%0 Journal Article
%T Probabilistic Solution of the General Robin Boundary Value Problem on Arbitrary Domains
%A Khalid Akhlil
%J International Journal of Stochastic Analysis
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/163096
%X 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 £¿, 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. 1. Introduction The classical Robin boundary conditions on a smooth domain of ( ) is giving by where is the outward normal vector field on the boundary and a positive bounded Borel measurable function defined on . The probabilistic treatment of Robin boundary value problems has been considered by many authors [1¨C4]. The first two authors considered bounded -domains since the third considered bounded domains with Lipschitz boundary, and the study of [4] was concerned with -domains but with smooth measures instead of . If one wants to generalize the probabilistic treatment to a general domain, a difficulty arise when we try to get a diffusion process representing Neumann¡¯s boundary conditions. In fact, the Robin boundary conditions (1.1) are nothing but a perturbation of , which represent Neumann¡¯s boundary conditions, by the measure , where is the surface measure. Consequently, the associated diffusion process is the reflecting Brownian motion killed by a certain additive functional, and the semigroup generated by the Laplacian with classical Robin boundary conditions is then giving by where is a reflecting Brownian motion (RBM) and is the boundary local time, which corresponds to by Revuz correspondence. It is clear that the smoothness of the domain in classical Robin boundary value problem follows the smoothness of the domains where RBM is constructed (see [5¨C10] and references therein for more details about RBM). In [6], the RBM is defined to be the Hunt process associated with the form defined on by where is assumed to be bounded with Lipschitz boundary so that the Dirichlet form is regular. If is an arbitrary domain, then the Dirichlet form needs not to be regular, and to not to lose the generality we consider , the closure of in . The domain is so defined to insure the Dirichlet form to be regular. Now, if we perturb the Neumann boundary conditions by Borel¡¯s positive measure [11¨C13], we get the Dirichlet form defined on by In the case of ( bounded with Lipschitz boundary), (1.4) is the form associated with
%U http://www.hindawi.com/journals/ijsa/2012/163096/