Branching brownian motion with an inhomogeneous breeding potential
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, p. 793-801
Cet article concerne un mouvement brownien branchant (BBM) en deux particules avec un taux β|y|p pour une particule située en y∈ℝ, avec une constante β>0. Il est connu que pour p>2, le nombre de particules explose presque sûrement en temps fini, alors que pour p=2 le nombre de particules explose en moyenne en temps fini bien qu'il reste fini presque sûrement à tout moment. Nous définissons la particule la plus à droite Rt comme le supremum des positions spatiales des particules vivant à l'instant t et étudions les asymptotiques de Rt quand t tend vers l'infini. Dans le cas d'une reproduction à taux constant β, l'asymptotique linéaire de Rt est bien connue. Ici, nous trouvons des résultats asymptotiques pour Rt dans le cas où p∈(0, 2]. Contrastant avec les asymptotiques linéaires du BBM standard, nous trouvons des asymptotiques polynomiales de degré arbitrairement grand quand p croit vers 2, et une limite non triviale pour lnRt quand p=2. Nos preuves s'appuient sur certaines martingales positives et des changements de mesures.
This article concerns branching brownian motion (BBM) with dyadic branching at rate β|y|p for a particle with spatial position y∈ℝ, where β>0. It is known that for p>2 the number of particles blows up almost surely in finite time, while for p=2 the expected number of particles alive blows up in finite time, although the number of particles alive remains finite almost surely, for all time. We define the right-most particle, Rt, to be the supremum of the spatial positions of the particles alive at time t and study the asymptotics of Rt as t→∞. In the case of constant breeding at rate β the linear asymptotic for Rt is long established. Here, we find asymptotic results for Rt in the case p∈(0, 2]. In contrast to the linear asymptotic in standard BBM we find polynomial asymptotics of arbitrarily high order as p↑2, and a non-trivial limit for lnRt when p=2. Our proofs rest on the analysis of certain additive martingales, and related spine changes of measure.
@article{AIHPB_2009__45_3_793_0,
     author = {Harris, J. W. and Harris, S. C.},
     title = {Branching brownian motion with an inhomogeneous breeding potential},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     pages = {793-801},
     doi = {10.1214/08-AIHP300},
     zbl = {1183.60029},
     mrnumber = {2548504},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_3_793_0}
}
Harris, J. W.; Harris, S. C. Branching brownian motion with an inhomogeneous breeding potential. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 793-801. doi : 10.1214/08-AIHP300. http://www.numdam.org/item/AIHPB_2009__45_3_793_0/

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