Positive self-similar Markov processes obtained by resurrection

TL;DR

This paper investigates positive self-similar Markov processes created by resurrecting stable processes at their exit times, analyzing their long-term behavior, absorption conditions, and jump kernel estimates.

Contribution

It introduces a method to construct resurrected stable processes using the Lamperti transform and characterizes their long-term dynamics and jump behaviors.

Findings

Conditions for absorption at zero in finite time

Existence criteria for recurrent extensions

Sharp estimates of jump kernels for resurrected processes

Abstract

In this paper we study positive self-similar Markov processes obtained by (partially) resurrecting a strictly $\alpha$-stable process at its first exit time from $(0,\infty)$. We construct those processes by using the Lamperti transform. We explain their long term behavior and give conditions for absorption at 0 in finite time. In case the process is absorbed at 0 in finite time, we give a necessary and sufficient condition for the existence of a recurrent extension. The motivation to study resurrected processes comes from the fact that their jump kernels may explode at zero. We establish sharp two-sided jump kernel estimates for a large class of resurrected stable processes.