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International Journal of Differential Equations
Volume 2010, Article ID 104505, 29 pages
Review Article

The 𝑀 -Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey

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

Received 13 September 2009; Accepted 8 November 2009

Academic Editor: Fawang Liu

Copyright © 2010 Francesco Mainardi 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.


In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as 𝑀 -Wright function, entering as a probability density in a relevant class of self-similar stochastic processes that we generally refer to as time-fractional diffusion processes. Indeed, the master equations governing these processes generalize the standard diffusion equation by means of time-integral operators interpreted as derivatives of fractional order. When these generalized diffusion processes are properly characterized with stationary increments, the 𝑀 -Wright function is shown to play the same key role as the Gaussian density in the standard and fractional Brownian motions. Furthermore, these processes provide stochastic models suitable for describing phenomena of anomalous diffusion of both slow and fast types.