Coupling of probability distributions

A coupling of probability distributions $\mu$ and $\nu$ is a joint distribution of $(X,Y)$ whose marginals are $\mu$ and $\nu$.

A maximal coupling makes $\mathbb{P}(X\ne Y)$ as small as possible among all couplings of its two marginals.

For a metric $\rho$, the first Wasserstein distance is

A Markov kernel is Wasserstein contractive with rate $\alpha >0$ when $W_{\rho }(\mu P,\nu P)\le e^{-\alpha }{W}_{\rho }(\mu ,\nu )$ for all probability distributions $\mu ,\nu$.