TY - JOUR
T1 - Green function estimates for relativistic stable processes in half-space-like open sets
AU - Chen, Zhen Qing
AU - Kim, Panki
AU - Song, Renming
N1 - Funding Information:
The second author’s research was supported by National Research Foundation of Korea Grant funded by the Korean Government ( 2010-0028007 ).
Funding Information:
The first author’s research was partially supported by NSF Grant DMS-0600206 and DMS-0906743 .
PY - 2011/5
Y1 - 2011/5
N2 - 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.
AB - 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.
KW - Exit time
KW - Green function
KW - Lévy system
KW - Relativistic stable process
KW - Symmetric α-stable process
KW - Uniform Harnack inequality
KW - Uniform boundary Harnack principle
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U2 - 10.1016/j.spa.2011.01.004
DO - 10.1016/j.spa.2011.01.004
M3 - Article
AN - SCOPUS:79952625890
VL - 121
SP - 1148
EP - 1172
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
SN - 0304-4149
IS - 5
ER -