Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Critical branching Brownian motion with absorption: Particle configurations

Julien Berestycki, Nathanaël Berestycki, and Jason Schweinsberg

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider critical branching Brownian motion with absorption, in which there is initially a single particle at $x>0$, particles move according to independent one-dimensional Brownian motions with the critical drift of $-\sqrt{2}$, and particles are absorbed when they reach zero. Here we obtain asymptotic results concerning the behavior of the process before the extinction time, as the position $x$ of the initial particle tends to infinity. We estimate the number of particles in the system at a given time and the position of the right-most particle. We also obtain asymptotic results for the configuration of particles at a typical time.

Résumé

Nous considérons un mouvement brownien branchant avec absorption critique, issu d’une particule en $x>0$, dans lequel les particules se déplacent selon des mouvement browniens réels indépendants avec une dérive critique de $-\sqrt{2}$, et sont absorbées en zero. Nous obtenons des résultats asymptotiques sur le comportement de ce processus avant son extinction, quand la position $x$ de la particule initiale tend vers l’infini. En particulier nous obtenons des éstimées sur le nombre de particules dans le système, la position de la particule la plus à droite, et la configuration des particules à un instant typique.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 51, Number 4 (2015), 1215-1250.

Dates
Received: 28 September 2013
Revised: 13 March 2014
Accepted: 14 March 2014
First available in Project Euclid: 21 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1445432041

Digital Object Identifier
doi:10.1214/14-AIHP613

Mathematical Reviews number (MathSciNet)
MR3414446

Zentralblatt MATH identifier
1329.60300

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J25: Continuous-time Markov processes on general state spaces

Keywords
Branching Brownian motion Critical phenomena Yaglom limit laws

Citation

Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason. Critical branching Brownian motion with absorption: Particle configurations. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1215--1250. doi:10.1214/14-AIHP613. https://projecteuclid.org/euclid.aihp/1445432041


Export citation

References

  • [1] E. Aïdékon, J. Berestycki, E. Brunet and Z. Shi. Branching Brownian motion seen from its tip. Probab. Theory Related Fields 157 (2013) 405–451.
  • [2] L.-P. Arguin, A. Bovier and N. Kistler. The extremal process of branching Brownian motion. Probab. Theory Related Fields 157 (2013) 535–574.
  • [3] A. Asselah, P. Ferrari and P. Groisman. Quasi-stationary distributions and Fleming–Viot processes in finite spaces. J. Appl. Probab. 48 (2011) 322–332.
  • [4] A. Asselah, P. Ferrari, P. Groisman and M. Jonckheere. Fleming–Viot selects the minimal quasi-stationary distribution: The Galton–Watson case. Ann. Inst. Henri Poincaré. To appear, 2015. Available at arXiv:1206.6114.
  • [5] J. Berestycki, N. Berestycki and J. Schweinsberg. The genealogy of branching Brownian motion with absorption. Ann. Probab. 41 (2013) 527–618.
  • [6] J. Berestycki, N. Berestycki and J. Schweinsberg. Critical branching Brownian motion with absorption: Survival probability. Probab. Theory Related Fields 160 (2014) 489–520.
  • [7] M. Bramson. Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44 (285) (1983) iv+190.
  • [8] E. Brunet, B. Derrida, A. H. Mueller and S. Munier. Noisy traveling waves: Effect of selection on genealogies. Europhys. Lett. 76 (2006) 1–7.
  • [9] E. Brunet, B. Derrida, A. H. Mueller and S. Munier. Effect of selection on ancestry: An exactly soluble case and its phenomenological generalization. Phys. Rev. E (3) 76 (2007) 041104.
  • [10] K. Burdzy, R. Holyst, D. Ingerman and P. March. Configurational transition in a Fleming–Viot-type model and probabilistic interpretation of Laplacian eigenfunctions. J. Phys. A: Math. Gen. 29 (1996) 2633–2642.
  • [11] K. Burdzy, R. Holyst and P. March. A Fleming–Viot particle representation of the Dirichlet Laplacian. Comm. Math. Phys. 214 (2000) 679–703.
  • [12] I. Grigorescu and M. Kang. Hydrodynamic limit for a Fleming–Viot type system. Stochastic Process. Appl. 110 (2004) 111–143.
  • [13] I. Grigorescu and M. Kang. Immortal particle for a catalytic branching process. Probab. Theory Related Fields 153 (2011) 333–361.
  • [14] F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik. A short proof of the logarithmic Bramson correction in Fisher–KPP equations. Netw. Heterog. Media 8 (2013) 275–289.
  • [15] F. Hamel, J. Nolen, J.-M. Roquejoffre and L. Ryzhik. The logarithmic delay of KPP fronts in a periodic medium. Preprint, arXiv:1211.6173.
  • [16] J. W. Harris and S. C. Harris. Survival probabilities for branching Brownian motion with absorption. Electron. Commun. Probab. 12 (2007) 81–92.
  • [17] J. W. Harris, S. C. Harris and A. E. Kyprianou. Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: One-sided traveling waves. Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 125–145.
  • [18] S. C. Harris and M. I. Roberts. The unscaled paths of branching Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 579–608.
  • [19] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Springer, New York, 2002.
  • [20] H. Kesten. Branching Brownian motion with absorption. Stochastic Process. Appl. 7 (1978) 9–47.
  • [21] S. P. Lalley and Y. Shao. On the maximal displacement of a critical branching random walk. Probab. Theory Related Fields 162 (2015) 71–96.
  • [22] P. Maillard. Speed and fluctuations of $N$-particle branching Brownian motion with spatial selection. Preprint, arXiv:1304.0562.
  • [23] J. Neveu. Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes, 1987 223–241. E. Çinlar, K. L. Chung and R. K. Getoor (Eds). Prog. Probab. Statist. 15. Birkhäuser, Boston, 1988.
  • [24] S. Sawyer. Branching diffusion processes in population genetics. Adv. in Appl. Probab. 8 (1976) 659–689.
  • [25] S. Martinez and J. San Martin. Quasi-stationary distributions for a Brownian motion with drift and associated limit laws. J. Appl. Probab. 31 (4) (1994) 911–920.
  • [26] A. M. Yaglom. Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.) 56 (1947) 795–798.