Optimal mass transport and symmetric representations of their cost functions

Abstract

We consider Monge–Kantorovich problems corresponding to general cost functions \(c(x,y)\) but with symmetry constraints on a Polish space \(X\times X\). Such couplings naturally generate anti-symmetric Hamiltonians on \(X\times X\) that are \(c\)-convex with respect to one of the variables. In particular, if \(c\) is differentiable with respect to the first variable on an open subset \(X\) in \( \mathbb {R}^d\), we show that for every probability measure \(\mu \) on \(X\), there exists a symmetric probability measure \(\pi _0\) on \(X\times X\) with marginals \(\mu \), and an anti-symmetric Hamiltonian \(H\) such that \(\nabla _2H(y, x)=\nabla _1c(x,y)\) for \( \pi _0\)-almost all \((x,y) \in X \times X.\) If \(\pi _0\) is supported on a graph \((x, Sx)\), then \(S\) is necessarily a \(\mu \)-measure preserving involution (i.e., \(S^2=I\)) and \(\nabla _2H(x, Sx)=\nabla _1c(Sx,x)\) for \(\mu \)-almost all \(x \in X.\) For monotone cost functions such as those given by \(c(x,y)=\langle x, u(y)\rangle \) or \(c(x,y)=-|x-u(y)|^2\) where \(u\) is a monotone operator, \(S\) is necessarily the identity yielding a classical result by Krause, namely that \(u(x)=\nabla _2H(x, x)\) where \(H\) is anti-symmetric and concave-convex.