Ricci curvature of Markov chains on metric spaces

Ollivier, Yann

doi:10.1016/j.jfa.2008.11.001

Cited by 619 publications

(889 citation statements)

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“…(Actually the latter only yields an inequality, as there might be better transference plans than parallel transport. But parallel transport gives the exact value of the Wasserstein distance at this order [Oll09]. )…”

Section: Nmentioning

confidence: 96%

“…Interestingly, these groups turn out to be, in some sense, "generic" among discrete groups [Gro87,Oll05]. This time, let us begin with the discrete case, and the notion of coarse Ricci curvature used by the author [Oll07,Oll09]. Another approach introduced by Sturm [Stu06] and Lott-Villani [LV09], based on displacement convexity of entropy, is also presented below.…”

Section: Discrete Curvaturesmentioning

confidence: 99%

“…Note that we have computed κ(x, y) only for adjacent points x and y; but, as mentioned before, this is enough to imply the same lower bound on curvature for all pairs of points, by the triangle inequality for W 1 [Oll07,Oll09].…”

Section: Nmentioning

confidence: 99%

“…We mentioned above numerous more sophisticated examples from Markov chains or statistical physics; see also the examples in [Oll09].…”

Section: Nmentioning

confidence: 99%

“…This is indeed the case. For δ-hyperbolic groups, for instance, with b x the uniform measure on a ball of radius ≫ δ around x, one can check [Oll09] that coarse Ricci curvature is indeed negative (and actually takes the smallest possible value). For spaces with curvature bounded below in the sense of Alexandrov, equipped with their (continuous-time) canonical diffusion semigroup (see e.g.…”

Section: Nmentioning

confidence: 99%

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A visual introduction to Riemannian curvatures and some discrete generalizations

Ollivier¹

2013

CRM Proceedings and Lecture Notes

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Abstract. We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the strategies used to define analogues of curvature in non-smooth or discrete spaces, beginning with Alexandrov curvature and δ-hyperbolic spaces, and insisting on various notions of generalized Ricci curvature, which we briefly compare.The first part of this text covers in a hopefully intuitive and visual way some of the usual objects of Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, the Riemann tensor, Bianchi identities... For each of those we try to provide one or several pictures that convey the meaning (or at least one possible interpretation) of the formal definition.Next, we consider the problem of defining analogues of curvature for nonsmooth or discrete objets. This is in the spirit of "synthetic" or "coarse" geometry, in which large-scale properties of metric spaces are investigated instead of fine smallscale properties; one of the aims being to be impervious to small perturbations of the underlying space. Motivation for these non-smooth or discrete extensions often comes from various other fields of mathematics, such as group theory or optimal transport. From this viewpoint is emerging a theory of metric measure spaces, see for instance [Gro99].Thus we cover the definitions and mention some applications of δ-hyperbolic spaces and Alexandrov curvature (generalizing sectional curvature), and of two notions of generalized Ricci curvature: displacement convexity of entropy introduced by Sturm and Lott-Villani (following Renesse-Sturm and others), and coarse Ricci curvature as used by the author. We mention new theorems obtained for the original Riemannian case thanks to this more general viewpoint. We also briefly discuss the differences between these two approaches to Ricci curvature.Note that scalar curvature is not covered, for lack of the author's competence about recent developments and applications in this field.

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Section: Nmentioning

confidence: 96%

Section: Discrete Curvaturesmentioning

confidence: 99%

Section: Nmentioning

confidence: 99%

“…We mentioned above numerous more sophisticated examples from Markov chains or statistical physics; see also the examples in [Oll09].…”

Section: Nmentioning

confidence: 99%

Section: Nmentioning

confidence: 99%

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A visual introduction to Riemannian curvatures and some discrete generalizations

Ollivier¹

2013

CRM Proceedings and Lecture Notes

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Heat Flow on Time‐Dependent Metric Measure Spaces and Super‐Ricci Flows

Kopfer

Sturm

2018

Comm Pure Appl Math

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We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness, and regularity. Of particular interest are properties that characterize the underlying space as a super-Ricci flow as previously introduced by the second author [51]. Our main result yields the equivalence of F dynamic convexity of the Boltzmann entropy on the (time-dependent) L 2 -Wasserstein space, F monotonicity of L 2 -Kantorovich-Wasserstein distances under the dual heat flow acting on probability measures (backward in time), F gradient estimates for the heat flow acting on functions (forward in time), and F a Bochner inequality involving the time-derivative of the metric. Moreover, we characterize the heat flow on functions as the unique forward EVI flow for the (time-dependent) energy in L 2 -Hilbert space and the dual heat flow on probability measures as the unique backward EVI flow for the (timedependent) Boltzmann entropy in L 2 -Wasserstein space.

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Positively curved graphs

Yancey

2020

Journal of Graph Theory

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This paper consists of two halves. In the first half of the paper, we consider real-valued functions f whose domain is the vertex set of a graph G and that are Lipschitz with respect to the graph distance. By placing a uniform distribution on the vertex set, we treat as a random variable. We investigate the link between the isoperimetric function of G and the functions f that have maximum variance or meet the bound established by the subgaussian inequality. We present several results Question 1.2 (Alon et al. [2], p. 416). Let X be a variance-optimal function over G. Is it true for d sufficiently large and r, ℓ in appropriate ranges that |B rOur initial suspicion was that the question would have a positive answer, based on the following intuition. Constructions of functions that maximize variance whose expectations are 0 will evenly spread values toward ∞ and −∞ as possible. Because X is Lipschitz we have that B r (S ℓ ) ⊆ S ℓ+r . The hope is that V(G d ) − S ℓ + r, X = S −(ℓ + r), −X is relatively large.A positive answer to the question would have significance. Let X 1 , X 2 be variance-optimal over G 1 , G 2 , respectively. Let H = G 1 ◻ G 2 , and define variable Y over H as Y(u 1 , u 2 ) = X 1 (u 1 ) + X 2 (u 2 ). Bobkov YANCEY | 541 X .Once again we are able to use our statements describing optimal functions to characterize the optimal functions of specific examples.Theorem 2.11. Conjecture 1.5 is true.All of our discussion so far generalizes in the obvious way to arbitrary finite metric spaces. An example is the symmetric group S d on d elements equipped with the Hamming distance, which is d H (π, π′) = |{i : π(i) ≠ π′(i)}|. Bobkov et al [9] bounded σ S d . YANCEY | 543

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Ricci curvature of Markov chains on metric spaces

Cited by 619 publications

References 37 publications

A visual introduction to Riemannian curvatures and some discrete generalizations

A visual introduction to Riemannian curvatures and some discrete generalizations

Heat Flow on Time‐Dependent Metric Measure Spaces and Super‐Ricci Flows

Positively curved graphs

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