Anomalous diffusion limit for a kinetic equation with a thermostatted interface

TL;DR

This paper analyzes the long-time, large-space limit of scaled kinetic equations with interface conditions, showing convergence to a fractional heat equation related to symmetric stable processes with reflection, transmission, and absorption.

Contribution

It extends previous work by deriving an anomalous diffusion limit for kinetic equations with degenerate absorption and scattering at an interface, leading to a fractional heat equation.

Findings

Limit is the unique solution of a fractional in space heat equation.

The process involves reflection, transmission, and killing at the interface.

Results generalize previous non-degenerate absorption cases.

Abstract

We consider the limit of solutions of scaled linear kinetic equations with a reflection-transmission-absorption condition at the interface. Both the coefficient describing the probability of absorption and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat equation that corresponds to the Kolmogorov equation for a symmetric stable process, which is reflected, or transmitted while crossing the interface and is killed upon the first hitting of the interface. The results of the paper are related to the work in [KOR20], where the case of a non-degenerate probability of absorption has been considered.