On multidimensional diffusion processes with jumps
Toshihiro Uemura · Osaka City University (Osaka City University) · 2014
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.