Global Heat Kernel Estimate for Relativistic Stable Processes in Exterior Open Sets

Abstract

In this paper, sharp two-sided estimates for the transition densities of relativistic $\alpha$-stable processes with mass $m\in (0,1]$ in $C^{1,1}$ exterior open sets are established for all time $t>0$. These transition densities are also the Dirichlet heat kernels of $m-(m^{2/\alpha }-\Delta {)}^{\alpha /2}$ with $m\in (0,1]$ in $C^{1,1}$ exterior open sets. The estimates are uniform in $m$ in the sense that the constants are independent of $m\in (0,1]$. As a corollary of our main result, we establish sharp two-sided Green function estimates for relativistic $\alpha$-stable processes with mass $m\in (0,1]$ in $C^{1,1}$ exterior open sets.

26 pages

References in corpus (1)

• Symmetric Jump Processes and their Heat Kernel Estimates