Accelerating Brownian motion: A fractional dynamics approach to fast diffusion

Superdiffusion in the sub-ballistic regime with a non-diverging mean-squared displacement is studied on the basis of a linear, fractional kinetic equation with constant coefficients which is non-local in time and leads to an exponential tail of the corresponding probability density function. It is shown that sub-ballistic superdiffusion can be regarded as ballistic motion with a memory, much as slow diffusion can be thought of as a random walk with a memory. This suggests that fractional kinetic equations are useful in describing both sub-and superdiffusion processes.