The Annals of Probability

Sharp heat kernel estimates for relativistic stable processes in open sets

Zhen-Qing Chen, Panki Kim, and Renming Song

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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 − (m2/α − Δ)α/2] in C1,1 open sets. Here m > 0 and α ∈ (0, 2). The estimates are uniform in m ∈ (0, M] for each fixed M > 0. Letting m ↓ 0, we recover the Dirichlet heat kernel estimates for Δα/2 := −(−Δ)α/2 in C1,1 open sets obtained in [14]. Sharp two-sided estimates are also obtained for Green functions of relativistic stable processes in bounded C1,1 open sets.

Article information

Source
Ann. Probab., Volume 40, Number 1 (2012), 213-244.

Dates
First available in Project Euclid: 3 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1325605002

Digital Object Identifier
doi:10.1214/10-AOP611

Mathematical Reviews number (MathSciNet)
MR2917772

Zentralblatt MATH identifier
1235.60101

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 60J75: Jump processes
Secondary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}

Keywords
Symmetric α-stable process relativistic stable process heat kernel transition density Green function exit time Lévy system parabolic Harnack inequality

Citation

Chen, Zhen-Qing; Kim, Panki; Song, Renming. Sharp heat kernel estimates for relativistic stable processes in open sets. Ann. Probab. 40 (2012), no. 1, 213--244. doi:10.1214/10-AOP611. https://projecteuclid.org/euclid.aop/1325605002


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