Density Behaviour Related To Levy Processes

Let p(t)(x), f(t)(x) and (q)t*(x) be the densities at time t of a real Levy process, its running supremum and the entrance law of the reflected excursions at the infimum. We provide relationships between the asymptotic behaviour of P-t(x), f(t)(x) and 4(x), when t is small and x is large. Then for large x, these asymptotic behaviours are compared to this of the density of the Levy measure. We show in particular that, under mild conditions, if p(t)(x) is comparable to t nu(x), as t -> 0 and x -> oc, then so is f(t)(x).