The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\nu,\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\pi\in\Pi(\nu,\mu)$ be a coupling of $\nu$ and $\mu$, let $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ be the coordinate projections, and let $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $\eta\in L^{2}(\mu;\mathbb{R}^{d})$ be square-integrable vector fields, the norms $\lVert q\rVert_{\nu}$ and $\lVert\eta\rVert_{\mu}$ being those fixed there. For a point $z\in\mathbb{R}^{d+d}$ write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$, and for representatives of $q$ and $\eta$ let

$$f_{q,\eta}:\mathbb{R}^{d+d}\to\mathbb{R},\qquad f_{q,\eta}(z)=\lVert q(x)-\eta(y)\rVert^{2}.$$

Integrals against $\pi$ are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii), and $\mathrm{id}$ is the identity map of $\mathbb{R}^{d}$. In this statement the letter $q$ denotes a vector field; the dimension written $q$ in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i) and in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i) is not used. Then the following hold.

1. (The discrepancy along a coupling)(i) The function $f_{q,\eta}$ is nonnegative and Borel, its integral satisfies

$$\int_{\mathbb{R}^{d+d}}\lVert q(x)-\eta(y)\rVert^{2}\,\pi(dz)\le2\,\lVert q\rVert_{\nu}^{2}+2\,\lVert\eta\rVert_{\mu}^{2}<\infty ,$$

and this integral does not depend on the representatives of $q$ and $\eta$ chosen. It is called the discrepancy of $q$ and $\eta$ along $\pi$.

2. (The diagonal coupling)(ii) If $\mu=\nu$, so that $\eta\in L^{2}(\nu;\mathbb{R}^{d})$ and the difference $q-\eta$ is formed in that space, and $\pi=(\mathrm{id},\mathrm{id})_{\#}\nu$, then

$$\int_{\mathbb{R}^{d+d}}\lVert q(x)-\eta(y)\rVert^{2}\,\pi(dz)=\lVert q-\eta\rVert_{\nu}^{2}.$$