[Paper Review] Hitting Probabilities of Finite Points for One-Dimensional Lévy Processes

Kohki Iba|arXiv (Cornell University)|Feb 10, 2026

Probability and Risk Models0 citations

TL;DR

The paper derives explicit formulas for the probability that a one-dimensional Lévy process first hits a specific point among a finite set, and expresses the Q-matrix of the trace process in terms of the renormalized zero resolvent.

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.

Motivation & Objective

• Motivate the study of multi-point hitting probabilities for one-dimensional Lévy processes where jumps make such probabilities meaningful beyond Brownian motion.
• Derive an explicit formula for the probability that the process starting from x hits a particular a_i first among a finite set A_n.
• Show that these probabilities can be represented solely by the renormalized zero resolvent h.
• Extend the two-point result to the multi-point case and relate to the Q-matrix of the trace process on A_n.

Proposed method

• Utilize renormalized zero resolvent h, defined via limiting resolvent differences (recurrence) or via P(T_0=∞) in the transient case.
• Apply excursion theory and local time to connect hitting events with excursion measures n^{a} and the Green matrix.
• Derive explicit probability formulas for T_{a_i}=T_{A_n} in terms of h (and κ in the transient/Poissonized setting).
• Express the multi-point hitting probabilities as a linear system involving pairwise hitting probabilities, solved by inverting a structured matrix (Theorem 1.2).
• Obtain Q-matrix entries q_{i,j} in terms of -q_{i,i} and P_{a_i}(T_{a_j}<T_{A_nackslashirst a_i}), using excursion measures (Theorem 1.4).
• Provide concrete examples (Brownian, stable, spectrally negative) to illustrate the formulas.

Experimental results

Research questions

• RQ1 What is the explicit form of the probability that a one-dimensional Lévy process first hits a specific point among a finite set A_n?
• RQ2 How can these multi-point hitting probabilities be represented using the renormalized zero resolvent h?
• RQ3 How can the Q-matrix of the trace process on A_n be expressed in terms of h and excursion measures?
• RQ4 Do these results extend naturally from the two-point case to multiple points, and how is the associated linear system solved?
• RQ5 What are concrete expressions for Q in key process families (Brownian, stable, spectrally negative)?

Key findings

• An explicit formula for P_x(T_{a_i}=T_{A_n}) is obtained in terms of the renormalized zero resolvent h (and κ in the transient case).
• Multi-point hitting probabilities are given by inverting an (n−1)×(n−1) matrix built from pairwise hitting probabilities (Theorem 1.2).
• The Q-matrix of the trace process on A_n has entries q_{i,j} = −q_{i,i} P_{a_i}(T_{a_j}=T_{A_n\{a_i}}), giving a concrete link between excursions and the embedded chain (Theorem 1.4).
• In recurrent settings, h is derived from the zero-resolvent limit r_q(0)−r_q(−x); in transient settings, h(x) = lim_{q→0+}(r_q(0)−r_q(−x)) = (1/κ) P_x(T_0=∞).
• The paper provides explicit expressions for Q in Brownian motion, strictly α-stable, and spectrally negative Lévy processes (Examples 5.1–5.3).
• Connections are drawn to Getoor’s results on trace processes and resovlents, clarifying that all quantities can be represented through h.