Two important extensions alter the set of admissible couplings in different directions. Multi-marginal transport couples more than two laws simultaneously. Martingale transport keeps two marginals but imposes a conditional-mean constraint. Their compactness arguments resemble classical transport, whereas their geometry and dual variables are genuinely different.
The multi-marginal Kantorovich problem
Definition. Let \(X_i\) be Polish spaces, let \(\mu_i\in\mathcal P(X_i)\), and let \(c:X_1\times\cdots\times X_N\to(-\infty,\infty]\). Write \[ \Pi(\mu_1,\ldots,\mu_N) = \{\gamma\in\mathcal P(X_1\times\cdots\times X_N): (p_i)_\#\gamma=\mu_i,\ 1\le i\le N\}. \] The multi-marginal transport value is \[ \mathsf{MMOT}_c(\mu_1,\ldots,\mu_N) = \inf_{\gamma\in\Pi(\mu_1,\ldots,\mu_N)} \int c(x_1,\ldots,x_N)\,\mathrm{d}\gamma. \]
A plan \(\gamma\) describes a joint law with all prescribed one-coordinate laws. Pairwise optimal couplings generally cannot be assembled independently: their shared coordinates must be compatible with one common joint distribution.
Existence. Suppose \(c\) is lower semicontinuous and \[ c(x_1,\ldots,x_N)\ge \sum_{i=1}^N a_i(x_i) \] for continuous functions \(a_i\in L^1(\mu_i)\). If the value is finite, an optimal multi-marginal plan exists. In particular, the conclusion holds whenever \(c\) is nonnegative and lower semicontinuous.
Proof. For compact sets \(K_i\subset X_i\), \[ \gamma\!\left((K_1\times\cdots\times K_N)^c\right) \le \sum_{i=1}^N\mu_i(K_i^c) \] for every admissible \(\gamma\). Tightness of the marginals therefore gives uniform tightness of \(\Pi(\mu_1,\ldots,\mu_N)\). A minimizing sequence has a weakly convergent subsequence. Coordinate projections are continuous, so the limit retains every marginal. Apply the Portmanteau theorem to \(c-\sum_i a_i\), then restore the fixed integrals \(\int a_i\,\mathrm{d}\mu_i\).
Duality and splitting sets
Multi-marginal Kantorovich duality. On compact \(X_i\), for continuous \(c\), \[ \inf_{\gamma\in\Pi(\mu_1,\ldots,\mu_N)}\int c\,\mathrm{d}\gamma = \sup_{\sum_i\varphi_i(x_i)\le c(x_1,\ldots,x_N)} \sum_{i=1}^N\int\varphi_i\,\mathrm{d}\mu_i. \] The dual contains one potential for each marginal.
Weak duality follows by integrating the pointwise inequality. Strong duality is the same separation principle as in the two-marginal problem, applied to the linear marginal map \(\gamma\mapsto((p_1)_\#\gamma,\ldots,(p_N)_\#\gamma)\). Under the displayed assumptions, normalized maximizing potentials may be chosen and compactness yields dual attainment.
Definition. A set \(\Gamma\subset X_1\times\cdots\times X_N\) is a \(c\)-splitting set if there are functions \(\varphi_i\) such that \[ \sum_i\varphi_i(x_i)\le c(x_1,\ldots,x_N) \quad\text{everywhere},\qquad \sum_i\varphi_i(x_i)=c(x_1,\ldots,x_N) \quad\text{on }\Gamma. \]
Generalized cyclical monotonicity. If \(\Gamma\) is \(c\)-splitting, \(x^1,\ldots,x^m\in\Gamma\), and \(\sigma_1,\ldots,\sigma_N\) are permutations of \(\{1,\ldots,m\}\), then \[ \sum_{k=1}^m c(x_1^k,\ldots,x_N^k) \le \sum_{k=1}^m c(x_1^{\sigma_1(k)},\ldots,x_N^{\sigma_N(k)}). \]
Proof. Equality on \(\Gamma\) changes the left side into \(\sum_{i,k}\varphi_i(x_i^k)\). Permuting the \(k\)-indices does not change this sum. Applying the splitting inequality to each recombined tuple gives the right side. An optimal plan is concentrated on a splitting set whenever dual optimizers exist.
Quadratic interaction and Wasserstein barycenters
Let \(\lambda_i>0\), \(\sum_i\lambda_i=1\), and put \[ \bar x=\sum_{i=1}^N\lambda_i x_i,\qquad c_\lambda(x_1,\ldots,x_N) =\sum_i\lambda_i|x_i-\bar x|^2 =\sum_{i<j}\lambda_i\lambda_j|x_i-x_j|^2. \] Thus the cost measures dispersion around the Euclidean barycenter of each transported tuple.
Barycenter equivalence. For \(\mu_i\in\mathcal P_2(\mathbb R^d)\), \[ \inf_{\gamma\in\Pi(\mu_1,\ldots,\mu_N)} \int c_\lambda\,\mathrm{d}\gamma = \inf_{\nu\in\mathcal P_2(\mathbb R^d)} \sum_{i=1}^N\lambda_iW_2^2(\mu_i,\nu). \] If \(\gamma^\star\) is optimal, then \(\nu^\star=(\sum_i\lambda_i p_i)_\#\gamma^\star\) is a Wasserstein barycenter.
Proof. Rewrite the right side as an infimum over joint laws of \((X_1,\ldots,X_N,Y)\) with \(X_i\sim\mu_i\): \[ \inf\mathbb E\sum_i\lambda_i|X_i-Y|^2. \] Conditional on \(X_1,\ldots,X_N\), the unique minimizer in \(Y\) is \(Y=\sum_i\lambda_iX_i\), and the minimum is \(c_\lambda(X_1,\ldots,X_N)\). Conversely, glue optimal couplings between any candidate \(\nu\) and the \(\mu_i\) conditionally on their common \(\nu\)-coordinate. The two infima are therefore equal.
When is a multi-marginal optimizer a graph?
In two-marginal quadratic transport, absolute continuity of one marginal produces a unique Monge map. For \(N\ge3\), pairwise twist is not enough. A useful replacement is twist on splitting sets: for fixed \(x_1\), the map \[ (x_2,\ldots,x_N)\longmapsto D_{x_1}c(x_1,\ldots,x_N) \] must be injective when restricted to every \(c\)-splitting set.
Graph theorem. Assume \(c\) is differentiable in \(x_1\), twisted on splitting sets, and regular enough for the first dual potential to be differentiable \(\mu_1\)-almost everywhere. If \(\mu_1\) is absolutely continuous, then every optimizer is concentrated on a graph \[ (x_1,T_2(x_1),\ldots,T_N(x_1)). \] Consequently the optimizer is unique.
Proof. On a splitting set supporting an optimizer, equality of the dual constraint gives, at differentiability points, \[ \nabla\varphi_1(x_1)=D_{x_1}c(x_1,x_2,\ldots,x_N). \] For fixed \(x_1\), twist on splitting sets permits at most one tuple \((x_2,\ldots,x_N)\). Disintegration over \(\mu_1\) therefore has Dirac conditional laws. If \(\gamma\) and \(\widetilde\gamma\) were two optimizers, their average would also be optimal and hence would also have Dirac conditional laws. But its conditional law is \(\tfrac12\delta_{(T_2,\ldots,T_N)}+\tfrac12\delta_{(\widetilde T_2,\ldots,\widetilde T_N)}\); it is Dirac only when the two tuples agree \(\mu_1\)-almost everywhere. Thus \(\gamma=\widetilde\gamma\).
Martingale couplings and convex order
Definition. For \(\mu,\nu\in\mathcal P_1(\mathbb R^d)\), a coupling \(\pi\in\Pi(\mu,\nu)\) is a martingale coupling if \[ \int (y-x)f(x)\,\mathrm{d}\pi(x,y)=0 \] for every bounded measurable scalar function \(f\). Equivalently, under \((X,Y)\sim\pi\), \(\mathbb E[Y\mid X]=X\). Denote the set by \(\Pi_M(\mu,\nu)\).
Definition. We say that \(\mu\) is below \(\nu\) in convex order, written \(\mu\preceq_{\mathrm{cx}}\nu\), if \[ \int\phi\,\mathrm{d}\mu\le\int\phi\,\mathrm{d}\nu \] for every finite convex \(\phi\) of at most linear growth. Affine test functions show that the two measures must have the same mean.
Strassen's theorem. \[ \Pi_M(\mu,\nu)\ne\varnothing \quad\Longleftrightarrow\quad \mu\preceq_{\mathrm{cx}}\nu. \]
Proof. If \((X,Y)\) is a martingale, conditional Jensen gives \[ \phi(X)=\phi(\mathbb E[Y\mid X]) \le\mathbb E[\phi(Y)\mid X], \] and integration proves convex order. For the converse, consider the convex set of second marginals generated by kernels \(K(x,\mathrm{d}y)\) satisfying \(\int y\,K(x,\mathrm{d}y)=x\). If \(\nu\) were outside its weakly closed, tight subset with controlled first moments, separation would produce a continuous function \(g\) whose expectation under \(\nu\) is smaller than under every reachable second marginal. Minimizing \(\int g\,\mathrm{d}K(x,\cdot)\) subject to mean \(x\) gives the convex envelope \(g^{**}(x)\). Hence separation would imply \(\int g\,\mathrm{d}\nu<\int g^{**}\,\mathrm{d}\mu\), while convex order and \(g^{**}\le g\) imply the reverse inequality. Thus \(\nu\) is reachable. Approximation extends the argument from compact support to \(\mathcal P_1\).
Example. Let \(\mu=\delta_0\) and \(\nu=\tfrac12\delta_{-a}+\tfrac12\delta_a\). Then \(\mu\preceq_{\mathrm{cx}}\nu\), and the only coupling is \[ \pi=\tfrac12\delta_{(0,-a)}+\tfrac12\delta_{(0,a)}. \] It is a martingale because the conditional mean of \(Y\) is zero. The second law is a mean-preserving spread of the first.
Martingale optimal transport duality
For a lower semicontinuous cost \(c\), the primal problem is \[ \mathsf{MOT}_c(\mu,\nu) = \inf_{\pi\in\Pi_M(\mu,\nu)}\int c(x,y)\,\mathrm{d}\pi. \] The martingale restriction introduces a vector-valued multiplier \(h(x)\).
Martingale duality. Suppose \(\mu,\nu\) have compact support, \(\mu\preceq_{\mathrm{cx}}\nu\), and \(c\) is continuous on the product of their convex hulls. Then \[ \mathsf{MOT}_c(\mu,\nu) = \sup_{\varphi,\psi,h} \left\{ \int\varphi\,\mathrm{d}\mu+\int\psi\,\mathrm{d}\nu: \varphi(x)+\psi(y)+h(x)\cdot(y-x)\le c(x,y) \right\}, \] where the supremum may be taken over continuous \(\varphi,\psi\) and continuous \(h\) with values in \(\mathbb R^d\).
Weak duality. Integrate the constraint against \(\pi\in\Pi_M(\mu,\nu)\). The marginal terms become the two displayed integrals, while \[ \int h(x)\cdot(y-x)\,\mathrm{d}\pi=0 \] by the martingale property. For strong duality, include the conditional-moment functional in the linear marginal operator and separate its compact convex image from any value strictly below the primal optimum. Compact support makes all constraint functions bounded and makes the feasible image closed, so the separating functional has precisely the form \((\varphi,\psi,h)\) above.
The term \(h(x)\cdot(y-x)\) has no counterpart in ordinary transport. In robust pricing it is the gain of a self-financing linear hedge, while \(\varphi\) and \(\psi\) are static positions depending on the initial and terminal states.
One-dimensional structure
On \(\mathbb R\), convex order can be checked through call functions: \[ \mu\preceq_{\mathrm{cx}}\nu \quad\Longleftrightarrow\quad \int(x-k)_+\,\mathrm{d}\mu(x) \le \int(x-k)_+\,\mathrm{d}\nu(x) \quad\text{for every }k, \] together with equality of means. Canonical couplings such as the left-curtain coupling transport each infinitesimal source interval to at most two target locations and are selected by a martingale analogue of monotonicity. For suitable Spence--Mirrlees costs, this coupling is the unique optimizer.
Stability
Closedness of the martingale constraint. Suppose \(\pi_n\in\Pi_M(\mu_n,\nu_n)\), \(\pi_n\Rightarrow\pi\), and the first moments are uniformly integrable. Then \(\pi\in\Pi_M(\mu,\nu)\), where \(\mu,\nu\) are its marginals.
Proof. For bounded continuous \(f\), truncate \(y-x\) at radius \(R\). Weak convergence passes the bounded continuous truncated integral to the limit. Uniform integrability makes the truncation error vanish uniformly as \(R\to\infty\). Hence \(\int(y-x)f(x)\,\mathrm{d}\pi=0\). A monotone-class argument extends from continuous \(f\) to bounded measurable \(f\).
Existence and stability of martingale optimizers then follow by the direct method. The essential extra hypothesis is first-moment control: weak convergence alone does not preserve conditional means.