Quenched law of large numbers for branching brownian motion in a random medium

János Engländer

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 3, page 490-518
  • ISSN: 0246-0203

Abstract

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We study a spatial branching model, where the underlying motion is d-dimensional (d≥1) brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result of this paper is the quenched law of large numbers for the population for all d≥1. We also show that the branching brownian motion with mild obstacles spreads less quickly than ordinary branching brownian motion by giving an upper estimate on its speed. When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the quenched local growth that is independent of the poissonian intensity. More general offspring distributions (beyond the dyadic one considered in the main theorems) as well as mild obstacle models for superprocesses are also discussed.

How to cite

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Engländer, János. "Quenched law of large numbers for branching brownian motion in a random medium." Annales de l'I.H.P. Probabilités et statistiques 44.3 (2008): 490-518. <http://eudml.org/doc/77980>.

@article{Engländer2008,
abstract = {We study a spatial branching model, where the underlying motion is d-dimensional (d≥1) brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result of this paper is the quenched law of large numbers for the population for all d≥1. We also show that the branching brownian motion with mild obstacles spreads less quickly than ordinary branching brownian motion by giving an upper estimate on its speed. When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the quenched local growth that is independent of the poissonian intensity. More general offspring distributions (beyond the dyadic one considered in the main theorems) as well as mild obstacle models for superprocesses are also discussed.},
author = {Engländer, János},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {poissonian obstacles; branching brownian motion; random environment; fecundity selection; radial speed; wavefronts in random medium; random KPP equation; Poissonian obstacles; branching Brownian motion},
language = {eng},
number = {3},
pages = {490-518},
publisher = {Gauthier-Villars},
title = {Quenched law of large numbers for branching brownian motion in a random medium},
url = {http://eudml.org/doc/77980},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Engländer, János
TI - Quenched law of large numbers for branching brownian motion in a random medium
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 3
SP - 490
EP - 518
AB - We study a spatial branching model, where the underlying motion is d-dimensional (d≥1) brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result of this paper is the quenched law of large numbers for the population for all d≥1. We also show that the branching brownian motion with mild obstacles spreads less quickly than ordinary branching brownian motion by giving an upper estimate on its speed. When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the quenched local growth that is independent of the poissonian intensity. More general offspring distributions (beyond the dyadic one considered in the main theorems) as well as mild obstacle models for superprocesses are also discussed.
LA - eng
KW - poissonian obstacles; branching brownian motion; random environment; fecundity selection; radial speed; wavefronts in random medium; random KPP equation; Poissonian obstacles; branching Brownian motion
UR - http://eudml.org/doc/77980
ER -

References

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