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

Panki Kim, Renming Song and Zoran Vondraček

Stochastic Processes and their Applications, 2014, vol. 124, issue 1, 235-267

Abstract: 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 Harnack 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.

Keywords: Lévy processes; Subordinate Brownian motions; Harmonic functions; Boundary Harnack principle; Poisson kernel; Heat kernel; Green function (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (8)

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DOI: 10.1016/j.spa.2013.07.007

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