Regularity of Local times associated to Volterra-Levy processes and path-wise regularization of stochastic differential equations

We investigate the space-time regularity of the local time associated to Volterra-Levy processes, including Volterra processes driven by $alpha$-stable processes for $alphain(0,2]$. We show that the spatial regularity of the local time for Volterra-Levy process is $P$-a.s. inverse proportionally to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbaPtion of ODEs by a Volterra-Levy process which has sufficiently regular local time. Following along the lines of [15], we show existence, uniqueness and differentiablility of the flow associated to such equations.