On A Mixture Of Brenier and Strassen Theorems

TL;DR

This paper characterizes optimal transport plans involving a deterministic gradient map followed by a martingale coupling, connecting classical optimal transport with martingale theory and contraction properties.

Contribution

It introduces a novel characterization of optimal plans combining Brenier's theorem with Strassen's martingale coupling, extending classical optimal transport results.

Findings

Optimal plans are compositions of a gradient map and a martingale coupling.

Connections established between optimal transport and Caffarelli's contraction theorem.

New insights into transport costs involving martingale structures.

Abstract

We give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in [33]. Optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling. We also establish some connections with Caffarelli's contraction theorem [14].