACES

We consider three models of evolving interfaces intimately related to the weakly asymmetric simple exclusion process with N particles on a finite lattice of 2N sites. Our MODEL 1 defines an evolving bridge on [0, 1], our MODEL 1-W an evolving excursion on [0, 1] while our MODEL 2 consists of an evolving pair of non-crossing bridges on [0, 1]. Based on the observation that the invariant measures of the dynamics depend on the area under- or between- the interface(s), we characterise the scaling limits of the invariant measures when the asymmetry of the exclusion process scales like N− 3 2. Then, we show that the scaling limits of the dynamics themselves are expressed in terms of variants of the stochastic heat equation. In particular, in MODEL 1-W we obtain the well-studied reflected stochastic heat equation introduced by Nualart and Pardoux