Single integro-differential wave equation for L\'evy walk

TL;DR

This paper derives a comprehensive integro-differential wave equation for the probability density of a one-dimensional Lévy walk, capturing non-Markovian effects and generalizing the telegraph equation without large-scale approximations.

Contribution

It introduces a novel integro-differential wave equation for Lévy walks that accounts for memory effects and broad running time distributions, extending previous models.

Findings

Derived a general wave equation valid for any running time PDF.

Included non-Markovian cases with gamma and power-law distributions.

Generalized the telegraph equation for Lévy walks.

Abstract

The integro-differential wave equation for the probability density function for a classical one-dimensional L\'evy walk with continuous sample paths has been derived. This equation involves a classical wave operator together with memory integrals describing the spatio-temporal coupling of the L\'evy walk. It is valid for any running time PDF and it does not involve any long-time large-scale approximations. It generalizes the well-known telegraph equation obtained from the persistent random walk. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and power-law distributed random times.