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

We consider a one-dimensional diffusion in a stable L\'evy environment. We show that the normalized local time process refocused at the bottom of the standard 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 processes conditioned to stay positive. To prove this result, we show that the law of the standard valley is close to a two-sided L\'evy process conditioned to stay positive. We also obtain the limit law of the supremum of the normalized local time. This result has been obtained by Andreoletti and Diel in the case of a Brownian environment.