Sharp Heat Kernel Estimates for Relativistic Stable Processes in Open Sets

Mathematics – Probability

Scientific paper

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40 page, no figure

Scientific paper

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

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