RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT

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

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: favorite sites; local time supremum; diffusion in a random potential

Domaines: Probabilités [math.PR]

Pierre Andreoletti, Grégoire Vechambre, Alexis Devulder. RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT. 2015. ⟨hal-01152982v1⟩