Pagnini, Gianni The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes. (English) Zbl 1312.33061 Fract. Calc. Appl. Anal. 16, No. 2, 436-453 (2013). Summary: The leading role of a special function of the Wright-type, referred to as M-Wright or Mainardi function, within a parametric class of self-similar stochastic processes with stationary increments, is surveyed. This class of processes, known as generalized grey Brownian motion, provides models for both fast and slow anomalous diffusion. In view of a subordination-type formula involving M-Wright functions, these processes emerge to have all finite moments and be uniquely defined by their mean and auto-covariance structure like Gaussian processes. The corresponding master equation is shown to be a fractional differential equation in the Erdélyi-Kober sense and the diffusive process is named Erdélyi-Kober fractional diffusion. In Appendix, an historical overview on the M-Wright function is reported. Cited in 17 Documents MSC: 33E20 Other functions defined by series and integrals 26A33 Fractional derivatives and integrals 44A35 Convolution as an integral transform 60G18 Self-similar stochastic processes 60G22 Fractional processes, including fractional Brownian motion 33E30 Other functions coming from differential, difference and integral equations Keywords:M-Wright function; fractional diffusion; generalized grey Brownian motion; Erdélyi-Kober fractional diffusion PDFBibTeX XMLCite \textit{G. Pagnini}, Fract. Calc. Appl. Anal. 16, No. 2, 436--453 (2013; Zbl 1312.33061) Full Text: DOI References: [1] B. Baeumer, M.M. Meerschaert, E. Nane, Brownian subordinators and fractional Cauchy problems. Trans. Amer. Math. Soc. 361, No 7 (2009), 3915-3930. http://dx.doi.org/10.1090/S0002-9947-09-04678-9; · Zbl 1186.60079 [2] R. Balescu, V-Langevin equations, continuous time random walks and fractional diffusion. 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