Accessibility, Martin boundary and minimal thinness for Feller processes in metric measure spaces

TL;DR

This paper investigates the Martin boundary at infinity for purely discontinuous Feller processes in metric measure spaces, establishing conditions for minimality and the local nature of minimal thinness.

Contribution

It provides new results on the structure of Martin boundaries at infinity and finite points for Feller processes, highlighting the local property of minimal thinness.

Findings

Unique Martin boundary point at infinity when accessible

Minimality of boundary points under certain conditions

Minimal thinness is a local property

Abstract

In this paper we study the Martin boundary at infinity for a large class of purely discontinuous Feller processes on metric measure spaces. We show that if $\infty$ is accessible from an open set $D$, then there is only one Martin boundary point of $D$ associated with it, and this point is minimal. We also prove the analogous result for finite boundary points. As a consequence, we show that minimal thinness of a set is a local property.