Issue |
ESAIM: COCV
Volume 30, 2024
|
|
---|---|---|
Article Number | 25 | |
Number of page(s) | 24 | |
DOI | https://doi.org/10.1051/cocv/2024013 | |
Published online | 09 April 2024 |
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
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 Ck+1 with k ∈ 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. 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.
Mathematics Subject Classification: 49Q22 / 49K40 / 35A15
Key words: Entropic optimal transport / Schrödinger map / Wasserstein gradient flows
© The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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