The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. Let $V$ be a confining potential on $\mathbb{R}^{d}$ with $N$-particle potential $V_{N}$, a confining potential on $\mathbb{R}^{dN}$ by The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §confining; let $\lambda_{0},\sigma,\theta\in\mathbb{R}$ be positive, let $p\in\mathbb{N}$, let $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$ with $N$-particle common-noise matrix $\Gamma_{N}\in\mathcal{M}_{p\times dN}(\mathbb{R})$, and let $c:\mathbb{R}^{dN}\to\mathbb{R}$ be bounded and Borel, hence integrable with respect to every probability measure on $\mathbb{R}^{dN}$. 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 $\sigma$, formed at the configuration level, and let $\xi_{P}$ be the score of $P\in\mathcal{D}_{N,\Sigma}$, at the configuration level. 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. (The operator)¶ The lifted $N$-particle Hamilton-Jacobi operator with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, control cost $\theta$, common-noise matrix $\Gamma$ and running cost $c$ is the Langevin Hamilton-Jacobi operator with common noise, at the configuration level, with potential $V_{N}$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma_{N}$, control cost $\theta$ and running cost $P\mapsto\int_{\mathbb{R}^{dN}}c\,dP$ on $\mathcal{P}_{2}(\mathbb{R}^{dN})$. By that clause its value at $(P,r,q,Y)$, with $(P,q)$ in the bundle $\mathcal{V}(\mathcal{D}_{N,\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(dN)$, is

2. (The equation)¶ The lifted $N$-particle Hamilton-Jacobi equation is $F_{N}(P,r,q,Y)=0$. Its classical and viscosity solutions, subsolutions and supersolutions are those of The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space §equation, at the configuration level, for these data; viscosity ones are functions on $\mathcal{D}_{N}$.