Hitting Probabilities of Finite Points for One-Dimensional L\'{e}vy Processes

TL;DR

This paper derives explicit formulas for hitting probabilities of finite points by one-dimensional Lévy processes and uses them to compute the Q-matrix of the trace process, involving the renormalized zero resolvent.

Contribution

It provides explicit formulas for hitting probabilities and the Q-matrix of the trace process for one-dimensional Lévy processes, advancing understanding of their boundary behaviors.

Findings

Explicit hitting probability formulas involving the renormalized zero resolvent.

Derived explicit Q-matrix entries for the trace process on finite sets.

Applicable to analyzing boundary behaviors of Lévy processes.

Abstract

For a one-dimensional L\'{e}vy process, we derive an explicit formula for the probability of first hitting a specified point among a fixed finite set. Moreover, using this formula, we obtain an explicit expression for each entry of the $Q$-matrix of the trace process on the finite set. These formulas involve solely the renormalized zero resolvent.