Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection. (English) Zbl 1009.60047

Zambotti, Lorenzo

Probab. Theory Relat. Fields 123, No. 4, 579-600 (2002).

Let \(K_\alpha\), \(\alpha\geq 0\), be the set of paths from \([0,1]\) into \([-\alpha,\infty)\). Integration by parts formulae are proved for the integration on \(K_0\) with respect to the Bessel bridge of dimension 3, and for the integration on \(K_\alpha\) with respect to the Brownian bridge. These formulae involve boundary terms with boundary measures \(\sigma_0\) and \(\sigma_\alpha\) which are explicited. They can be applied to the study of a family (indexed by \(\alpha\)) of stochastic partial differential equations (SPDE) with reflection which was introduced by Nualart and Pardoux. For instance, the solution of the SPDE can be viewed as the diffusion associated to a symmetric Dirichlet form on \(K_\alpha\); the reflection term of the SPDE and the boundary measure \(\sigma_\alpha\) can also be related.

Reviewer: Jean Picard (Aubiere)

MSC:

60H07	Stochastic calculus of variations and the Malliavin calculus
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60J55	Local time and additive functionals
31C25	Dirichlet forms

Keywords:

integration by parts; stochastic partial differential equations with reflection; additive functionals; Dirichlet processes