Anomalous Diffusion: Deterministic and Stochastic Perspectives

Normal diffusion arises in a natural way from random walks with uncorrelated steps of bounded variance, or, in the deterministic setting, from wandering trajectories of a chaotic map. There are many ways in which such a picture fails, and, for instance, the variance of the traveller’s position does not grow linearly with time. We review the basic mechanisms that induce deviations from normal transport (long waiting times, broad step length distributions, intermittency, topological issues), and we describe how their origin can be traced back in stochastic and deterministic settings, illustrating a few techniques that allow for a quantitative analysis of such anomalies.