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International Journal of Stochastic Analysis
Volume 2012, Article ID 163096, 17 pages
http://dx.doi.org/10.1155/2012/163096
Research Article

Probabilistic Solution of the General Robin Boundary Value Problem on Arbitrary Domains

Institut für Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany

Received 13 April 2012; Revised 30 August 2012; Accepted 5 December 2012

Academic Editor: Donal O'Regan

Copyright © 2012 Khalid Akhlil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using a capacity approach and the theory of the measure’s perturbation of the Dirichlet forms, we give the probabilistic representation of the general Robin boundary value problems on an arbitrary domain Ω, involving smooth measures, which give rise to a new process obtained by killing the general reflecting Brownian motion at a random time. We obtain some properties of the semigroup directly from its probabilistic representation, some convergence theorems, and also a probabilistic interpretation of the phenomena occurring on the boundary.