The Quadratic Wasserstein Distance on Euclidean Space

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$, and let $\mu,\nu$ belong to the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ of probability measures with finite second moment. The set $\Pi(\mu,\nu)$ of couplings of $\mu$ and $\nu$ is nonempty by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (i), and the quadratic cost $I(\pi)$ of each $\pi\in\Pi(\mu,\nu)$ is finite by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (iii), hence a nonnegative real number. Hence the set $\{I(\pi):\pi\in\Pi(\mu,\nu)\}$ is a nonempty set of real numbers bounded below by $0$, and has a unique greatest lower bound $\inf\{I(\pi):\pi\in\Pi(\mu,\nu)\}$ by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below, which is nonnegative, $0$ being a lower bound.

(The quadratic Wasserstein distance)(i) The quadratic Wasserstein distance between $\mu$ and $\nu$ is

$$W_{2}(\mu,\nu)=\sqrt{\inf\{I(\pi):\pi\in\Pi(\mu,\nu)\}}\ ,$$

the nonnegative square root of that greatest lower bound; thus $W_{2}(\mu,\nu)$ is a nonnegative real number with $W_{2}(\mu,\nu)^{2}\le I(\pi)$ for every $\pi\in\Pi(\mu,\nu)$.