Hitting times of points and intervals for symmetric L\'{e}vy processes

TL;DR

This paper derives sharp bounds for the tail distribution of hitting times for points and intervals in symmetric Lévy processes, under weak scaling conditions, and applies these to estimate transition densities after hitting a set.

Contribution

It provides new sharp bounds for hitting time tail functions and transition densities for symmetric Lévy processes under weak scaling assumptions.

Findings

Sharp bounds for hitting time tail functions are established.

Optimal estimates for transition densities after hitting a set are derived.

Results apply to processes with weak type scaling on the characteristic exponent.

Abstract

For one-dimensional symmetric L\'{e}vy processes, which hit every point with positive probability, we give sharp bounds for the tail function of 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 optimal estimates of the transition density of the process killed after hitting B.