Sharp stability of Brenier maps via quantitative regularity of potentials
Résumé
We study the stability of optimal transport maps for the quadratic cost c(x, y) = ∥x -y∥^2 on R^d with respect to Wassersein-1 perturbations of the target measure. Existing results establish Hölder stability estimates, but the best-known exponent (1/6) does not match the theoretical lower bound (1/4). In this paper, we close this gap by proving a sharp 1/4-Hölder stability estimate when the source probability distribution has a bounded density. Our approach relies on analyzing the strong convexity of the Kantorovich functional through its associated Bregman-Kantorovich and Jensen-Kantorovich divergences. We show that quantitative bounds on the singularities of convex functions yield meaningful lower bounds on these divergences involving distances between transport maps. Furthermore, these methods also yield quantitative convergence rates for the entropic Brenier map towards the unregularized Brenier map as the regularization parameter vanishes, requiring only mild assumptions on the measures.
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