Product Fields and the Projection onto One-Particle Tangent Fields

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, whose probability space $(\Omega,\mathcal{F},P)$ is not used (the letter $P$ below denotes a probability measure on a configuration space), let $N\in\mathbb{N}$. Block maps $\mathfrak{p}_{k}$ and product maps $h^{\oplus}$ are those of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points with $q=p=d$, and $P^{[1]}$ is the one-particle marginal with $q=d$. For $\rho\in\mathcal{P}(\mathbb{R}^{m})$ the space $L^{2}(\rho;\mathbb{R}^{r})$ with $\langle\cdot,\cdot\rangle_{\rho}$ and $\lVert\cdot\rVert_{\rho}$ is that of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields; for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $T_{\mu}$ is its tangent space, and $C_{c}^{\infty}(\mathbb{R}^{d})$ and the gradients $\nabla\psi$ are those of Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients. Let $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$; then $P^{[1]}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments.

1. (Product fields)¶ For $g\in L^{2}(P^{[1]};\mathbb{R}^{d})$, the product field $g^{\oplus}\in L^{2}(P;\mathbb{R}^{dN})$ is the class of the product map $\tilde g^{\oplus}$ of a Borel representative $\tilde g$ of $g$. It does not depend on the representative and has $\lVert g^{\oplus}\rVert_{P}^{2}=N\lVert g\rVert_{P^{[1]}}^{2}$: the map $\tilde g^{\oplus}$ is Borel by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map, and its square integral and the null sets are controlled by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §product-maps.

2. (Projection)¶ For $D\in L^{2}(P;\mathbb{R}^{dN})$, the one-particle projection $\Pi_{P}(D)$ is the unique $\zeta\in T_{P^{[1]}}$ with

It exists and is unique by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §representation: the right-hand side is linear in $\psi$ and, by The Cauchy-Schwarz Inequality in a Real Inner Product Space and clause 1, at most $N^{-1/2}\lVert D\rVert_{P}\lVert\nabla\psi\rVert_{P^{[1]}}$ in absolute value.