Random Flights and Anomalous Diffusion: A Non-Markovian Take on Lorentz Processes

TL;DR

This paper investigates non-Markovian random-flight processes emerging from Lorentz models with infinite free-flight times, revealing superdiffusive behavior and fractional kinetic equations through advanced scaling limits and semigroup techniques.

Contribution

It introduces a novel non-Markovian framework for Lorentz processes with infinite expectation of free-flight times, deriving fractional kinetic equations and new scaling limits.

Findings

Convergence to non-Markovian superdiffusive processes

Derivation of fractional kinetic equations in time and space

Development of a mixture of Feller semigroups technique

Abstract

We study Lorentz processes in two different settings. Both cases are characterized by infinite expectation of the free-flight times, contrary to what happens in the classical Gallavotti-Spohn models. Under a suitable Boltzmann-Grad type scaling limit, they converge to non-Markovian random-flight processes with superdiffusive behavior. A further scaling limit yields another non Markovian process, i.e., a superdiffusion obtained by a suitable time-change of Brownian motion. Furthermore, we obtain the governing equations for our random flights and anomalous diffusion, which represent a non-local counterpart for the linear-Boltzmann and diffusion equations arising in the classical theory. It turns out that these equations have the form of fractional kinetic equations in both time and space. To prove these results, we develop a technique based on mixtures of Feller semigroups.