Occupation densities for SPDEs 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, where $\eta$ is a reflecting measure on $[0,\infty)\times(0,1)$ which forces the continuous function u, defined on $[0,\infty)\times[0,1]$, to remain nonnegative and $\eta$ has support in the set of zeros of u. 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 behavior 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.