The Mollified Mean-Field Cost and the Mollified N-Particle Cost of a Local Coupling

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 $\mathcal{G}_{\Phi,\varepsilon}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be the mollified density cost with integrand $\Phi$, kernel $\eta$ and scale $\varepsilon$. 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; by Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, read with $m=d$, $f$ is Borel and integrable with respect to every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. For $x\in\mathbb{R}^{dN}$ the empirical measure $\mu^{N}_{x}$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$ by Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §moment; and $N$, as a real factor, is its image under the canonical map.

1. (Mollified mean-field cost)¶ The mollified mean-field cost with data $f$, $\Phi$, $\eta$ and $\varepsilon$ is the function

2. (Mollified $N$-particle cost)¶ The mollified $N$-particle cost with data $f$, $\Phi$, $\eta$ and $\varepsilon$ is the function

with $g_{\varepsilon}$ the mollified mean-field cost of clause 1.