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

Author

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  • Garbaczewski, Piotr
  • Stephanovich, Vladimir
  • Kȩdzierski, Dariusz

Abstract

We show that, under suitable confinement conditions, the ordinary Fokker–Planck equation may generate non-Gaussian heavy-tailed probability density functions (pdfs) (like, for example, Cauchy or more general Lévy stable distributions) in its long-time asymptotics. In fact, all heavy-tailed pdfs known in the literature can be obtained this way. For the underlying diffusion-type processes, our main focus is on their transient regimes and specifically the crossover features, when an initially infinite number of pdf moments decreases to a few or none at all. The time dependence of the variance (if in existence), ∼tγ with 0<γ<2, may in principle be interpreted as a signature of subdiffusive, normal diffusive or superdiffusive behavior under confining conditions; the exponent γ 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.

Suggested Citation

  • Garbaczewski, Piotr & Stephanovich, Vladimir & Kȩdzierski, Dariusz, 2011. "Heavy-tailed targets and (ab)normal asymptotics in diffusive motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(6), pages 990-1008.
    • Handle: RePEc:eee:phsmap:v:390:y:2011:i:6:p:990-1008
    DOI: 10.1016/j.physa.2010.11.041
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