Scores on the Configuration Space: the Score of a Tensor Power is the Product Field of the Score, and the Score of the One-Particle Marginal is the Projection of the Score

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $N\in\mathbb{N}$. Tensor powers $\rho^{\otimes N}$ are those of The Tensor Power of a Probability Measure on Euclidean Space §tensor, one-particle marginals $P^{[1]}$ those of The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal (with $q=d$), and product fields $g^{\oplus}$ and projections $\Pi_{P}$ those of Product Fields and the Projection onto One-Particle Tangent Fields. For $m\in\{d,dN\}$ and $\mu\in\mathcal{P}_{2}(\mathbb{R}^{m})$, the tangent space $T_{\mu}$, finite Fisher information, the score $\xi_{\mu}$ and the Fisher information $\mathcal{I}(\mu)$ are those definitions read with $m$ in place of $d$: they are stated for the arbitrary dimension $d\ge1$ of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data and use nothing of it beyond $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, so they apply in every dimension $m$. $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{m})$ is the set of measures in $\mathcal{P}_{2}(\mathbb{R}^{m})$ with finite Fisher information. For $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $P^{[1]}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments. The letter $P$ denotes a probability measure on $\mathbb{R}^{dN}$; the probability space $(\Omega,\mathcal{F},P)$ of the setting is not used.

1. (Tensor powers)¶ For $\rho\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, $\rho^{\otimes N}\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{dN})$, $\xi_{\rho^{\otimes N}}=(\xi_{\rho})^{\oplus}$ and $\mathcal{I}(\rho^{\otimes N})=N\,\mathcal{I}(\rho)$.

2. (One-particle marginals)¶ For $P\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{dN})$, $P^{[1]}\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, $\xi_{P^{[1]}}=\Pi_{P}(\xi_{P})$ and $N\,\mathcal{I}(P^{[1]})\le\mathcal{I}(P)$. In particular $\langle\xi_{P},g^{\oplus}\rangle_{P}=N\,\langle\xi_{P^{[1]}},g\rangle_{P^{[1]}}$ for every $g\in T_{P^{[1]}}$.