Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates

Kuwada, Kazumasa

doi:10.1007/s00526-014-0781-2

Cited by 17 publications

(24 citation statements)

References 46 publications

(81 reference statements)

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“…Indeed, this proof will yield a slightly stronger statement: the equivalence of the respective estimates for given pairs s; t. See also [31] for a related result. Indeed, this proof will yield a slightly stronger statement: the equivalence of the respective estimates for given pairs s; t. See also [31] for a related result.…”

Section: Duality Between Transport and Gradient Estimates In The Casementioning

confidence: 85%

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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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Section: Duality Between Transport and Gradient Estimates In The Casementioning

confidence: 85%

“…Here we present a direct, much simpler proof in the particular case N D 1. Indeed, this proof will yield a slightly stronger statement: the equivalence of the respective estimates for given pairs s; t. See also [31] for a related result.…”

Section: Duality Between Transport and Gradient Estimates In The Casementioning

confidence: 85%

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

Kopfer

Sturm

2018

Comm Pure Appl Math

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“…As noted in Section 1, this result has a particular new flavor. Indeed, the recent results (1.2) and (1.3) in [9,22,27] present a dimensional correction term for the contraction property, but for solutions at different times only. If the approaches for these inequalities are slightly different, then it would be of interest to obtain a dimensional correction term for our contraction also at different times.…”

Section: Contraction Property Under Cd(r N)mentioning

confidence: 99%

Dimensional contraction via Markov transportation distance

Bolley¹,

Gentil

Guillin³

2014

Journal of the London Mathematical Society

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It is now well known that curvature conditionsà la Bakry-Émery are equivalent to contraction properties of the heat semigroup with respect to the classical quadratic Wasserstein distance. However, this curvature condition may include a dimensional correction which up to now had not induced any strenghtening of this contraction. We first consider the simplest example of the Euclidean heat semigroup, and prove that indeed it is so. To consider the case of a general Markov semigroup, we introduce a new distance between probability measures, based on the semigroup, and adapted to it. We prove, in the setting of a compact Riemannian manifold, that this Markov transportation distance satisfies the same properties as the Wasserstein distance does in the specific case of the Euclidean heat semigroup, namely dimensional contraction properties and Evolution Variational Inequalities.

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“…For instance in [Kuw13,BGL15] the authors prove that if M is a n-dimensional Riemannian manifold with a non-negative Ricci curvature, then for any f, g probability densities with respect to dx, and any s, t 0, W 2 2 (P s f dx, P t gdx) ≤ W 2 2 (f dx, gdx) + 2n( √ s − √ t) 2 , ∀s, t 0,…”

Section: Introductionmentioning

confidence: 99%

“…• In [Kuw13], K. Kuwada proves that the Ricci curvature is bounded from below by R ∈ R if and only if for every s, t 0,…”

Section: Introductionmentioning

confidence: 99%

Dimensional Contraction in Wasserstein Distance for Diffusion Semigroups on a Riemannian Manifold

Gentil

2015

Potential Anal

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We prove a refined contraction inequality for diffusion semigroups with respect to the Wasserstein distance on a compact Riemannian manifold taking account of the dimension. The result generalizes in a Riemannian context, the dimensional contraction established in [BGG13] for the Euclidean heat equation. It is proved by using a dimensional coercive estimate for the Hodge-de Rham semigroup on 1-forms.

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Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates

Cited by 17 publications

References 46 publications

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

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

Dimensional contraction via Markov transportation distance

Dimensional Contraction in Wasserstein Distance for Diffusion Semigroups on a Riemannian Manifold

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