Fractional-diffusion-advection limit of a kinetic model

TL;DR

This paper rigorously derives a fractional diffusion-advection equation from a kinetic transport model with fat-tailed equilibrium and directional bias, using entropy bounds and advanced limit methods.

Contribution

It introduces a rigorous derivation of a fractional diffusion-advection limit from a kinetic model with new analytical techniques.

Findings

Derivation of fractional diffusion with advection from kinetic equations.

Application of entropy inequalities and novel limit methods.

Validation of the macroscopic limit for models with fat-tailed distributions.

Abstract

A fractional diffusion equation with advection term is rigorously derived from a kinetic transport model with a linear turning operator, featuring a fat-tailed equilibrium distribution and a small directional bias due to a given vector field. The analysis is based on bounds derived by relative entropy inequalities and on two recently developed approaches for the macroscopic limit: a Fourier-Laplace transform method for spatially homogeneous data and the so called moment method, based on a modified test function.