Wasserstein distance ($W_{\rho }(\mu ,\nu )$)

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 $\mu ,\nu$.