Oscillation of harmonic functions for subordinate Brownian motion and its applications

TL;DR

This paper establishes oscillation estimates for nonnegative harmonic functions of subordinate Brownian motion and applies these results to prove a form of the relative Fatou theorem, enhancing understanding of boundary behaviors in probabilistic potential theory.

Contribution

It provides a new oscillation estimate for harmonic functions of subordinate Brownian motion and offers a probabilistic proof of a relative Fatou theorem in bounded kappa-fat open sets.

Findings

Oscillation estimate for harmonic functions of subordinate Brownian motion

Probabilistic proof of the relative Fatou theorem

Almost everywhere existence of non-tangential limits

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.