A PROOF OF THE CAFFARELLI CONTRACTION THEOREM VIA ENTROPIC REGULARIZATION

Max Fathi (1) , Nathael Gozlan (2) , Maxime Prod’homme (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

We give a new proof of the Caffarelli contraction theorem, which states that the Brenier optimal transport map sending the standard Gaussian measure onto a uniformly log-concave probability measure is Lipschitz. The proof combines a recent variational characterization of Lipschitz transport map by the second author and Juillet with a convexity property of optimizers in the dual formulation of the entropy-regularized optimal transport (or Schrödinger) problem.

Mots clés: Optimal transport; Caffarelli Contraction Theorem; Schroedinger bridges

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]; Optimisation et contrôle [math.OC]

Max Fathi, Nathael Gozlan, Maxime Prod’homme. A PROOF OF THE CAFFARELLI CONTRACTION THEOREM VIA ENTROPIC REGULARIZATION. Calculus of Variations and Partial Differential Equations, 2020, 59 (3), pp.96. ⟨10.1007/s00526-020-01754-0⟩. ⟨hal-02096009v2⟩