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

Maillard, Pascal

Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, pp. 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 $c\ge c_0$. 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.

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.

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