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 ρ and a probability measure μ on Rd, which we denote Tμ. Assuming that the source density ρ is bounded from above and below on a compact convex set, we prove that the map μ↦Tμ is bi-Hölder 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 W2,ρ(μ,ν)=‖Tμ−Tν‖L2(ρ,Rd) is bi-Hölder equivalent to the 2-Wasserstein distance on such sets, justifying its use in applications.