A Brenier-Strassen Theorem on CAT(kappa) Spaces

Nathael Gozlan; Hugo Malamut; Shin-Ichi Ohta

probability theory

A Brenier-Strassen Theorem on CAT(kappa) Spaces

arXiv:2607.26671

summary

The paper generalizes the Brenier‑Strassen theorem to spaces with curvature bounded above, proving existence and uniqueness of a Wasserstein‑2 projection onto measures dominated in convex order on CAT(0) and CAT(κ) spaces, and characterizing the optimal coupling via Lipschitz maps.

Abstract

We extend the Brenier-Strassen theorem about projections in convex order to non-flat spaces with curvature bounded from above. Precisely, for probability measures $I^¼$ , $I^½$ of finite second moment on a complete separable CAT(0) space, we prove that $I^¼$ admits a unique W 2 -projection \barÎ¼ to the set of probability measures dominated by $I^½$ in convex order. Moreover, the unique optimal coupling from $I^¼$ to \barÎ¼ is induced by a 1-Lipschitz map, without any absolute-continuity assumption on $I^¼$ . Our proof identifies the projection problem with a weak optimal transport problem whose cost is the squared distance to the set of convex means. We also establish a localized version on CAT(kappa) spaces with kappa \geq 0, where the optimal map is 1/2-H{Ã¶}lder continuous. Finally, we give a Strassen-type characterization of one-step barycentric martingales on proper CAT(0) spaces.

Topics & keywords

#optimal transport #cat(kappa) spaces #convex order #martingale transport #metric geometryW2 projection1-Lipschitz mapbarycentric martingaleStrassen theoremHölder continuity