A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators

DOI10.3934/KRM.2020016zbMATH Open1447.58032arXiv1905.07042OpenAlexW3014266616MaRDI QIDQ2197873

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Publication date: 1 September 2020

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Abstract: The aim of this paper is to offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Our approach rests on a commutation relationship for linear operators known as intertwining, and we utilize this identity to transfer spectral information from a known, reference semigroup $ildeP=(e^{-tildemathbfA}{)}_{tgeq0}$ to a target semigroup $P$ which is the object of study. This allows us to obtain conditions under which $P$ satisfies a hypocoercive estimate with exponential decay rate given by the spectral gap of $ildemathbfA$. Along the way we also develop a functional calculus involving the non-self-adjoint resolution of identity induced by the intertwining relations. We apply these results in a general Hilbert space setting to two cases: degenerate, hypoelliptic Ornstein-Uhlenbeck semigroups on $mathbb{R}^d$, and non-local Jacobi semigroups on $[0,1{]}^d$, which have been recently introduced and studied for $d=1$. In both cases we obtain hypocoercive estimates and are able to explicitly identify the hypocoercive constants

Full work available at URL: https://arxiv.org/abs/1905.07042

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