Dirichlet problem for integro-differential operators

In this paper, we study the Dirichlet problem for the following integro-differential operator on bounded (not necessarily connected) open sets in R^d: [Formula presented] where A(x)=(a_ij(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 the Kato class K_d, and J(x, y) ≥ 0 is a measurable symmetric non-trivial kernel on R^d×R^d bounded above by cmax{|x−y|^−(d+α),|x−y|^−(d+β)} for some 0 < β ≤ α < 2 and c > 0. We show that there is a Feller process X on R^d having strong Feller property associated with the non-local operator L. We further show that for any bounded open set D in R^d that is regular with respect to the Feller process X and for every bounded function φ on D^c that is continuous on ∂D, the Dirichlet problem for L on D has a unique weak solution on R^d that is continuous on D‾. Moreover, the solution can be represented in terms of the associated Feller process.