Heat Flow on Alexandrov Spaces

Gigli, Nicola; Kuwada, Kazumasa; Ohta, Shin ichi

doi:10.1002/cpa.21431

Cited by 102 publications

(141 citation statements)

References 38 publications

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“…As a further step we consider the distance induced by the bilinear form E Remark 6.6 In [22] the techniques of [19,2] are applied to a case slightly different than the one considered here. The starting point of [22] is a Dirichlet form E on a measure space (X, m) and X is endowed with the distance d E .…”

Section: Dirichlet Form and Brownian Motionmentioning

confidence: 99%

“…Assuming compactness of (X, d E ), K-geodesic convexity of Ent m in P 2 (X) with cost function c = d 2 E , doubling, weak (1, 2)-Poincaré inequality and the validity of the so-called Newtonian property, the authors prove that the L 2 (X, m) heat flow induced by E coincides with H t . The authors also analyze some consequences of this identification, as Bakry-Emery estimates and the short time asymptotic of the heat kernel (a theme discussed neither here nor in [19]). As a consequence of [22,Theorem 5.1] and [2, Theorem 9.3] the Dirichlet form coincides with the Cheeger energy of (X, d E , m) (because their flows coincide).…”

Section: Dirichlet Form and Brownian Motionmentioning

confidence: 99%

“…Having (1.2) at our disposal at the level of measures, it is easy to obtain fundamental solutions, integral representation formulas, regularizing and contractivity properties of the heat flow, which exhibits a strong Feller regularization from L ∞ (X, m) to Lipschitz. Denoting by W 1,2 (X, d, m) ⊂ L 2 (X, m) the finiteness domain of Ch, the identification of the L 2 -gradient flow of Ch and of the W 2 -gradient flow of Ent m in conjunction with the K-contractivity of W 2 along the heat flow, yields, as in [19], the Bakry-Emery estimate…”

Section: Introductionmentioning

confidence: 99%

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Metric measure spaces with Riemannian Ricci curvature bounded from below

Ambrosio¹,

Gigli²,

Savaré³

2014

Duke Math. J.

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442

688

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In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X, d, m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, globalto-local and local-to-global properties. In these spaces, that we call RCD(K, ∞) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-Emery estimates and the L ∞ − Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger's relaxed slope and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincaré and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.

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Section: Dirichlet Form and Brownian Motionmentioning

confidence: 99%

Section: Dirichlet Form and Brownian Motionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Metric measure spaces with Riemannian Ricci curvature bounded from below

Ambrosio¹,

Gigli²,

Savaré³

2014

Duke Math. J.

Self Cite

442

688

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“…Another one initiated by the seminal work of Jordan et al [26] is to consider heat ow as the gradient ow of the relative entropy in the L -Wasserstein space. These interpretations and their equivalence were generalized to various settings ( [52], [39], [24], [18], [44], [35], [23], [10] etc.) including singular spaces without di erentiable structures.…”

Section: On the Curvature And Heat Flow On Hamiltonian Systemsmentioning

confidence: 99%

On the Curvature and Heat Flow on Hamiltonian Systems

Ohta

2014

Analysis and Geometry in Metric Spaces

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Abstract:We develop the di erential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation. We prove appropriate generalizations of the Bochner-Weitzenböck formula and Laplacian comparison theorem, and study the heat ow.Keywords: Hamiltonian, curvature, comparison theorem, heat ow MSC: Primary: 53C21; Secondary: 58J35, 49Q20 DOI 10.2478/agms-2014-0003 Received August 20, 2013 accepted February 14, 2014 IntroductionThe aim of this article is to apply the recently developed technique in Finsler geometry to the study of Hamiltonian systems. A Finsler manifold carries a (Minkowski) norm on each tangent space. Although Finsler manifolds form a much wider class than Riemannian manifolds, the notion of curvature makes sense and we can consider various comparison theorems similarly to the Riemannian case (see, e.g., [54]). Especially, the weighted Ricci curvature introduced by the author [41] has fruitful applications including the curvature-dimension condition ([41]), Laplacian comparison theorem for the natural nonlinear Laplacian ([44]), Bochner-Weitzenböck formula and gradient estimates ([46]), and generalizations of the CheegerGromoll splitting theorem ([43]). To be precise, the weighted Ricci curvature is de ned for a pair consisting of a Finsler manifold and a measure on it, and our Laplacian depends on the choice of the measure (see Subsections 2.3, 4.1 for details).Then, it is natural to expect that the theory of curvature can be applied beyond Finsler manifolds, and a class of manifolds M endowed with Lagragians L or, equivalently, Hamiltonians H is a natural choice. In fact, on the one hand, we know that Agrachev and Gamkrelidze [3] (see also [2]) have developed the theory of curvature operator for Hamiltonian systems in connection with optimal control theory (see [1] for a dynamical application). On the other hand, optimal transport theory (which is related to the curvature-dimension condition) for Lagrangian cost functions has been well investigated ([13], [19], [58]). Furthermore, Lee [31] recently showed a Riccati equation (see also [4], [5], [32] for the sub-Riemannian case) as well as convexity estimates for entropy functionals along smooth optimal transports for general (time-dependent) Hamiltonian systems, by means of the curvature operator. His uni ed approach recovers both the curvature-dimension condition (CD(K, ∞) and CD( , N) to be precise) for Riemannian or Finsler manifolds and various monotonicity formulas along ows in Riemannian metrics related to the Ricci ow.Our Hamiltonian will always be time-independent and non-negative. Compared with the Finsler situation, the lack of the homogeneity causes many di culties, for example, we need to take care of the di erence between the Lagrangian and the Hamiltonian (they coincide as functions via the Legendre transform in the Finsler case). Nevertheless, by combining the Riccati equation and the technique in the Finsler case, we prove ...

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“…For spaces with curvature bounded below in the sense of Alexandrov, equipped with their (continuous-time) canonical diffusion semigroup (see e.g. [GKO]), a result of Gigli, Kuwada and Ohta [GKO,Oht09a] (combined with [Pet,ZZ10]) implies that if Alexandrov curvature is bounded below, then coarse Ricci curvature is bounded below by the corresponding value.…”

Section: Nmentioning

confidence: 99%

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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Heat Flow on Alexandrov Spaces

Cited by 102 publications

References 38 publications

Metric measure spaces with Riemannian Ricci curvature bounded from below

Metric measure spaces with Riemannian Ricci curvature bounded from below

On the Curvature and Heat Flow on Hamiltonian Systems

A visual introduction to Riemannian curvatures and some discrete generalizations

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