Optimally Coupled Pairs of Square-Integrable Random Vectors

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $X,Y\in L^{2}(\Omega;\mathbb{R}^{d})$, the space of classes of square-integrable random vectors with its norm $\lVert\cdot\rVert_{L^{2}}$ and the law $\mathcal{L}(X)$ of a class, and let $W_{2}$ be the quadratic Wasserstein distance on $\mathcal{P}_{2}(\mathbb{R}^{d})$. The joint law $\mathcal{L}(X,Y)$ is a coupling of $\mathcal{L}(X)$ and $\mathcal{L}(Y)$ with quadratic cost $\lVert X-Y\rVert_{L^{2}}^{2}$, by that clause.

(Optimally coupled pair)(i) The pair $(X,Y)$ is optimally coupled if $\mathcal{L}(X,Y)$ is an optimal coupling of $\mathcal{L}(X)$ and $\mathcal{L}(Y)$, that is, by Optimal Coupling of Two Probability Measures with Finite Second Moment (i) and The Joint Law of Two Classes of Square-Integrable Random Vectors: Marginals, Cost, and Invariance Under Shifts by a Vector Field (ii), if

$$\lVert X-Y\rVert_{L^{2}}^{2}=W_{2}\bigl(\mathcal{L}(X),\mathcal{L}(Y)\bigr)^{2}.$$