Existence of quasi-stationary distributions for spectrally positive L\'evy processes on the half-line

TL;DR

This paper characterizes the existence and properties of quasi-stationary distributions for spectrally positive Lévy processes on the half-line, linking them to exit times, Laplace exponents, and scale functions.

Contribution

It provides a comprehensive characterization of quasi-stationary distributions, including their existence, multiplicity, and conditions for the Yaglom limit in spectrally positive Lévy processes.

Findings

Existence of quasi-stationary distributions is linked to exponential integrability of exit times.

There are infinitely many quasi-stationary distributions if they exist.

Conditions are provided under which the minimal quasi-stationary distribution is the Yaglom limit.

Abstract

For spectrally positive L\'evy processes killed on exiting the half-line, existence of a quasi-stationary distribution is characterized by the exponential integrability of the exit time, the Laplace exponent and the non-negativity of the scale functions. It is proven that if there is a quasi-stationary distribution, there are necessarily infinitely many ones and the set of quasi-stationary distributions is characterized. A sufficient condition for the minimal quasi-stationary distribution to be the Yaglom limit is given.