A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics

Abstract

In this paper we present a general mathematical construction that allows us to define a parametric class of $H$-sssi stochastic processes (self-similar with stationary increments), which have marginal probability density function that evolves in time according to a partial integro-differential equation of fractional type. This construction is based on the theory of finite measures on functional spaces. Since the variance evolves in time as a power function, these $H$-sssi processes naturally provide models for slow and fast anomalous diffusion. Such a class includes, as particular cases, fractional Brownian motion, grey Brownian motion and Brownian motion.