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Clement Mouhot - One of the best experts on this subject based on the ideXlab platform.

  • exponential stability of slowly decaying solutions to the kinetic fokker planck equation
    Archive for Rational Mechanics and Analysis, 2016
    Co-Authors: Stephane Mischler, Clement Mouhot
    Abstract:

    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 [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. 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 W1.

  • exponential stability of slowly decaying solutions to the kinetic fokker planck equation
    arXiv: Analysis of PDEs, 2014
    Co-Authors: Stephane Mischler, Clement Mouhot
    Abstract:

    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 [10]. 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 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 perturba- tion" 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 es- timates for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance W\_1.

Stephane Mischler - One of the best experts on this subject based on the ideXlab platform.

  • exponential stability of slowly decaying solutions to the kinetic fokker planck equation
    Archive for Rational Mechanics and Analysis, 2016
    Co-Authors: Stephane Mischler, Clement Mouhot
    Abstract:

    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 [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. 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 W1.

  • exponential stability of slowly decaying solutions to the kinetic fokker planck equation
    arXiv: Analysis of PDEs, 2014
    Co-Authors: Stephane Mischler, Clement Mouhot
    Abstract:

    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 [10]. 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 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 perturba- tion" 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 es- timates for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance W\_1.

J M A M Vanneerven - One of the best experts on this subject based on the ideXlab platform.

  • exponential stability of operators and operator semigroups
    Journal of Functional Analysis, 1995
    Co-Authors: J M A M Vanneerven
    Abstract:

    Extending earlier results of Datko, Pazy, and Littman on C0-Semigroups, and of Przyluski and Weiss on operators, we prove the following. Let T be a bounded linear operator on a Banach space X and let r(T) denotes its spectral radius. Let E be a Banach function space over N with the property that limn→∞||χ {0,...,n−1}||E=∞. If for each x ∈ X and x* ∈ X* the map n ↦ 〈x*, Tnx;〉 belongs to E, then r(T) 0 for all t > 0. If ∑∞n =0 φ(|〈x*, Tnx;〉 < ∞ for all ||x|| ||x*|| ≤ 1, then r(T) < 1. Assuming a Δ2-condition on φ, a further improvement is obtained. For locally bounded semigroups T = {T(t)} t ≥ 0, we obtain similar results in terms of the maps t ↦ ||T (t) x||.

Bernd Straub - One of the best experts on this subject based on the ideXlab platform.

  • filter products of c0 semigroups and ultraproduct representations for lie groups
    Journal of Functional Analysis, 2004
    Co-Authors: Pierrealain Cherix, Michael Cowling, Bernd Straub
    Abstract:

    We provide a new approach to filter products of C0-Semigroups and prove a spectral theorem for the generator and its filter product. In a similar fashion, we construct ultraproducts of strongly continuous unitary representations of locally compact groups and study spectral theoretic connections between the representations and their ultraproducts. In the case of Lie groups, our investigations are extended to the infinitesimal representation.

Zhichao Shao - One of the best experts on this subject based on the ideXlab platform.