Displacement smoothness of entropic optimal transport

¹ Ceremade, Université Paris Dauphine, PSL, 7577 Paris, and Inria-Paris, Mokaplan, France
² Institute of Mathematics, École polytechnique fédérale de Lausanne (EPFL), Station Z, CH-1015 Lausanne, Switzerland
³ Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75006 Paris, France

^* Corresponding author: carlier@ceremade.dauphine.fr

Received: 4 October 2022
Accepted: 27 February 2024

Abstract

The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schr¨odinger map. We prove that when the cost function is C^k+1 with k ∈ ℕ^* then this map is Lipschitz continuous from the L2-Wasserstein space to the space of C^k functions. Our result holds on compact domains and covers the multi-marginal case. We also include regularity results under negative Sobolev metrics weaker than Wasserstein under stronger smoothness assumptions on the cost. 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.