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

TL;DR

This paper derives precise two-sided estimates for the transition densities and Green functions of relativistic stable processes with mass in exterior open sets, providing uniform bounds independent of the mass parameter.

Contribution

The paper introduces sharp two-sided estimates for transition densities and Green functions of relativistic stable processes in exterior domains, uniformly in the mass parameter.

Findings

Established sharp two-sided heat kernel estimates for all time.

Derived Green function estimates with uniform bounds in mass.

Extended results to $C^{1,1}$ 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.