Characterisation of optimal solutions to second-order Beckmann problem through bimartingale couplings and leaf decompositions

TL;DR

This paper characterizes the solutions to a three-marginal optimal transport problem, linking bimartingale couplings, leaf decompositions, and dual problem solutions, extending classical results like Strassen's theorem.

Contribution

It introduces the concept of bimartingale couplings, establishes their existence conditions, and decomposes the problem into simpler leaf-based subproblems.

Findings

Existence of bimartingale couplings under convex-concave order.

Dual problem attains a smooth optimal solution.

Decomposition into leaf-based subproblems with explicit solutions.

Abstract

We completely characterise the optimal solutions for the three-marginal optimal transport problem - introduced in [K. Bolbotowski, G. Bouchitt\'e, Kantorovich-Rubinstein duality theory for the Hessian, 2024, preprint], and whose relaxation is the second-order Beckmann problem - for arbitrary pairs $\mu,\nu\in\mathcal{P}_2(\mathbb{R}^n)$ of absolutely continuous measures with common barycentre such that there exists an optimal plan with absolutely continuous third marginal. In our work, we define the concept of bimartingale couplings for a pair of measures and establish several equivalent conditions that ensure such couplings exist. One of these conditions is that the pair is ordered according to the convex-concave order, thereby generalising the classical Strassen theorem. Another equivalent condition is that the dual problem associated with the second-order Beckmann problem attains its optimum at a $\mathcal{C}^{1,1}(\mathbb{R}^n)$ function with isometric derivative. We prove that the problem for $\mu,\nu$ completely decomposes into a collection of simpler problems on the leaves of the $1$-Lipschitz derivative $Du$ of an optimal solution $u\in\mathcal{C}^{1,1}(\mathbb{R}^n)$ for the dual problem. On each such, leaf the solution is expressed in terms of bimartingale couplings between conditional measures of $\mu,\nu$, where the conditioning is defined relative to the foliation induced by $Du$.