The generalized Cattaneo (telegrapher's) equation and corresponding random walks

TL;DR

This paper investigates generalized Cattaneo equations, identifying conditions for their solutions to be valid probability distributions and classifying the resulting diffusion processes as normal or anomalous, using random walk models.

Contribution

It provides a comprehensive analysis of generalized Cattaneo equations, establishing criteria for probability solutions and linking them to specific diffusion behaviors.

Findings

Conditions for solutions to be probability distributions identified

Classification of diffusion as normal, super-, or subdiffusion

Derivations using continuous time and persistent random walk approaches

Abstract

The various types of generalized Cattaneo, called also telegrapher's equation, are studied. We find conditions under which solutions of the equations considered so far can be recognized as probability distributions, \textit{i.e.} are normalizable and non-negative on their domains. Analysis of the relevant mean squared displacements enables us to classify diffusion processes described by such obtained solutions and to identify them with either ordinary or anomalous super- or subdiffusion. To complete our study we analyse derivations of just considered examples the generalized Cattaneo equations using the continuous time random walk and the persistent random walk approaches.