[Analyse spectrale uniforme des équations de Fokker-Planck discrète, fractionnaire et classique]
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.
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.
Accepté le :
Publié le :
DOI : 10.5802/jep.46
Keywords: Fokker-Planck equation, fractional Laplacian, spectral gap, exponential rate of convergence, long-time asymptotic, semigroup, dissipativity
Mot clés : Équation de Fokker-Planck, laplacien fractionnaire, trou spectral, taux de convergence exponentiel, asymptotique en temps long, semi-groupe, dissipativité
@article{JEP_2017__4__389_0,
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},
volume = {4},
year = {2017},
doi = {10.5802/jep.46},
mrnumber = {3623358},
zbl = {06754331},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jep.46/}
}
TY - JOUR AU - Mischler, Stéphane AU - Tristani, Isabelle TI - Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations JO - Journal de l’École polytechnique — Mathématiques PY - 2017 SP - 389 EP - 433 VL - 4 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.46/ DO - 10.5802/jep.46 LA - en ID - JEP_2017__4__389_0 ER -
%0 Journal Article %A Mischler, Stéphane %A Tristani, Isabelle %T Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations %J Journal de l’École polytechnique — Mathématiques %D 2017 %P 389-433 %V 4 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.46/ %R 10.5802/jep.46 %G en %F JEP_2017__4__389_0
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://www.numdam.org/articles/10.5802/jep.46/
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