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.
Citation
Toshihiro Uemura. "On multidimensional diffusion processes with jumps." Osaka J. Math. 51 (4) 969 - 993, October 2014.
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