The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, with the second moment $M_{2}$ and the Wasserstein distance $W_{2}$, and let $L^{2}(\mu;\mathbb{R}^{d})$ be the space of square-integrable vector fields against $\mu$, with norm $\lVert\cdot\rVert_{\mu}$. Let $\mathrm{id}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the identity map, whose class belongs to $L^{2}(\mu;\mathbb{R}^{d})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (ii) and is again written $\mathrm{id}$. Push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), $\Pi(\mu,\nu)$ is the set of couplings of $\mu$ and $\nu$, and for $a\in\mathbb{R}^{d}$ the translation $\tau_{a}$ of $\mathbb{R}^{d}$ is that fixed there, a Borel map. Then the following hold.

1. (Push-forward by a Borel map, and square-integrability)(i) Let $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Borel and write $\lVert S\rVert^{2}$ for the map $x\mapsto\lVert S(x)\rVert^{2}$, which is Borel and nonnegative by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv). Then

$$M_{2}(S_{\#}\mu)=\int_{\mathbb{R}^{d}}\lVert S\rVert^{2}\,d\mu$$

in $[0,\infty]$. Consequently, if $\int_{\mathbb{R}^{d}}\lVert S\rVert^{2}\,d\mu<\infty$, then $S_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, the class of $S$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$ and is again written $S$, and $\mathrm{id}-S\in L^{2}(\mu;\mathbb{R}^{d})$; and if $S_{\#}\mu=\nu$, then $\int_{\mathbb{R}^{d}}\lVert S\rVert^{2}\,d\mu=M_{2}(\nu)<\infty$, so the same conclusions hold. In particular, for every $a\in\mathbb{R}^{d}$ the translate $(\tau_{a})_{\#}\mu$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$, with

$$M_{2}\bigl((\tau_{a})_{\#}\mu\bigr)\le2\,M_{2}(\mu)+2\,\lVert a\rVert^{2}.$$

2. (The transport cost of an optimal map)(ii) Let $T$ be an optimal map from $\mu$ to $\nu$. Then

$$\lVert\mathrm{id}-T\rVert_{\mu}^{2}=W_{2}(\mu,\nu)^{2}.$$

3. (Uniqueness of the class)(iii) Suppose that the ordered pair $(\mu,\nu)$ is uniquely mapped, and let $T$ and $T'$ be optimal maps from $\mu$ to $\nu$. Then the classes of $T$ and of $T'$ in $L^{2}(\mu;\mathbb{R}^{d})$ are equal; consequently so are the classes of $\mathrm{id}-T$ and of $\mathrm{id}-T'$.