# Fractional diffusion limit for a kinetic equation with an interface

## TL;DR

This paper derives a fractional diffusion limit for a kinetic equation with an interface, modeling a microscopic oscillator chain, revealing how interface interactions influence superdiffusive behavior in the long-term dynamics.

## Contribution

It introduces a new fractional heat equation with interface conditions as the limit of a kinetic model with reflection, transmission, and absorption effects.

## Key findings

- The kinetic equation exhibits superdiffusive behavior without the interface.
- The long-time limit is a fractional heat equation with interface conditions.
- The limit process involves a stable process with absorption, reflection, or transmission at the interface.

## Abstract

We consider the limit of a linear kinetic equation, with reflection-transmission-absorption at an interface, with a degenerate scattering kernel. The equation arise from a microscopic chain of oscillators in contact with a heat bath. In the absence of the interface, the solutions exhibit a superdiffusive behavior in the long time limit. With the interface, the long time limit is the unique solution of a version of the fractional in space heat equation, with reflection-transmission-absorption at the interface. The limit problem corresponds to a certain stable process that is either absorbed, reflected, or transmitted upon crossing the interface.