Realization of Levy flights as continuous processes

TL;DR

This paper presents a method to model Levy flights as continuous stochastic processes using generalized Langevin and Fokker-Planck equations, facilitating analysis in complex media.

Contribution

It introduces a novel realization of Levy flights as continuous processes via explicit derivation of generalized equations, extending their applicability.

Findings

Derived generalized Langevin equation for Levy flights

Formulated generalized Fokker-Planck equation for Levy flights

Applicable to inhomogeneous media and systems with boundaries

Abstract

On the basis of multivariate Langevin processes we present a realization of Levy flights as a continuous process. For the simple case of a particle moving under the influence of friction and a velocity dependent stochastic force we explicitly derive the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Levy flights. Our procedure is similar to the treatment of the Kramers-Fokker Planck equation in the Smoluchowski limit. The proposed approach forms a feasible way of tackling Levy flights in inhomogeneous media or systems with boundaries what is up to now a challenging problem.