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

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

In Euclidean space, push an optimal transport plan $\pi$ forward by $(x,y)\mapsto (1-t)x+ty$. The resulting probability measures form a constant speed curve for the p-Wasserstein distance:If the plan is induced by a transport map $T$, this becomes the displayed title formula.

On a Polish space $(X,\rho )$, probability measures with finite first moments satisfyThe supremum runs over functions with Lipschitz constant at most one. Fixing $f(x_0)=0$ ensures integrability from the moment assumption.

For continuous functions $f,g$ on a compact metric space satisfying $f(x)+g(y)\le d(x,y)$, set $h(x)={\min }_y(d(x,y)-g(y))$. The triangle inequality makes $h$ $1$-Lipschitz, while the constraint gives $f\le h$ and evaluation at $y=x$ gives $g\le -h$. Thus the feasible pair $(h,-h)$ has at least as large an objective against any two probability measures. This explains why metric optimal transport duality reduces to a single $1$-Lipschitz function.

For $1\le p<\infty$ and probability measures with finite $p$th absolute moments,where $\rho$ is the ground metric and $\pi$ ranges over transport plans. For $p=2$ this is the second Wasserstein distance.

For probability measures with finite second moments on a metric space,

For probability measures ${\mu }_n,\mu$ on the real line with finite second moments, $W_2({\mu }_n,\mu )\to 0$ exactly when ${\mu }_n$ converges weakly to $\mu$ and their second moments converge. The common-quantile coupling is optimal and satisfies

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 $\mu ,\nu$.