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

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 (2009), \url{http://arxiv.org/abs/0908.1083v1}] and [E. A\"{\i}dékon (2009), \url{http://arxiv.org/abs/0911.0877v1}] 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.
Type de document :
Pré-publication, Document de travail
31 pages. 2011
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https://hal.archives-ouvertes.fr/hal-00472913
Contributeur : Pascal Maillard <>
Soumis le : jeudi 1 décembre 2011 - 16:53:18
Dernière modification le : samedi 29 avril 2017 - 01:07:08
Document(s) archivé(s) le : vendredi 2 mars 2012 - 02:35:36

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absorbed_particles.pdf
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  • HAL Id : hal-00472913, version 3

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Pascal Maillard. The number of absorbed individuals in branching Brownian motion with a barrier. 31 pages. 2011. <hal-00472913v3>

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