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

Sébastien Gadat; Laurent Miclo

doi:10.3934/krm.2013.6.317

Article Contents

2013, Volume 6, Issue 2: 317-372. Doi: 10.3934/krm.2013.6.317

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

Sébastien Gadat^1, and
Laurent Miclo^1,

1.
Institut Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne F-31062 Toulouse Cedex 9

Received: February 2012

Revised: October 2012

Published: June 2013

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35K65; Secondary: 35H10, 47A10, 15A18, 15A60, 37A30, 60J60.

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