Continuous time multidimensional Markovian description of Levy walks

TL;DR

This paper introduces a multidimensional Markovian model for Levy walks using continuous trajectories, linking system parameters to Levy exponents and explaining long displacements via velocity fluctuations.

Contribution

It generalizes previous models to describe Levy processes with a Markovian framework, enabling analysis of inhomogeneous media and boundaries.

Findings

Relationship between system parameters and Levy exponent established

Large velocity fluctuations cause long displacements

Model overcomes non-Markovian issues in Levy walks

Abstract

The paper presents a multidimensional model for nonlinear Markovian random walks that generalizes one we developed previously (Phys. Rev. E v.79, 011110, 2009) in order to describe the Levy type stochastic processes in terms of continuous trajectories of walker motion. This approach may open a way to treat Levy flights or Levy random walks in inhomogeneous media or systems with boundaries in the future. The proposed model assumes the velocity of a wandering particle to be affected by a linear friction and a nonlinear Langevin force whose intensity is proportional to the magnitude of the velocity for its large values. Based on the singular perturbation technique the corresponding Fokker-Planck equation is analyzed and the relationship between the system parameters and the Levy exponent is found. Following actually the previous paper we demonstrate also that anomalously long displacements of the wandering particle are caused by extremely large fluctuations in the particle velocity whose duration is determined by the system parameters rather than the duration of the observation interval. In this way we overcome the problem of ascribing to Levy random walks non-Markov properties.