A skew stochastic heat equation

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.