The Mollified N-Particle Cost Dominates N Times the Mollified Mean-Field Cost of the One-Particle Marginal

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. The letter $\eta$, which denotes vector fields in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields, denotes here a mollifier kernel. Let $\eta$ be a mollifier kernel of radius $1$ on $\mathbb{R}^{d}$, let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$, let $L\in\mathbb{R}$ be nonnegative, let $\Phi$ be a convex Lipschitz integrand with constant $L$, and let $f:\mathbb{R}^{d}\to\mathbb{R}$ be bounded and uniformly continuous for the Euclidean distance and the metric of The Absolute Value Metric on the Real Line. Let $g_{\varepsilon}$ be the mollified mean-field cost and $c_{N,\varepsilon}$ the mollified $N$-particle cost with data $f$, $\Phi$, $\eta$ and $\varepsilon$, and let $N$, as a real factor, be its image under the canonical map. For $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, the one-particle marginal $P^{[1]}$, read with dimension parameter $d$ as in N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles, belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments, and $c_{N,\varepsilon}$ is Borel and integrable with respect to $P$ by The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost §regularity and Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, read with $m=dN$.

(Cost domination)¶ For every $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$,