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 before time $t$ in an a 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, and provide a central limit theorem for the number of valleys visited up to time $t$. The proof relies on a Williams' decomposition of the trajectory of $\wk$ in the neighborhood of local minima, with the help of results of Faggionato \cite{Faggionato}, and on a precise analysis of exponential functionals of $W_{\kappa}$ and of 3-dimensional $(-\kappa/2)$-drifted Bessel processes.

Keywords: Diffusion in a random potential; Localization; Aging; Renewal theorem

Domains: Probability [math.PR]

Pierre Andreoletti, Alexis Devulder. Localization and number of visited valleys for a transient diffusion in random environment. 2013. ⟨hal-00908626v1⟩