The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of 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 $N\in\mathbb{N}$ and $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, and write $\mu=P^{[1]}$ for its one-particle marginal. Product fields $g^{\oplus}$ and the projection $\Pi_{P}$ are those of Product Fields and the Projection onto One-Particle Tangent Fields, with the notation fixed there.

1. (Pairing)¶ For $D\in L^{2}(P;\mathbb{R}^{dN})$ and $g\in T_{\mu}$, $\langle D,g^{\oplus}\rangle_{P}=N\,\langle\Pi_{P}(D),g\rangle_{\mu}$.

2. (Linearity and contraction)¶ $\Pi_{P}:L^{2}(P;\mathbb{R}^{dN})\to T_{\mu}$ is linear, and $N\lVert\Pi_{P}(D)\rVert_{\mu}^{2}\le\lVert D\rVert_{P}^{2}$ for every $D\in L^{2}(P;\mathbb{R}^{dN})$.

3. (Product fields)¶ For $g\in T_{\mu}$, $\Pi_{P}(g^{\oplus})=g$.