The number of absorbed individuals in branching Brownian motion with a barrier

TL;DR

This paper analyzes the distribution of absorbed individuals in a supercritical branching Brownian motion with a barrier, providing asymptotic probabilities and extending previous conjectures on branching processes.

Contribution

It offers new asymptotic results for the number of absorbed individuals in branching Brownian motion with a barrier, especially at critical drift, improving prior conjectures.

Findings

Asymptotic behavior of P(Z_x=n) as n→∞ for supercritical case

Extension of results to the critical drift case with deterministic reproduction

Improved understanding of total progeny in branching random walks

Abstract

We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$. At the point $x > 0$, we add an absorbing barrier, i.e.\ individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift $c_0$, such that this process becomes extinct almost surely if and only if $c \ge c_0$. In this case, if $Z_x$ denotes the number of individuals absorbed at the barrier, we give an asymptotic for $P(Z_x=n)$ as $n$ goes to infinity. If $c=c_0$ and the reproduction is deterministic, this improves upon results of L. Addario-Berry and N. Broutin (2011) and E. A\"{\i}d\'ekon (2010) on a conjecture by David Aldous about the total progeny of a branching random walk. The main technique used in the proofs is analysis of the generating function of $Z_x$ near its singular point 1, based on classical results on some complex differential equations.