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Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation. (English) Zbl 1338.35430
Summary: The aim of the present paper is twofold:
1.
We carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [M. P. Gualdani et al., “Factorization of non-symmetric operators and exponential H-Theorem”, Preprint, arXiv:1006.5523]. We extend the method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators written as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another – smaller or larger – Banach space under the condition that a certain iterate of the “mild perturbation” part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way. The cornerstone of our approach is a factorization argument, reminiscent of the Dyson series.
2.
We apply this method to the kinetic Fokker-Planck equation when the spatial domain is either the torus with periodic boundary conditions, or the whole space with a confinement potential. We then obtain spectral gap estimates for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance \(W_1\).

MSC:
35Q84 Fokker-Planck equations
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35B65 Smoothness and regularity of solutions to PDEs
35P15 Estimates of eigenvalues in context of PDEs
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