Buser's inequality on infinite graphs

Liu, Shuang

doi:10.1016/j.jmaa.2019.03.023

Cited by 9 publications

(8 citation statements)

References 22 publications

(30 reference statements)

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“…Buser inequality is the reverse inequality which has been proven classically under non-negative Ricci curvature [Bus82]. Recently, there has been significant interest to obtain analogous results in the discrete setting (see [LMP19,LP18,Liu19,KKRT16] for Bakry Emery curvature, [Mün19] for Ollivier curvature, and [EF18] for entropic curvature) Proposition 7.1. Suppose K, T > 0, ρ ∈ C(V ), G a mwg with Deg max < ∞ and CD(ρ, ∞), and…”

Section: Buser's Inequality and Curvature In Kato Classmentioning

confidence: 99%

Spectrally positive Bakry-Émery Ricci curvature on graphs

Münch

Rose

2020

Journal de Mathématiques Pures et Appliquées

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We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that L + 2 Ric is a positive operator where L is the graph Laplacian. Assuming that the negative part of the Ricci curvature is small in Kato sense, we prove diameter bounds, elliptic Harnack inequality and Buser inequality. This article seems to be the first one establishing these results while allowing for some negative curvature. * MPI Leipzig, muench@mis.mpg.de † MPI Leipzig, crose@mis.mpg.de

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Section: Buser's Inequality and Curvature In Kato Classmentioning

confidence: 99%

Spectrally positive Bakry-Émery Ricci curvature on graphs

Münch

Rose

2020

Journal de Mathématiques Pures et Appliquées

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“…(ii) Importantly, the CD Υ pκ, nq condition (or more generally CD Υ pκ, F q with F prq " 1 n r 2 as r Ñ 0 `) implies the Bakry-Émery condition CDpκ, nq. This fact relies on the identities (19) lim…”

Section: The CD υ Condition With Finite Dimension and Some Examplesmentioning

confidence: 99%

“…(iii) The property (19) has an important consequence for CD-functions that behave like a power-type function near the origin. Indeed, if CD Υ pκ, F q holds with κ P R and F prq " r 1`δ as r Ñ 0 `for some δ ą 0, then we infer from ( 19) that for any f P ℓ 8 pXq with ´Lf pxq ą 0, x P X, we have at x…”

Section: The CD υ Condition With Finite Dimension and Some Examplesmentioning

confidence: 99%

Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains

Weber¹

2020

Preprint

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In the setting of reversible continuous-time Markov chains, the CD Υ condition has been shown recently to be a consistent analogue to the Bakry-Émery condition in the diffusive setting in terms of proving Li-Yau inequalities under a finite dimension term and proving the modified logarithmic Sobolev inequality under a positive curvature bound. In this article we examine the case where both is given, a finite dimension term and a positive curvature bound. For this purpose we introduce the CD Υ pκ, F q condition, where the dimension term is expressed by a so called CD-function F . We derive functional inequalities relating the entropy to the Fisher information, which we will call entropy-information inequalities. Further, we deduce applications of entropy-information inequalities such as ultracontractivity bounds, exponential integrability of Lipschitz functions, finite diameter bounds and a modified version of the celebrated Nash inequality.

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“…By (5.8) we recover (by different methods) exactly the same diameter bound as in [18], where there it is assumed on the one hand only CD(κ, n) but on the other hand that the underlying graph to L is locally finite and satisfies the completeness assumption and non-degeneracy of the vertex measure. Note that by [19,Theorem 2.2] the latter boils down to the case of finite graphs since κ > 0. Hence, although the curvature-dimension condition of Corollary 5.5 is more restrictive, the setting where it applies can be expected to be more general compared to the one of [18].…”

Section: Entropy-information Inequalities For Markov Chainsmentioning

confidence: 99%

Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains

Weber

2021

Electron. J. Probab.

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In the setting of reversible continuous-time Markov chains, the CDΥ condition has been shown recently to be a consistent analogue to the Bakry-Émery condition in the diffusive setting in terms of proving Li-Yau inequalities under a finite dimension term and proving the modified logarithmic Sobolev inequality under a positive curvature bound. In this article we examine the case where both is given, a finite dimension term and a positive curvature bound. For this purpose we introduce the CDΥ(κ, F ) condition, where the dimension term is expressed by a so called CD-function F . We derive functional inequalities relating the entropy to the Fisher information, which we will call entropy-information inequalities. Further, we deduce applications of entropyinformation inequalities such as ultracontractivity bounds, exponential integrability of Lipschitz functions, finite diameter bounds and a modified version of the celebrated Nash inequality.

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Buser's inequality on infinite graphs

Cited by 9 publications

References 22 publications

Spectrally positive Bakry-Émery Ricci curvature on graphs

Spectrally positive Bakry-Émery Ricci curvature on graphs

Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains

Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains

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