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International Journal of Stochastic Analysis
Volume 2012 (2012), Article ID 427383, 14 pages
Research Article

Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion

1CRS4, Centro Ricerche Studi Superiori e Sviluppo in Sardegna, Polaris Building 1, 09010 Pula (CA), Italy
2CRESME Research S.p.A., Viale Gorizia 25C, 00199 Roma, Italy
3Department of Physics, University of Bologna and INFN, Via Irnerio 46, 40126 Bologna, Italy

Received 31 May 2012; Accepted 11 September 2012

Academic Editor: Ciprian A. Tudor

Copyright © 2012 Gianni Pagnini et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function) emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.