# American Institute of Mathematical Sciences

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 and 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), 727-759. doi: 10.1016/j.jfa.2007.11.002. [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), 1-42. [3] Persi Diaconis and Laurent Miclo, On the spectral analysis of second-order Markov chains, preprint, 2010. [4] Jean Dolbeault, Clément Mouhot and Christian Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,, 2010. Available from: , (). [5] Jean-Pierre Eckmann and Martin Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Communication in Mathematical Physics, 212 (2000), 105-164. doi: 10.1007/s002200000216. [6] Jean-Pierre Eckmann and Martin Hairer, Spectral properties of hypoelliptic operators, Communication in Mathematical Physics, 235 (2003), 233-253. doi: 10.1007/s00220-003-0805-9. [7] William Feller, "An Introduction To Probability Theory And Its Applications. Vol. II," Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. [8] Brian C. Hall, "Lie Groups, Lie Algebras, and Representations. An Elementary Introduction," Graduate Texts in Mathematics, 222, Springer-Verlag, New York, 2003. [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), 95-118. doi: 10.1016/j.jfa.2006.11.013. [10] Bernard Helffer and Francis Nier, "Hypoelliptic Estimates And Spectral Theory For Fokker-Planck Operators And Witten Laplacians," Lecture Notes in Mathematics, 1862, Springer-Verlag, Berlin, 2005. [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), 151-218. doi: 10.1007/s00205-003-0276-3. [12] Lars Hörmander, Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171. [13] Nobuyuki Ikeda and Shinzo Watanabe, "Stochastic Differential Equations And Diffusion Processes," North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo, 1989. [14] Tosio Kato, "Perturbation Theory For Linear Operators," Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. [15] , Otared Kavian,, Personal communication., (). [16] Anna Lee, Centro-Hermitian and skew-centro-Hermitian matrices, Linear Algebra and its Application, 29 (1980), 205-210. doi: 10.1016/0024-3795(80)90241-4. [17] Michela Ottobre, Grigorios Pavliotis and Karel Pravda-Starov, Exponential return to equilibrium for hypoelliptic quadratic systems, Journal of Functional Analysis, 262 (2012), 4000-4039. doi: 10.1016/j.jfa.2012.02.008. [18] Luc Rey-Bellet and Lawrence Thomas, Fluctuations of the entropy production in anharmonic chains, Annales de l'Institut Henri Poincaré, 3 (2002), 483-502. doi: 10.1007/s00023-002-8625-6. [19] Hannes Risken, "The Fokker-Planck Equation. Methods of Solution and Applications," Second edition, Springer Series in Synergetics, 18, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61544-3. [20] Gábor Szegõ, "Orthogonal Polynomials," Fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. [21] Cédric Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5. [22] James R. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors, Amer. Math. Monthly, 92 (1985), 711-717. doi: 10.2307/2323222. [23] Kōsaku Yosida, "Functional Analysis," Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

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##### 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), 727-759. doi: 10.1016/j.jfa.2007.11.002. [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), 1-42. [3] Persi Diaconis and Laurent Miclo, On the spectral analysis of second-order Markov chains, preprint, 2010. [4] Jean Dolbeault, Clément Mouhot and Christian Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,, 2010. Available from: , (). [5] Jean-Pierre Eckmann and Martin Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Communication in Mathematical Physics, 212 (2000), 105-164. doi: 10.1007/s002200000216. [6] Jean-Pierre Eckmann and Martin Hairer, Spectral properties of hypoelliptic operators, Communication in Mathematical Physics, 235 (2003), 233-253. doi: 10.1007/s00220-003-0805-9. [7] William Feller, "An Introduction To Probability Theory And Its Applications. Vol. II," Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. [8] Brian C. Hall, "Lie Groups, Lie Algebras, and Representations. An Elementary Introduction," Graduate Texts in Mathematics, 222, Springer-Verlag, New York, 2003. [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), 95-118. doi: 10.1016/j.jfa.2006.11.013. [10] Bernard Helffer and Francis Nier, "Hypoelliptic Estimates And Spectral Theory For Fokker-Planck Operators And Witten Laplacians," Lecture Notes in Mathematics, 1862, Springer-Verlag, Berlin, 2005. [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), 151-218. doi: 10.1007/s00205-003-0276-3. [12] Lars Hörmander, Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171. [13] Nobuyuki Ikeda and Shinzo Watanabe, "Stochastic Differential Equations And Diffusion Processes," North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo, 1989. [14] Tosio Kato, "Perturbation Theory For Linear Operators," Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. [15] , Otared Kavian,, Personal communication., (). [16] Anna Lee, Centro-Hermitian and skew-centro-Hermitian matrices, Linear Algebra and its Application, 29 (1980), 205-210. doi: 10.1016/0024-3795(80)90241-4. [17] Michela Ottobre, Grigorios Pavliotis and Karel Pravda-Starov, Exponential return to equilibrium for hypoelliptic quadratic systems, Journal of Functional Analysis, 262 (2012), 4000-4039. doi: 10.1016/j.jfa.2012.02.008. [18] Luc Rey-Bellet and Lawrence Thomas, Fluctuations of the entropy production in anharmonic chains, Annales de l'Institut Henri Poincaré, 3 (2002), 483-502. doi: 10.1007/s00023-002-8625-6. [19] Hannes Risken, "The Fokker-Planck Equation. Methods of Solution and Applications," Second edition, Springer Series in Synergetics, 18, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61544-3. [20] Gábor Szegõ, "Orthogonal Polynomials," Fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. [21] Cédric Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00567-5. [22] James R. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors, Amer. Math. Monthly, 92 (1985), 711-717. doi: 10.2307/2323222. [23] Kōsaku Yosida, "Functional Analysis," Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
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