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arXiv · 1004.1426

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

Pascal Maillard
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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\"ıdékon (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.

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Pascal Maillard. 2011-11-24. The number of absorbed individuals in branching Brownian motion with a barrier. https://doi.org/10.1214/11-aihp451

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