McCann's Theorem on the Flat Torus: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map with a Periodic Potential

In the setting of The Flat Torus: Standing Notation, used with $n=d$ for a natural number $d$ with $1\le d$, and of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $\mathcal{P}(\mathbb{T}^{d})$ be as in Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures and $Q$ the half-open unit cell; let optimal maps on the torus, uniquely mapped pairs on the torus and the displacement field $v_{T}$ of a Borel map $T$ be as defined there, and let regular points of $\mathbb{R}^{d}$ be as defined there. Let $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ and assume that $\mu$ is absolutely continuous. For Borel $T,T':\mathbb{R}^{d}\to\mathbb{R}^{d}$ the set $\{x:T(x)\ne T'(x)\}$ belongs to $\mathcal{B}(\mathbb{R}^{d})$, as in the preamble of Brenier's Theorem: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map. Then the following hold.

1. (Unique optimal map)¶ The pair $(\mu,\nu)$ is uniquely mapped on the torus, and it has an optimal map on the torus with values in $Q$. If $T$ and $T'$ are optimal maps on the torus from $\mu$ to $\nu$, then $\mu(\{x:T(x)\ne T'(x)\})=0$.

2. (Periodic potential)¶ Let $T$ be an optimal map on the torus from $\mu$ to $\nu$. There are a $\mathbb{Z}^{d}$-periodic function $\varphi:\mathbb{R}^{d}\to\mathbb{R}$, Lipschitz with constant $\sqrt{d}/2$, such that $\psi(x)=\tfrac12\lVert x\rVert^{2}-\varphi(x)$ is convex on $\mathbb{R}^{d}$, and a set $D\in\mathcal{B}(\mathbb{R}^{d})$ with $\mu(D)=1$, such that for every $x\in D$: $\psi$ is differentiable at $x$, the point $T(x)-x$ is regular, and the subdifferential of $\psi$ is

$$\partial_{\mathbb{R}^{d}}\psi(x)=\{x+v_{T}(x)\}.$$

3. (Inverse map)¶ Assume moreover that $\nu$ is absolutely continuous, let $T$ be an optimal map on the torus from $\mu$ to $\nu$ and $S$ one from $\nu$ to $\mu$. Then there is a set $D'\in\mathcal{B}(\mathbb{R}^{d})$ with $\mu(D')=1$ such that $S(T(x))=x$ and $v_{S}(T(x))=-v_{T}(x)$ for every $x\in D'$.