On hyperboundedness and spectrum of Markov operators

Miclo, Laurent

doi:10.1007/s00222-014-0538-8

Cited by 63 publications

(83 citation statements)

References 35 publications

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“…As a consequence of a result by Miclo [41], we have that gap(−∆ m ) > 0 if ∆ m is ergodic and M m is hyperbounded, that is, if there exists p > 2 such that M m is bounded from L 2 (X, ν) to L p (X, ν). If we have that m x ≪ ν, i.e., m x = f x ν with f x ∈ L 1 (X, ν), and we assume that…”

Section: Indeed By Theorem 33 For Any Borel Set a ⊂ Xmentioning

confidence: 79%

The heat flow on metric random walk spaces

Mazón

Solera

Toledo

2020

Journal of Mathematical Analysis and Applications

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In this paper we study the Heat Flow on Metric Random Walk Spaces, which unifies into a broad framework the heat flow on locally finite weighted connected graphs, the heat flow determined by finite Markov chains and some nonlocal evolution problems. We give different characterizations of the ergodicity and we also prove that a metric random walk space with positive Ollivier-Ricci curvature is ergodic. Furthermore, we prove a Cheeger inequality and, as a consequence, we show that a Poincaré inequality holds if and only if an isoperimetrical inequality holds. We also study the Bakry-Émery curvature-dimension condition and its relation with functional inequalities like the Poincaré inequality and the transport-information inequalities. µ * m(A) := X m x (A)dµ(x), for all Borel sets A ⊂ X,

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Section: Indeed By Theorem 33 For Any Borel Set a ⊂ Xmentioning

confidence: 79%

The heat flow on metric random walk spaces

Mazón

Solera

Toledo

2020

Journal of Mathematical Analysis and Applications

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“…One could also obtain similar lower bounds for σ k from the higher order Cheeger inequality for λ k [22,10], which should be compared with the higher order Cheeger type inequality proved in [14]. The lower bounds for σ k given in [17,14] depends on the global geometry of the manifold and not only the geometry of the manifold near the boundary.…”

Section: Introductionmentioning

confidence: 79%

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Colbois

Girouard

Hassannezhad

2020

Journal of Functional Analysis

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Given two compact Riemannian manifolds M 1 and M 2 such that their respective boundaries Σ 1 and Σ 2 admit neighbourhoods Ω 1 and Ω 2 which are isometric, we prove the existence of a constant C such that |σ k (M 1 ) − σ k (M 2 )| ≤ C for each k ∈ N. The constant C depends only on the geometry of Ω 1 ∼ = Ω 2 . This follows from a quantitative relationship between the Steklov eigenvalues σ k of a compact Riemannian manifold M and the eigenvalues λ k of the Laplacian on its boundary. Our main result states that the difference |σ k − √ λ k | is bounded above by a constant which depends on the geometry of M only in a neighbourhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant C is given explicitly in terms of bounds on the geometry of Ω 1 ∼ = Ω 2 .1991 Mathematics Subject Classification. 35P15 (primary), 58C40, 35P20 (secondary).1 The notation O(k −∞ ) designates a quantity which tends to zero faster than any power of k.

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“…Thus, according to [18,Corollary 1.3] (see also [14,Theorem 1]), the defective logSobolev inequality implies the desired log-Sobolev inequality. Then the proof is finished.…”

Section: Perturbation Argumentmentioning

confidence: 96%

Functional inequalities for convolution probability measures

Wang¹,

Wang²

2016

Ann. Inst. H. Poincaré Probab. Statist.

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Let µ and ν be two probability measures on R d , where µ(dx) = e −V (x) dx R d e −V (x) dx for some V ∈ C 1 (R d ). Explicit sufficient conditions on V and ν are presented such that µ * ν satisfies the log-Sobolev, Poincaré and super Poincaré inequalities. In particular, if V (x) = λ|x| 2 for some λ > 0 and ν(e λθ|·| 2 ) < ∞ for some θ > 1, then µ * ν satisfies the log-Sobolev inequality. This improves and extends the recent results on the log-Sobolev inequality derived in [20] for convolutions of the Gaussian measure and compactly supported probability measures. On the other hand, it is well known that the log-Sobolev inequality for µ * ν implies ν(e ε|·| 2 ) < ∞ for some ε > 0.Des conditions explicites suffisantes sur V et ν sont présentées telles que µ * ν satisfait des inégalités de Sobolev logarithmique, de Poincaré et de superPoincaré. En particulier, si V (x) = λ|x| 2 pour quelque λ > 0 et ν(e λθ|·| 2 ) < ∞ avec θ > 1, alors µ * ν satisfait l'inégalité de Sobolev logarithmique. Cela améliore et etend des résultats récents sur l'inégalité de Sobolev logarithmique obtenus dans [20] pour des convolutions de la mesure de Gauss et des mesures de probabilitéà support compact. D'autre part, il est bien connu que l'inégalité de Sobolev logarithmique pour µ * ν implique ν(e ε|·| 2 ) < ∞ pour quelque ε > 0.

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On hyperboundedness and spectrum of Markov operators

Cited by 63 publications

References 35 publications

The heat flow on metric random walk spaces

The heat flow on metric random walk spaces

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Functional inequalities for convolution probability measures

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