Recurrent extensions of self-similar Markov processes and Cram\'er's condition II

TL;DR

This paper establishes that positive self-similar Markov processes can be extended to recurrent processes leaving zero continuously if and only if their underlying Lévy processes satisfy Cramér's condition, linking process behavior to Lévy properties.

Contribution

It provides a necessary and sufficient condition, based on Cramér's condition, for extending self-similar Markov processes to recurrent processes that leave zero continuously.

Findings

Extension exists if and only if Lévy process satisfies Cramér's condition

Characterizes the behavior of self-similar Markov processes near zero

Links Lévy process properties to process recurrence and boundary behavior

Abstract

We prove that a positive self-similar Markov process $(X,\mathbb{P})$ that hits 0 in a finite time admits a self-similar recurrent extension that leaves 0 continuously if and only if the underlying L\'{e}vy process satisfies Cram\'{e}r's condition.