Heavy-tailed targets and (ab)normal asymptotics in diffusive motion

TL;DR

This paper explores how confining conditions influence the emergence of heavy-tailed probability density functions in jump-type and diffusion processes, revealing complex transient behaviors and the absence of universal diffusion exponents.

Contribution

It demonstrates that confining regimes can produce non-Gaussian heavy-tailed asymptotics in the Fokker-Planck framework and analyzes transient crossover phenomena in diffusive processes.

Findings

Heavy-tailed pdfs can arise in confined systems.

Transient regimes show crossover features in moments.

No universal diffusion exponent hierarchy is observed.

Abstract

We investigate temporal behavior of probability density functions (pdfs) of paradigmatic jump-type and continuous processes that, under confining regimes, share common heavy-tailed asymptotic (target) pdfs. Namely, we have shown that under suitable confinement conditions, the ordinary Fokker-Planck equation may generate non-Gaussian heavy-tailed pdfs (like e.g. Cauchy or more general L\'evy stable distribution) in its long time asymptotics. For diffusion-type processes, our main focus is on their transient regimes and specifically the crossover features, when initially infinite number of the pdf moments drops down to a few or none at all. The time-dependence of the variance (if in existence), $\sim t^{\gamma}$ with $0<\gamma <2$, in principle may be interpreted as a signature of sub-, normal or super-diffusive behavior under confining conditions; the exponent $\gamma $ is generically well defined in substantial periods of time. However, there is no indication of any universal time rate hierarchy, due to a proper choice of the driver and/or external potential.