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We study the convergence in distribution of the supremum of the local timeand of the favorite site for a transient diffusion in a spectrally negativeL\'evy potential. To do so, we study the h-valleys of a spectrally negativeL\'evy process, and we prove in partiular that the renormalized sequence of theh-minima converges to the jumping times sequence of a standard Poisson process.

Path decomposition of a spectrally negative L\'evy process, and local time of a diffusion in this environment