Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**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*