Plans and Their Velocity Fields: the Marginal, the Composition Isometry, the Pairing, the Velocity Shift and the Lift

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $\pi$ be a plan, with the coordinate projections $\mathrm{pr}_{1},\mathrm{pr}_{2}$ of $\mathbb{R}^{d+d}$, the variables $x,p$ and the integral notation fixed there, and let

$$\nu=\pi^{(1)}$$

be its first marginal. The spaces $L^{2}(\nu;\mathbb{R}^{d})$ and $L^{2}(\pi;\mathbb{R}^{d})$ are those of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii). Then the following hold.

1. (The marginal and the coordinate fields)(i) $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, the classes of $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ belong to $L^{2}(\pi;\mathbb{R}^{d})$, and

$$\lVert\mathrm{pr}_{1}\rVert_{\pi}^{2}=M_{2}(\nu),\qquad\lVert\mathrm{pr}_{2}\rVert_{\pi}^{2}=\int\lVert p\rVert^{2}\,\pi(dz)<\infty .$$

2. (The composition isometry)(ii) For $\eta\in L^{2}(\nu;\mathbb{R}^{d})$ the composition $\eta\circ\mathrm{pr}_{1}$ of a representative of $\eta$ with $\mathrm{pr}_{1}$ is a Borel map $\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ whose class in $L^{2}(\pi;\mathbb{R}^{d})$ does not depend on the representative chosen; this class is written $\eta\circ\mathrm{pr}_{1}$. The map $\eta\mapsto\eta\circ\mathrm{pr}_{1}$ from $L^{2}(\nu;\mathbb{R}^{d})$ to $L^{2}(\pi;\mathbb{R}^{d})$ is linear and preserves inner products,

$$\langle\eta\circ\mathrm{pr}_{1},\eta'\circ\mathrm{pr}_{1}\rangle_{\pi}=\langle\eta,\eta'\rangle_{\nu},\qquad\lVert\eta\circ\mathrm{pr}_{1}\rVert_{\pi}=\lVert\eta\rVert_{\nu}\qquad(\eta,\eta'\in L^{2}(\nu;\mathbb{R}^{d})),$$

and $\mathrm{id}\circ\mathrm{pr}_{1}=\mathrm{pr}_{1}$, where $\mathrm{id}$ is the identity map of $\mathbb{R}^{d}$.

3. (The pairing of a field with the velocity)(iii) For $\eta\in L^{2}(\nu;\mathbb{R}^{d})$ the function $z\mapsto\eta(x)\cdot p$ is Borel and integrable with respect to $\pi$, and

$$\int\eta(x)\cdot p\,\pi(dz)=\langle\eta\circ\mathrm{pr}_{1},\mathrm{pr}_{2}\rangle_{\pi},\qquad\Bigl|\int\eta(x)\cdot p\,\pi(dz)\Bigr|\le\lVert\eta\rVert_{\nu}\sqrt{\int\lVert p\rVert^{2}\,\pi(dz)} ;$$

in particular the integral depends only on the class of $\eta$.

4. (The velocity shift)(iv) Let $\eta\in L^{2}(\nu;\mathbb{R}^{d})$, let $t\in\mathbb{R}$, and let

$$\Theta^{\eta}_{t}=\bigl(\mathrm{pr}_{1},\ \mathrm{pr}_{2}+t\,(\eta\circ\mathrm{pr}_{1})\bigr):\mathbb{R}^{d+d}\to\mathbb{R}^{d+d}$$

be the pairing formed from a representative of $\eta$, a Borel map. Then $(\Theta^{\eta}_{t})_{\#}\pi$ is a plan with first marginal $\nu$, it does not depend on the representative of $\eta$ chosen, $(\Theta^{\eta}_{0})_{\#}\pi=\pi$, and for every $\eta'\in L^{2}(\nu;\mathbb{R}^{d})$

$$\int\eta'(x)\cdot p\,d\bigl((\Theta^{\eta}_{t})_{\#}\pi\bigr)=\int\eta'(x)\cdot p\,\pi(dz)+t\,\langle\eta',\eta\rangle_{\nu},$$ $$\int\lVert p\rVert^{2}\,d\bigl((\Theta^{\eta}_{t})_{\#}\pi\bigr)=\int\lVert p\rVert^{2}\,\pi(dz)+2t\int\eta(x)\cdot p\,\pi(dz)+t^{2}\lVert\eta\rVert_{\nu}^{2},$$

where $2=1+1$.

5. (The lift)(v) Let $Y,Y'\in L^{2}(\Omega;\mathbb{R}^{d})$ be such that the law of the pairing $(Y,Y')$ of representatives of them is $\pi$. Then $\mathcal{L}(Y)=\nu$ and, for every $\eta\in L^{2}(\nu;\mathbb{R}^{d})$,

$$\int\eta(x)\cdot p\,\pi(dz)=\langle\eta\circ Y,Y'\rangle_{L^{2}},\qquad\int\lVert p\rVert^{2}\,\pi(dz)=\lVert Y'\rVert_{L^{2}}^{2},$$

the class $\eta\circ Y\in L^{2}(\Omega;\mathbb{R}^{d})$ being that of Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i).