Hitting probabilities for L\'{e}vy processes on the real line

TL;DR

This paper derives precise estimates for the probability that a Lévy process hits a bounded interval on the real line, using a global scale invariant Harnack inequality under certain conditions on the process's characteristic exponent.

Contribution

It introduces sharp two-sided estimates for hitting probabilities of Lévy processes and establishes a global scale invariant Harnack inequality for a broad class of such processes.

Findings

Sharp two-sided estimates for hitting probabilities.

Asymptotic behavior of hitting times characterized.

Applicable to a wide class of Lévy processes.

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 L\'{e}vy processs which satisfy these assumptions.