Anomalous processes with general waiting times: functionals and multi-point structure

TL;DR

This paper studies complex anomalous diffusion processes with general waiting times, providing a comprehensive characterization including functionals and multi-point correlations, useful for analyzing experimental data.

Contribution

It introduces a framework for generalized anomalous diffusion processes with arbitrary waiting time distributions, including analytical expressions for correlation functions.

Findings

Derived closed-form two-point correlation functions.

Captured crossover between different scaling regimes.

Provided a representation linking to normal diffusive processes.

Abstract

Many transport processes in nature exhibit anomalous diffusive properties with non-trivial scaling of the mean square displacement, e.g., diffusion of cells or of biomolecules inside the cell nucleus, where typically a crossover between different scaling regimes appears over time. Here, we investigate a class of anomalous diffusion processes that is able to capture such complex dynamics by virtue of a general waiting time distribution. We obtain a complete characterization of such generalized anomalous processes, including their functionals and multi-point structure, using a representation in terms of a normal diffusive process plus a stochastic time change. In particular, we derive analytical closed form expressions for the two-point correlation functions, which can be readily compared with experimental data.