SELF-SIMILAR STOCHASTIC PROCESSES

This work is concerned with the analysis of self-similar stochastic pro-cesses, where statistical self-similarity refers to invariance in distribution under a suitable change of scale. In particular, emphasis is placed upon those processes which, being non-local in time, can display long memory and/or long-range correlation effects, and are connected to anomalous diffu-sion. Whereas normal diffusion is characterized by a mean-square displace-ment which is asymptotically linear in time according to Einstein law, in anomalous diffusion, a phenomenon met in a number of physical, chemical and biological systems, we find appreciable deviations from this law. Fractional Brownian motion (fBm) and time fractional diffusion are both self-similar stochastic processes, that display deviations from Einstein law. The first one is Gaussian, non-stationary, with stationary increments; the second is non-Gaussian, non-stationary with increments that are also non-stationary. The class of self-similar stochastic processes is related, via a simple one-to-one connection, called Lamperti transformation, to anothe