2015/10/31 by Marcin Magdziarz, Tomasz Zorawik · 1 citation
Biochemistry, Genetics and Molecular Biology · Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Applied mathematics #Complex Systems and Time Series Analysis #Derivative (finance) #Diffusion and Search Dynamics #Hypergeometric distribution #Hypergeometric function #Mathematical analysis #Mathematics #Monte Carlo method #Physics #Pure mathematics #Random walk #Statistical physics #Stochastic processes and statistical mechanics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.94.022130
arxiv created 2015/12/11 · openalex publication_date 2016/08/22 · arxiv updated 2016/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Lévy walks have proved to be useful models of stochastic dynamics with a number of applications in the modeling of real-life phenomena. In this paper we derive explicit formulas for densities of the two- (2D) and three-dimensional (3D) ballistic Lévy walks, which are most important in applications. 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 us to efficiently evaluate them numerically. The theoretical results agree perfectly with Monte Carlo simulations.