LANGEVIN-EQUATIONS FOR CONTINUOUS-TIME LEVY FLIGHTS

We consider the combined effects of a power law Levy 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 Levy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder. For d below the critical dimension d(c) = 2f - 2 the disorder is relevant, corresponding to a nontrivial fixed point for the force correlation function.