Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Optimal survival strategy for branching Brownian motion in a Poissonian trap field

Mehmet Öz and János Engländer

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Abstract

We study a branching Brownian motion $Z$ with a generic branching law, evolving in $\mathbb{R}^{d}$, where a field of Poissonian traps is present. Each trap is a ball with constant radius. The traps are hard in the sense that the process is killed instantly once it enters the trap field. We focus on two cases of Poissonian fields, a uniform field and a radially decaying field, and consider an annealed environment. Using classical results on the convergence of the speed of branching Brownian motion, we establish precise annealed results on the population size of $Z$, given that it avoids the trap field, while staying alive up to time $t$. The results are stated so that each gives an ‘optimal survival strategy’ for $Z$. As corollaries of the results concerning the population size, we prove several other optimal survival strategies concerning the range of $Z$, and the size and position of clearings in $\mathbb{R}^{d}$. We also prove a result about the hitting time of a single trap by a branching system (Lemma 1), which may be useful in a completely generic setting too.

Inter alia, we answer some open problems raised in (Markov Process. Related Fields 9 (2003) 363–389).

Résumé

Nous étudions un mouvement brownien branchant $Z$ ayant une loi de branchement générique et évoluant dans $\mathbb{R}^{d}$, où se trouve un champ de pièges poissonniens. Chaque piège est constitué d’une boule de rayon constant. Les pièges sont durs, au sens où le processus est tué instantanément dès qu’il pénètre dans l’un des pièges. Nous nous concentrons sur deux cas particuliers de champs poissonniens, un champ uniforme et un champ décroissant radialement, et nous considérons un environnement annealed. En utilisant des résultats classiques sur la convergence de la vitesse du mouvement brownien branchant, nous établissons des résultats annealed précis sur la taille de la population décrite par $Z$, conditionnellement à ce qu’il évite l’ensemble des pièges et reste en vie jusqu’au temps $t$. Les résultats sont formulés de sorte que chacun d’entre eux donne une ‘stratégie optimale de survie’ pour $Z$. En corollaires de ces résultats, nous démontrons l’optimalité de plusieurs autres stratégies concernant le support de $Z$ jusqu’au temps $t$ et la taille et la position de clairières dans $\mathbb{R}^{d}$. Nous démontrons également un résultat sur le temps d’atteinte d’un seul piège par un système branchant (Lemme 1), qui pourra aussi être utile dans un cadre totalement générique.

Au passage, nous apportons une réponse à plusieurs questions ouvertes formulées dans (Markov Process. Related Fields 9 (2003) 363–389).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 4 (2019), 1890-1915.

Dates
Received: 25 October 2017
Revised: 28 July 2018
Accepted: 24 September 2018
First available in Project Euclid: 8 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1573203618

Digital Object Identifier
doi:10.1214/18-AIHP937

Mathematical Reviews number (MathSciNet)
MR4029143

Zentralblatt MATH identifier
07161494

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 Optimal survival strategy

Citation

Öz, Mehmet; Engländer, János. Optimal survival strategy for branching Brownian motion in a Poissonian trap field. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 4, 1890--1915. doi:10.1214/18-AIHP937. https://projecteuclid.org/euclid.aihp/1573203618


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