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Hitting Probabilities of Finite Points for One-Dimensional Lévy Processes

Kohki Iba

TL;DR

This work derives explicit formulas for the probability that a one-dimensional Lévy process starting at $x$ first hits a specified point among a finite set $A_n$, expressing these hitting probabilities solely through the renormalized zero resolvent $h$. It extends classic two-point hitting results to multi-point configurations and connects them to the $Q$-matrix of the trace process on $A_n$, under recurrent and transient regimes. The approach combines excursion theory, local time, and resolvent analysis, yielding concrete representations for Brownian motion, stable processes, and spectrally negative Lévy processes. The results provide practical tools for computing hitting probabilities and trace-process dynamics in one dimension, with explicit illustrations in Section 5.

Abstract

For a one-dimensional Lévy 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.

Hitting Probabilities of Finite Points for One-Dimensional Lévy Processes

TL;DR

This work derives explicit formulas for the probability that a one-dimensional Lévy process starting at first hits a specified point among a finite set , expressing these hitting probabilities solely through the renormalized zero resolvent . It extends classic two-point hitting results to multi-point configurations and connects them to the -matrix of the trace process on , under recurrent and transient regimes. The approach combines excursion theory, local time, and resolvent analysis, yielding concrete representations for Brownian motion, stable processes, and spectrally negative Lévy processes. The results provide practical tools for computing hitting probabilities and trace-process dynamics in one dimension, with explicit illustrations in Section 5.

Abstract

For a one-dimensional Lévy 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 -matrix of the trace process on the finite set. These formulas involve solely the renormalized zero resolvent.
Paper Structure (11 sections, 11 theorems, 54 equations)

This paper contains 11 sections, 11 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Preliminaries
  4. Lévy Process and Resolvent Density
  5. Local Time and Excursion
  6. Renormalized Zero Resolvent
  7. Recurrent Case.
  8. Transient Case
  9. Hitting Probabilities of Finite Points
  10. The $Q$-matrix of the trace process
  11. Examples

Key Result

Theorem 1.1

We assume that either of the following two conditions is satisfied: Then, the following hold:

Theorems & Definitions (26)

  • Theorem 1.1: Lemma 3.5 and Lemma 9.3 of Takeda-Yano TY
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Example 2.1: Brownian motion
  • Example 2.2: Strictly $\alpha$-stable process
  • Example 2.3: Spectrally negative Lévy process
  • Theorem 2.4: Theorem 15.1 of Tsukada Tukada
  • Theorem 2.5: Lemma 3.5 and Theorem 3.8 of Takeda-Yano TY
  • ...and 16 more