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

Using a capacity approach, and the theory of measure's perturbation of Dirichlet forms, we give the probabilistic representation of the General Robin boundary value problems on an arbitrary domain $\Omega$, involving smooth measures, which give arise 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, and some convergence theorems, and also a probabilistic interpretation of the phenomena occurring on the boundary.