Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts

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 $q,N,p\in\mathbb{N}$. Block maps $\mathfrak{p}_{k}$, product maps $h^{\oplus}$ and diagonal points $a^{\oplus}$ are those of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks and the clauses just cited, finite products are written $\prod_{k=1}^{N}$, tensor powers $\rho^{\otimes N}$ those of The Tensor Power of a Probability Measure on Euclidean Space §tensor and one-particle marginals $P^{[1]}$ those of The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal; $\rho\boxtimes\rho'$ and the pairing $(u,v)$ of maps are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, and $\tau_{a}$ is the translation by $a$. Let $\rho\in\mathcal{P}(\mathbb{R}^{q})$ and $P\in\mathcal{P}(\mathbb{R}^{qN})$.

1. (Particle laws)¶ $(\mathfrak{p}_{k})_{\#}\rho^{\otimes N}=\rho$ for every $k\in[N]$, and $(\mathfrak{p}_{k},\mathfrak{p}_{l})_{\#}\rho^{\otimes N}=\rho\boxtimes\rho$ for all $k,l\in[N]$ with $k\ne l$.

2. (Product integrals)¶ For bounded Borel $f_{1},\dots,f_{N}:\mathbb{R}^{q}\to\mathbb{R}$,

3. (Push-forwards)¶ For Borel $h:\mathbb{R}^{q}\to\mathbb{R}^{p}$, $(h^{\oplus})_{\#}\rho^{\otimes N}=(h_{\#}\rho)^{\otimes N}$ and $\bigl((h^{\oplus})_{\#}P\bigr)^{[1]}=h_{\#}\bigl(P^{[1]}\bigr)$.

4. (One-particle marginal of a tensor power)¶ $(\rho^{\otimes N})^{[1]}=\rho$.

5. (Second moments)¶ $M_{2}(P^{[1]})=\frac{1}{N}M_{2}(P)$ and $M_{2}(\rho^{\otimes N})=N\,M_{2}(\rho)$ in $[0,\infty]$. Hence $P^{[1]}\in\mathcal{P}_{2}(\mathbb{R}^{q})$ when $P\in\mathcal{P}_{2}(\mathbb{R}^{qN})$, and $\rho^{\otimes N}\in\mathcal{P}_{2}(\mathbb{R}^{qN})$ exactly when $\rho\in\mathcal{P}_{2}(\mathbb{R}^{q})$.

6. (Product maps)¶ For Borel $g:\mathbb{R}^{q}\to\mathbb{R}^{p}$,

and if $Z\in\mathcal{B}(\mathbb{R}^{q})$ satisfies $P^{[1]}(Z)=0$ then $P\bigl(\bigcup_{k=1}^{N}\mathfrak{p}_{k}^{-1}(Z)\bigr)=0$.

7. (Diagonal shifts)¶ For $a\in\mathbb{R}^{q}$, $\bigl((\tau_{a^{\oplus}})_{\#}P\bigr)^{[1]}=(\tau_{a})_{\#}P^{[1]}$ and $(\tau_{a^{\oplus}})_{\#}\rho^{\otimes N}=\bigl((\tau_{a})_{\#}\rho\bigr)^{\otimes N}$.