The kinetic Fokker–Planck equation with weak confinement force

Cao, Chuqi

doi:10.4310/cms.2019.v17.n8.a9

Cited by 19 publications

(11 citation statements)

References 16 publications

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“…We also look at Harris type theorems with weaker controls on moments to give analogues of all our theorems when the confining potential is weaker and give algebraic rates of convergence with rates depending on the assumption we make on the confining potential. Subgeometric convergence for kinetic Fokker-Planck equations with weak confinement has been shown in [19,1,13]. To our knowledge this is the only work showing this type of convergence in a quantitative way for the equations we present.…”

Section: Introductionsupporting

confidence: 53%

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Hypocoercivity of linear kinetic equations via Harris's Theorem

Cañizo¹,

Cao²,

Evans³

et al. 2020

Kinetic &Amp; Related Models

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We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the toruswith a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively L 1 or weighted L 1 norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem. Contents

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Section: Introductionsupporting

confidence: 53%

“…Lemma 3.3 (Doeblin condition for the linear relaxation Botzmann equation on the torus). For any t * > 0 there exist constants α, δ L > 0 (depending on t * ) such that any solution f to equation (13) with initial condition…”

Section: On the Torusmentioning

confidence: 99%

Hypocoercivity of linear kinetic equations via Harris's Theorem

Cañizo¹,

Cao²,

Evans³

et al. 2020

Kinetic &Amp; Related Models

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“…Recent related results for non-conservative semigroups have been reported by Bansaye, Cloez, and Gabriel (2019), and applications to models for the electrical activity of groups of neurons can be found in Dumont and Gabriel (2017); Cañizo and Yoldaş (2019). Recent works dealing with applications to Fokker-Planck equations and related models are due to Hu and Wang (2019); Eberle, Guillin, and Zimmer (2019); Cao (2019) and Lafleche (2020). We also mention applications to the study of hypocoercivity for kinetic equations and fragmentation-type equations Cañizo et al (2020b,a).…”

Section: Previous Contributionsmentioning

confidence: 84%

Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups

Cañizo¹,

Mischler²

2021

Preprint

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We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discretetime and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation arguments, valid for both geometric and subgeometric cases with essentially the same ideas. In particular, we present very simple new proofs of the geometric case.

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“…Moreover, hypocoercivity can be measured in many different ways, such as entropy, ϕentropies, total variation, Wasserstein distance, L 2 norms, H k norms, or more general weighted Sobolev norms, see for instance [5], [9], [3], [41], [16], [21], [7], [24](see also related work [22] and [23]), [25] and the references therein. We also call attentions to [39], [18], [10], [11], [12], [15], [14] and [30] for some recent results concerning hypocoercivity with various kinds of potentials. In particular, we highlight [8] and [37] for some results concerning singular potentials.…”

Section: Introductionmentioning

confidence: 99%

Hypocoercivity and global hypoellipticity for the kinetic Fokker-Planck equation in $H^k$ spaces

Zhang

2020

Preprint

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The purpose of this paper is to extend the hypocoercivity results for the kinetic Fokker-Planck equation in H 1 space in Villani's memoir [45] to higher order Sobolev spaces. As in the L 2 and H 1 setting, there is lack of coercivity in H k for the associated operator. To remedy this issue, we shall modify the usual H k norm with certain well-chosen mixed terms and with suitable coefficients which are constructed by induction on k. In parallel, a similar strategy but with coefficients depending on time (c.f. [34]), usually referred as Hérau's method, can be employed to prove global hypoellipticity in H k . The exponents in our regularity estimates are optimal in short time. Moreover, as in our recent work [30], the general results here can be applied in the mean-field setting to get estimates independent of the dimension; in particular, an application to the Curie-Weiss model is presented.

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The kinetic Fokker–Planck equation with weak confinement force

Cited by 19 publications

References 16 publications

Hypocoercivity of linear kinetic equations via Harris's Theorem

Hypocoercivity of linear kinetic equations via Harris's Theorem

Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups

Hypocoercivity and global hypoellipticity for the kinetic Fokker-Planck equation in $H^k$ spaces

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