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 processes which satisfy these assumptions.

Yaglom limit for unimodal Lévy processes

“…If d = 1, then the assumption A1 implies global scale invariant Harnack inequality for the process due to Grzywny and Kwaśnicki [28, Theorem 1.9 and Remark 1.10 e)]. With this in hand one can repeat the proof of Corrolary 5.6 and the upper bound in Corrolary 5.5 in Grzywny, Leżaj and Miśta [29] to get the claim. Proposition 2.3.…”

First exit times from a bounded interval for Lévy processes

“…The results seem to be valuable in themselves because they break a bit of the barrier of using fluctuation theory for people from potential theory and shed some light on the general situation. We use them to derive estimates and asymptotics of the tail probability of the first hitting time of a point or an interval in the forthcoming paper [12]. The explicit formula for the mean of the first exit time from a bounded interval is very rare.…”