The Mean-Field Limit of the Lifted N-Particle Hamilton-Jacobi Equation with a Mollified Local Coupling

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. The letter $N$, fixed in N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles as the number of particles, is used here as a running index: for each $N\in\mathbb{N}$, the configuration space $\mathbb{R}^{dN}$, the mollified $N$-particle cost, the configuration-level Langevin pair, the lifted $N$-particle equation and its solution $U_{N}$ below are those formed in that setting with this $N$, and in real arithmetic $N$ is its image under the canonical map, with multiplicative inverse $N^{-1}$. 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; the letter $\sigma$ denotes the noise intensity, and 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; the letter $\theta$ denotes the control cost, and tolerances are written $\vartheta$.

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 let $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$ be a real $p\times d$ matrix. Let $L\in\mathbb{R}$ be nonnegative, let $\Phi$ be a convex Lipschitz integrand with constant $L$, let $\eta$ be a mollifier kernel of radius $1$ on $\mathbb{R}^{d}$, 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 $(\varepsilon_{N})_{N\in\mathbb{N}}$ be real numbers with $0<\varepsilon_{N}\le1$ for every $N\in\mathbb{N}$ such that $\varepsilon_{N}$ tends to $0$ and $N\varepsilon_{N}^{d}$ tends to infinity, in the following sense: for every positive $\tau\in\mathbb{R}$ there is $N_{0}\in\mathbb{N}$ with $\varepsilon_{N}\le\tau$ for every $N\in\mathbb{N}$ with $N_{0}\le N$; and for every positive $M\in\mathbb{R}$ there is $N_{0}\in\mathbb{N}$ with $M\le N\,\varepsilon_{N}^{d}$ for every $N\in\mathbb{N}$ with $N_{0}\le N$, the power being that of Natural Number Power of an Element of a Field.

Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Langevin free-energy pair with potential $V$ and noise intensity $\sigma$, and let $|s|$ be the absolute value of $s\in\mathbb{R}$. The function $\nu\mapsto\int_{\mathbb{R}^{d}}f\,d\nu$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$ is bounded and uniformly continuous for $W_{2}$ by Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, read with $m=d$; let $u:\mathcal{D}\to\mathbb{R}$ be the unique bounded viscosity solution of the Langevin Hamilton-Jacobi equation with common noise and density cost with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma$, control cost $\theta$, running cost $\nu\mapsto\int_{\mathbb{R}^{d}}f\,d\nu$ and integrand $\Phi$, which exists and is unique by Well-Posedness of the Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost: Existence and Uniqueness of a Bounded Viscosity Solution §existence and Well-Posedness of the Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost: Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness.

For each $N\in\mathbb{N}$, let $c_{N,\varepsilon_{N}}$ be the mollified $N$-particle cost with data $f$, $\Phi$, $\eta$ and $\varepsilon_{N}$, bounded and uniformly continuous on $\mathbb{R}^{dN}$ by The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost §regularity; let $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ be the Langevin free-energy pair with the $N$-particle potential $V_{N}$ and noise intensity $\sigma$, formed at the configuration level, as in The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space; and let $U_{N}:\mathcal{D}_{N}\to\mathbb{R}$ be the unique bounded viscosity solution of the lifted $N$-particle Hamilton-Jacobi equation with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, control cost $\theta$, common-noise matrix $\Gamma$ and running cost $c_{N,\varepsilon_{N}}$, which exists and is unique by Well-Posedness of the Lifted N-Particle Hamilton-Jacobi Equation: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §existence and Well-Posedness of the Lifted N-Particle Hamilton-Jacobi Equation: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness. Tensor powers $\mu^{\otimes N}$ and one-particle marginals $P^{[1]}$ are those of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles; for $\mu\in\mathcal{D}$, $\mu^{\otimes N}\in\mathcal{D}_{N}$ by The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §tensor, and for $P\in\mathcal{D}_{N}$, $P^{[1]}\in\mathcal{D}$ by The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §marginal.

1. (Convergence at tensor powers)¶ For all positive $R,\vartheta\in\mathbb{R}$ there is $N_{1}\in\mathbb{N}$ such that

2. (Lower bound at one-particle marginals)¶ For all positive $R,\vartheta\in\mathbb{R}$ there is $N_{1}\in\mathbb{N}$ such that