Quantitative stability of optimal transport maps under variations of the target measure

We study the quantitative stability of the quadratic optimal transport map between a fixed probability density p and a probability measure mu on R-d, which we denote T-mu. Assuming that the source density p is bounded from above and below on a compact convex set, we prove that the map mu 7! T is bi-Holder continuous on large families of probability measures, such as the set of probability measures whose moment of order p > d is bounded by some constant. These stability estimates show that the linearized optimal transport metric W-2,W-rho(mu,v) = ||T-mu -T-v||L-2(rho,R-d) is bi-Holder equivalent to the 2-Wasserstein distance on such sets, justifying its use in applications.