A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium

TL;DR

This paper studies the long-term behavior of kinetic equations with heavy-tailed equilibria, demonstrating exponential convergence to equilibrium for the kinetic Lévy-Fokker-Planck equation using adapted hypocoercivity methods.

Contribution

It introduces adapted hypocoercivity techniques to analyze the exponential convergence of solutions for kinetic Lévy-Fokker-Planck equations with heavy-tailed equilibria.

Findings

Solutions converge exponentially fast to equilibrium

Adapted hypocoercivity techniques handle non-local operators

Regularization properties are characterized for heavy-tailed cases

Abstract

In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic L\'evy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local L\'evy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation.