Method of calculating densities for isotropic L\'evy Walks

TL;DR

This paper derives explicit formulas for the asymptotic densities of multidimensional isotropic Lévy walks, revealing elementary functions for odd dimensions and fractional derivatives of hypergeometric functions for even dimensions, aiding numerical computation.

Contribution

It introduces explicit formulas for densities of multidimensional isotropic Lévy walks, including undershooting and overshooting cases, with novel expressions depending on the parity of the dimension.

Findings

Densities are elementary functions for odd dimensions.

Densities are fractional derivatives of hypergeometric functions for even dimensions.

Formulas enable efficient numerical evaluation.

Abstract

We provide explicit formulas for asymptotic densities of $d$-dimensional isotropic L\'evy walks, when $d>1$. The densities of multidimensional undershooting and overshooting L\'evy walks are presented as well. Interestingly, when the number of dimensions is odd the densities of all these L\'evy walks are given by elementary functions. When $d$ is even, we can express the densities as fractional derivatives of hypergeometric functions, which makes an efficient numerical evaluation possible.