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.