Dirichlet problem for diffusions with jumps

In this paper, we study Dirichlet problem for non-local operator on bounded domains in ${\mathbb R}^d$ $${\cal L}u = {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) + \int_{{\mathbb R}^d} (u(y)-u(x) ) J(x, dy) ,$$ where $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on ${\mathbb R}^d$ that is uniformly elliptic and bounded, $b$ is an ${\mathbb R}^d$-valued function so that $|b|^2$ is in some Kato class ${\mathbb K}_d$, for each $x\in {\mathbb R}^d$, $J(x, dy)$ is a finite measure on ${\mathbb R}^d$ so that $x\mapsto J(x, {\mathbb R}^d)$ is in the Kato class ${\mathbb K}_d$. We show there is a unique Feller process $X$ having strong Feller property associated with ${\cal L}$, which can be obtained from the diffusion process having generator ${\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x)$ through redistribution. We further show that for any bounded connected open subset $D\subset{\mathbb R}^d$ that is regular with respect to the Laplace operator $\Delta$ and for any bounded continuous function $\varphi$ on $D^c$, the Dirichlet problem ${\cal L} u=0$ in $D$ with $u=\varphi$ on $D^c$ has a unique bounded continuous weak solution on ${\mathbb R}^d$. This unique weak solution can be represented in terms of the Feller process associated with ${\cal L}$.