Almost Sure Behavior for the Local Time of a Diffusion in a Spectrally Negative Levy Environment

We study the almost sure asymptotic behavior of the supremum of the local time for a transient sub-ballistic diffusion in a spectrally negative Levy environment. More precisely, we provide the proper renormalizations for the extremely large values of the supremum of the local time. This is done by establishing a-so far unknown-connection between the latter and the exponential functional of a Levy process conditioned to stay positive, which allows us to use the properties of such exponential functionals to characterize the sought behavior. It appears from our results that the renormalization of the extremely large values of the supremum of the local time is determined by the asymptotic behavior of the Laplace exponent of the Levy environment and is surprisingly greater than the renormalization that was previously known for the recurrent case. Our results show moreover a rich variety of behaviors, which is a new phenomenon, as it does not occur in the discrete setting.