Hitting probabilities for Lévy processes on the real line

Abstract

We prove sharp two-sided estimates on the tail probability of the first hitting time of bounded interval as well as its asymptotic behaviour for general non-symmetric processes which satisfy an integral conditionTo this end, we first prove and then apply the global scale invariant Harnack inequality.Results are obtained under certain conditions on the characteristic exponent. We provide a wide class of Lévy processs which satisfy these assumptions.