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We consider a one-dimensional diffusion in a stable L\'evy environment. Weshow that the normalized local time process refocused at the bottom of thestandard valley with height $\log t$, $(L_X(t,\mathfrak m_{\log t}+x)/t,x\in\R)$, converges in law to a functional of two independent L\'evy processesconditioned to stay positive. To prove this result, we show that the law of thestandard valley is close to a two-sided L\'evy process conditioned to staypositive. We also obtain the limit law of the supremum of the normalized localtime. This result has been obtained by Andreoletti and Diel in the case of aBrownian environment.

Local time of a diffusion in a stable L\'evy environment