Number of particles absorbed in a BBM on the extinction event

TL;DR

This paper analyzes the number of particles absorbed in a branching Brownian motion, focusing on the asymptotic behavior of the probability distribution of absorbed particles across different drift regimes, especially near the critical drift.

Contribution

It completes the understanding of absorbed particle counts in BBM by studying the case where the drift exceeds the critical value, providing asymptotics and conditions for the radius of convergence.

Findings

The radius of convergence increases with drift until a critical point.

Finer asymptotics reveal three regimes based on the drift relative to the critical value.

Conditions for the finiteness of the critical drift are established.

Abstract

We consider a branching Brownian motion which starts from $0$ with drift $\mu \in \mathbb{R}$ and we focus on the number $Z_x$ of particles killed at $-x$, where $x>0$. Let us call $\mu_0$ the critical drift such that there is a positive probability of survival if and only if $\mu>-\mu_0$. Maillard \cite{maillard2013number} and Berestycki et al. \cite{berestycki2015branching} have study $Z_x$ in the case $\mu \leq -\mu_0$ and $\mu\geq \mu_0$ respectively. We complete the picture by considering the case where $\mu>-\mu_0$ on the extinction event. More precisely we study the asymptotic of $q_i(x):=\mathbb{P}\left(Z_x=i,\zeta_x<\infty\right)$. We show that the radius of convergence $R(\mu)$ of the corresponding power series increases as $\mu$ increases, up until $\mu=\mu_c\in [-\mu_0,+\infty]$ after which it is constant. We also give a necessary and sufficient condition for $\mu_c<+\infty$. In addition, finer asymptotics are also obtained, which highlight three different regimes depending on $\mu<\mu_c$, $\mu=\mu_c$ or $\mu>\mu_c$.