Dirichlet problem for diffusions with jumps

In this paper, we study Dirichlet problem on bounded open sets in R^d for non-local operators of the following form Lu=div(A(x)∇u(x))+b(x)⋅∇u(x)+∫R^d(u(y)−u(x))J(x,dy), where A(x)=(aij(x))_1≤i,j≤d is a measurable d×d matrix-valued function on R^d that is uniformly elliptic and bounded, b is an R^d-valued function so that |b|² is in some Kato class K_d, J(x,dy) is a finite measure on R^d for each x∈R^d so that x↦J(x,R^d) is in the Kato class K_d. We show there is a unique Feller process X having strong Feller property associated with L, which can be obtained from the diffusion process having generator div(A(x)∇u(x))+b(x)⋅∇u(x) through redistribution. We further show that for any bounded connected open subset D⊂R^d that is regular with respect to the Laplace operator Δ and for any bounded continuous function φ on D^c, the Dirichlet problem Lu=0 in D with u=φ on D^c has a unique bounded continuous weak solution on R^d. This unique weak solution can be represented in terms of the Feller process associated with L.