Sub-ballistic random walk in Dirichlet environment

Élodie Bouchet (1)

1 ICJ - Institut Camille Jordan

We consider random walks in Dirichlet environment (RWDE) on $\Z ^d$, for $ d \geq 3 $, in the sub-ballistic case. We associate to any parameter $ (\alpha_1, \dots, \alpha _{2d}) $ of the Dirichlet law a time-change to accelerate the walk. We prove that the continuous-time accelerated walk has an absolutely continuous invariant probability measure for the environment viewed from the particle. This allows to characterize directional transience for the initial RWDE. It solves as a corollary the problem of Kalikow's $0-1$ law in the Dirichlet case in any dimension. Furthermore, we find the polynomial order of the magnitude of the original walk's displacement.

Mots clés: Random walk in random environment; Dirichlet distribution; Reinforced random walks; Invariant measure viewed from the particle

Domaines: Probabilités [math.PR]

Élodie Bouchet. Sub-ballistic random walk in Dirichlet environment. Electronic Journal of Probability, 2013, 18, pp.Article 58. ⟨10.1214/EJP.v18-2109⟩. ⟨hal-00701531⟩