Slow Lévy flights

Among Markovian processes, the hallmark of Lévy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Lévy laws, as well as Gaussian distributions, can also be the limit distributions of processes with long-range memory that exhibit very slow diffusion, logarithmic in time. These processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the central limit theorem is modified in this context, keeping the usual distinction between analytic and nonanalytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.

Figure 1

(a) Schematic view of a process relocating at a constant rate $q$ to sites occupied at previous times, these times being chosen stochastically. The numbers label the beginning and end of each excursion. Each end is followed by the beginning of the next excursion (arrow). (b) Two simulated trajectories corresponding to Lévy excursions with $p\left(\ell \right)\sim 1/{\left|\ell \right|}^{1+\mu }$, relocation rate $q=0.05$, and relocation kernel given by Eq. (8) (panels at the same scale).

Figure 2

Preferential visit model in one dimension. (a) Here $\langle |X_t{{|}^{\nu }\rangle }^{\mu /\nu }$, obtained from simulations with different $\mu$ and $\nu$ (averages over $5\times {10}^5$ runs), is proportional to $lnt$ as expected. (b) Mean and variance of $X_t$ for a NN walk with bias $\alpha$ in rule (i). Colored solid lines are simulations and dark dashed lines, theory. (c) Normal diffusion for spatially uniform relocations.