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Sharp heat kernel estimates for relativistic stable processes in open sets. (English) Zbl 1235.60101

Summary: 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 [the authors, J. Eur. Math. Soc. (JEMS) 12, No. 5, 1307–1329 (2010; Zbl 1203.60114)]. Sharp two-sided estimates are also obtained for Green functions of relativistic stable processes in bounded \(C^{1,1}\) open sets.

MSC:

60J35 Transition functions, generators and resolvents
47G20 Integro-differential operators
60J75 Jump processes (MSC2010)
47D07 Markov semigroups and applications to diffusion processes

Citations:

Zbl 1203.60114
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