Mid-term report of project "Anomalous transport in complex systems: stochastic modeling and statistical data analysis"

Anomalous diffusion is defined by the nonlinear growth of the fluctuations' variance with time and it is often related to long-range memory. Many complex systems are supposed to trigger long-range memory dynamical structures, as long-range correlations are usually observed. However, apparent long-range correlations can arise non only as a consequence of long-range memory dynamics, but also due to inhomogeneities in the set of interacting units/particles in the system. Fractional diffusion equation (FDE) is a evolution equation for the 1-point 1-time probability density function of a fluctuating/diffusing variable and is a ubiquitously used model for anomalous diffusion. Following the alternative assumption that long-range memory could be related to heterogeneity of the complex medium, we derive a class of stochastic models, based of Gaussian processes, that are compatible with the fractional diffusion equation (FDE). Thus, we prove that FDE can emerge also as a consequence of heterogeneity and is not only due to long-range memory in the system, such as in the Continuous Time Random Walk (CTRW) model. In fact, even if CTRW is driven by renewal events, destroying memory at some crucial time points, it is also characterized by inter-event long-life coherent structures determining the emergence of long-range correlations in the system.