Abstract: 

We characterise the optimal solutions for the three-marginal optimal transport problem introduced in [K. Bolbotowski, G. Bouchitte, Kantorovich–Rubinstein duality theory for the Hessian, 2024, arXiv:2308.00811], and whose relaxation is the second-order Beckmann problem.

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 C1,1(Rn) function with isometric derivative.

For pairs μ, ν ∈ P2(Rn) of absolutely continuous measures with common barycentre such that there exists an optimal plan with absolutely continuous third marginal, we show 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 ∈ C1,1(Rn) 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.