Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups

Röckner, Michael; Wang, Feng‐Yu

doi:10.1006/jfan.2001.3776

Cited by 157 publications

(224 citation statements)

References 24 publications

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“…This result is actually true and shown (using another route) in [18] It is nevertheless interesting, at least for counter examples to know some sufficient conditions for the usual (SGP). If N = 1 a necessary and sufficient condition was obtained by Muckenhoupt (see [3] chapter 6).…”

Section: Examples: Rmentioning

confidence: 52%

“…The usual spectral gap (or Poincaré) inequality will be denoted by (SGP). A weaker one introduced by Röckner and Wang (see [18]) called the weak spectral gap property (WSGP) is discussed in [1] and in section 5 of [6]. In particular (DLSI)+(WSGP) implies (TLSI) originally due to Mathieu ([17]) is shown in [6] Proposition 5.13.…”

Section: Some Notation and General Resultsmentioning

confidence: 99%

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Hypercontractivity for perturbed diffusion semigroups

Cattiaux¹

2005

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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Abstract. µ being a nonnegative measure satisfying some Log-Sobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some Log-Sobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary condition are given and examples are explicitly studied.Résumé. µétant une mesure positive satisfaisant une inégalité de Sobolev logarithmique, nous donnons des conditions sur F pour que la mesure de Boltzmann ν = e −2F µ satisfasseégalement une telle inégalité (améliorant et complétant ainsi la dernière partie de [6]). Les conditions obtenues sont illustrées par des exemples.

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Section: Examples: Rmentioning

confidence: 52%

Section: Some Notation and General Resultsmentioning

confidence: 99%

Hypercontractivity for perturbed diffusion semigroups

Cattiaux¹

2005

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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“…Therefore, we can apply Theorem 2.2 of [22] (see also [30] and [6]) and we get the following algebraic rate of convergence…”

Section: Weakly Converges Toũ(t) As N → ∞mentioning

confidence: 99%

“…The proof includes a result on the algebraic L 2 -convergence rate of the semi-group (Section 4.4). The key point is the derivation of a Nash type inequality which provides an estimate for convergence rates slower than exponential ( [22], [6], [30]). The diffusion coefficient is given by an infrared regularization of the thermal conductivity obtained in [4], [5], with a proper renormalization (13).…”

Section: Introductionmentioning

confidence: 99%

Energy Diffusion in Harmonic System with Conservative Noise

Basile

Olla

2014

J Stat Phys

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“…Next, we consider the weak Poincaré inequality introduced in [23], which describes the general convergence rate of the associated semigroup:…”

Section: The Dirichlet Formsmentioning

confidence: 99%

Functional Inequalities for Particle Systems on Polish Spaces

Röckner

Wang

2006

Potential Anal

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Various Poincaré-Sobolev type inequalities are studied for a reaction-diffusion model of particle systems on Polish spaces. The systems we consider consist of finite particles which are killed or produced at certain rates, while particles in the system move on the Polish space interacting with one another (i.e. diffusion). Thus, the corresponding Dirichlet form, which we call reaction-diffusion Dirichlet form, consists of two parts: the diffusion part induced by certain Markov processes on the product spaces E n (n ≥ 1) which determine the motion of particles, and the reaction part induced by a Q-process on Z + and a sequence of reference probability measures, where the Q-process determines the variation of the number of particles and the reference measures describe the locations of newly produced particles. We prove that the validity of Poincaré and weak Poincaré inequalities are essentially due to the pure reaction part, i.e. either of these inequalities holds if and only if it holds for the pure reaction Dirichlet form, or equivalently, for the corresponding Q-process. But under a mild condition, stronger inequalities rely on both parts: the reaction-diffusion Dirichlet form satisfies a super Poincaré inequality (e.g. the log-Sobolev inequality) if and only if so do both the corresponding Q-process and the diffusion part. Explicit estimates of constants in the inequalities are derived. Finally, some specific examples are presented to illustrate the main results.

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Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups

Cited by 157 publications

References 24 publications

Hypercontractivity for perturbed diffusion semigroups

Hypercontractivity for perturbed diffusion semigroups

Energy Diffusion in Harmonic System with Conservative Noise

Functional Inequalities for Particle Systems on Polish Spaces

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