Local extinction in continuous-state branching processes with immigration

TL;DR

This paper studies the zero sets of continuous-state branching processes with immigration, revealing they are infinitely divisible regenerative sets and providing detailed insights based on their branching and immigration mechanisms.

Contribution

It demonstrates that zero sets of CBI processes are infinitely divisible regenerative sets and connects their structure to the underlying mechanisms.

Findings

Zero sets are infinitely divisible regenerative sets

Construction via Mandelbrot's random cutouts

Detailed characterization using branching and immigration mechanisms

Abstract

The purpose of this article is to observe that the zero sets of continuous-state branching processes with immigration (CBI) are infinitely divisible regenerative sets. Indeed, they can be constructed by the procedure of random cutouts introduced by Mandelbrot in 1972. We then show how very precise information about the zero sets of CBI can be obtained in terms of the branching and immigrating mechanism.