Geodesic
Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations
[Analyse spectrale uniforme des équations de Fokker-Planck discrète, fractionnaire et classique]
Mischler, Stéphane1 ; Tristani, Isabelle2
1 Université Paris-Dauphine, Institut Universitaire de France (IUF), PSL Research University, CNRS, UMR [7534], CEREMADE, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
2 Université Paris-Dauphine, PSL Research University, CNRS, UMR [7534], CEREMADE, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France.
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 389-433.

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In this paper, we investigate the spectral analysis and long time asymptotic convergence of semigroups associated to discrete, fractional and classical Fokker-Planck equations in some regime where the corresponding operators are close. We successively deal with the discrete and the classical Fokker-Planck model, the fractional and the classical Fokker-Planck model and finally the discrete and the fractional Fokker-Planck model. In each case, we prove uniform spectral estimates using perturbation and/or enlargement arguments.

Dans cet article, nous nous intéressons à l’analyse spectrale et au comportement asymptotique en temps long des semi-groupes associés aux équations de Fokker-Planck discrète, fractionnaire et classique dans des régimes où les opérateurs correspondants sont proches. Nous traitons successivement les modèles de Fokker-Planck discret et classique, puis fractionnaire et classique et enfin discret et fractionnaire. Dans chaque cas, nous démontrons des estimations spectrales uniformes en utilisant des arguments de perturbation et/ou d’élargissement.

Reçu le :
Accepté le :
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DOI : 10.5802/jep.46
Classification : 47G20, 35B40, 35Q84
Keywords: Fokker-Planck equation, fractional Laplacian, spectral gap, exponential rate of convergence, long-time asymptotic, semigroup, dissipativity
Mots-clés : Équation de Fokker-Planck, laplacien fractionnaire, trou spectral, taux de convergence exponentiel, asymptotique en temps long, semi-groupe, dissipativité

Mischler, Stéphane 1 ; Tristani, Isabelle 2

1 Université Paris-Dauphine, Institut Universitaire de France (IUF), PSL Research University, CNRS, UMR [7534], CEREMADE, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
2 Université Paris-Dauphine, PSL Research University, CNRS, UMR [7534], CEREMADE, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France.
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Mischler, St\'ephane and Tristani, Isabelle},
     title = {Uniform semigroup spectral analysis of the~discrete, fractional and classical {Fokker-Planck} equations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {389--433},
     publisher = {\'Ecole polytechnique},
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Mischler, Stéphane; Tristani, Isabelle. Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 389-433. doi : 10.5802/jep.46. http://geodesic.mathdoc.fr/articles/10.5802/jep.46/

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