Oscillation of harmonic functions for subordinate Brownian motion and its applications
Abstract
In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X in bounded kappa-fat open set; if u is a positive harmonic function with respect to X in a bounded kappa-fat open set D and h is a positive harmonic function in D vanishing on D^c, then the non-tangential limit of u/h exists almost everywhere with respect to the Martin-representing measure of h.
Keywords
Cite
@article{arxiv.1208.5196,
title = {Oscillation of harmonic functions for subordinate Brownian motion and its applications},
author = {Panki Kim and Yunju Lee},
journal= {arXiv preprint arXiv:1208.5196},
year = {2012}
}
Comments
24pages. To appear in Stochastic Processes and their Applications (http://www.journals.elsevier.com/stochastic-processes-and-their-applications)