Martin boundary for some symmetric L\'evy processes

TL;DR

This paper investigates the Martin boundary for a broad class of symmetric Lévy processes, establishing conditions under which boundary points correspond uniquely to minimal Martin boundary points.

Contribution

It provides a new characterization of the Martin boundary for symmetric Lévy processes in open sets with $ ext{κ}$-fat boundary points.

Findings

Unique Martin boundary point for $ ext{κ}$-fat boundary points

Martin boundary point is minimal in these cases

Results extend understanding of boundary behavior for Lévy processes

Abstract

In this paper we study the Martin boundary of open sets with respect to a large class of purely discontinuous symmetric L\'evy processes in ${\mathbb R}^d$. We show that, if $D\subset {\mathbb R}^d$ is an open set which is $\kappa$-fat at a boundary point $Q\in \partial D$, then there is exactly one Martin boundary point associated with $Q$ and this Martin boundary point is minimal.