Hitting Probabilities of Finite Points for One-Dimensional Lévy Processes

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.

💡 Research Summary

The paper addresses the problem of determining, for a one‑dimensional Lévy process (X_t), which point among a finite set (A_n={a_1,\dots,a_n}) is hit first. Under the standing assumptions that either (A) the characteristic exponent (\Psi) satisfies (\int_0^\infty \frac{d\lambda}{q+\Psi(\lambda)}<\infty) for every (q>0) (which implies non‑compound‑Poisson and regularity of 0) or that the process is transient and satisfies the weaker conditions (A1) and (A2), the authors introduce the renormalized zero resolvent \