N -Branching random walk with α-stable spine

Bastien Mallein (1, 2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 DMA - Département de Mathématiques et Applications - ENS Paris

We consider a branching-selection particle system on the real line, introduced by Brunet and Derrida in [7]. 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 [2], assuming the tails of the displacement decays at exponential rate; Bérard and Maillard [3] 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 α-stable random walk.

Mots clés: selection; small deviations; stable process; branching random walk

Domaines: Probabilités [math.PR]

Bastien Mallein. N -Branching random walk with α-stable spine. Theory of Probability and Its Applications c/c of Teoriia Veroiatnostei i Ee Primenenie, 2017, 62 (2), pp.365--392. ⟨hal-01322452⟩