Correlated continuous-time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics

TL;DR

This paper introduces a generalized correlated continuous-time random walk model that combines scale-invariance with long-range memory, capturing diverse anomalous diffusion behaviors observed in experimental data.

Contribution

It develops a comprehensive stochastic model integrating features of Levy flights, subdiffusive CTRWs, and fractional Brownian motion, with exact solutions for probability densities.

Findings

Different parameter combinations yield similar probability density shapes.

The model reproduces key features of experimental single particle tracking data.

It unifies various anomalous diffusion mechanisms in a single framework.

Abstract

Standard continuous time random walk (CTRW) models are renewal processes in the sense that at each jump a new, independent pair of jump length and waiting time are chosen. Globally, anomalous diffusion emerges through action of the generalized central limit theorem leading to scale-free forms of the jump length or waiting time distributions. Here we present a modified version of recently proposed correlated CTRW processes, where we incorporate a power-law correlated noise on the level of both jump length and waiting time dynamics. We obtain a very general stochastic model, that encompasses key features of several paradigmatic models of anomalous diffusion: discontinuous, scale-free displacements as in Levy flights, scale-free waiting times as in subdiffusive CTRWs, and the long-range temporal correlations of fractional Brownian motion (FBM). We derive the exact solutions for the single-time probability density functions and extract the scaling behaviours. Interestingly, we find that different combinations of the model parameters lead to indistinguishable shapes of the emerging probability density functions and identical scaling laws. Our model will be useful to describe recent experimental single particle tracking data, that feature a combination of CTRW and FBM properties.