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Convergence to equilibrium for linear Fokker-Planck equations
Isabelle Tristani1
1 Centre de Mathématiques Laurent Schwartz École Polytechnique, CNRS, Université Paris-Saclay 91128 Palaiseau Cedex France
Séminaire Laurent Schwartz — EDP et applications (2015-2016), Talk no. 10, 14 p.
  • Abstract

In this note, 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 and the fractional and the classical Fokker-Planck model. In each case, we present results of uniform convergence to equilibrium based on perturbation and/or enlargement arguments and obtained in collaboration with S. Mischler in [7].

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Published online: 2016-10-12
DOI: 10.5802/slsedp.83
Author's affiliations:
Isabelle Tristani 1

1 Centre de Mathématiques Laurent Schwartz École Polytechnique, CNRS, Université Paris-Saclay 91128 Palaiseau Cedex France
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@article{SLSEDP_2015-2016____A10_0,
     author = {Isabelle Tristani},
     title = {Convergence to equilibrium for linear {Fokker-Planck} equations},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:10},
     pages = {1--14},
     publisher = {Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique},
     year = {2015-2016},
     doi = {10.5802/slsedp.83},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.83/}
}
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AU  - Isabelle Tristani
TI  - Convergence to equilibrium for linear Fokker-Planck equations
JO  - Séminaire Laurent Schwartz — EDP et applications
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SP  - 1
EP  - 14
PB  - Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.83/
DO  - 10.5802/slsedp.83
LA  - en
ID  - SLSEDP_2015-2016____A10_0
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%0 Journal Article
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%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:10
%D 2015-2016
%P 1-14
%I Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique
%U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.83/
%R 10.5802/slsedp.83
%G en
%F SLSEDP_2015-2016____A10_0
Isabelle Tristani. Convergence to equilibrium for linear Fokker-Planck equations. Séminaire Laurent Schwartz — EDP et applications (2015-2016), Talk no. 10, 14 p. doi : 10.5802/slsedp.83. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.83/
  • References
  • Cited by

[1] Gentil, I., and Imbert, C. The Lévy-Fokker-Planck equation: Φ-entropies and convergence to equilibrium. Asymptot. Anal. 59, 3-4 (2008), 125–138.

[2] Gualdani, M. P., Mischler, S., and Mouhot, C. Factorization of non-symmetric operators and exponential H-Theorem. (2013) . | HAL

[3] Mischler, S. Semigroups in Banach spaces, factorisation and spectral analysis. Work in progress.

[4] Mischler, S., and Mouhot, C. Exponential stability of slowly decaying solutions to the Kinetic-Fokker-Planck equation. (2014) , to appear in Arch. Rational Mech. Anal. | HAL

[5] Mischler, S., and Mouhot, C. Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres. Comm. Math. Phys. 288, 2 (2009), 431–502.

[6] Mischler, S., and Scher, J. Semigroup spectral analysis and growth-fragmentation equation. (2012) , to appear in Annales de l’Institut Henri Poincaré, Analyse Non Linéaire. | HAL

[7] Mischler, S., and Tristani, I. Uniform semigroup spectral analysis of the discrete, fractional & classical Fokker-Planck equations. (2015) . | HAL

[8] Mouhot, C. Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Comm. Math. Phys. 261, 3 (2006), 629–672.

[9] Tristani, I. Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting. (2013) , to appear in J. Funct. Anal. | HAL

[10] Tristani, I. Fractional Fokker-Planck equation. Commun. Math. Sci. 13, 5 (2015), 1243–1260.

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