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 language | English |
|---|---|
| Pages (from-to) | 1148-1172 |
| Number of pages | 25 |
| Journal | Stochastic Processes and their Applications |
| Volume | 121 |
| Issue number | 5 |
| DOIs | |
| Publication status | Published - May 2011 |
| Externally published | Yes |
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