A unifying approach to diffusive transport in annealed heterogeneous media

Abstract

We introduce the concept of Randomly Modulated Gaussian Processes as a unifying framework for elaborating, analyzing and classifying anomalous diffusion models in annealed heterogeneous media. This formulation incorporates correlations in the displacements together with correlated fluctuations of their amplitudes. Most known models of anomalous diffusion (including continuous-time random walk, fractional Brownian motion, and L\'evy flights) and random diffusivity can be described and further generalized within this framework. Moreover, the unified view identifies the main statistical properties to be probed experimentally for a reliable classification of diffusive dynamics. The proposed matrix formulation facilitates the computation of the first four moments and allows for a systematic statistical characterization of the considered processes. The necessary and sufficient conditions are provided for the emergence of anomalous diffusion. General expressions for the non-Gaussian parameter, the ergodicity breaking parameter and the covariance of squared increments are derived. An expression for the characteristic function and the codifference (i.e., a generalized measure of correlations) are obtained and used to study the special cases of L\'evy flights and Laplace motion with correlated displacements. Potential applications of this framework for systematic analysis and biophysical interpretations of experimental single-particle trajectories are discussed.