Occupation densities for SPDE's with reflection

Abstract

We consider the solution (u,\eta) of the white-noise driven stochastic partial differential equation with reflection on the space interval [0,1] introduced by Nualart and Pardoux. First, we prove that at any fixed time t>0, the measure \eta([0,t]\times d\theta) is absolutely continuous w.r.t. the Lebesgue measure d\theta on (0,1). We characterize the density as a family of additive functionals of u, and we interpret it as a renormalized local time at 0 of (u(t,\theta))_{t\geq 0}. Finally we study the behaviour of \eta at the boundary of [0,1]. The main technical novelty is a projection principle from the Dirichlet space of a Gaussian process, vector-valued solution of a linear SPDE, to the Dirichlet space of the process u.