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

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.