Heat kernels for reflected diffusions with jumps on inner uniform domains

Abstract:

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain $D$ in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When $D$ is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration is symmetric reflected diffusions with jumps on $D$, whose infinitesimal generators are non-local (pseudo-differential) operators $\mathcal {L}$ on $D$ of the form \[ \mathcal {L} u(x)\! =\!\frac 12 \!\sum _{i, j=1}^d\! \frac {\partial }{\partial x_i}\! \left (\!\!a_{ij}(x) \frac {\partial u(x)}{\partial x_j}\!\right ) \!+ \lim _{\varepsilon \downarrow 0}\! \int _{\{y\in D: \, \rho _D(y, x)>\varepsilon \}}\!\! (u(y)-u(x)) J(x, y)\, dy \] satisfying “Neumann boundary condition”. Here, $\rho _D(x,y)$ is the length metric on $D$, $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on $D$ that is uniformly elliptic and bounded, and \[ J(x,y)≔\frac {1}{\Phi (\rho _D(x,y))} \int _{[\alpha _1, \alpha _2]} \frac {c(\alpha , x,y)} {\rho _D(x,y)^{d+\alpha }} \,\nu (d\alpha ), \] where $\nu$ is a finite measure on $[\alpha _1, \alpha _2] \subset (0, 2)$, $\Phi$ is an increasing function on $[ 0, \infty )$ with $c_1e^{c_2r^{\beta }} \le \Phi (r) \le c_3 e^{c_4r^{\beta }}$ for some $\beta \in [0,\infty ]$, and $c(\alpha , x, y)$ is a jointly measurable function that is bounded between two positive constants and is symmetric in $(x, y)$.