Fractional Hypocoercivity

TL;DR

This paper studies the long-term behavior of kinetic equations with fat-tailed equilibria, revealing how fractional diffusion governs decay rates through hypocoercivity methods.

Contribution

It introduces an $ ext{L}^2$-hypocoercivity framework for kinetic equations with fractional diffusion limits, extending classical approaches to anomalous diffusion scenarios.

Findings

Established decay rates consistent with fractional diffusion limits.

Developed an $ ext{L}^2$-hypocoercivity approach for kinetic equations.

Linked kinetic decay behavior to fractional Nash inequalities.

Abstract

This paper is devoted to kinetic equations without confinement. We investigate the large time behaviour induced by collision operators with fat tailed local equilibria. Such operators have an anomalous diffusion limit. In the appropriate scaling, the macroscopic equation involves a fractional diffusion operator so that the optimal decay rate is determined by a fractional Nash type inequality. At kinetic level we develop an $\mathrm L^2$-hypocoercivity approach and establish a rate of decay compatible with the fractional diffusion limit.