Optimal Transport Old and New

Textbook

Authors: Cédric Villani | Publisher: Springer, 2009 | Series: Grundlehren der mathematischen Wissenschaften, Vol. 338

Overview

The comprehensive reference on optimal transport theory, covering Monge-Kantorovich theory, Wasserstein geometry, displacement interpolation, regularity of transport maps, Otto calculus, gradient flows, and synthetic Ricci curvature bounds. ~990 pages across 30 chapters and three parts. Based on 2005 Saint-Flour lectures.

Topics Studied

Diffusion Bridge (studied 22-03-2026)

Chapters read: Ch. 1 (pp. 17-31), Ch. 5 (pp. 63-103), Ch. 6 (pp. 105-123), Ch. 7 (pp. 125-173)

Key Definitions

• Definition 1.1 (p. 17): coupling — Joint measure $\pi$ on $\mathcal{X}\times \mathcal{Y}$ with marginals $\mu$ and $\nu$
• Definition 1.2 (p. 18): deterministic coupling — Coupling concentrated on graph of $T$; transport map $T_{\#}\mu =\nu$
• Definition 5.1 (p. 64): c-cyclical monotonicity — $\sum c(x_i,y_i)\le \sum c(x_i,y_{i+1})$ for all cycles; optimality condition
• Definition 5.2 (pp. 66-67): c-convexity — $\psi (x)={\sup }_y(\zeta (y)-c(x,y))$; c-transform ${\psi }^c(y)={\inf }_x(\psi (x)+c(x,y))$; for quadratic cost, reduces to ordinary convexity
• Definition 6.1 (p. 105): Wasserstein distance — $W_p(\mu ,\nu )=({\inf }_{\pi }\int d(x,y{)}^{p}d\pi {)}^{1/p}$
• Definition 6.4 (pp. 106-107): Wasserstein space — $P_p(\mathcal{X})=\{\mu :\int d(x_0,x{)}^{p}d\mu <\infty \}$ with metric $W_p$
• Definition 7.20 (p. 138): dynamical optimal coupling — Random action-minimizing curve whose endpoints are optimally coupled
• Definition 7.33 (p. 156): Hamilton-Jacobi semigroup — Forward $H_{+}^{s,t}\psi (y)={\inf }_x(\psi (x)+c^{s,t}(x,y))$; backward $H_{-}^{t,s}\varphi (x)={\sup }_y(\varphi (y)-c^{s,t}(x,y))$

Key Theorems

• Theorem 5.10 (pp. 70-71): Kantorovich duality — ${\min }_{\pi }\int cd\pi ={\sup }_{\varphi -\psi \le c}(\int \varphi d\nu -\int \psi d\mu )$; optimality $⟺$ c-cyclical monotonicity $⟺$ concentrated on ${\partial }_c\psi$
• Theorem 5.30 (p. 96): existence of OT maps — If ${\partial }_c\psi (x)$ is a singleton $\mu$-a.e., then the unique optimal coupling is deterministic $T=\nabla \psi$. For quadratic cost + absolutely continuous $\mu$, this gives Brenier’s theorem
• Theorem 6.9 (p. 108): $W_p$ metrizes weak convergence in $P_p(\mathcal{X})$
• Theorem 6.18 (pp. 116-117): $P_p(\mathcal{X})$ is Polish (complete, separable) when $\mathcal{X}$ is Polish
• Theorem 7.21 (p. 139): displacement interpolation — Equivalence of three characterizations: (i) law of random optimal geodesic, (ii) additive cost-splitting over time subdivisions, (iii) action-minimizing curve in $P(\mathcal{X})$
• Corollary 7.22 (pp. 139-140): Displacement interpolation $=$ geodesic in $(P_p,W_p)$; constant-speed: $W_p({\mu }_s,{\mu }_t)=|t-s|W_p({\mu }_0,{\mu }_1)$
• Corollary 7.23 (p. 140): Uniqueness of displacement interpolation when OT plan and minimizing curves are unique
• Theorem 7.30(iv) (p. 151): Non-crossing: optimal paths cannot cross at intermediate times — $[{\gamma }_t={\tilde{\gamma }}_t]⟹[\gamma =\tilde{\gamma }]$ a.s.
• Corollary 7.32 (p. 152): Nonbranching of $\mathcal{X}$ is inherited by $P_p(\mathcal{X})$
• Theorem 7.36 (p. 158): Interpolation of dual prices along displacement interpolation via Hamilton-Jacobi semigroups

Key Examples

• Example 7.2 (pp. 127-128): For strictly convex $L(v)$ in ${\mathbb{R}}^n$, Jensen forces constant-velocity straight lines ${\gamma }_t=(1-t)x+ty$. The fundamental “straightness” argument
• Example 7.4 (p. 128): Power-law $L=|v{|}^p$ on Riemannian manifold: minimizers are constant-speed geodesics (zero covariant acceleration)
• Example 7.35 (p. 157): For quadratic Hamiltonian $H=|p{|}^2/2$, optimal velocity field $v=\nabla S$ where $S$ solves Hamilton-Jacobi ${\partial }_{t}S+|\nabla S{|}^2/2=0$
• Exercise 7.39 (p. 159): Displacement interpolation between balls is always a ball; between ellipsoids is always an ellipsoid

Key Aphorisms

• “A geodesic in the space of laws is the law of a geodesic” (p. 139) — encapsulates the entire displacement interpolation theory

Atomic Notes

• displacement interpolation
• Benamou-Brenier formula
• Wasserstein distance
• Kantorovich duality
• c-cyclical monotonicity
• Brenier theorem

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