Maximal displacement of a branching random walk in time-inhomogeneous environment

Bastien Mallein (1, 2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 LAGA - Laboratoire Analyse, Géométrie et Applications

Consider a branching random walk evolving in a macroscopic time-inhomogeneous environment, that scales with the length $n$ of the process under study. We compute the first two terms of the asymptotic of the maximal displacement at time $n$. The coefficient of the first (ballistic) order is obtained as the solution of an optimization problem, while the second term, of order $n^{1/3}$, comes from time-inhomogeneous random walk estimates, that may be of independent interest. This result partially answers a conjecture of Fang and Zeitouni. Same techniques are used to obtain the asymptotic of other quantities, such as the consistent maximal displacement.

Mots clés: Branching process; Time-inhomogeneous environment; Branching random walk; maximal displacement

Domaines: Probabilités [math.PR]

Bastien Mallein. Maximal displacement of a branching random walk in time-inhomogeneous environment. Stochastic Processes and their Applications, 2015, 125 (10), pp.3958-4019. ⟨10.1016/j.spa.2015.05.011⟩. ⟨hal-00874541v3⟩