Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Conditional speed of branching Brownian motion, skeleton decomposition and application to random obstacles

Mehmet Öz, Mine Çağlar, and János Engländer

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Abstract

We study a branching Brownian motion $Z$ in $\mathbb{R}^{d}$, among obstacles scattered according to a Poisson random measure with a radially decaying intensity. Obstacles are balls with constant radius and each one works as a trap for the whole motion when hit by a particle. Considering a general offspring distribution, we derive the decay rate of the annealed probability that none of the particles of $Z$ hits a trap, asymptotically in time $t$. This proves to be a rich problem motivating the proof of a more general result about the speed of branching Brownian motion conditioned on non-extinction. We provide an appropriate “skeleton” decomposition for the underlying Galton–Watson process when supercritical and show that the “doomed” particles do not contribute to the asymptotic decay rate.

Résumé

On étudie un mouvement brownien branchant $Z$ dans $\mathbb{R}^{d}$ qui se déplace parmi des obstacles qui sont dispersés par rapport à une mesure aléatoire de Poisson d’une intensité déclinant radialement. Les obstacles sont des boules de rayon constant, et lorsqu’une particule rencontre un tel obstacle, tout le mouvement s’arrête. En considérant une distribution générale pour le nombre de descendants, on calcule le taux de décroissance de la probabilité <annealed> que toutes les particules de $Z$ évitent les pièges, asymptotiquement en temps. Cela se révèle être un problème riche qui motive la preuve d’un résultat plus général concernant la vitesse du mouvement brownien branchant conditionné à survivre. Dans le cas sur-critique on fournit une décomposition <en squelette> appropriée pour le processus de Galton–Watson sous-jacent, et on montre que les particules <condamnées> ne contribuent pas au taux de décroissance asymptotique.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 842-864.

Dates
Received: 6 September 2014
Revised: 8 December 2015
Accepted: 6 January 2016
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897748

Digital Object Identifier
doi:10.1214/16-AIHP739

Mathematical Reviews number (MathSciNet)
MR3634277

Zentralblatt MATH identifier
06729835

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K37: Processes in random environments 60F10: Large deviations

Keywords
Branching Brownian motion Poissonian traps Random environment Hard obstacles Rightmost particle

Citation

Öz, Mehmet; Çağlar, Mine; Engländer, János. Conditional speed of branching Brownian motion, skeleton decomposition and application to random obstacles. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 842--864. doi:10.1214/16-AIHP739. https://projecteuclid.org/euclid.aihp/1491897748


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