Sharp heat kernel estimates for relativistic stable processes in open sets

Abstract

In this paper, we establish sharp two-sided estimates for the transition densities of relativistic stable processes [i.e., for the heat kernels of the operators $m-(m^{2/\alpha }-\Delta {)}^{\alpha /2}$] in $C^{1,1}$ open sets. Here $m>0$ and $\alpha \in (0,2)$. The estimates are uniform in $m\in (0,M]$ for each fixed $M>0$. Letting $m\downarrow 0$, we recover the Dirichlet heat kernel estimates for ${\Delta }^{\alpha /2}:=-(-\Delta {)}^{\alpha /2}$ in $C^{1,1}$ open sets obtained in [14]. Sharp two-sided estimates are also obtained for Green functions of relativistic stable processes in bounded $C^{1,1}$ open sets.

Published in at http://dx.doi.org/10.1214/10-AOP611 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Heat kernel estimates for the fractional Laplacian with Dirichlet conditions
• Symmetric Jump Processes and their Heat Kernel Estimates

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