Path decomposition of a spectrally negative Lévy process, and local time of a diffusion in this environment

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