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

Krzysztof J. Ciosmak

TL;DR

The paper addresses the complete characterization of optimal plans for the three-marginal OT problem with a Hessian (second-order Beckmann) relaxation, under absolute continuity of marginals and a common barycentre. It introduces bimartingale couplings and leaf decompositions via a dual optimizer with a 1-Lipschitz derivative that is an isometry on leaves, reducing the global problem to leaf-wise problems on conditional marginals. The core results show that CC order and bimartingale couplings on mutually orthogonal subspaces fully describe optimal plans, with a precise leaf-wise reconstruction formula using a transform $R_S$. This framework connects to optimal grillage designs in 2D and suggests avenues for polar-set analysis and absolute continuity questions, offering a fine-grained, measurable decomposition of optimal solutions.

Abstract

We completely characterise the optimal solutions for the three-marginal optimal transport problem - introduced in [K. Bolbotowski, G. Bouchitté, Kantorovich-Rubinstein duality theory for the Hessian, 2024, preprint], and whose relaxation is the second-order Beckmann problem - for arbitrary pairs $μ,ν\in\mathcal{P}_2(\mathbb{R}^n)$ 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 $μ,ν$ 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 $μ,ν$, where the conditioning is defined relative to the foliation induced by $Du$.

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

TL;DR

The paper addresses the complete characterization of optimal plans for the three-marginal OT problem with a Hessian (second-order Beckmann) relaxation, under absolute continuity of marginals and a common barycentre. It introduces bimartingale couplings and leaf decompositions via a dual optimizer with a 1-Lipschitz derivative that is an isometry on leaves, reducing the global problem to leaf-wise problems on conditional marginals. The core results show that CC order and bimartingale couplings on mutually orthogonal subspaces fully describe optimal plans, with a precise leaf-wise reconstruction formula using a transform . This framework connects to optimal grillage designs in 2D and suggests avenues for polar-set analysis and absolute continuity questions, offering a fine-grained, measurable decomposition of optimal solutions.

Abstract

We completely characterise the optimal solutions for the three-marginal optimal transport problem - introduced in [K. Bolbotowski, G. Bouchitté, Kantorovich-Rubinstein duality theory for the Hessian, 2024, preprint], and whose relaxation is the second-order Beckmann problem - for arbitrary pairs 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 function with isometric derivative. We prove that the problem for completely decomposes into a collection of simpler problems on the leaves of the -Lipschitz derivative of an optimal solution for the dual problem. On each such, leaf the solution is expressed in terms of bimartingale couplings between conditional measures of , where the conditioning is defined relative to the foliation induced by .

Paper Structure

This paper contains 36 sections, 18 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Classical Kantorovich--Rubinstein duality and related decomposition into transport rays
  3. Optimality conditions
  4. Transport rays
  5. Decomposition
  6. Monotone order
  7. Further partition
  8. Characterisation of optimal transports
  9. Kantorovich--Rubinstein duality for the Hessian
  10. Second-order Beckmann problem
  11. Dual problem and optimality conditions
  12. Leaf decompositions
  13. Bimartingale couplings and convex-concave order
  14. Covariance matrices
  15. Characterisation of optimal plans
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $\mu,\nu\in\mathcal{P}_2(\mathbb{R}^n)$ be absolutely continuous with respect to the Lebesgue measure and have common barycentre. Suppose that there exists an optimal plan $\sigma\in\Sigma(\mu,\nu)$ with absolutely continuous third marginal. Then there exists a partition of $\mathbb{R}^n$, up to such that for $\theta$-almost every $\mathcal{S}$: Moreover a plan $\sigma\in \mathcal{P}((\mathbb

Theorems & Definitions (50)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • proof : Proof of Theorem \ref{['thm:deco']}
  • Theorem 2.6
  • Theorem 2.7
  • ...and 40 more