Displacement smoothness of entropic optimal transport

The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schr center dot odinger map. We prove that when the cost function is Ck+1 with k is an element of 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.