Global uniform boundary Harnack principle with explicit decay rate and its application

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:124

Article

Kim, Panki^1,2  Song, Renming³  Vondracek, Zoran⁴ 

[1] Seoul Natl Univ, Dept Math Sci, Seoul 151747, South Korea

[2] Seoul Natl Univ, Res Inst Math, Seoul 151747, South Korea

[3] Univ Illinois, Dept Math, Urbana, IL 61801 USA

[4] Univ Zagreb, Dept Math, Zagreb 41000, Croatia

关键词: Levy processes;    Subordinate Brownian motions;    Harmonic functions;    Boundary Harnack principle;    Poisson kernel;    Heat kernel;    Green function;   

DOI  :  10.1016/j.spa.2013.07.007

来源: Elsevier

【 摘 要 】

In this paper, we consider a large class of subordinate Brownian motions X via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions at zero and at infinity. We first discuss how such conditions govern the behavior of the subordinator and the corresponding subordinate Brownian motion for both large and small time and space. Then we establish a global uniform boundary Harnack principle in (unbounded) open sets for the subordinate Brownian motion. When the open set satisfies the interior and exterior ball conditions with radius R > 0, we get a global uniform boundary Harnack principle with explicit decay rate. Our boundary Hamack principle is global in the sense that it holds for all R > 0 and the comparison constant does not depend on R, and it is uniform in the sense that it holds for all balls with radii r <= R and the comparison constant depends neither on D nor on r. As an application, we give sharp two-sided estimates for the transition densities and Green functions of such subordinate Brownian motions in the half-space. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free