Resumen de Occupation densities for SPDEs with reflection

Lorenzo Zambotti

• We consider the solution (u,η) of the white-noise driven stochastic partial differential equation with reflection on the space interval [0,1] introduced by Nualart and Pardoux, where η is a reflecting measure on [0,∞)×(0,1) which forces the continuous function u, defined on [0,∞)×[0,1], to remain nonnegative and η has support in the set of zeros of u. First, we prove that at any fixed time t>0, the measure η([0,t]×dθ) is absolutely continuous w.r.t. the Lebesgue measure dθ 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,θ))t≥0. Finally, we study the behavior of η 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.