Reflecting Brownian snake and a Neumann–Dirichlet problem

The paper deals with a path-valued Markov process: the reflecting Brownian snake. It is a particular case of the path-valued process previously introduced by Le Gall. Here the spatial motion is a reflecting Brownian motion in a domain D of Rd. Using this probabilistic tool, we construct an explicit function v solution of an integral equation which is, under some hypotheses on the regularity of v, equivalent to a semi-linear partial differential equation in D with some mixed Neumann–Dirichlet conditions on the boundary. When the hypotheses on v are not satisfied, we prove that v is still solution of a weak formulation of the Neumann–Dirichlet problem.