Slow L\'evy flights

TL;DR

This paper demonstrates that processes with long-range memory can exhibit superdiffusive Le9vy flight behavior or Gaussian limits, with anomalously slow, logarithmic diffusion, challenging traditional views of diffusion processes.

Contribution

It introduces a new class of long-memory processes that produce superdiffusive or Gaussian limits with logarithmic diffusion, extending the Central Limit Theorem in this context.

Findings

Le9vy laws and Gaussians can be limit distributions of long-memory processes.

Anomalous, logarithmic diffusion arises from frequent revisits to previously visited sites.

A modified Central Limit Theorem and a fluctuation-dissipation relation are derived.

Abstract

Among Markovian processes, the hallmark of L\'evy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that L\'evy laws, as well as Gaussians, 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 non-analytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.