Almost sure convergence for stochastically biased random walks on trees

Gabriel Faraud (1) , Yueyun Hu (1) , Zhan Shi (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We are interested in the biased random walk on a supercritical Galton- Watson tree in the sense of Lyons (Ann. Probab. 18:931-958, 1990) and Lyons, Pemantle and Peres (Probab. Theory Relat. Fields 106:249-264, 1996), and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system's non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)3, converges almost surely to a known positive constant.

Mots clés: Biased random walk on a Galton-Watson tree; Branching random walk; Slow movement; Random walk in a random environment

Domaines: Probabilités [math.PR]

Gabriel Faraud, Yueyun Hu, Zhan Shi. Almost sure convergence for stochastically biased random walks on trees. Probability Theory and Related Fields, 2012, 154, pp.621-660. ⟨10.1007/s00440-011-0379-y⟩. ⟨hal-00608163⟩