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 (v), $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ 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 (iv).

1. (Coupling)(i) 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 $\nu$:

$$(\mathrm{pr}_{1})_{\#}\pi=\mu,\qquad(\mathrm{pr}_{2})_{\#}\pi=\nu ,$$

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)(ii) The quadratic cost of a coupling $\pi\in\Pi(\mu,\nu)$ is

$$I(\pi)=\int_{\mathbb{R}^{d+d}}\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert^{2}\,\pi(dz)\in[0,\infty],$$

the integral, in the sense of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii), 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 (v).