The structure of entrance and exit at infinity for time-changed L\'evy processes

TL;DR

This paper investigates the boundary behavior at infinity for time-changed Lévy processes, providing sharp conditions for entrance and exit, and introduces a generalized scaling property applicable to all such processes.

Contribution

It introduces a generalized scaling property for time-changed Lévy processes, enabling analysis of boundary points at infinity with minimal model assumptions.

Findings

Sharp conditions for entrance at infinity

Conditions for regular boundary points at infinity

A generalized scaling property for all time-changed Lévy processes

Abstract

Studying the behaviour of Markov processes at boundary points of the state space has a long history, dating back all the way to William Feller. With different motivations in mind entrance and exit questions have been explored for different discontinuous Markov processes in the past two decades. Proofs often use time-change techniques and rely on problem specific knowledge such branching or scaling properties. In this article we ask how far techniques can be pushed with as little as possible model assumptions. We give sharp conditions on time-changed L\'evy processes to allow entrance and regular boundary point at infinity. The main tool we introduce is a generalised scaling property that holds for all time-changed L\'evy processes and can be used to extend scaling arguments for self-similar Markov processes.