Speaker
Description
Classical localisation in optimal transport decomposes transport into monotone couplings along transport rays. In this talk, I will present a second-order analogue: transport rays are replaced by the leaves of the 1-Lipschitz gradient Du of an optimal dual potential u in C^(1,1)(R^n), while monotone couplings are replaced by bimartingale couplings. This framework applies to the three-marginal optimal transport problem introduced by Bolbotowski and Bouchitté, whose relaxation is the second-order Beckmann problem.
We introduce bimartingale couplings and characterise their existence through a convex–concave order condition, extending Strassen’s theorem beyond convex order. This condition is also equivalent to the existence of an optimiser with isometric gradient for the dual problem associated with the second-order Beckmann problem.
For absolutely continuous measures with a common barycentre, we prove that the existence of one optimal plan with an absolutely continuous third marginal implies that every optimal plan decomposes along the leaves of Du. On each leaf, all optimal plans are completely characterised by bimartingale couplings between the corresponding conditional measures.
Without the absolute continuity assumption on the third marginal, a more general geometric decomposition persists: optimal plans are mixtures of plans concentrated on triples (x,y,z) with x in S_1, y in S_2, and z is in the intersection of S_1 and S_2, where S_1, and S_2 are neighbouring leaves of Du.