---
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 condition $$\int_0^{\infty} \frac{d\xi}{1+{\operatorname{Re}}\psi(\xi)}<\infty.$$ To 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.'
address:
- |
     Tomasz Grzywny\
    Wydzia[ł]{} Matematyki, Politechnika Wroc[ł]{}awska\
    Wyb. Wyspiańskiego 27\
    50-370 Wroc[ł]{}aw\
    Poland
- |
     Łukasz Leżaj\
    Wydzia[ł]{} Matematyki, Politechnika Wroc[ł]{}awska\
    Wyb. Wyspiańskiego 27\
    50-370 Wroc[ł]{}aw\
    Poland
- |
     Maciej Miśta\
    Wydzia[ł]{} Matematyki, Politechnika Wroc[ł]{}awska\
    Wyb. Wyspiańskiego 27\
    50-370 Wroc[ł]{}aw\
    Poland
author:
- Tomasz Grzywny
- Łukasz Leżaj
- Maciej Miśta
bibliography:
- 'bibfile.bib'
title: Hitting probabilities for Lévy processes on the real line
---

[^1]

Introduction
============

The aim of this paper is to discuss the distribution of the first hitting time of the point or the bounded interval for non-symmetric Lévy processes which satisfy the following condition: $$\int_0^{\infty} \frac{d\xi}{1+{\operatorname{Re}}\psi(\xi)} < \infty.$$ Such condition implies that $0$ is regular for itself. Under some regularity assumptions we prove sharp two-sided estimates on the tail probability 