Symmetric Monge-Kantorovich problems and polar decompositions of vector fields

الأصل · EN

For any given integer N≥ 2, we show that every bounded measurable vector field from a bounded domain Ω into ᵈ is N-cyclically monotone up to a measure preserving N-involution. The proof involves the solution of a multidimensional symmetric Monge-Kantorovich problem, which we first study in the case of a general cost function on a product domain Ωⁿ. The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually N-1 of them). In this case, we show that the supremum over all probability measures on Ωⁿ which are invariant under cyclic permutations and with a given first marginal μ, is attained on a probability measure that is supported on the graph of a function of the form x→ (x, Sx, S²x,..., Sⁿ⁻¹x), where S is a μ-measure preserving transformation on Ω such that Sⁿ=I a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are N-cyclically antisymmetric.