On the real line, order determines optimality. Pairing equal quantile levels produces a coupling with no crossings, and every crossing can be uncrossed without increasing a convex distance cost. This yields explicit formulas unavailable in higher dimensions.
Distribution and quantile functions
Definition. For \(\mu\in\mathcal P(\mathbb R)\), define \[ F_\mu(x)=\mu((-\infty,x]),\qquad F_\mu^{-1}(u)=\inf\{x:F_\mu(x)\ge u\},\quad 0<u<1. \] The generalized inverse \(F_\mu^{-1}\) is nondecreasing and left-continuous.
Quantile representation. If \(U\sim\operatorname{Unif}(0,1)\), then \(F_\mu^{-1}(U)\sim\mu\).
Proof. For every \(x\), \[ F_\mu^{-1}(u)\le x\quad\Longleftrightarrow\quad u\le F_\mu(x). \] Therefore \[ \mathbb P(F_\mu^{-1}(U)\le x)=\mathbb P(U\le F_\mu(x))=F_\mu(x), \] which identifies the distribution.
The quantile coupling
Definition. The monotone or comonotone coupling of \(\mu,\nu\) is \[ \pi^\uparrow=(F_\mu^{-1},F_\nu^{-1})_\#\lambda_{(0,1)}. \] If \(\mu\) has no atoms, the monotone transport map is \[ T(x)=F_\nu^{-1}(F_\mu(x)), \qquad T_\#\mu=\nu. \]
When \(\mu\) has atoms, the formula \(F_\nu^{-1}\circ F_\mu\) may fail to split them and need not push \(\mu\) to \(\nu\). The quantile coupling remains valid because the auxiliary uniform variable supplies the required randomization inside jumps.
The uncrossing inequality
Monge inequality. Let \(h:\mathbb R\to\mathbb R\) be convex. If \(x_1\le x_2\) and \(y_1\le y_2\), then \[ h(x_1-y_1)+h(x_2-y_2) \le h(x_1-y_2)+h(x_2-y_1). \]
Proof. Put \(a=x_1-y_2\), \(b=x_1-y_1\), \(c=x_2-y_2\), and \(d=x_2-y_1\). Then \(a\le b,c\le d\) and \(a+d=b+c\). Thus \(b,c\) are less spread than \(a,d\) with the same sum. Convexity, equivalently monotonicity of secant slopes, gives \(h(b)+h(c)\le h(a)+h(d)\).
The right-hand side pairs the smaller source with the larger target and the larger source with the smaller target. The lemma says that removing such a crossing cannot increase any convex displacement cost.
Optimality of monotone rearrangement
Monotone rearrangement theorem. Let \(c(x,y)=h(x-y)\) with \(h\) convex and lower semicontinuous. Among all \(\pi\in\Pi(\mu,\nu)\), the quantile coupling \(\pi^\uparrow\) minimizes \(\int c\,\mathrm{d}\pi\). If \(h\) is strictly convex and \(\mu\) is nonatomic, the optimal plan is unique and is induced by \(T=F_\nu^{-1}\circ F_\mu\), up to \(\mu\)-null sets.
Proof. For finitely supported measures with rational masses, split atoms into equal subatoms. Any coupling becomes a convex combination of matchings. Repeatedly apply the uncrossing inequality to each inverted pair; after finitely many swaps the matching is monotone and its cost has not increased. The unique monotone matching pairs equal ranks. Approximate general \(\mu,\nu\) weakly by quantized measures and use lower semicontinuity and stability of optimal costs. Strict convexity makes every genuine crossing strictly more expensive. A noncrossing plan with nonatomic first marginal has conditionals concentrated at the monotone map, giving uniqueness.
Explicit Wasserstein formulas
Corollary. For \(1\le p<\infty\) and \(\mu,\nu\in\mathcal P_p(\mathbb R)\), \[ W_p^p(\mu,\nu) =\int_0^1|F_\mu^{-1}(u)-F_\nu^{-1}(u)|^p\,\mathrm{d}u. \] For compactly supported measures, \[ W_\infty(\mu,\nu) =\operatorname*{ess\,sup}_{0<u<1} |F_\mu^{-1}(u)-F_\nu^{-1}(u)|. \]
Gaussian laws. If \(\mu=N(m_0,\sigma_0^2)\) and \(\nu=N(m_1,\sigma_1^2)\) with \(\sigma_0,\sigma_1>0\), then \[ T(x)=m_1+\frac{\sigma_1}{\sigma_0}(x-m_0), \qquad W_2^2(\mu,\nu)=(m_0-m_1)^2+(\sigma_0-\sigma_1)^2. \] The formula follows by writing both quantiles with the same standard normal quantile and integrating.
Empirical measures. If \[ \mu_n=\frac1n\sum_{i=1}^n\delta_{x_i},\qquad \nu_n=\frac1n\sum_{i=1}^n\delta_{y_i}, \] and \(x_{(1)}\le\cdots\le x_{(n)}\), \(y_{(1)}\le\cdots\le y_{(n)}\), then \[ W_p^p(\mu_n,\nu_n)=\frac1n\sum_{i=1}^n|x_{(i)}-y_{(i)}|^p. \] Thus one-dimensional empirical transport reduces to sorting.
Geodesics on the line
The quantile representation also linearizes Wasserstein geodesics: \[ F_{\mu_t}^{-1}(u)=(1-t)F_{\mu_0}^{-1}(u)+tF_{\mu_1}^{-1}(u). \] Consequently \(\mathcal P_2(\mathbb R)\) embeds isometrically as the closed convex cone of nondecreasing functions in \(L^2(0,1)\). This special Hilbertian description explains why one-dimensional barycenters and asymptotics are unusually explicit.