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2003

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August

  

Reviews of topical problems


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.

Text can be downloaded in Russian. English translation is available on IOP Science.
PACS: 02.50.−r, 05.40.Fb, 05.40.Jc (all)
DOI: 10.1070/PU2003v046n08ABEH001324
URL: https://ufn.ru/en/articles/2003/8/c/
Citation: Uchaikin V V "Self-similar anomalous diffusion and Levy-stable laws" Phys. Usp. 46 821–849 (2003)
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Оригинал: Учайкин В В «Автомодельная аномальная диффузия и устойчивые законы» УФН 173 847–876 (2003); DOI: 10.3367/UFNr.0173.200308c.0847

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