TY - JOUR

T1 - Two-sided Green function estimates for killed subordinate Brownian motions

AU - Kim, Panki

AU - Song, Renming

AU - Vondraček, Zoran

N1 - Funding Information:
The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2011-0001251). The third author was supported in part by the MZOS grant 037-0372790-2801.

PY - 2012/5

Y1 - 2012/5

N2 - A subordinate Brownian motion is a Lévy process that can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is-φ(-Δ), where φ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with φ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of independent relativistic stable processes, and much more. We give sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded k-fat open set D. When D is a bounded C 1, 1 open set, we establish an explicit form of the estimates in terms of the distance to the boundary. As a consequence of such sharp Green function estimates, we obtain a boundary Harnack principle in C 1, 1 open sets with explicit rate of decay.

AB - A subordinate Brownian motion is a Lévy process that can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is-φ(-Δ), where φ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with φ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of independent relativistic stable processes, and much more. We give sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded k-fat open set D. When D is a bounded C 1, 1 open set, we establish an explicit form of the estimates in terms of the distance to the boundary. As a consequence of such sharp Green function estimates, we obtain a boundary Harnack principle in C 1, 1 open sets with explicit rate of decay.

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U2 - 10.1112/plms/pdr050

DO - 10.1112/plms/pdr050

M3 - Article

AN - SCOPUS:84861064750

SN - 0024-6115

VL - 104

SP - 927

EP - 958

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 5

ER -