First passage leapovers of L\'evy flights and the proper formulation of absorbing boundary conditions

TL;DR

This paper introduces a new approach using the first-hit distribution to accurately formulate absorbing boundary conditions for Lévy flights, enabling analytical solutions for first passage times and positions.

Contribution

It demonstrates that employing the first-hit distribution as an absorbing sink maintains the tractability of Lévy flight dynamics and extends results to various potentials.

Findings

First-hit distribution effectively models absorbing boundaries in Lévy flights.

Analytical expressions for first passage time and position are derived.

Results apply to Lévy flights with arbitrary skewness and certain potentials.

Abstract

An important open problem in the theory of L\'evy flights concerns the analytically tractable formulation of absorbing boundary conditions. Although numerical studies using the correctly defined nonlocal approach have yielded substantial insights regarding the statistics of first passage, the resultant modifications to the dynamical equations hinder the detailed analysis possible in the absence of these conditions. In this study it is demonstrated that using the first-hit distribution, related to the first passage leapover, as the absorbing sink preserves the tractability of the dynamical equations for a particle undergoing L\'evy flight. In particular, knowledge of the first-hit distribution is sufficient to fully determine the first passage time and position density of the particle, without requiring integral truncation or numerical simulations. In addition, we report on the first-hit and leapover properties of first passages and arrivals for L\'evy flights of arbitrary skew parameter, and extend these results to L\'evy flights in a certain ubiquitous class of potentials satisfying an integral condition.