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We study the almost sure asymptotic behavior of the supremum of the localtime for a transient diffusion in a spectrally negative L\'evy environment.More precisely, we provide the proper renormalizations for the extremely largeand the extremely small values of the supremum of the local time. Inparticular, in the sub-ballistic regime, we link the almost sure behavior ofthe supremum of the local time with the left asymptotic tail of the exponentialfunctional of the environment conditioned to stay positive, which allows to usethe properties of such exponential functionals to characterize the soughtbehavior. It appears from our results that the renormalization of the extremelylarge values of the supremum of the local time in the sub-ballistic regime isdetermined by the asymptotic behavior of the Laplace exponent of the L\'evyenvironment, and is surprisingly greater than the renormalization that waspreviously known for the recurrent case. For the renormalization of theextremely small values of the supremum of the local time in the sub-ballisticregime, our results show that there is only one possible renormalization,whatever is the choice of the L\'evy environment.

Almost sure behavior for the local time of a diffusion in a spectrally negative L\'evy environment