2002/04/25 by Lorenzo Zambotti, Zambotti, Lorenzo
Economics, Econometrics and Finance · Mathematics · #60H15 #60J55 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:60H15 #msc:60J55
paper · pdf · doi:10.48550/arxiv.math/0204313
arxiv created 2002/04/25 · openalex publication_date 2002/04/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. 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 dθon (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 behaviour of ηat 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.