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.
Accepted:
Published online:
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
Stéphane Mischler 1; Isabelle Tristani 2
@article{JEP_2017__4__389_0, author = {St\'ephane Mischler and Isabelle Tristani}, 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}, zbl = {06754331}, mrnumber = {3623358}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.46/} }
TY - JOUR AU - Stéphane Mischler AU - Isabelle Tristani 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 - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.46/ DO - 10.5802/jep.46 LA - en ID - JEP_2017__4__389_0 ER -
%0 Journal Article %A Stéphane Mischler %A Isabelle Tristani %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 École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.46/ %R 10.5802/jep.46 %G en %F JEP_2017__4__389_0
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] - “Uniqueness and long time asymptotic for the parabolic-parabolic Keller-Segel equation” (2016), to appear in Comm. Partial Differential Equations, hal-01011361 | Zbl
[2] - “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] - “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] - “The Lévy-Fokker-Planck equation: -entropies and convergence to equilibrium”, Asymptot. Anal. 59 (2008) no. 3-4, p. 125-138 | Zbl
[5] - “Factorization of non-symmetric operators and exponential -Theorem” (2013), hal-00495786 | Zbl
[6] - “The Fokker-Planck equation with subcritical confinement force” (2015), hal-01241680
[7] - “Semigroups in Banach spaces, factorisation and spectral analysis”, work in progress
[8] - “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] - “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] - “On a kinetic Fitzhugh-Nagumo model of neuronal network”, Comm. Math. Phys. 342 (2016) no. 3, p. 1001-1042 | DOI | MR | Zbl
[11] - “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] - “On a linear runs and tumbles equation”, Kinet. and Relat. Mod. 10 (2017) no. 3, p. 799-822, hal-01272429 | DOI | MR | Zbl
[13] - “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] - “Fractional Fokker-Planck equation”, Commun. Math. Sci. 13 (2015) no. 5, p. 1243-1260 | DOI | MR | Zbl
[15] - “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] - “A perturbation theorem for the essential spectral radius of strongly continuous semigroups”, Monatsh. Math. 90 (1980) no. 2, p. 153-161 | DOI | MR | Zbl
Cited by Sources: