Quasi-stationary distributions associated with explosive CSBP

Cyril Labbé

doi:10.1214/ecp.v18-2508

Abstract

We characterize all the quasi-stationary distributions and the Q process associated with a continuous state branching process that explodes in finite time. We also provide a rescaling for the continuous state branching process conditioned on non-explosion when the branching mechanism is regularly varying at 0.

Highlights

Continuous-state branching processes (CSBP) are [0, ∞]-valued Markov processes that describe the evolution of the size of a continuous population
We characterise all the quasi-stationary distributions and the Q-process associated with a continuous state branching process that explodes in finite time
We provide a rescaling for the continuous state branching process conditioned on nonexplosion when the branching mechanism is regularly varying at 0

Summary

Introduction

Continuous-state branching processes (CSBP) are [0, ∞]-valued Markov processes that describe the evolution of the size of a continuous population. They have been introduced by Jirina [6] and Lamperti [10]. Critical ), the convexity of Ψ implies q = 0 and the process is almost surely absorbed at 0. The extinction time T0 is almost surely finite iff. On the explosion event {T = T∞}, the explosion time T∞ is almost surely finite iff 0+. The goal of the present paper is to investigate the QSD associated with a CSBP that explodes in finite time almost surely.

A brief review of the literature: the extinction case

Main results: the explosive case

The discrete case

Quasi-stationary distributions and Q-process in the explosive case