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

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.