Geodesic
$N$-Branching random walk with $\alpha$-stable spine
Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 2, pp. 365-392 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a branching-selection particle system on the real line, introduced by Brunet and Derrida in [Phys. Rev. E, 56 (1997), pp. 2597–2604]. In this model the size of the population is fixed to a constant $N$. At each step individuals in the population reproduce independently, making children around their current position. Only the $N$ rightmost children survive to reproduce at the next step. Bérard and Gouéré studied the speed at which the cloud of individuals drifts in [Comm. Math. Phys., 298 (2010), pp. 323–342], assuming the tails of the displacement decays at exponential rate; Bérard and Maillard [Electron. J. Probab., 19 (2014), 22] took interest in the case of heavy tail displacements. We take interest in an intermediate model, considering branching random walks in which the critical “spine” behaves as an $\alpha$-stable random walk.
Keywords: branching random walk, selection
Mots-clés : stable distribution. 
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B. Mallein. $N$-Branching random walk with $\alpha$-stable spine. Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 2, pp. 365-392. http://geodesic.mathdoc.fr/item/TVP_2017_62_2_a6/

[1] J. Bérard, J.-B. Gouéré, “Brunet–Derrida behavior of branching-selection particle systems on the line”, Comm. Math. Phys., 298:2 (2010), 323–342 | DOI | MR | Zbl

[2] J. Bérard, P. Maillard, “The limiting process of {$N$}-particle branching random walk with polynomial tails”, Electron. J. Probab., 19 (2014), 22, 17 pp. | DOI | MR | Zbl

[3] J. D. Biggins, A. E. Kyprianou, “Fixed points of the smoothing transform: the boundary case”, Electron. J. Probab., 10 (2005), paper No 17, 609–631 | DOI | MR | Zbl

[4] É. Brunet, B. Derrida, “Shift in the velocity of a front due to a cutoff”, Phys. Rev. E (3), 56:3, part A (1997), 2597–2604 | DOI | MR

[5] É. Brunet, B. Derrida, A. H. Mueller, S. Munier, “Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization”, Phys. Rev. E (3), 76:4 (2007), 041104, 20 pp. | DOI | MR

[6] W. Feller, An introduction to probability theory and its applications, v. II, 2nd ed., John Wiley Sons, Inc., New York–London–Sydney, 1971, xxiv+669 pp. | MR | MR | Zbl | Zbl

[7] K. Fleischmann, V. Wachtel, “Lower deviation probabilities for supercritical Galton–Watson processes”, Ann. Inst. H. Poincaré Probab. Statist., 43:2 (2007), 233–255 | DOI | MR | Zbl

[8] N. Gantert, Y. Hu, Z. Shi, “Asymptotics for the survival probability in a killed branching random walk”, Ann. Inst. H. Poincaré Probab. Statist., 47:1 (2011), 111–129 | DOI | MR | Zbl

[9] B. Jaffuel, “The critical barrier for the survival of branching random walk with absorption”, Ann. Inst. H. Poincaré Probab. Statist., 48:4 (2012), 989–1009 | DOI | MR | Zbl

[10] J.-P. Kahane, J. Peyrière, “Sur certaines martingales de Benoit Mandelbrot”, Adv. in Math., 22:2 (1976), 131–145 | DOI | MR | Zbl

[11] R. Lyons, “A simple path to Biggins' martingale convergence for branching random walk”, Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., 84, Springer, New York, 1997, 217–221 | DOI | MR | Zbl

[12] R. Lyons, R. Pemantle, Y. Peres, “Conceptual proofs of {$L\log L$} criteria for mean behavior of branching processes”, Ann. Probab., 23:3 (1995), 1125–1138 | DOI | MR | Zbl

[13] B. Mallein, Branching random walk with selection at critical rate, 2015, 29 pp., arXiv: 1502.07390

[14] A. A. Mogul'skii, “Small deviations in a space of trajectories”, Theory Probab. Appl., 19:4 (1975), 726–736 | DOI | MR | Zbl

[15] J. Peyrière, “Turbulence et dimension de Hausdorff”, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 567–569 | MR | Zbl

[16] Yu. V. Prokhorov, “Convergence of random processes and limit theorems in probability theory”, Theory Probab. Appl., 1:2 (1956), 157–214 | DOI | MR | Zbl