Non-local in time telegraph equations and very slowly growing variances

TL;DR

This paper investigates non-local in time telegraph equations, analyzing the asymptotic variance behavior of associated stochastic processes, and introduces methods to construct examples with slowly growing variances, relevant for sub-diffusion modeling.

Contribution

It develops a new approach to analyze variance growth in non-local telegraph equations and constructs novel examples with slow variance growth, extending previous results.

Findings

Variance can grow very slowly at large and short times.

New examples of processes with slowly growing variances are constructed.

The approach can be adapted to define new operators for sub-diffusion.

Abstract

In this paper we consider a class of non-local in time telegraph equations. Recently, it has been proved that the fundamental solutions of such equations can be interpreted as the probability density function of a stochastic process. We study the asymptotic behavior of the variance of this process at large and short times. In this context, we develop a method to construct new examples such the variance has a slowly growth behavior, extending the earlier results. Finally, we show that our approach can be adapted to define new integro-differential operators which are interesting in sub-diffusion processes.