Cross-Level Inequalities between the Shifted Lifted N-Particle and Mean-Field Langevin Operators at Tensor Powers and through One-Particle Marginals

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. Let $V$ be a confining potential on $\mathbb{R}^{d}$, let $\lambda_{0},\sigma\in\mathbb{R}$ be positive, let $\theta\in\mathbb{R}$ satisfy $0<\theta\le1$, let $p\in\mathbb{N}$ and $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$, let $c:\mathbb{R}^{dN}\to\mathbb{R}$ be bounded and Borel, and let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$. Let $F_{N}$ be the lifted $N$-particle Hamilton-Jacobi operator with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, control cost $\theta$, common-noise matrix $\Gamma$ and running cost $c$, over the score domain $\mathcal{D}_{N,\Sigma}$ of the configuration-level Langevin pair $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ named there, and let $F$ be the Langevin Hamilton-Jacobi operator with common noise with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma$, control cost $\theta$ and running cost $g$, over the score domain $\mathcal{D}_{\Sigma}$ of the Langevin pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ named there. For positive $\delta\in\mathbb{R}$, $F^{+}_{N,\delta}$ and $F^{+}_{\delta}$ are the $\delta$-shifts $F^{+}_{\delta}$ of $F_{N}$ relative to the configuration-level pair and of $F$ relative to the particle-level pair. For $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $\Pi_{P}$ is the one-particle projection and $h^{\oplus}$ the product field of $h\in L^{2}(P^{[1]};\mathbb{R}^{d})$; $a^{\oplus}$ is the diagonal point of $a\in\mathbb{R}^{d}$. Tensor powers and one-particle marginals of points of the score domains lie in the other score domain by The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §tensor and The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §marginal. 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.

Let $\delta\in\mathbb{R}$ satisfy $0<\delta<1$, and let $\mathbb{Y}_{N}\in\mathcal{S}(dN)$ and $\mathbb{Y}\in\mathcal{S}(d)$ satisfy $a^{\oplus}\cdot(\mathbb{Y}_{N}a^{\oplus})=N\,a\cdot(\mathbb{Y}a)$ for every $a\in\mathbb{R}^{d}$.

1. (At tensor powers)¶ Suppose that $g$ is the tensor-averaged cost $\tilde{c}$ of $c$. Then for every $\mu\in\mathcal{D}_{\Sigma}$, every $r\in\mathbb{R}$ and every $G\in L^{2}(\mu^{\otimes N};\mathbb{R}^{dN})$,

2. (Through one-particle marginals)¶ Let $P\in\mathcal{D}_{N,\Sigma}$ satisfy $N\,g(P^{[1]})\le\int_{\mathbb{R}^{dN}}c\,dP$. Then for every $r\in\mathbb{R}$ and every $h\in T_{P^{[1]}}$,