Curvature and transport inequalities for Markov chains in discrete spaces

Fathi, Max; Shu, Yan

doi:10.3150/16-bej892

Cited by 40 publications

(52 citation statements)

References 35 publications

(73 reference statements)

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“…Diameter bounds under a positive average Ollivier curvature can be found in [Pae12]. Diameter bounds under uniformly positive Bakry-Emery curvature are proven in [FS18;Hor+14;LMP16]. In this article we show that if the vertex degree is bounded and the curvature decays not faster than 1/R, then the graph is finite (see Theorem 4.15 and Corollary 4.16).…”

Section: Introductionmentioning

confidence: 77%

Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds

Münch

Wojciechowski

2019

Advances in Mathematics

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Discrete time random walks on a finite set naturally translate via a oneto-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representation of Ollivier curvature using the graph Laplacian. Using the Laplacian as a generator of a continuous time Markov chain, we connect Ollivier curvature with the heat equation which is strongly related to continuous time random walks. In particular, we prove that a lower bound on the Ollivier curvature is equivalent to a certain Lipschitz decay of solutions to the heat equation. This is a discrete analogue to a celebrated Ricci curvature lower bound characterization by Renesse and Sturm. Our representation of Ollivier curvature via the Laplacian allows us to deduce a Laplacian comparison principle by which we prove non-explosion and improved diameter bounds.Date: December 5, 2017. 5.1. Discrete and continuous time Markov kernels 30 5.2. Another Ricci curvature characterization 32

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Section: Introductionmentioning

confidence: 77%

Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds

Münch

Wojciechowski

2019

Advances in Mathematics

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“…In particular, Kantorovich type duality formulas are obtained [33,Theorem 9.6] under the assumption that c is convex with respect to the p variable (and some additional mild regularity conditions). We refer to [22,31,32,66,68] for works directly connected to [33] and to [65] for an up-to-date survey of applications of weak transport costs to concentration of measure. Besides their many applications in the field of functional inequalities and concentration of measure, it turns out that weak transport costs are also interesting in themselves as a natural generalization of the transportation problem.…”

Section: More About Weak Optimal Transport Costsmentioning

confidence: 99%

“…In particular, Kantorovich type duality formulas are obtained [, Theorem 9.6] under the assumption that

c

is convex with respect to the

p

variable (and some additional mild regularity conditions). We refer to for works directly connected to and to for an up‐to‐date survey of applications of weak transport costs to concentration of measure.…”

Section: Introductionmentioning

confidence: 99%

On a mixture of Brenier and Strassen Theorems

Gozlan

Juillet

2020

Proc. London Math. Soc.

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We give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in Gozlan, Roberto, Samson and Tetali (J. Funct. Anal. 273 (2017) 3327–3405). Optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling. We also establish some connections with Caffarelli's contraction theorem (Caffarelli, Comm. Math. Phys. 214 (2000) 547–563).

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“…In the next result we show that the Bakry-Émery curvature-dimension condition BE(K, ∞) with K > 0 implies a transport-information inequality, result that was obtained for the particular case of Markov chains in discrete spaces in [25]. Proof.…”

Section: Having In Mind Thatmentioning

confidence: 67%

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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Curvature and transport inequalities for Markov chains in discrete spaces

Cited by 40 publications

References 35 publications

Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds

Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds

On a mixture of Brenier and Strassen Theorems

The heat flow on metric random walk spaces

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