If I have two probability distributions $\mu$ and $\nu$ defined on $X$ and $Y$ respectively, then the *$p$-th Wasserstein distance* between the two of them is defined as $$W\_p(\mu,\nu) = \left(\inf\_{\pi\in\Pi(\mu,\nu)}\int\_{X\times Y}d(x,y)^{p}\,\mathrm{d}\pi(x,y)\right)^{1/p}
 $$
where $\Pi(\mu,\nu)$ is the set of all distributions on $X\times Y$ whose marginals are $\mu$ and $\nu$ respectively and $p\geq 1$. My question is: is the limiting behavior of this quantity as $p\to\infty$ defined? Does it have a name? 



Is there an $\infty$ version of the Wasserstein distance between two distributions?
I see no reason why $W\_\infty := \inf\_{\pi\in\Pi}||d||\_{L^\infty(\pi)}$ wouldn't fit. This is not to claim that $\lim\_{p\to\infty} W\_p=W\_\infty$ (which I believe is true anyway), but at least it exists, although with properties different from the $W\_p$s, $p<\infty$.


