Boundary Harnack Principle for Subordinate Brownian Motions

TL;DR

This paper proves a boundary Harnack principle for a broad class of subordinate Brownian motions in certain fat open sets, and characterizes their Martin boundary in relation to the Euclidean boundary.

Contribution

It establishes a boundary Harnack principle for subordinate Brownian motions in $ ext{kappa}$-fat open sets and identifies their Martin boundary with the Euclidean boundary.

Findings

Boundary Harnack principle proven for subordinate Brownian motions.

Martin boundary coincides with Euclidean boundary for these processes.

Applicable to a class including mixtures of symmetric stable processes.

Abstract

We establish a boundary Harnack principle for a large class of subordinate Brownian motion, including mixtures of symmetric stable processes, in bounded $\kappa$-fat open set (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded $\kappa$-fat open sets with respect to these processes with their Euclidean boundary.