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

In the setting of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: 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

be its first marginal. The spaces $L^{2}(\nu;\mathbb{R}^{d})$ and $L^{2}(\pi;\mathbb{R}^{d})$ are those of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation §fields. Then the following hold.

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

2. (The composition isometry)¶ 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,

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)¶ 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

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

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

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 $\eta'\in L^{2}(\nu;\mathbb{R}^{d})$

where $2=1+1$.

5. (The lift)¶ 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 $\eta\in L^{2}(\nu;\mathbb{R}^{d})$,

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 §composition.