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

Maillard, Pascal

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 428-455.

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 $c\ge c_0$. 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.

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 [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.

Classification : Primary 60J80, secondary, 34M35
Mots clés : branching brownian motion, Galton-Watson process, Briot-Bouquet equation, FKPP equation, travelling wave, singularity analysis of generating functions

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