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47
Heat flow on Alexandrov spaces
, 2012
"... We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L²space produces the same evolution as the gradient flow of the relative entropy in the L²Wasserstein space. This means that the heat flow is well defined by either one of the t ..."
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Cited by 42 (15 self)
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We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L²space produces the same evolution as the gradient flow of the relative entropy in the L²Wasserstein space. This means that the heat flow is well defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as BakryÉmery gradient estimates and the Γ2condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift.
Ricci curvature of finite Markov chains via convexity of the entropy.
 Arch. Rational Mech. Anal.,
"... Abstract. We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the ..."
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Cited by 24 (6 self)
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Abstract. We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by BakryÉmery and OttoVillani. Furthermore we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube.
Finsler interpolation inequalities
, 2009
"... We extend CorderoErausquin, McCann and Schmuckenschläger’s Riemannian BorellBrascampLieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvaturedimension condition and a certain lower Ricci curvature bound. We also prove a ne ..."
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Cited by 17 (3 self)
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We extend CorderoErausquin, McCann and Schmuckenschläger’s Riemannian BorellBrascampLieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvaturedimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest.
BakryÉmery curvaturedimension condition and Riemannian Ricci curvature bounds
, 2014
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First variation formula in Wasserstein spaces over compact Alexandrov spaces
, 2010
"... We extend results proven by the second author ([Oh]) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces X with curvature bounded below: the gradient flow of a geodesically convex functional on the quadratic Wasserstein space (P(X), W2) satisfies the evolution variational ..."
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Cited by 10 (7 self)
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We extend results proven by the second author ([Oh]) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces X with curvature bounded below: the gradient flow of a geodesically convex functional on the quadratic Wasserstein space (P(X), W2) satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance. 1
Optimal transport and Ricci curvature In Finsler Geometry
, 2010
"... This is a survey article on recent progress (in [Oh3], [OS]) of the theory of weighted Ricci curvature in Finsler geometry. Optimal transport theory plays an impressive role as is developed in the Riemannian case by Lott, Sturm and Villani. ..."
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Cited by 6 (3 self)
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This is a survey article on recent progress (in [Oh3], [OS]) of the theory of weighted Ricci curvature in Finsler geometry. Optimal transport theory plays an impressive role as is developed in the Riemannian case by Lott, Sturm and Villani.