Slow Lévy flights

Abstract

Among Markovian processes, the hallmark of Lévy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Lévy 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.

6 pages, 2 figures

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• Non-stationary Markovian Replication Process causing Diverse Diffusions
• Large deviations principle for a stochastic process with random reinforced relocations
• Ultra slow sub-logarithmic diffusion of a sluggish random walker subject to resetting with memory
• Random walkers on a deformable medium
• Collective learning from individual experiences and information transfer during group foraging