Probabilistic approach to the Dirichlet problem of perturbed stable processes

Let X_t be a symmetric stable process of index α, 0<α<2, in R^d (d≧2). In this paper we deal with the perturbation of X_t by a multiplicative functional of the following form: {Mathematical expression} with F being a function on R^d×R^d satisfying certain conditions. First we prove the following gauge theorem: If D is a bounded open domain of R^d, then the function g(x)=E^x{M(τ_D)} is either identically infinite on D or bounded on D, where τ_D is the first exit time from D. Then we formulate the Dirichlet problem associated with the perturbed symmetric stable process by using Dirichlet form theory. Finally we apply the gauge theorem to prove the existence and uniqueness of solutions to the Dirichlet problem mentioned above.