Brenier theorem

Brenier’s theorem establishes that the optimal transport map for quadratic cost between absolutely continuous measures on ${\mathbb{R}}^n$ is the gradient of a convex function. This is the foundational result connecting optimal transport to convex analysis, and it guarantees that OT paths are straight lines in any dimension.

For $\mu ,\nu \in P_2({\mathbb{R}}^n)$ with $\mu$ absolutely continuous w.r.t. Lebesgue, the unique optimal coupling for cost $c(x,y)=|x-y{|}^2/2$ (or equivalently $c(x,y)=-x\cdot y$) is deterministic: $\pi =(\text{Id},T{)}_{\#}\mu$ where $T=\nabla \phi$ for a convex function $\phi$. The convexity of $\phi$ ensures $T$ is a monotone map: $(T(x)-T(y))\cdot (x-y)\ge 0$.

The proof follows from the Kantorovich duality machinery: for quadratic cost, $c$-convexity reduces to ordinary convexity (Particular Case 5.3 of Villani), so the $c$-subdifferential ${\partial }_c\psi (x)$ becomes the ordinary subdifferential of a convex function. By Alexandrov’s theorem, a convex function is differentiable a.e., so ${\partial }_c\psi (x)$ is a singleton $\mu$-a.e. when $\mu \ll \text{Leb}$. Theorem 5.30 then delivers existence and uniqueness.

Key Details

• $T=\nabla \phi$ with $\phi$ convex: the OT map is a monotone operator
• Uniqueness: the optimal coupling is the unique one concentrated on ${\partial }_c\psi =\{(x,\nabla \phi (x))\}$
• The displacement interpolation $x_t=(1-t)x+t\nabla \phi (x)$ traces straight lines because each particle follows a constant-velocity path
• The map $x\mapsto (1-t)x+t\nabla \phi (x)=\nabla ((1-t)|x{|}^2/2+t\phi (x))$ is injective (gradient of strictly convex function), so paths don’t cross
• This is why the OT solution satisfies the rectified flow straightness condition $S(Z)=0$: with the OT coupling, the conditional expectation $\mathbb{E}[X_1-X_0\mid X_t=z]$ is deterministic (no averaging needed)
• On Riemannian manifolds: $T={\exp }_x(-\nabla \psi (x))$ where $\psi$ is $c$-convex for $c=d^2/2$ (McCann 2001)

Textbook References

Optimal Transport Old and New (Villani, 2009)

• Particular Case 5.3 (p. 67): For $c(x,y)=-x\cdot y$, $c$-convexity $=$ ordinary convexity, $c$-transform $=$ Legendre transform
• Theorem 5.10 (pp. 70-71): Kantorovich duality — optimality $⟺$ $c$-cyclical monotonicity $⟺$ concentrated on ${\partial }_c\psi$
• Theorem 5.30 (p. 96): If ${\partial }_c\psi (x)$ is a singleton $\mu$-a.e., the unique optimal coupling is deterministic $T(x)\in {\partial }_c\psi (x)$
• Bibliographical notes (pp. 98-99): For quadratic cost, expand $|x-y{|}^2/2=|x{|}^2/2-x\cdot y+|y{|}^2/2$, absorb quadratic terms, reduce to $c=-x\cdot y$

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