Green function estimates for relativistic stable processes in half-space-like open sets

Zhen Qing Chen, Panki Kim*, Renming Song

*Corresponding author for this work

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Abstract

In this paper, we establish sharp two-sided estimates for the Green functions of relativistic stable processes (i.e. Green functions for non-local operators m-(m2/α-Δ)α/2) in half-space-like C1,1 open sets. The estimates are uniform in m∈(0,M] for each fixed M ∈ (0,∞). When m ↓ 0, our estimates reduce to the sharp Green function estimates for -(-Δ) α/2 in such kind of open sets that were obtained recently in Chen and Tokle [12]. As a tool for proving our Green function estimates, we show that a boundary Harnack principle for Xm, which is uniform for all m ∈ (0,∞), holds for a large class of non-smooth open sets.

Original languageEnglish
Pages (from-to)1148-1172
Number of pages25
JournalStochastic Processes and their Applications
Volume121
Issue number5
DOIs
Publication statusPublished - May 2011
Externally publishedYes

Keywords

  • Exit time
  • Green function
  • Lévy system
  • Relativistic stable process
  • Symmetric α-stable process
  • Uniform Harnack inequality
  • Uniform boundary Harnack principle

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Chen, Z. Q., Kim, P., & Song, R. (2011). Green function estimates for relativistic stable processes in half-space-like open sets. Stochastic Processes and their Applications, 121(5), 1148-1172. https://doi.org/10.1016/j.spa.2011.01.004