Reflecting Brownian snake and a Neumann-Dirichlet problem

[摘要] 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 R-d. Using this probabilistic tool, we construct an explicit function upsilon solution of an integral equation which is, under some hypotheses on the regularity of upsilon, equivalent to a semi-linear partial differential equation in D with some mixed Neumann-Dirichlet conditions on the boundary. When the hypotheses on upsilon are not satisfied, we prove that upsilon is still solution of a weak formulation of the Neumann-Dirichlet problem. (C) 2000 Elsevier Science B.V. All rights reserved.

[发布日期] 2000-10-01

[关键词] Brownian snake;reflecting Brownian motion;semi-linear partial differential equations;Neumann problem