A Quenched Functional Central Limit Theorem for Random Walks in Random Environments under $(T)_\gamma$

Elodie Bouchet (1) , Christophe Sabot (1) , Renato Soares dos Santos (2)

1 PSPM - Probabilités, statistique, physique mathématique
2 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]

We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently by Rassoul-Agha and Seppäläinen in [10] and Berger and Zeitouni in [2] under the assumption of large finite moments for the regeneration time. In this paper, with the extra $(T)_{\gamma}$ condition of Sznitman we reduce the moment condition to ${\Bbb E}(\tau^2(\ln \tau)^{1+m})<+\infty$ for $m>1+1/\gamma$, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.

Mots clés: ballisticity condition.; ballisticity condition; Random walk in random environment; quenched central limit theorem

Domaines: Probabilités [math.PR]

Elodie Bouchet, Christophe Sabot, Renato Soares dos Santos. A Quenched Functional Central Limit Theorem for Random Walks in Random Environments under $(T)_\gamma$. Stochastic Processes and their Applications, 2016, 126 (4), pp.1206-1225. ⟨10.1016/j.spa.2015.10.015⟩. ⟨hal-01065997⟩