Recurrent extensions of real-valued self-similar Markov processes

TL;DR

This paper establishes a precise criterion for extending real-valued self-similar Markov processes to be recurrent and continuous at zero, expanding understanding beyond positive cases and including new analysis of stable Lévy processes.

Contribution

It provides a necessary and sufficient condition for the existence of recurrent extensions of real-valued self-similar Markov processes, generalizing previous positive-only results and analyzing new classes like stable Lévy processes.

Findings

Derived a condition using Lamperti-Kiu representation

Extended previous results to real-valued processes

Analyzed recurrent extension of stable Lévy processes

Abstract

Let $X=(X_t, t\geq 0)$ be a self-similar Markov process taking values in $\mathbb{R}$ such that the state 0 is a trap. In this paper, we present a necessary and sufficient condition for the existence of a self-similar recurrent extension of $X$ that leaves 0 continuously. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. Our results extend those of Fitzsimmons (2006) and Rivero (2005, 2007) where the existence and uniqueness of a recurrent extension for positive self similar Markov processes were treated. In particular, we describe the recurrent extension of a stable L\'evy process which to the best of our knowledge has not been studied before.