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

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 Lévy 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.

Major revision

References in corpus (3)

• Maximum entropy principle and power-law tailed distributions
• Lévy flights in inhomogeneous environments
• A nonlinear drift which leads to $\kappa$-generalized distributions

Cited by in corpus (4)

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• Lévy flights confinement in a parabolic potential and fractional quantum oscillator
• Levy targeting and the principle of detailed balance
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