June  2013, 6(2): 317-372. doi: 10.3934/krm.2013.6.317

Spectral decompositions and $\mathbb{L}^2$-operator norms of toy hypocoercive semi-groups

1. 

Institut Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne F-31062 Toulouse Cedex 9, France, France

Received  February 2012 Revised  October 2012 Published  February 2013

For any $a>0$, consider the hypocoercive generators $y∂_x+a∂_y^2-y∂_y$ and $y∂_x-ax∂_y+∂_y^2-y∂_y$, respectively for $(x,y)\in\mathbb{R}/(2\pi\mathbb{Z})\times\mathbb{R}$ and $(x,y)\in\mathbb{R}\times\mathbb{R}$. The goal of the paper is to obtain exactly the $\mathbb{L}^2(\mu_a)$-operator norms of the corresponding Markov semi-group at any time, where $\mu_a$ is the associated invariant measure. The computations are based on the spectral decomposition of the generator and especially on the scalar products of the eigenvectors. The motivation comes from an attempt to find an alternative approach to classical ones developed to obtain hypocoercive bounds for kinetic models.
Citation: Sébastien Gadat, Laurent Miclo. Spectral decompositions and $\mathbb{L}^2$-operator norms of toy hypocoercive semi-groups. Kinetic & Related Models, 2013, 6 (2) : 317-372. doi: 10.3934/krm.2013.6.317
References:
[1]

Dominique Bakry, Patrick Cattiaux and Arnaud Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré,, Journal of Functional Analysis, 254 (2008), 727.  doi: 10.1016/j.jfa.2007.11.002.  Google Scholar

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Laurent Desvillettes and Cédric Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation,, Communications on Pure and Applied Mathematics, 54 (2001), 1.   Google Scholar

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Persi Diaconis and Laurent Miclo, On the spectral analysis of second-order Markov chains,, preprint, (2010).   Google Scholar

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Jean Dolbeault, Clément Mouhot and Christian Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,, 2010. Available from: , ().   Google Scholar

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Jean-Pierre Eckmann and Martin Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators,, Communication in Mathematical Physics, 212 (2000), 105.  doi: 10.1007/s002200000216.  Google Scholar

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Jean-Pierre Eckmann and Martin Hairer, Spectral properties of hypoelliptic operators,, Communication in Mathematical Physics, 235 (2003), 233.  doi: 10.1007/s00220-003-0805-9.  Google Scholar

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William Feller, "An Introduction To Probability Theory And Its Applications. Vol. II,", Second edition, (1971).   Google Scholar

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Brian C. Hall, "Lie Groups, Lie Algebras, and Representations. An Elementary Introduction," Graduate Texts in Mathematics, 222,, Springer-Verlag, (2003).   Google Scholar

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Frédéric Hérau, Short and long time behaviour of the Fokker-Planck equation in a confining potential and applications,, Journal of Functional Analysis, 244 (2007), 95.  doi: 10.1016/j.jfa.2006.11.013.  Google Scholar

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Bernard Helffer and Francis Nier, "Hypoelliptic Estimates And Spectral Theory For Fokker-Planck Operators And Witten Laplacians,", Lecture Notes in Mathematics, 1862 (2005).   Google Scholar

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Frédéric Hérau and Francis Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential,, Archive for Rational Mechanics and Analysis, 171 (2004), 151.  doi: 10.1007/s00205-003-0276-3.  Google Scholar

[12]

Lars Hörmander, Hypoelliptic second order differential equations,, Acta Mathematica, 119 (1967), 147.   Google Scholar

[13]

Nobuyuki Ikeda and Shinzo Watanabe, "Stochastic Differential Equations And Diffusion Processes,", North-Holland Mathematical Library, 24 (1989).   Google Scholar

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Tosio Kato, "Perturbation Theory For Linear Operators,", Reprint of the 1980 edition, (1980).   Google Scholar

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Anna Lee, Centro-Hermitian and skew-centro-Hermitian matrices,, Linear Algebra and its Application, 29 (1980), 205.  doi: 10.1016/0024-3795(80)90241-4.  Google Scholar

[17]

Michela Ottobre, Grigorios Pavliotis and Karel Pravda-Starov, Exponential return to equilibrium for hypoelliptic quadratic systems,, Journal of Functional Analysis, 262 (2012), 4000.  doi: 10.1016/j.jfa.2012.02.008.  Google Scholar

[18]

Luc Rey-Bellet and Lawrence Thomas, Fluctuations of the entropy production in anharmonic chains,, Annales de l'Institut Henri Poincaré, 3 (2002), 483.  doi: 10.1007/s00023-002-8625-6.  Google Scholar

[19]

Hannes Risken, "The Fokker-Planck Equation. Methods of Solution and Applications,", Second edition, 18 (1989).  doi: 10.1007/978-3-642-61544-3.  Google Scholar

[20]

Gábor Szegõ, "Orthogonal Polynomials,", Fourth edition, (1975).   Google Scholar

[21]

Cédric Villani, Hypocoercivity,, Mem. Amer. Math. Soc., 202 (2009).  doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[22]

James R. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors,, Amer. Math. Monthly, 92 (1985), 711.  doi: 10.2307/2323222.  Google Scholar

[23]

Kōsaku Yosida, "Functional Analysis,", Reprint of the sixth (1980) edition, (1980).   Google Scholar

show all references

References:
[1]

Dominique Bakry, Patrick Cattiaux and Arnaud Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré,, Journal of Functional Analysis, 254 (2008), 727.  doi: 10.1016/j.jfa.2007.11.002.  Google Scholar

[2]

Laurent Desvillettes and Cédric Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation,, Communications on Pure and Applied Mathematics, 54 (2001), 1.   Google Scholar

[3]

Persi Diaconis and Laurent Miclo, On the spectral analysis of second-order Markov chains,, preprint, (2010).   Google Scholar

[4]

Jean Dolbeault, Clément Mouhot and Christian Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,, 2010. Available from: , ().   Google Scholar

[5]

Jean-Pierre Eckmann and Martin Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators,, Communication in Mathematical Physics, 212 (2000), 105.  doi: 10.1007/s002200000216.  Google Scholar

[6]

Jean-Pierre Eckmann and Martin Hairer, Spectral properties of hypoelliptic operators,, Communication in Mathematical Physics, 235 (2003), 233.  doi: 10.1007/s00220-003-0805-9.  Google Scholar

[7]

William Feller, "An Introduction To Probability Theory And Its Applications. Vol. II,", Second edition, (1971).   Google Scholar

[8]

Brian C. Hall, "Lie Groups, Lie Algebras, and Representations. An Elementary Introduction," Graduate Texts in Mathematics, 222,, Springer-Verlag, (2003).   Google Scholar

[9]

Frédéric Hérau, Short and long time behaviour of the Fokker-Planck equation in a confining potential and applications,, Journal of Functional Analysis, 244 (2007), 95.  doi: 10.1016/j.jfa.2006.11.013.  Google Scholar

[10]

Bernard Helffer and Francis Nier, "Hypoelliptic Estimates And Spectral Theory For Fokker-Planck Operators And Witten Laplacians,", Lecture Notes in Mathematics, 1862 (2005).   Google Scholar

[11]

Frédéric Hérau and Francis Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential,, Archive for Rational Mechanics and Analysis, 171 (2004), 151.  doi: 10.1007/s00205-003-0276-3.  Google Scholar

[12]

Lars Hörmander, Hypoelliptic second order differential equations,, Acta Mathematica, 119 (1967), 147.   Google Scholar

[13]

Nobuyuki Ikeda and Shinzo Watanabe, "Stochastic Differential Equations And Diffusion Processes,", North-Holland Mathematical Library, 24 (1989).   Google Scholar

[14]

Tosio Kato, "Perturbation Theory For Linear Operators,", Reprint of the 1980 edition, (1980).   Google Scholar

[15]

, Otared Kavian,, Personal communication., ().   Google Scholar

[16]

Anna Lee, Centro-Hermitian and skew-centro-Hermitian matrices,, Linear Algebra and its Application, 29 (1980), 205.  doi: 10.1016/0024-3795(80)90241-4.  Google Scholar

[17]

Michela Ottobre, Grigorios Pavliotis and Karel Pravda-Starov, Exponential return to equilibrium for hypoelliptic quadratic systems,, Journal of Functional Analysis, 262 (2012), 4000.  doi: 10.1016/j.jfa.2012.02.008.  Google Scholar

[18]

Luc Rey-Bellet and Lawrence Thomas, Fluctuations of the entropy production in anharmonic chains,, Annales de l'Institut Henri Poincaré, 3 (2002), 483.  doi: 10.1007/s00023-002-8625-6.  Google Scholar

[19]

Hannes Risken, "The Fokker-Planck Equation. Methods of Solution and Applications,", Second edition, 18 (1989).  doi: 10.1007/978-3-642-61544-3.  Google Scholar

[20]

Gábor Szegõ, "Orthogonal Polynomials,", Fourth edition, (1975).   Google Scholar

[21]

Cédric Villani, Hypocoercivity,, Mem. Amer. Math. Soc., 202 (2009).  doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[22]

James R. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors,, Amer. Math. Monthly, 92 (1985), 711.  doi: 10.2307/2323222.  Google Scholar

[23]

Kōsaku Yosida, "Functional Analysis,", Reprint of the sixth (1980) edition, (1980).   Google Scholar

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