From a kinetic equation to a diffusion under an anomalous scaling

TL;DR

This paper demonstrates that a linear Boltzmann equation with a Markov process involving heavy-tailed waiting times converges under anomalous scaling to a diffusion process, linking kinetic equations to anomalous diffusion.

Contribution

It provides a rigorous derivation of diffusion limits from a kinetic equation with infinite variance waiting times, revealing anomalous scaling behavior.

Findings

Y(t) converges to a two-dimensional Brownian motion

Rescaled Boltzmann solutions converge to a diffusion equation

Establishes a link between kinetic equations and anomalous diffusion

Abstract

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t), i(t), Y(t)), where (K(t), i(t)) is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance, and Y(t) is an additive functional of K(t). We prove that under an anomalous rescaling Y converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately rescaled solution of the Boltzmann equation converges to a diffusion equation.