  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.
