The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost

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 $L$, let $\mathcal{G}_{\Phi,\varepsilon}$ be the mollified density cost with integrand $\Phi$, kernel $\eta$ and scale $\varepsilon$, and let $f:\mathbb{R}^{d}\to\mathbb{R}$ be bounded and uniformly continuous, with a bound $b$. Let $g_{\varepsilon}$ be the mollified mean-field cost and $c_{N,\varepsilon}$ the mollified $N$-particle cost with data $f$, $\Phi$, $\eta$ and $\varepsilon$. Uniform continuity of a real function on $\mathbb{R}^{m}$ refers to the Euclidean distance and on $\mathcal{P}_{2}(\mathbb{R}^{d})$ to $W_{2}$, with the metric of The Absolute Value Metric on the Real Line on $\mathbb{R}$. By claim 1 below and Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, read with $m=dN$, $c_{N,\varepsilon}$ is bounded and Borel, so that its tensor-averaged cost $\tilde{c}_{N,\varepsilon}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ is defined. Tensor powers $\mu^{\otimes N}$, one-particle marginals $P^{[1]}$, empirical measures $\mu^{N}_{x}$ and block maps $\mathfrak{p}_{k}$ are those of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles. Let $\lambda_{d}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$ and $\kappa_{d}=\lambda_{d}(\bar{B}(0,1))$, where $0$ is the zero vector of $\mathbb{R}^{d}$, the normalising constant of closed balls; let $\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$ and the density cost $\mathcal{G}_{\Phi}$ be those of The Density Cost of a Convex Lipschitz Integrand, and $M_{2}(\mu)$ the second moment of $\mu$. Natural numbers occurring as real factors are read through the canonical map, powers with natural exponent are those of Natural Number Power of an Element of a Field, $t^{-1}$ is the multiplicative inverse of a real $t\ne0$, $2=1+1$, and $\sqrt{t}$ is the nonnegative square root of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces. Then the following hold.

1. (Regularity)¶ $|g_{\varepsilon}(\nu)|\le b+L$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and $g_{\varepsilon}$ is uniformly continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$; $|c_{N,\varepsilon}(x)|\le N(b+L)$ for every $x\in\mathbb{R}^{dN}$, and $c_{N,\varepsilon}$ is uniformly continuous on $\mathbb{R}^{dN}$.

2. (The tensor-averaged cost)¶ The functions $x\mapsto\int_{\mathbb{R}^{d}}f\,d\mu^{N}_{x}$ and $x\mapsto\mathcal{G}_{\Phi,\varepsilon}(\mu^{N}_{x})$ on $\mathbb{R}^{dN}$ are Borel, the first with absolute value at most $b$ and the second with values in $[0,L]$. For every $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$,

and for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$,

3. (Defect against the density cost)¶ Suppose that $\varepsilon\le1$, let $S$ be a positive real number with $\eta(z)\le S$ for every $z\in\mathbb{R}^{d}$ (one exists by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel), let $\mu\in\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$, and let $R$ be a real number with $1<R$. Then