Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation
Résumé
The aim of the present paper is twofold:
(1) We carry on with developing an abstract method for deriving growth estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [8]. We extend the method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators writing as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a growth estimate in some reference space then it satisfies the same growth estimate in another – smaller or larger – Banach space under the condition that a certain iterate of the regularizing 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 for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance W_1.
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