2 The Kantorovich–Rubinstein \(W_1\) distance
Solutions depend stably on their data, and the right distance is the Wasserstein-\(1\) metric. The development defines \(W_1\) by its dual sup-formula (Definition 1.4 of the paper), which makes the non-expansion estimates — the workhorses of the dynamical chapters — immediate. The primal (coupling) face, on which Dobrushin’s argument runs, is recovered by Kantorovich–Rubinstein duality: the hard direction is built from finite transport problems by cone separation, with no compactness of the ambient space and no attained optimum anywhere — every bound in the development is \(\varepsilon \)-optimal.
The cost-generic Wasserstein functional \(W_c(\mu ,\nu ) := \sup \{ \int \varphi \, \mathrm{d}\mu - \int \psi \, \mathrm{d}\nu \} \) over the \(c\)-admissible dual pairs (\(\varphi (x) - \psi (y) \le c(x,y)\)). The \(W_1\) distance is its specialization to the metric ground cost \(c = \mathrm{dist}\).
\(W_c(\mu ,\mu ) = 0\).
The diagonal pairing \(\varphi =\psi \) realizes the value \(0\), while any \(c\)-admissible pair gives \(\int \varphi \, \mathrm{d}\mu -\int \varphi \, \mathrm{d}\mu =0\); hence the supremum is exactly \(0\).
Symmetry: \(W_c(\mu ,\nu ) = W_c(\nu ,\mu )\) for a symmetric cost.
Negating the dual functional and swapping the admissible pair \((\varphi ,\psi )\mapsto (\psi ,\varphi )\) exchanges the roles of \(\mu \) and \(\nu \) while preserving admissibility, so the two suprema coincide.
The triangle inequality for \(W_c\).
Given admissible pairs for \((\mu ,\eta )\) and \((\eta ,\nu )\), their sum is admissible for \((\mu ,\nu )\) and the terms tested against \(\eta \) telescope; taking suprema gives subadditivity.
Non-expansion of \(W_c\) under a \(1\)-Lipschitz pushforward: \(W_c(T_\# \mu , T_\# \nu ) \le W_c(\mu ,\nu )\).
If \(T\) is \(1\)-Lipschitz then \(\varphi \circ T\) is \(c\)-admissible whenever \(\varphi \) is; substituting into the dual supremum for \(T_\# \mu ,T_\# \nu \) recovers the dual functional for \(\mu ,\nu \), so the value cannot increase.
For probability measures \(\mu , \nu \) with finite first moment,
the dual face of Definition 1.4 of the paper. The primal (coupling) face and the equality between them are the duality theorem closing this chapter. Neither optimum is attained anywhere in the development — every bound is \(\varepsilon \)-optimal.
\(W_1(\mu ,\mu ) = 0\).
Specialize the cost-generic identity to the metric ground cost \(c=\mathrm{dist}\).
Symmetry of \(W_1\).
Specialize the cost-generic symmetry to the symmetric cost \(c=\mathrm{dist}\).
The triangle inequality for \(W_1\).
Specialize the cost-generic triangle inequality to \(c=\mathrm{dist}\).
Non-expansion of \(W_1\) under a \(1\)-Lipschitz pushforward — the workhorse estimate for transporting mass along flows.
Specialize the cost-generic non-expansion: a \(1\)-Lipschitz test composed with a \(1\)-Lipschitz map is again \(1\)-Lipschitz.
Finiteness: if \(\mu \) and \(\nu \) have finite first moment then \(W_1(\mu ,\nu ) {\lt} \infty \).
A \(1\)-Lipschitz test normalized by \(\varphi (0)=0\) obeys \(|\varphi (x)|\le \| x\| \), so the dual functional is bounded by \(\int \| x\| \, \mathrm{d}\mu +\int \| x\| \, \mathrm{d}\nu \).
A coupling \(\pi \) of \(\mu \) and \(\nu \): a probability measure on the product space whose marginals are \(\mu \) and \(\nu \). The set of couplings is the domain of the primal face of \(W_1\).
The Monge–Kantorovich (primal) cost: \(\inf _\pi \int \mathrm{dist}\, \mathrm{d}\pi \) over couplings \(\pi \) of \(\mu \) and \(\nu \).
\(W_1\) expressed via the infimum over couplings — the primal face of Definition 1.4.
The easy direction of duality: the dual value is at most any coupling cost.
For any coupling \(\pi \) and \(1\)-Lipschitz \(\varphi \), \(\int \varphi \, \mathrm{d}\mu -\int \varphi \, \mathrm{d}\nu =\int (\varphi (x)-\varphi (y))\, \mathrm{d}\pi \le \int \mathrm{dist}(x,y)\, \mathrm{d}\pi \); take the supremum over \(\varphi \) and the infimum over \(\pi \).
The hard direction of duality: the optimal coupling cost is at most the dual value.
By reduction to finite transport problems: restrict to finite-range approximations, exhibit an \(\varepsilon \)-optimal coupling for each finite problem via cone-separation (the hyperplane separation theorem, Farkas duality), and transfer the bound to arbitrary marginals by a disintegration and triangle argument. No compactness of the ambient space is used, and no optimum is ever attained.
At the metric ground cost, for probability measures with finite first moment on a Polish-type space, the dual and primal faces agree: \(W_1(\mu ,\nu ) = \inf _\pi \int \mathrm{dist}\, \mathrm{d}\pi \). This is the bridge on which the Dobrushin argument crosses between the two faces of Definition 1.4.
Combine the easy inequality (dual \(\le \) any coupling cost) with the hard inequality (optimal coupling cost \(\le \) dual); equality is Kantorovich–Rubinstein duality.