Spectral decompositions and l2-operator norms of toy hypocoercive models

Sébastien Gadat and Laurent Miclo
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Sébastien Gadat: TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement
Laurent Miclo: TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement

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Abstract: For any a > 0, consider the hypocoercive generators y partial derivative(x) + a partial derivative(2)(y) - y partial derivative(y) and y partial derivative(x) - ax partial derivative(y) + partial derivative(2)(y) - y partial derivative(y), respectively for (x, y) is an element of R/(2 pi Z) x R and (x,y) is an element of R x R. The goal of the paper is to obtain exactly the 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 fi nd an alternative approach to classical ones developed to obtain hypocoercive bounds for kinetic models.

Keywords: Convergence to equilibrium; Hypocoercive Markovian semi-groups; Spectral decompositions; Ornstein-Ulhenbeck generator; L-2-operator norms; Kinetic evolution equations (search for similar items in EconPapers)
Date: 2013-02
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Published in Kinetic and Related Models , 2013, 6 (2), pp.317-372. ⟨10.3934/krm.2013.6.317⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-05474346

DOI: 10.3934/krm.2013.6.317

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