Persistence in Lévy-flight anomalous diffusion

The evolution of the number of persistent sites in a field governed by Lévy-flight anomalous diffusion is characterized. It is shown that, as in the case of ordinary diffusion, the number of persistent sites exhibits a long-time power-law decay. For the case of white-noise initial conditions, the exponent in this power-law decay can be numerically found from an algebraic equation as a function of the Lévy exponent γ. As expected, the decay is faster as the transport mechanism becomes more efficient, i.e., as γ decreases. Numerical simulations that validate the analytical results are also presented.