Densities of L\'evy walks and the corresponding fractional equations

TL;DR

This paper derives explicit formulas for Levy walk densities, covering jump-first and wait-first scenarios, solving fractional differential equations, and providing efficient numerical evaluation methods validated by simulations.

Contribution

It provides explicit formulas for Levy walk densities involving fractional derivatives and Meijer G functions, enhancing analytical and numerical understanding.

Findings

Derived explicit Levy walk densities for different scenarios

Connected densities to fractional differential equations with fractional derivatives

Validated formulas with Monte Carlo simulations

Abstract

In this paper we derive explicit formulas for the densities of Levy walks. Our results cover both jump-first and wait-first scenarios. The obtained densities solve certain fractional differential equations involving fractional material derivative operators. In the particular case, when the stability index is rational, the densities can be represented as an integral of Meijer G function. This allows to efficiently evaluate them numerically. Our results show perfect agreement with the Monte Carlo simulations.