Cross-Level Comparison through One-Particle Marginals: N Times a Mean-Field Subsolution Lies below a Lifted N-Particle Supersolution under Cost Domination

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 $0<\theta\le1$, and let $p\in\mathbb{N}$ and $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$. Let $c:\mathbb{R}^{dN}\to\mathbb{R}$ and $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be uniformly continuous, for the Euclidean distance and for $W_{2}$ respectively and the metric of The Absolute Value Metric on the Real Line, and let $b\in\mathbb{R}$ satisfy $|c(x)|\le b$ and $|g(\nu)|\le b$ for every $x\in\mathbb{R}^{dN}$ and $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The $N$-particle equation is 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 $c$, whose viscosity sub- and supersolutions are functions on the domain $\mathcal{D}_{N}$ of the configuration-level Langevin pair named there; the mean-field equation is the Langevin Hamilton-Jacobi equation with common noise with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma$, control cost $\theta$ and running cost $g$, whose viscosity sub- and supersolutions are functions on the domain $\mathcal{D}$ of the Langevin pair named there. 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. Assume the cost domination¶

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.

1. (Comparison)¶ Let $v:\mathcal{D}\to\mathbb{R}$ be a viscosity subsolution of the mean-field equation that is bounded above, and let $U:\mathcal{D}_{N}\to\mathbb{R}$ be a viscosity supersolution of the $N$-particle equation that is bounded below. Then

2. (Solutions)¶ The mean-field equation has a unique bounded viscosity solution $\bar{u}$, by Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution §existence and Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness, and the $N$-particle equation has a unique bounded viscosity solution $U_{N}$, 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. They satisfy $N\,\bar{u}(P^{[1]})\le U_{N}(P)$ for every $P\in\mathcal{D}_{N}$.