Lipschitz Continuity of the Schrödinger Map in Entropic Optimal Transport

Carlier, Guillaume; Chizat, Lenaic; Laborde, Maxime

Carlier, Guillaume; Chizat, Lenaic; Laborde, Maxime (2022), Lipschitz Continuity of the Schrödinger Map in Entropic Optimal Transport. https://basepub.dauphine.psl.eu/handle/123456789/23105

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Type

Document de travail / Working paper

Date

2022

Series title

Cahier de recherche CEREMADE, Université Paris Dauphine-PSL

Published in

Paris

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Author(s)

Carlier, Guillaume
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Chizat, Lenaic

Laborde, Maxime
Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]

Abstract (EN)

The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schrödinger map. We prove that when the cost function is Ck+1 with k in N* then this map is Lipschitz continuous from the L2-Wasserstein space to the space of Ck functions. Our result holds on compact domains and covers the multi-marginal case. As applications, we prove displacement smoothness of the entropic optimal transport cost and the well-posedness of certain Wasserstein gradient flows involving this functional, including the Sinkhorn divergence and a multi-species system.

Subjects / Keywords

Entropic Optimal Transport; Schrödinger map; Wasserstein gradient flows