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

By Panki Kim, Renming Song, Z. Vondravcek. 2012

85 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/9e9c6c342ea859969068f926f2a8180686ea9bc2

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Citations in the corpus

Dirichlet heat kernel for unimodal Lévy processes (2014, methodology): “Gradual extensions were then obtained for generators of many subordinate Brownian motions satisfying scaling conditions [14, 16, 15, 17], and for processes with comparable Lévy measure [33].”
Barriers, exit time and survival probability for unimodal Lévy processes (2013, background): “[17,36] and Sect.”