The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\pi\in\Pi(\mu,\nu)$ be a coupling, with quadratic cost $I(\pi)$, a nonnegative real number 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). Let $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ be the coordinate projections, and for $z\in\mathbb{R}^{d+d}$ write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$. The spaces $L^{2}(\mu;\mathbb{R}^{d})$ and $L^{2}(\pi;\mathbb{R}^{d})$ of square-integrable vector fields, the latter read with $d+d$ in place of $q$ and $d$ in place of $r$, are those fixed there, with inner products $\langle\cdot,\cdot\rangle_{\mu}$, $\langle\cdot,\cdot\rangle_{\pi}$ and norms $\lVert\cdot\rVert_{\mu}$, $\lVert\cdot\rVert_{\pi}$; $\mathrm{id}$ is the identity map of $\mathbb{R}^{d}$, which is Borel, being continuous; push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv); and $m(\mu)\in\mathbb{R}^{d}$ is the mean of $\mu$. Then the following hold.

1. (The displacement pairing)(i) Let $\eta\in L^{2}(\mu;\mathbb{R}^{d})$. For every representative of $\eta$ the function $\mathbb{R}^{d+d}\to\mathbb{R}$ with value $\eta(x)\cdot(y-x)$ at $z$ is Borel and integrable with respect to $\pi$, and its integral does not depend on the representative chosen. That integral,

$$\mathcal{J}(\eta,\pi)=\int_{\mathbb{R}^{d+d}}\eta(x)\cdot(y-x)\,\pi(dz)\in\mathbb{R},$$

is called the displacement pairing of $\eta$ along $\pi$.

2. (Bound)(ii) For every $\eta\in L^{2}(\mu;\mathbb{R}^{d})$,

$$\bigl|\mathcal{J}(\eta,\pi)\bigr|\le\lVert\eta\rVert_{\mu}\,\sqrt{I(\pi)} .$$

3. (Linearity in the field)(iii) For all $\eta,\xi\in L^{2}(\mu;\mathbb{R}^{d})$ and all $a,b\in\mathbb{R}$,

$$\mathcal{J}(a\,\eta+b\,\xi,\pi)=a\,\mathcal{J}(\eta,\pi)+b\,\mathcal{J}(\xi,\pi).$$

4. (Displacement couplings)(iv) Let $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Borel and satisfy $\int_{\mathbb{R}^{d}}\lVert S-\mathrm{id}\rVert^{2}\,d\mu<\infty$, where $S-\mathrm{id}$ denotes the map $x\mapsto S(x)-x$; it is Borel because each of its components is a difference of Borel real functions, hence Borel by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, and a map into $\mathbb{R}^{d}$ with Borel components is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii). Then the class of $S-\mathrm{id}$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$ and is again written $S-\mathrm{id}$, the push-forward $S_{\#}\mu$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$, the coupling $\pi_{S}=(\mathrm{id},S)_{\#}\mu$ belongs to $\Pi(\mu,S_{\#}\mu)$ with

$$I(\pi_{S})=\lVert S-\mathrm{id}\rVert_{\mu}^{2},$$

and for every $\eta\in L^{2}(\mu;\mathbb{R}^{d})$,

$$\mathcal{J}(\eta,\pi_{S})=\langle\eta,\,S-\mathrm{id}\rangle_{\mu}.$$

5. (Constant fields)(v) Let $c\in\mathbb{R}^{d}$ and let $\eta$ be the class in $L^{2}(\mu;\mathbb{R}^{d})$ of the constant map with value $c$, which is Borel, being continuous, and square-integrable against $\mu$. Then

$$\mathcal{J}(\eta,\pi)=c\cdot\bigl(m(\nu)-m(\mu)\bigr).$$