Boundary Harnack principle and Martin boundary at infinity for subordinate Brownian motions

TL;DR

This paper investigates the Martin boundary at infinity for subordinate Brownian motions, establishing the boundary Harnack principle for unbounded sets and characterizing the Martin boundary at infinity for sets with a specific geometric property.

Contribution

It proves the boundary Harnack principle at infinity for subordinate Brownian motions and introduces the concept of -fatness at infinity, characterizing the Martin boundary at infinity.

Findings

Boundary Harnack principle at infinity holds for unbounded open sets.

Martin boundary at infinity consists of exactly one point for -fat sets.

The single Martin boundary point is minimal.

Abstract

In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of $\kappa$-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.