Explicit Densities of Multidimensional L\'evy Walks

TL;DR

This paper derives explicit formulas for the asymptotic densities of 2D and 3D ballistic Lévy walks, revealing elementary functions in 3D and hypergeometric functions in 2D, with results validated by simulations.

Contribution

It provides the first explicit formulas for densities of multidimensional Lévy walks, including elementary solutions in 3D and fractional derivatives in 2D.

Findings

Densities in 3D are elementary functions.

2D densities involve hypergeometric functions and fractional derivatives.

Results match Monte-Carlo simulations.

Abstract

We provide explicit formulas for asymptotic densities of the 2- and 3-dimensional ballistic L\'evy walks. It turns out that in the 3D case the densities are given by elementary functions. The densities of the 2D L\'evy walks are expressed in terms of hypergeometric functions and the right-side Riemann-Liouville fractional derivative which allows to efficiently evaluate them numerically. The theoretical results agree with Monte-Carlo simulations. The obtained functions solve certain differential equations with the fractional material derivative.