From semi-Markov random evolutions to scattering transport and superdiffusion

TL;DR

This paper investigates semi-Markov driven random evolutions on Banach spaces, generalizes the telegraph equation, and analyzes scattering transport models with superdiffusive limits, revealing new connections between stochastic processes and transport phenomena.

Contribution

It introduces a semi-Markov framework for random evolutions, generalizes classical equations, and establishes superdiffusive limits for scattering transport models with infinite mean flight times.

Findings

Expectation of semi-Markov evolutions solves abstract Cauchy problems

Generalization of the telegraph equation to semi-Markov perturbations

Scaling limits of scattering transport models lead to superdiffusive processes

Abstract

We here study random evolutions on Banach spaces, driven by a class of semi-Markov processes. The expectation (in the sense of Bochner) of such evolutions is shown to solve some abstract Cauchy problems. Further, the abstract telegraph (damped wave) equation is generalized to the case of semi-Markov perturbations. A special attention is devoted to semi-Markov models of scattering transport processes which can be represented through these evolutions. In particular, we consider random flights with infinite mean flight times which turn out to be governed by a semi-Markov generalization of a linear Boltzmann equation; their scaling limit is proved to converge to superdiffusive transport processes.