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Infinite divisibility of hitting times for symmetric stable processes

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Contributed by Min WANG ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

Let TaT_a be the first hitting time of a nonzero point aa by a symmetric strictly α\alpha-stable L\'evy process, where 1<α<21<\alpha<2. We give a computer-assisted classification of its ordinary additive infinite divisibility. There is a unique threshold α∗\alpha_*, with 1.6670659903<α∗<1.66706599151.6670659903<\alpha_*<1.6670659915, such that TaT_a is infinitely divisible if and only if 1<α≤α∗1<\alpha\le\alpha_*. The threshold is characterized by a unique zero contact of the canonical inverse logarithmic-derivative kernel. The proof combines explicit analytic estimates, rigorous interval covers, Gaussian necessary conditions and Rouch\'e's theorem. In particular, a fully analytic argument proves failure of infinite divisibility for every 15/8≤α<215/8\le\alpha<2. Scalar bounds at the two time ends reduce the compact interval computations. The finite computations and their acceptance conditions are supplied as ancillary material.

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Version history

  1. v1Submitted by Min WANGInitial depositCurrentOct 29, 2026