Explicit Densities of Multidimensional L\'evy Walks

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.