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Self-similar anomalous diffusion and Levy-stable laws


Ul'yanovsk State University, ul. L. Tolstogo 42, Ulyanovsk, 432700, Russian Federation

Stochastic principles for constructing the process of anomalous diffusion are considered, and corresponding models of random processes are reviewed. The self-similarity and the independent-increments principles are used to extend the notion of diffusion process to the class of Levy-stable processes. Replacing the independent-increments principle with the renewal principle allows us to take the next step in generalizing the notion of diffusion, which results in fractional-order partial space-time differential equations of diffusion. Fundamental solutions to these equations are represented in terms of stable laws, and their relationship to the fractality and memory of the medium is discussed. A new class of distributions, called fractional stable distributions, is introduced.

Fulltext pdf (634 KB)
Fulltext is also available at DOI: 10.1070/PU2003v046n08ABEH001324
PACS: 02.50.−r, 05.40.Fb, 05.40.Jc (all)
DOI: 10.1070/PU2003v046n08ABEH001324
URL: https://ufn.ru/en/articles/2003/8/c/
000187205200003
Citation: Uchaikin V V "Self-similar anomalous diffusion and Levy-stable laws" Phys. Usp. 46 821–849 (2003)
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Îðèãèíàë: Ó÷àéêèí Â Â «Àâòîìîäåëüíàÿ àíîìàëüíàÿ äèôôóçèÿ è óñòîé÷èâûå çàêîíû» ÓÔÍ 173 847–876 (2003); DOI: 10.3367/UFNr.0173.200308c.0847

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