Semi-Markov processes, integro-differential equations and anomalous diffusion-aggregation

TL;DR

This paper explores semi-Markov processes linked to integro-differential equations, modeling anomalous diffusion with position-dependent trapping effects, and analyzes how these traps influence the process's asymptotic behavior and diffusion properties.

Contribution

It establishes a connection between variable order fractional diffusion equations and semi-Markov processes with position-dependent trapping, extending understanding of anomalous diffusion in disordered media.

Findings

Processes tend to localize near regions minimizing rithmetical(lpha(x))

Traps cause the process to be non-diffusive, spending negligible time outside certain neighborhoods

The process converges in probability to the region of minimal lpha(x) under specific conditions.

Abstract

In this article integro-differential Volterra equations whose convolution kernel depends on the vector variable are considered and a connection of these equations with a class of semi-Markov processes is established. The variable order $\alpha(x)$-fractional diffusion equation is a particular case of our analysis and it turns out that it is associated with a suitable (non-independent) time-change of the Brownian motion. The resulting process is semi-Markovian and its paths have intervals of constancy, as it happens for the delayed Brownian motion, suitable to model trapping effects induced by the medium. However in our scenario the interval of constancy may be position dependent and this means traps of space-varying depth as it happens in a disordered medium. The strength of the trapping is investigated by means of the asymptotic behaviour of the process: it is proved that, under some technical assumptions on $\alpha(x)$, traps make the process non-diffusive in the sense that it spends a negligible amount of time out of a neighborhood of the region $\text{argmin}(\alpha(x))$ to which it converges in probability under some more restrictive hypotheses on $\alpha(x)$.