We consider the combined effects of a power law Lévy 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 Lévy 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 ${\mathit{d}}_{\mathit{c}}$=2f-2 the disorder is relevant, corresponding to a nontrivial fixed point for the force correlation function.

• Received 9 February 1994

DOI:https://doi.org/10.1103/PhysRevE.50.1657