Characterization of absolutely continuous curves in Wasserstein spaces

Lisini, Stefano

doi:10.1007/s00526-006-0032-2

Cited by 97 publications

(108 citation statements)

References 24 publications

Supporting

Mentioning

106

Contrasting

Order By: Relevance

“…functionals the descending slope is an upper gradient, in particular the property we shall need is 25) for all locally absolutely continuous curves y : [0, ∞) → D(E). A metric gradient flow for the K-geodesically convex functional E is a locally absolutely continuous curve y : [0, ∞) → D(E) along which (2.25) holds as an equality and moreover |ẏ t | = |∇ − E|(y t ) for a.e.…”

Section: Convex Functionals: Gradient Flows Entropy and The Cd(k ∞mentioning

confidence: 99%

See 1 more Smart Citation

Metric measure spaces with Riemannian Ricci curvature bounded from below

Ambrosio¹,

Gigli²,

Savaré³

2014

Duke Math. J.

445

688

View full text Add to dashboard Cite

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.

show abstract

Section: Convex Functionals: Gradient Flows Entropy and The Cd(k ∞mentioning

confidence: 99%

“…Now use [25] to get the existence of a plan π ∈ P(AC 2 ([0, 1]; X)) such that (e t ) ♯ π = µ t := ρ t m for all t ∈ [0, 1] and …”

Section: Proposition 410 (Properties Ofmentioning

confidence: 99%

Metric measure spaces with Riemannian Ricci curvature bounded from below

Ambrosio¹,

Gigli²,

Savaré³

2014

Duke Math. J.

445

688

View full text Add to dashboard Cite

show abstract

“…The superposition is represented by means of a probability measure on the space of continuous curves in R n G , concentrated on the subset AC 2 (I; R n G ). This point of view was studied in [24] for arbitrary metric spaces. We recall here the main result adapted to our purposes.…”

Section: Representation Of Absolutely Continuous Curves In Wassersteimentioning

confidence: 99%

Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

Lisini¹

2008

ESAIM: COCV

Self Cite

View full text Add to dashboard Cite

Abstract.We study existence and approximation of non-negative solutions of partial differential equations of the typewhere A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition,, we show that u is the "gradient flow" of φ with respect to the 2-Wasserstein distance between probability measures on the space R n , endowed with the Riemannian distance induced by A −1 . In the case of uniform convexity of V , long time asymptotic behaviour and decay rate to the stationary state for solutions of equation (0.1) are studied. A contraction property in Wasserstein distance for solutions of equation (0.1) is also studied in a particular case.Mathematics Subject Classification. 35K55, 35K15, 35B40.

show abstract

“…Theorem 2.3 (Superposition principle, [24]). Let (X, d) be a complete and separable metric space and let…”

Section: The Set Of Optimal Dynamical Plans From µ To µ Is Denoted Bymentioning

confidence: 99%

Angles between Curves in Metric Measure Spaces

Han

Mondino

2017

Analysis and Geometry in Metric Spaces

View full text Add to dashboard Cite

Abstract:The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on RCD * (K, N) metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.

show abstract

Characterization of absolutely continuous curves in Wasserstein spaces

Cited by 97 publications

References 24 publications

Metric measure spaces with Riemannian Ricci curvature bounded from below

Metric measure spaces with Riemannian Ricci curvature bounded from below

Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

Angles between Curves in Metric Measure Spaces

Contact Info

Product

Resources

About