On multidimensional diffusion processes with jumps

Abstract

Let $G$ be an open set of $\mathbb{R}^{d}$ ($d\ge 2$) and $dx$ denotes the Lebesgue measure on it. We construct a diffusion process with jumps associated with diffusion data (diffusion coefficients $\{a_{ij}(x)\}$, a drift coefficient $\{b_{i}(x)\}$ and a killing function $c(x)$) and a Lévy kernel $k(x,y)$ in terms of a lower bounded semi-Dirichlet form on $L^{2}(G;dx)$. When $G$ is the whole space, we allow that the diffusion coefficients may degenerate. We also show some Sobolev inequalities for the Dirichlet form and then show the absolute continuity of its resolvent.