Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost

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 $\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$. As in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ are the coordinate projections for the splitting $d+d$, and push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward.

1. (Coupling)¶ A coupling of $\mu$ and $\nu$ is a probability measure $\pi\in\mathcal{P}(\mathbb{R}^{d+d})$ whose push-forwards by the two projections are $\mu$ and $\nu$:

that is, $\pi(\mathrm{pr}_{1}^{-1}(A))=\mu(A)$ and $\pi(\mathrm{pr}_{2}^{-1}(B))=\nu(B)$ for all $A,B\in\mathcal{B}(\mathbb{R}^{d})$. The set of all couplings of $\mu$ and $\nu$ is denoted $\Pi(\mu,\nu)$.

2. (Quadratic cost)¶ The quadratic cost of a coupling $\pi\in\Pi(\mu,\nu)$ is

the integral, in the sense of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, of the nonnegative Borel function $z\mapsto\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert^{2}$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs.