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

Pascal Maillard

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 2, page 428-455
  • ISSN: 0246-0203

Abstract

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We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift c . At the point x g t ; 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 ≥ 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 [1] and E. Aïdékon [2] 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.

How to cite

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Maillard, Pascal. "The number of absorbed individuals in branching brownian motion with a barrier." Annales de l'I.H.P. Probabilités et statistiques 49.2 (2013): 428-455. <http://eudml.org/doc/271972>.

@article{Maillard2013,
abstract = {We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$. At the point $x&gt;0$, 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 [1] and E. Aïdékon [2] 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.},
author = {Maillard, Pascal},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {branching brownian motion; Galton–Watson process; Briot–Bouquet equation; FKPP equation; travelling wave; singularity analysis of generating functions; branching Brownian motion; Galton-Watson process; Briot-Bouquet equation},
language = {eng},
number = {2},
pages = {428-455},
publisher = {Gauthier-Villars},
title = {The number of absorbed individuals in branching brownian motion with a barrier},
url = {http://eudml.org/doc/271972},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Maillard, Pascal
TI - The number of absorbed individuals in branching brownian motion with a barrier
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 2
SP - 428
EP - 455
AB - We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$. At the point $x&gt;0$, 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 [1] and E. Aïdékon [2] 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.
LA - eng
KW - branching brownian motion; Galton–Watson process; Briot–Bouquet equation; FKPP equation; travelling wave; singularity analysis of generating functions; branching Brownian motion; Galton-Watson process; Briot-Bouquet equation
UR - http://eudml.org/doc/271972
ER -

References

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