Backward and Forward Wasserstein Projections in Stochastic Order

TL;DR

This paper investigates metric projections in Wasserstein space defined by stochastic orders, establishing dualities, characterizations, and decompositions, especially for convex and subharmonic orders, extending existing theories and introducing new results.

Contribution

It introduces new duality results and polar factorization theorems for Wasserstein projections under stochastic orders, including the first characterization in subharmonic order.

Findings

Established duality for backward and forward Wasserstein projections.

Proved Brenier-Strassen type polar factorization theorems.

Extended decomposition results to subharmonic order with volume distortion properties.

Abstract

We study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Dualities for backward and forward projections are established under general conditions. Dual optimal solutions and their characterizations require study on a case-by-case basis. Particular attention is given to convex order and subharmonic order. While backward and forward cones possess distinct geometric properties, strong connections between backward and forward projections can be obtained in the convex order case. Compared with convex order, the study of subharmonic order is subtler. In all cases, Brenier-Strassen type polar factorization theorems are proved, thus providing a full picture of the decomposition of optimal couplings between probability measures given by deterministic contractions (resp. expansions) and stochastic couplings. Our results extend to the forward convex order case the decomposition obtained by Gozlan and Juillet, which builds a connection with Caffarelli's contraction theorem. A further noteworthy addition to the early results is the decomposition in the subharmonic order case where the optimal mappings are characterized by volume distortion properties. To our knowledge, this is the first time in this occasion such results are available in the literature.