Asymptotics for the survival probability in a killed branching random walk
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 111-129.

Considérons une marche aléatoire branchante surcritique à temps discret. Nous nous intéressons à la probabilité qu'il existe un rayon infini du support de la marche aléatoire branchante, le long duquel elle croît plus vite qu'une fonction linéaire de pente γ - ε, où γ désigne la vitesse asymptotique de la position de la particule la plus à droite dans la marche aléatoire branchante. Sous des hypothèses générales peu restrictives, nous prouvons que, lorsque ε → 0, cette probabilité décroît comme exp{-(β+o(1)) / ε1/2}, où β est une constante strictement positive dont la valeur dépend de la loi de la marche aléatoire branchante. Dans le cas spécial où des variables aléatoires i.i.d. de Bernoulli(p) (avec 0 < p < ½) sont placées sur les arêtes d'un arbre binaire enraciné, ceci répond à une question ouverte de Robin Pemantle (Ann. Appl. Probab. 19 (2009) 1273-1291).

Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope γ - ε, where γ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when ε → 0, this probability decays like exp{-(β+o(1)) / ε1/2}, where β is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli(p) random variables (with 0 < p < ½) assigned on a rooted binary tree, this answers an open question of Robin Pemantle (see Ann. Appl. Probab. 19 (2009) 1273-1291).

DOI : 10.1214/10-AIHP362
Classification : 60J80
Mots clés : branching random walk, survival probability, maximal displacement
@article{AIHPB_2011__47_1_111_0,
     author = {Gantert, Nina and Hu, Yueyun and Shi, Zhan},
     title = {Asymptotics for the survival probability in a killed branching random walk},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {111--129},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {1},
     year = {2011},
     doi = {10.1214/10-AIHP362},
     mrnumber = {2779399},
     zbl = {1210.60093},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/10-AIHP362/}
}
TY  - JOUR
AU  - Gantert, Nina
AU  - Hu, Yueyun
AU  - Shi, Zhan
TI  - Asymptotics for the survival probability in a killed branching random walk
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2011
SP  - 111
EP  - 129
VL  - 47
IS  - 1
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/10-AIHP362/
DO  - 10.1214/10-AIHP362
LA  - en
ID  - AIHPB_2011__47_1_111_0
ER  - 
%0 Journal Article
%A Gantert, Nina
%A Hu, Yueyun
%A Shi, Zhan
%T Asymptotics for the survival probability in a killed branching random walk
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2011
%P 111-129
%V 47
%N 1
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/10-AIHP362/
%R 10.1214/10-AIHP362
%G en
%F AIHPB_2011__47_1_111_0
Gantert, Nina; Hu, Yueyun; Shi, Zhan. Asymptotics for the survival probability in a killed branching random walk. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 111-129. doi : 10.1214/10-AIHP362. http://www.numdam.org/articles/10.1214/10-AIHP362/

[1] D. J. Aldous. A Metropolis-type optimization algorithm on the infinite tree. Algorithmica 22 (1998) 388-412. | MR | Zbl

[2] J. D. Biggins. The first- and last-birth problems for a multitype age-dependent branching process. Adv. in Appl. Probab. 8 (1976) 446-459. | MR | Zbl

[3] J. D. Biggins and A. E. Kyprianou. Fixed points of the smoothing transform: The boundary case. Electron. J. Probab. 10 (2005) 609-631, Paper 17. | MR | Zbl

[4] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. | MR | Zbl

[5] É. Brunet and B. Derrida. Shift in the velocity of a front due to a cutoff. Phys. Rev. E 56 (1997) 2597-2604.

[6] B. Derrida and D. Simon. The survival probability of a branching random walk in presence of an absorbing wall. Europhys. Lett. 78 (2007), Paper 60006. | MR

[7] B. Derrida and D. Simon. Quasi-stationary regime of a branching random walk in presence of an absorbing wall. J. Stat. Phys. 131 (2008) 203-233. | MR | Zbl

[8] J. M. Hammersley. Postulates for subadditive processes. Ann. Probab. 2 (1974) 652-680. | MR | Zbl

[9] Y. Hu and Z. Shi. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 (2009) 742-789. | MR | Zbl

[10] K. Itô and H. P. Mckean Jr. Diffusion Processes and Their Sample Paths. Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer, Berlin, 1974. | MR | Zbl

[11] B. Jaffuel. The critical barrier for the survival of the branching random walk with absorption, 2009. Available at ArXiv math.PR/0911.2227.

[12] J.-P. Kahane and J. Peyrière. Sur certaines martingales de Mandelbrot. Adv. Math. 22 (1976) 131-145. | MR | Zbl

[13] H. Kesten. Branching Brownian motion with absorption. Stochastic Process. Appl. 7 (1978) 9-47. | MR | Zbl

[14] J. F. C. Kingman. The first birth problem for an age-dependent branching process. Ann. Probab. 3 (1975) 790-801. | MR | Zbl

[15] R. Lyons. A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes 217-221. K. B. Athreya and P. Jagers (Eds). IMA Volumes in Mathematics and Its Applications 84. Springer, New York, 1997. | MR | Zbl

[16] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of LlogL criteria for mean behavior of branching processes. Ann. Probab. 23 (1995) 1125-1138. | MR | Zbl

[17] C. Mcdiarmid. Minimal positions in a branching random walk. Ann. Appl. Probab. 5 (1995) 128-139. | MR | Zbl

[18] A. A. Mogulskii. Small deviations in the space of trajectories. Theory Probab. Appl. 19 (1974) 726-736. | MR | Zbl

[19] R. Pemantle. Search cost for a nearly optimal path in a binary tree. Ann. Appl. Probab. 19 (2009) 1273-1291. | MR | Zbl

Cité par Sources :