Pre-asymptotic analysis of Lévy flights

We study the properties of Lévy flights with index 0<α<2 at elapsed times smaller than those required for reaching the diffusive limit, and we focus on the bulk of the walkers' distribution rather than on its tails. On the basis of the analogs of the Kramers-Moyal expansion and of the Pawula theorem, we show that, for any α≤2/3, the bulk of the walkers' distribution occurs at wave-numbers greater than (2/α)1/(2α)≥1, and it remains non-self-similar for a time-scale longer than the Markovian time-lag of at least one order of magnitude. This result highlights the fact that for Lévy flights, the Markovianity time-lag is not the only time-scale of the process and indeed another and longer time-scale controls the transition to the familiar power-law regime in the final diffusive limit. The magnitude of this further time-scale is independent of the index α and may compromise the reliability of applications of Lévy flights to real world cases related with recurrence and transience as optimal searching, animal foraging, and site fidelity.