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A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium
[Une note sur l’hypocoercivité pour les équations cinétiques avec équilibres à queue lourde]
Comptes Rendus. Mathématique, Tome 358 (2020) no. 3, pp. 333-340.

Dans cet article, on s’intéresse au comportement en temps long d’équations cinétiques linéaires dont les équilibres locaux sont à queue lourde. Notre contribution principale concerne l’équation de Lévy–Fokker–Planck cinétique, pour laquelle nous adaptons des techniques d’hypocoercivité afin de démontrer la convergence exponentielle des solutions vers un équilibre global. En comparant au cas de l’équation de Fokker–Planck cinétique classique, les enjeux ici sont liés au manque de symétrie de l’opérateur non-local de Lévy–Fokker–Planck et à la compréhension de ses propriétés de régularisation. En complément de notre analyse, nous traitons également le cas de l’équation de BGK à queue lourde.

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.

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DOI : https://doi.org/10.5802/crmath.46
@article{CRMATH_2020__358_3_333_0,
     author = {Nathalie Ayi and Maxime Herda and H\'el\`ene Hivert and Isabelle Tristani},
     title = {A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {333--340},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {3},
     year = {2020},
     doi = {10.5802/crmath.46},
     language = {en},
}
Nathalie Ayi; Maxime Herda; Hélène Hivert; Isabelle Tristani. A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium. Comptes Rendus. Mathématique, Tome 358 (2020) no. 3, pp. 333-340. doi : 10.5802/crmath.46. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.46/

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