The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. Let $V$ be a confining potential on $\mathbb{R}^{d}$ with $N$-particle potential $V_{N}$, a confining potential on $\mathbb{R}^{dN}$ by The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §confining, and let $\sigma\in\mathbb{R}$ be positive. Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Langevin free-energy pair with potential $V$ and noise intensity $\sigma$, and $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ the one with potential $V_{N}$ and noise intensity $\sigma$, formed at the configuration level; both are penalty pairs by The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §pair, with translation Hessians of the penalty $H_{\mathcal{E}}$ and $H_{\mathcal{E}_{N}}$. For $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $\Pi_{P}$ is the one-particle projection and $g^{\oplus}$ the product field of $g\in L^{2}(P^{[1]};\mathbb{R}^{d})$; $a^{\oplus}$ is the diagonal point of $a\in\mathbb{R}^{d}$. The letter $\sigma$ denotes the noise intensity; the swap map written $\sigma$ in Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §swap is not used.

1. (Tensor powers)¶ For $\mu\in\mathcal{D}$, $\mu^{\otimes N}\in\mathcal{D}_{N}$ and $\mathcal{E}_{N}(\mu^{\otimes N})=N\,\mathcal{E}(\mu)$. If moreover $\mu\in\mathcal{D}_{\Sigma}$, then $\mu^{\otimes N}\in\mathcal{D}_{N,\Sigma}$ and $\Sigma_{N}(\mu^{\otimes N})=\Sigma(\mu)^{\oplus}$.

2. (One-particle marginals)¶ For $P\in\mathcal{D}_{N}$, $P^{[1]}\in\mathcal{D}$ and $N\,\mathcal{E}(P^{[1]})\le\mathcal{E}_{N}(P)$. If moreover $P\in\mathcal{D}_{N,\Sigma}$, then $P^{[1]}\in\mathcal{D}_{\Sigma}$ and $\Pi_{P}\bigl(\Sigma_{N}(P)\bigr)=\Sigma(P^{[1]})$.

3. (Diagonal translation Hessians)¶ For every $P\in\mathcal{D}_{N}$ and every $a\in\mathbb{R}^{d}$,