The number of absorbed individuals in branching brownian motion with a barrier
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, p. 428-455

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 cc 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.

Nous étudions le mouvement brownien branchant sur-critique sur la droite réelle, issu de l’origine et avec une dérive constante c. Au point x>0, nous ajoutons une barrière absorbante, c’est-à-dire les individus qui touchent la barrière sont tués instantanément et sans se reproduire. Il est connu qu’il existe une dérive critique c 0 tel que ce processus s’éteint presque surement si et seulement si cc 0 . Dans ce cas, si on note par Z x le nombre d’individus absorbés en la barrière, nous donnons un équivalent de P(Z x =n) quand n tend vers l’infini. Si c=c 0 et la reproduction est déterministe, ceci améliore des résultats de L. Addario-Berry et N. Broutin [1] et E. Aïdékon [2] sur une conjecture de David Aldous concernant la progéniture totale d’une marche aléatoire branchante. La technique principale utilisée dans les preuves est l’analyse de la fonction génératrice de Z x au voisinage de son point singulier 1, basée sur des résultats classiques concernant certaines équations differéntielles dans le champ complexe.

DOI : https://doi.org/10.1214/11-AIHP451
Classification:  Primary 60J80,  secondary,  34M35
Keywords: branching brownian motion, Galton-Watson process, Briot-Bouquet equation, FKPP equation, travelling wave, singularity analysis of generating functions
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     author = {Maillard, Pascal},
     title = {The number of absorbed individuals in branching brownian motion with a barrier},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {2},
     year = {2013},
     pages = {428-455},
     doi = {10.1214/11-AIHP451},
     zbl = {1281.60070},
     mrnumber = {3088376},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2013__49_2_428_0}
}
Maillard, Pascal. The number of absorbed individuals in branching brownian motion with a barrier. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, pp. 428-455. doi : 10.1214/11-AIHP451. http://www.numdam.org/item/AIHPB_2013__49_2_428_0/

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