Explicit Densities of Multidimensional Lévy Walks

Abstract

We provide explicit formulas for asymptotic densities of the 2- and 3-dimensional ballistic Lévy walks. It turns out that in the 3D case the densities are given by elementary functions. The densities of the 2D Lévy 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.

References in corpus (5)

• Understanding individual human mobility patterns
• The scaling laws of human travel
• Lévy walks
• Non-ergodic Intensity Correlation Functions for Blinking Nano Crystals
• Asymptotic densities of ballistic Lévy walks

Cited by in corpus (6)

• Superdiffusive dispersals impart the geometry of underlying random walks
• Extended Poisson-Kac theory: A unifying framework for stochastic processes with finite propagation velocity
• Finite-energy Lévy-type motion through heterogeneous ensemble of Brownian particles
• Method of calculating densities for isotropic Lévy Walks
• Large deviations of the ballistic Lévy walk model
• Asymptotic densities of planar Lévy walks: a non-isotropic case