Heat kernel estimates for anisotropic symmetric jump processes

Abstract: We show two-sided bounds of heat kernel for anisotropic non-singular symmetric pure jump Markov process whose jump kernel $J(x,y)$ is comparable to $\frac{{\bf 1}_{\mathcal{V}}(x-y)}{|x-y|^{d+\alpha}}$, where $\mathcal{V}$ is a union of symmetric cones, $0<\alpha<2$ and $x,y\in\mathbb{R}^d$.