Martin boundary of unbounded sets for purely discontinuous Feller processes

TL;DR

This paper investigates the Martin boundary for unbounded open sets in metric spaces related to purely discontinuous Feller processes, establishing conditions for the uniqueness and minimality of boundary points at infinity and finite points.

Contribution

It provides new results on the Martin boundary structure for purely discontinuous Feller processes in unbounded sets, linking boundary point minimality to accessibility of infinity.

Findings

Unique Martin boundary point at infinity under certain conditions

Minimality of the boundary point at infinity linked to its accessibility

Results extend to finite boundary points of the set

Abstract

In this paper, we study the Martin kernels of general open sets associated with inaccessible points for a large class of purely discontinuous Feller processes in metric measure spaces. Let $D$ be an unbounded open set. Infinity is accessible from $D$ if the expected exit time from $D$ is infinite, and inaccessible otherwise. We prove that under suitable assumptions there is only one Martin boundary point associated with infinity, and that this point is minimal if and only if infinity is accessible from $D$. Similar results are also proved for finite boundary points of $D$.