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 $N^{-\frac{3}{2}}$. 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.