Occupation densities for SPDEs with reflection

• 成果类型: Article
• 署名作者: Zambotti, L
• 署名单位: Scuola Normale Superiore di Pisa; University of Bielefeld
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1078415833
• 发表日期: 2004
• 页码: 191-215
• 关键词: integration; parts

摘要

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, infinity) x (0, 1) which forces the continuous function u, defined on [0, infinity) x [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] x dtheta) is absolutely continuous w.r.t. the Lebesgue measure dtheta 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))(tgreater than or equal to0). 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.