Localization and number of visited valleys for a transient diffusion in random environment

Pierre Andreoletti (1) , Alexis Devulder (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 LMV - Laboratoire de Mathématiques de Versailles

We consider a transient diffusion in a $(-\kappa/2)$-drifted Brownian potential $W_{\kappa}$ with $0<\kappa<1$. We prove its localization at time $t$ in the neighborhood of some random points depending only on the environment, which are the positive $h_t$-minima of the environment, for $h_t$ a bit smaller than $\log t$. We also prove an Aging phenomenon for the diffusion, a renewal theorem for the hitting time of the farthest visited valley, and provide a central limit theorem for the number of valleys visited up to time $t$.

Mots clés: Diffusion in a random potential; Localization; Aging; Renewal theorem

Domaines: Probabilités [math.PR]

Pierre Andreoletti, Alexis Devulder. Localization and number of visited valleys for a transient diffusion in random environment. 2014. ⟨hal-00908626v2⟩