Local time of a diffusion in a stable Levy environment

Consider a 1-D diffusion in a stable Levy environment. In this article, we prove that the normalized local time process recentred at the bottom of the standard valley with height logt, [image omitted], converges in law to a functional of two independent Levy processes, which are conditioned to stay positive. In the proof of the main result, we derive that the law of the standard valley is close to a two-sided Levy process conditioned to stay positive. Moreover, we compute the limit law of the supremum of the normalized local time. In the case of a Brownian environment, similar result to the ones proved here have been obtained by Andreoletti and Diel.