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Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. (English) Zbl 1143.82028
Summary: We study a parametric class of stochastic processes to model both fast and slow anomalous diffusions. This class, called generalized grey Brownian motion (ggBm), is made up of self-similar with stationary increments processes ($H$-sssi) and depends on two real parameters $\alpha \in$ (0, 2) and $\beta \in \left(0,1\right]$. It includes fractional Brownian motion when $\alpha \in \left(0,2\right)$ and $\beta =1$, and time-fractional diffusion stochastic processes when $\alpha =\beta \in \left(0,1\right)$. The latter have a marginal probability density function governed by time-fractional diffusion equations of order $\beta$. The ggBm is defined through the explicit construction of the underlying probability space. However, in this paper we show that it is possible to define it in an unspecified probability space. For this purpose, we write down explicitly all the finite-dimensional probability density functions. Moreover, we provide different ggBm characterizations. The role of the $M$-Wright function, which is related to the fundamental solution of the time-fractional diffusion equation, emerges as a natural generalization of the Gaussian distribution. Furthermore, we show that the ggBm can be represented in terms of the product of a random variable, which is related to the $M$-Wright function, and an independent fractional Brownian motion. This representation highlights the $H$-sssi nature of the ggBm and provides a way to study and simulate the trajectories. For this purpose, we developed a random walk model based on a finite difference approximation of a partial integro-differential equation of a fractional type.
##### MSC:
 82D10 Plasmas (statistical mechanics) 82C24 Interface problems (dynamic and non-equilibrium); diffusion-limited aggregation