Electronic Journal of Probability

Survival Probability of the Branching Random Walk Killed Below a Linear Boundary

Jean Bérard and Jean-Baptiste Gouéré

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We give an alternative proof of a result by N. Gantert, Y. Hu and Z. Shi on the asymptotic behavior of the survival probability of the branching random walk killed below a linear boundary, in the special case of deterministic binary branching and bounded random walk steps. Connections with the Brunet-Derrida theory of stochastic fronts are discussed.

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 14, 396-418.

Accepted: 18 February 2011
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Branching random walks survival probability

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Bérard, Jean; Gouéré, Jean-Baptiste. Survival Probability of the Branching Random Walk Killed Below a Linear Boundary. Electron. J. Probab. 16 (2011), paper no. 14, 396--418. doi:10.1214/EJP.v16-861. https://projecteuclid.org/euclid.ejp/1464820183

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  • E. Aidekon and S. Harris. Near-critical survival probability of branching Brownian motion with an absorbing barrier. Manuscript in preparation.
  • R. Benguria and M. C. Depassier. On the speed of pulled fronts with a cutoff. Phys. Rev. E, 75(5), 2007.
  • R. D. Benguria, M. C. Depassier, and M. Loss. Upper and lower bounds for the speed of pulled fronts with a cut-off. The European Physical Journal B - Condensed Matter and Complex Systems, 61:331–334, 2008.
  • Bérard, Jean; Gouéré, Jean-Baptiste. Brunet-Derrida behavior of branching-selection particle systems on the line. Comm. Math. Phys. 298 (2010), no. 2, 323–342.
  • N. Berestycki, J. Berestycki, and J. Schweinsberg. The genealogy of branching Brownian motion with absorption. arXiv:1001.2337, 2010.
  • N. Berestycki, J. Berestycki, and J. Schweinsberg. Survival of near-critical branching Brownian motion. arXiv:1009.0406, 2010.
  • Biggins, J. D. The first- and last-birth problems for a multitype age-dependent branching process. Advances in Appl. Probability 8 (1976), no. 3, 446–459.
  • J.D. Biggins. Branching out. arXiv:1003.4715, 2010.
  • Biggins, J. D.; Lubachevsky, Boris D.; Shwartz, Adam; Weiss, Alan. A branching random walk with a barrier. Ann. Appl. Probab. 1 (1991), no. 4, 573–581.
  • Brunet, E.; Derrida, B.; Mueller, A. H.; Munier, S. Noisy traveling waves: effect of selection on genealogies. Europhys. Lett. 76 (2006), no. 1, 1–7.
  • E. Brunet, B. Derrida, A. H. Mueller, and S. Munier. Phenomenological theory giving the full statistics of the position of fluctuating pulled fronts. Phys. Rev. E, 73(5):056126, May 2006.
  • Brunet, É.; Derrida, B.; Mueller, A. H.; Munier, S. Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization. Phys. Rev. E (3) 76 (2007), no. 4, 041104, 20 pp.
  • Brunet, Eric; Derrida, Bernard. Shift in the velocity of a front due to a cutoff. Phys. Rev. E (3) 56 (1997), no. 3, part A, 2597–2604.
  • Eric Brunet and Bernard Derrida. Microscopic models of traveling wave equations. Computer Physics Communications, 121-122:376–381, 1999.
  • Brunet, Éric; Derrida, Bernard. Effect of microscopic noise on front propagation. J. Statist. Phys. 103 (2001), no. 1-2, 269–282.
  • Conlon, Joseph G.; Doering, Charles R. On travelling waves for the stochastic Fisher-Kolmogorov-Petrovsky-Piscunov equation. J. Stat. Phys. 120 (2005), no. 3-4, 421–477.
  • Derrida, B.; Simon, D. The survival probability of a branching random walk in presence of an absorbing wall. Europhys. Lett. EPL 78 (2007), no. 6, Art. 60006, 6 pp.
  • Dumortier, Freddy; Popovič, Nikola; Kaper, Tasso J. The critical wave speed for the Fisher-Kolmogorov-Petrowskii-Piscounov equation with cut-off. Nonlinearity 20 (2007), no. 4, 855–877.
  • R. Durrett and D. Remenik. Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations. arXiv:0907.5180, to appear in Annals of Probability, 2009.
  • R.A. Fisher. The wave of advance of advantageous genes. Ann. Eugenics, 7:355–369, 1937.
  • N.Gantert, Yueyun Hu, and Zhan Shi. Asymptotics for the survival probability in a supercritical branching random walk. Ann. Inst. Henri Poincare Probab. Stat., 47(1):111–129, 2011.
  • Hammersley, J. M. Postulates for subadditive processes. Ann. Probability 2 (1974), 652–680.
  • B.Jaffuel. The critical barrier for the survival of the branching random walk with absorption. arXiv:0911.2227, 2009.
  • Kesten, Harry. Branching Brownian motion with absorption. Stochastic Processes Appl. 7 (1978), no. 1, 9–47.
  • Kingman, J. F. C. The first birth problem for an age-dependent branching process. Ann. Probability 3 (1975), no. 5, 790–801.
  • A.Kolmogorov, I.Petrovsky, and N.Piscounov. Etude de l'equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique. Bull. Univ. Etat Moscou Ser. Int. Sect. A Math. Mecan., 1(6):1–25, 1937.
  • C.Mueller, L.Mytnik, and J.Quastel. Effect of noise on front propagation in reaction-diffusion equations of KPP type. arXiv:0902.3423, To appear in Inventiones Math.
  • Mueller, C.; Mytnik, L.; Quastel, J. Small noise asymptotics of traveling waves. Markov Process. Related Fields 14 (2008), no. 3, 333–342.
  • Pemantle, Robin. Search cost for a nearly optimal path in a binary tree. Ann. Appl. Probab. 19 (2009), no. 4, 1273–1291.
  • Simon, Damien; Derrida, Bernard. Quasi-stationary regime of a branching random walk in presence of an absorbing wall. J. Stat. Phys. 131 (2008), no. 2, 203–233.