Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### A branching system of random walks in random environment

Auteur(s):

Code(s) de Classification MSC:

• 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
• 60K37 Processes in random environments

Résumé: We study a branching system of random walks in random environment. Particles reproduce with a fixed reproduction law, and move as one-dimensional random walks in a common random environment. Our model differs from that of branching random walk in random environment, in which particles move with a fixed transition probability and with a reproduction law depending on the locations. We assume that the branching mechanism is supercritical with mean $m>1$, and that the law of the random environment drives a random walk to $-\infty$. Our main result shows the existence of a critical value $m_c$ such that whenever $m>m_c$, there are particles going to $+\infty$ with positive speed (conditionally on the survival of the branching process), whereas for $m < m_c$, the system is bounded from the right. The exact value of $m_c$ is formulated in terms of the large deviation function for the random walk in random environment.

Mots Clés: Random walk in random environment ; branching random walk ; Galton--Watson tree

Date: 2003-06-27

Prépublication numéro: PMA-834