Self-similar Gaussian processes for modeling anomalous diffusion

Abstract

We study some Gaussian models for anomalous diffusion, which include the time-rescaled Brownian motion, two types of fractional Brownian motion, and models associated with fractional Brownian motion based on the generalized Langevin equation. Gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean-square displacement to vary with $t^{\alpha },$ $0<\mathrm{}\alpha <2.\mathrm{}$ However, these processes have different properties, thus indicating that the anomalous diffusion relation with a single parameter is insufficient to characterize the underlying mechanism. Although the two versions of fractional Brownian motion and time-rescaled Brownian motion all have the same probability distribution function, the Slepian theorem can be used to compare their first passage time distributions, which are different. Finally, in order to model anomalous diffusion with a variable exponent $\alpha \mathrm{}\left(t\right)\mathrm{}$ it is necessary to consider the multifractional extensions of these Gaussian processes.