Langevin equations for continuous time L\'{e}vy flights

TL;DR

This paper develops coupled Langevin equations to analyze the long-time behavior of Lévy flights with power-law step and waiting time distributions, revealing how external forces and disorder influence their dynamics.

Contribution

It introduces a Langevin equation framework for Lévy flights with power-law waiting times, incorporating external forces and analyzing disorder effects.

Findings

Dynamic exponent z equals f/g in certain regimes.

Disorder relevance depends on spatial dimension d.

Results are independent of weak quenched disorder presence.

Abstract

We consider the combined effects of a power law L\'{e}vy step distribution characterized by the step index $f$ and a power law waiting time distribution characterized by the time index $g$ on the long time behavior of a random walker. The main point of our analysis is a formulation in terms of coupled Langevin equations which allows in a natural way for the inclusion of external force fields. In the anomalous case for $f<2$ and $g<1$ the dynamic exponent $z$ locks onto the ratio $f/g$. Drawing on recent results on L\'{e}vy flights in the presence of a random force field we also find that this result is {\em independent} of the presence of weak quenched disorder. For $d$ below the critical dimension $d_c=2f-2$ the disorder is {\em relevant}, corresponding to a non trivial fixed point for the force correlation function.