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

Nathalie Ayi (1) , Maxime Herda (2, 3) , Hélène Hivert (4) , Isabelle Tristani (5)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 RAPSODI - Reliable numerical approximations of dissipative systems
4 ICJ - Institut Camille Jordan
5 DMA - Département de Mathématiques et Applications - ENS Paris

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 Lévy-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 Lévy-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.

Mots clés: Hypocoercivity; linear kinetic equations; Fokker-Planck operator; fractional diffusion; heavy-tailed distribution

Domaines: Equations aux dérivées partielles [math.AP]

Nathalie Ayi, Maxime Herda, Hélène Hivert, Isabelle Tristani. A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium. 2019. ⟨hal-02389146v1⟩