Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations
Journal de l’École polytechnique — Mathématiques, Volume 4 (2017), pp. 389-433.

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.

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Accepted:
Published online:
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
Mot clés : Équation de Fokker-Planck, laplacien fractionnaire, trou spectral, taux de convergence exponentiel, asymptotique en temps long, semi-groupe, dissipativité

Stéphane Mischler 1; Isabelle Tristani 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.
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Stéphane Mischler; Isabelle Tristani. Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations. Journal de l’École polytechnique — Mathématiques, Volume 4 (2017), pp. 389-433. doi : 10.5802/jep.46. https://jep.centre-mersenne.org/articles/10.5802/jep.46/

[1] K. Carrapatoso & S. Mischler - “Uniqueness and long time asymptotic for the parabolic-parabolic Keller-Segel equation” (2016), to appear in Comm. Partial Differential Equations, hal-01011361 | Zbl

[2] G. Egaña Fernández & S. Mischler - “Uniqueness and long time asymptotic for the Keller-Segel equation: the parabolic-elliptic case”, Arch. Rational Mech. Anal. 220 (2016) no. 3, p. 1159-1194 | DOI | MR | Zbl

[3] M. Escobedo, S. Mischler & M. Rodriguez Ricard - “On self-similarity and stationary problem for fragmentation and coagulation models”, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) no. 1, p. 99-125 | DOI | Numdam | MR | Zbl

[4] I. Gentil & C. Imbert - “The Lévy-Fokker-Planck equation: Φ-entropies and convergence to equilibrium”, Asymptot. Anal. 59 (2008) no. 3-4, p. 125-138 | Zbl

[5] M. P. Gualdani, S. Mischler & C. Mouhot - “Factorization of non-symmetric operators and exponential H-Theorem” (2013), hal-00495786 | Zbl

[6] O. Kavian & S. Mischler - “The Fokker-Planck equation with subcritical confinement force” (2015), hal-01241680

[7] S. Mischler - “Semigroups in Banach spaces, factorisation and spectral analysis”, work in progress

[8] S. Mischler & C. Mouhot - “Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres”, Comm. Math. Phys. 288 (2009) no. 2, p. 431-502 | DOI | MR | Zbl

[9] S. Mischler & C. Mouhot - “Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation”, Arch. Rational Mech. Anal. 221 (2016) no. 2, p. 677-723 | DOI | MR | Zbl

[10] S. Mischler, C. Quiñinao & J. Touboul - “On a kinetic Fitzhugh-Nagumo model of neuronal network”, Comm. Math. Phys. 342 (2016) no. 3, p. 1001-1042 | DOI | MR | Zbl

[11] S. Mischler & J. Scher - “Spectral analysis of semigroups and growth-fragmentation equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) no. 3, p. 849-898 | DOI | MR | Zbl

[12] S. Mischler & Q. Weng - “On a linear runs and tumbles equation”, Kinet. and Relat. Mod. 10 (2017) no. 3, p. 799-822, hal-01272429 | DOI | MR | Zbl

[13] C. Mouhot - “Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials”, Comm. Math. Phys. 261 (2006) no. 3, p. 629-672 | DOI | MR | Zbl

[14] I. Tristani - “Fractional Fokker-Planck equation”, Commun. Math. Sci. 13 (2015) no. 5, p. 1243-1260 | DOI | MR | Zbl

[15] I. Tristani - “Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting”, J. Funct. Anal. 270 (2016) no. 5, p. 1922-1970 | DOI | MR | Zbl

[16] J. Voigt - “A perturbation theorem for the essential spectral radius of strongly continuous semigroups”, Monatsh. Math. 90 (1980) no. 2, p. 153-161 | DOI | MR | Zbl

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