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.

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) BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks Оригинал: Учайкин В В «Автомодельная аномальная диффузия и устойчивые законы» УФН 173 847–876 (2003); DOI: 10.3367/UFNr.0173.200308c.0847