We consider a stochastic heat equation driven by a space-time white noise and
with a singular drift, where a local-time in space appears. The process we
study has an explicit invariant measure of Gibbs type, with a non-convex
potential. We obtain existence of a Markov solution, which is associated with
an explicit Dirichlet form. Moreover we study approximations of the stationary
solution by means of a regularization of the singular drift or by a
finite-dimensional projection