Global heat kernel estimates for symmetric jump processes

In this paper, we study sharp heat kernel estimates for a large class of symmetric jump-type processes in $\mathbb R^d$ for all $t>0$. A prototype of the processes under consideration are symmetric jump processes on $\mathbb R^d$ with jumping intensity \[ \frac {1}{\Phi (|x-y|)} \int _{[\alpha _1, \alpha _2]} \frac {c(\alpha , x,y)} {|x-y|^{d+\alpha }} \nu (d\alpha ) , \] where $\nu$ is a probability measure on $[\alpha _1, \alpha _2] \subset (0, 2)$, $\Phi$ is an increasing function on $[ 0, \infty )$ with $c_1e^{c_2r^{\beta }} \le \Phi (r) \le c_3 e^{c_4r^{\beta }}$ with $\beta \in (0,\infty )$, and $c(\alpha , x, y)$ is a jointly measurable function that is bounded between two positive constants and is symmetric in $(x, y)$. They include, in particular, mixed relativistic symmetric stable processes on $\mathbb {R}^d$ with different masses. We also establish the parabolic Harnack principle.