Displacement smoothness of entropic optimal transport

Understand

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 $\mathcal{C}^{k+1}$ with $k\in \mathbb{N}^*$ then this map is Lipschitz continuous from the $L^2$-Wasserstein space to the space of $\mathcal{C}^k$ 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.