Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation
By St'ephane Mischler, Clément Mouhot. Archive for Rational Mechanics and Analysis, 2014
88 citations recorded by the source corpus.
The aim of the present paper is twofold: 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 [10]. 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. 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 W1. 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 [10]. 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. 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 W1.
Subfields: Applied Mathematics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/31ee4845915c94c5f0d385c1ecf4386301e85015
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In Applied Mathematics
- Arnaud Guillin
- Raphael Zimmer
- Andreas Eberle
- Michael I. Jordan
- Peter L. Bartlett
- Xiang Cheng
- Niladri S. Chatterji
- Nathael Gozlan
In Statistical and Nonlinear Physics
Citations in the corpus
- Underdamped Langevin MCMC: A non-asymptotic analysis (2017, background): “The continuous-time process has been studied extensively [12, 13, 14, 15, 16, 17, 18, 19, 20].”
- Couplings and quantitative contraction rates for Langevin dynamics (2017, background): “Several analytic approaches to convergence to equilibrium for kinetic Fokker-Planck equations have been proposed during the last 15 years [10, 17, 27, 25, 26, 40, 42, 11, 7, 33, 19, 20, 3].”