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

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

2011-12-13

arxiv.org/abs/1112.2778v1

Mathematics

Probability

26 pages

Scientific paper

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.

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