Number of particles absorbed in a BBM on the extinction event

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 $\mu>\mu_c$.