Well-Posedness of the Lifted N-Particle Hamilton-Jacobi Equation: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution

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 $0<\theta\le1$, let $p\in\mathbb{N}$ and $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$, and let $c:\mathbb{R}^{dN}\to\mathbb{R}$ be uniformly continuous for the Euclidean distance and the metric of The Absolute Value Metric on the Real Line, with $b\in\mathbb{R}$ such that $|c(x)|\le b$ for every $x\in\mathbb{R}^{dN}$. Viscosity solutions, subsolutions and supersolutions 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 $c$ are those of that clause, functions on the domain $\mathcal{D}_{N}$ of the Langevin free-energy pair $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ named there; $\lambda_{0}^{-1}$ is the multiplicative inverse of $\lambda_{0}$, and uniform continuity on a subset of $\mathcal{D}_{N}$ refers to $W_{2}$ on $\mathcal{P}_{2}(\mathbb{R}^{dN})$ restricted to that subset. 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 $u,v:\mathcal{D}_{N}\to\mathbb{R}$ be a viscosity subsolution bounded above and a viscosity supersolution bounded below of that equation. Then $u(P)\le v(P)$ for every $P\in\mathcal{D}_{N}$.

2. (Existence)¶ There is a viscosity solution $u:\mathcal{D}_{N}\to\mathbb{R}$ of that equation with $-\lambda_{0}^{-1}b\le u(P)\le\lambda_{0}^{-1}b$ for every $P\in\mathcal{D}_{N}$.

3. (Uniqueness and continuity)¶ Any two bounded viscosity solutions of that equation are equal, and a bounded viscosity solution is uniformly continuous on $\{P\in\mathcal{D}_{N}:\mathcal{E}_{N}(P)\le a\}$ for every $a\in\mathbb{R}$.