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

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


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