Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition

Abstract : The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigroup, and quadratic Wasserstein distance contraction properties at different times. On the other hand, in a compact Riemannian manifold, it implies a same-time Wasserstein contraction property for this semigroup. In this work we generalize the latter result to metric measure spaces and more importantly prove the converse: contraction inequalities are equivalent to curvature-dimension conditions. Links with functional inequalities are also investigated.
Complete list of metadatas

Cited literature [28 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01220776
Contributor : Ivan Gentil <>
Submitted on : Monday, October 26, 2015 - 10:13:34 PM
Last modification on : Thursday, June 20, 2019 - 1:24:09 AM
Long-term archiving on : Wednesday, January 27, 2016 - 5:10:51 PM

Files

BGGK-contraction.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01220776, version 1
  • ARXIV : 1510.07793

Citation

François Bolley, Ivan Gentil, Arnaud Guillin, Kazumasa Kuwada. Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition. Annali della Scuola Normale Superiore di Pisa, 2018, 18 (4), pp.1-36. ⟨hal-01220776⟩

Share

Metrics

Record views

582

Files downloads

138