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Renewal structure and local time for diffusions in random environment

We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W\_κ$, with $ 0\textless{}κ\textless{}1$, and focus on the behavior of the local times $(\mathcal{L}(t,x),x)$ of $X$ before time $t\textgreater{}0$.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 L{é}vy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.