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

Gianni Pagnini, Antonio Mura and Francesco Mainardi

International Journal of Stochastic Analysis, 2012, vol. 2012, 1-14

Abstract:

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.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:427383

DOI: 10.1155/2012/427383

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