On multidimensional branching random walks in random environment

Francis Comets (1) , Serguei Popov (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 IME - Instituto de Matemática e Estatística

We study branching random walks in random i.i.d. environment in $\Z^d, d \geq 1$. For this model, the population size cannot decrease, and a natural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time.

Mots clés: shape theorem; recurrence; transience; subadditive ergodic theorem; nestling; hitting time

Domaines: Probabilités [math.PR]

Francis Comets, Serguei Popov. On multidimensional branching random walks in random environment. 2005. ⟨hal-00005865⟩