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Sur l'utilisation des relations d'entrelacement dans l'étude des générateurs de Markov auto-adjoints. Application aux inégalités spectrales et fonctionnelles et à l'analyse de sensibilité

Abstract : This thesis is part of a series of works carried out by Aldéric Joulin, Michel Bonnefont et alius, which aims at using intertwining relations to infer properties of some Markov generators. The present work deals specifically with three properties : Poincaré inequalities, logarithmic Sobolev inequalities and spectral estimates. Both above inequalities are widely use tools in infinite-dimensional analysis, that relate to the latter generators and underlying Boltzmann-Gibbs invariant distribution. In the first chapter, a method based on Feynman-Kac semigroups is proposed to infer new estimates, in relation to the logarithmic Sobolev inequality. The connexion between generators and stochastic processes is explored via a representation theorem for Feynman-Kac semigroups. In the second chapter, an algebraic approach to the estimation of eigenvalues of the aforementioned generators is discussed. This work echoes a related recent article by Emanuel Milman, in which he used optimal transport results in this very purpose. Multiplicities are addressed as well, in relation to the recent work of Franck Barthe and Boaz Klartag. In the last chapter, the relation between Poincaré inequalities and sensitivity analysis is investigated, particularly in order to compare two types of sensitivity indices. An estimation method related to this inequality is developed in dimension two, using finite elements methods.
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https://tel.archives-ouvertes.fr/tel-03727812
Contributor : Clément Steiner Connect in order to contact the contributor
Submitted on : Tuesday, July 19, 2022 - 4:34:00 PM
Last modification on : Friday, July 22, 2022 - 4:11:31 AM

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  • HAL Id : tel-03727812, version 1

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Clément Steiner. Sur l'utilisation des relations d'entrelacement dans l'étude des générateurs de Markov auto-adjoints. Application aux inégalités spectrales et fonctionnelles et à l'analyse de sensibilité. Mathématiques [math]. Université Toulouse 3 (Paul Sabatier), 2022. Français. ⟨tel-03727812⟩

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