Classical Solutions of the N-Particle Hamilton-Jacobi Equation Lift to Classical Solutions of the Lifted Equation, and a Bounded One is the Lifted 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 $\lambda_{0},\sigma,\theta\in\mathbb{R}$ be positive, let $p\in\mathbb{N}$ and $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$, let $c:\mathbb{R}^{dN}\to\mathbb{R}$ be bounded and continuous for the Euclidean distance and the metric of The Absolute Value Metric on the Real Line, and let $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ be the Langevin free-energy pair with potential $V_{N}$ and noise intensity $\sigma$ at the configuration level, as in The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space. Let $M\in\mathbb{R}$ be nonnegative and let $v:\mathbb{R}^{dN}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{dN}$ with $|\partial_{i}v(x)|\le M$ and $|\partial_{j}\partial_{i}v(x)|\le M$ for all $x\in\mathbb{R}^{dN}$ and $i,j\in[dN]$, and let $\varphi_{v}(P)=\int_{\mathbb{R}^{dN}}v\,dP$ for $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, an intrinsic test function on $\mathcal{P}_{2}(\mathbb{R}^{dN})$ at the configuration level by that clause, hence on $\mathcal{D}_{N,\Sigma}$ by Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §restriction. Classical sub- and supersolutions of the $N$-particle Hamilton-Jacobi equation and of the lifted $N$-particle Hamilton-Jacobi equation are taken with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, control cost $\theta$, common-noise matrix $\Gamma$ and running cost $c$, the former on $\mathbb{R}^{dN}$ and the latter on $\mathcal{D}_{N,\Sigma}$. 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. Then the following hold.

1. (Subsolutions)¶ If $v$ is a classical subsolution of the $N$-particle equation, then $\varphi_{v}$ is a classical subsolution of the lifted $N$-particle equation.

2. (Supersolutions)¶ If $v$ is a classical supersolution of the $N$-particle equation, then $\varphi_{v}$ is a classical supersolution of the lifted $N$-particle equation.

3. (Identification)¶ Suppose moreover that $\theta\le1$, that $c$ is uniformly continuous for the same metrics, that $v$ is bounded and that $v$ is a classical subsolution and a classical supersolution of the $N$-particle equation. Then the restriction of $\varphi_{v}$ to $\mathcal{D}_{N}$ is a viscosity solution of the lifted $N$-particle equation, and it is the only bounded one.