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 §cost-finite. 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 §pushforward; and $m(\mu)\in\mathbb{R}^{d}$ is the mean of $\mu$. Then the following hold.

1. (The displacement pairing)¶ 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,

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

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

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

4. (Displacement couplings)¶ 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, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and a map into $\mathbb{R}^{d}$ with Borel components is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. 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

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

5. (Constant fields)¶ 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