Quasi-stationary distributions associated with explosive CSBP

TL;DR

This paper characterizes all quasi-stationary distributions and the Q-process for explosive continuous state branching processes, providing insights into their behavior conditioned on non-explosion with specific rescaling methods.

Contribution

It introduces a complete characterization of quasi-stationary distributions and the associated Q-process for explosive CSBPs, including a rescaling approach for non-explosive conditioning.

Findings

All quasi-stationary distributions for explosive CSBPs are characterized.

A rescaling method is provided for conditioned non-explosion scenarios.

The Q-process associated with explosive CSBPs is explicitly described.

Abstract

We characterise 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.