RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT

Pierre Andreoletti (1) , Alexis Devulder (2) , Grégoire Vechambre (1)

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

We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W_\kappa$, with $ 0<\kappa<1$, and focus on the behavior of the local times $(\mathcal{L}(t,x),x)$ of $X$ before time $t>0$. In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable Lévy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.

Mots clés: renewal process; local time; diffusion; random potential

Domaines: Probabilités [math.PR]

Pierre Andreoletti, Alexis Devulder, Grégoire Vechambre. RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2016. ⟨hal-01152982v5⟩