Leapover lengths and first passage time statistics for L\'evy flights

TL;DR

This paper derives exact results for the first passage time and leapover length distributions of symmetric and one-sided Levy flights, revealing distinct power-law behaviors and confirming findings through simulations.

Contribution

It provides the first exact analytical expressions for leapover and first passage time statistics of Levy flights, highlighting differences based on symmetry.

Findings

Leapover lengths follow power-law distributions with index alpha (one-sided) and alpha/2 (symmetric).

First passage time distribution scales as a power-law with index 1/2 for symmetric LFs.

Simulations confirm the analytical results.

Abstract

Exact results for the first passage time and leapover statistics of symmetric and one-sided Levy flights (LFs) are derived. LFs with stable index alpha are shown to have leapover lengths, that are asymptotically power-law distributed with index alpha for one-sided LFs and, surprisingly, with index alpha/2 for symmetric LFs. The first passage time distribution scales like a power-law with index 1/2 as required by the Sparre Andersen theorem for symmetric LFs, whereas one-sided LFs have a narrow distribution of first passage times. The exact analytic results are confirmed by extensive simulations.