Symmetric Monge-Kantorovich problems and polar decompositions of vector fields

Abstract

For any given integer $N\geq 2$, we show that every bounded measurable vector field from a bounded domain $\Omega$ into $\R^d$ 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 $\Omega^N$. 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 $\Omega^N$ which are invariant under cyclic permutations and with a given first marginal $\mu$, is attained on a probability measure that is supported on the graph of a function of the form $x\to (x, Sx, S^2x,..., S^{N-1}x)$, where $S$ is a $\mu$-measure preserving transformation on $\Omega$ such that $S^N=I$ a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are $N$-cyclically antisymmetric.