  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.
