A remark on the optimal transport between two probability measures sharing the same copula

Abstract

We are interested in the Wasserstein distance between two probability measures on ${\mathbb{R}}^n$ sharing the same copula $C$. The image of the probability measure $dC$ by the vectors of pseudo-inverses of marginal distributions is a natural generalization of the coupling known to be optimal in dimension $n=1$. It turns out that for cost functions $c(x,y)$ equal to the $p$-th power of the $L^q$ norm of $x-y$ in ${\mathbb{R}}^n$, this coupling is optimal only when $p=q$ i.e. when $c(x,y)$ may be decomposed as the sum of coordinate-wise costs.