Number of particles absorbed in a BBM on the extinction event

Abstract

We consider a branching Brownian motion which starts from $0$ with drift $μ\in \mathbb{R}$ and we focus on the number $Z_x$ of particles killed at $-x$, where $x>0$. Let us call $μ_0$ the critical drift such that there is a positive probability of survival if and only if $μ>-μ_0$. Maillard \cite{maillard2013number} and Berestycki et al. \cite{berestycki2015branching} have study $Z_x$ in the case $μ\leq -μ_0$ and $μ\geq μ_0$ respectively. We complete the picture by considering the case where $μ>-μ_0$ on the extinction event. More precisely we study the asymptotic of $q_i(x):=\mathbb{P}\left(Z_x=i,ζ_x<\infty\right)$. We show that the radius of convergence $R(μ)$ of the corresponding power series increases as $μ$ increases, up until $μ=μ_c\in [-μ_0,+\infty]$ after which it is constant. We also give a necessary and sufficient condition for $μ_c<+\infty$. In addition, finer asymptotics are also obtained, which highlight three different regimes depending on $μ<μ_c$, $μ=μ_c$ or $μ>μ_c$.