Hitting Times of Points and Intervals for Symmetric Lévy Processes

Abstract

For one-dimensional symmetric Lévy processes, which hit every point with positive probability, we give sharp bounds for the tail function Px(TB>t), where TB is the first hitting time of B which is either a single point or an interval. The estimates are obtained under some weak type scaling assumptions on the characteristic exponent of the process. We apply these results to prove sharp two-sided estimates of the transition density of the process killed after hitting B.