A skew stochastic heat equation

TL;DR

This paper investigates a stochastic heat equation with a singular drift involving local-time, establishing existence of solutions, invariant measures, and approximation methods for the stationary state.

Contribution

It introduces a Markov solution framework for a stochastic heat equation with non-convex potential and local-time drift, including explicit invariant measures and approximation techniques.

Findings

Existence of a Markov solution with an explicit Dirichlet form.

Identification of an invariant Gibbs measure with non-convex potential.

Development of approximation methods via regularization and finite-dimensional projection.

Abstract

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.