Asymptotic behaviour of first passage time distributions for Lévy processes

Let X be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function c. As an analogue of the random walk results in Vatutin and Wachtel (Probab Theory Relat Fields 143(1-2):177-217, 2009) and Doney (Probab Theory Relat Fields 152(3-4):559-588, 2012), we study the local behaviour of the distribution of the lifetime ζ under the characteristic measure n of excursions away from 0 of the process X reflected in its past infimum, and of the first passage time of X below 0, T0 = inf {t>0:Xt <0}, under ℙx(·), for x > 0, in two different regimes for x, viz. x=o(c(·)) and x > D c(·), for some D > 0. We sharpen our estimates by distinguishing between two types of path behaviour, viz. continuous passage at T0 and discontinuous passage. In order to prove our main results we establish some sharp local estimates for the entrance law of the excursion process associated to X reflected in its past infimum. © 2012 Springer-Verlag.