Total progeny for spectrally negative branching L{\'e}vy processes with absorption

TL;DR

This paper analyzes the tail behavior of the total number of particles absorbed at zero in a spectrally negative branching Lévy process, revealing asymptotic properties in different regimes.

Contribution

It provides new asymptotic results for the distribution of total progeny in spectrally negative branching Lévy processes with absorption.

Findings

Derived tail asymptotics in subcritical regime

Established tail behavior in critical regime

Enhanced understanding of extinction dynamics

Abstract

We consider a spectrally negative branching L{\'e}vy process in which particles are killed upon crossing below zero. It is known that such a process becomes extinct almost surely if the drift toward -$\infty$ is sufficiently strong to counterbalance the reproduction rate. In this note, we study the tail asymptotics of the number of particles absorbed at the boundary during the lifetime of the process, in both the subcritical and critical regimes.