The N-Particle Dyson Hamilton-Jacobi Equation with a Mean-Field Running Cost

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level with particle dimension $d=1$, and in the setting of Second-Order Equations on Euclidean Open Sets. Let $N\ge2$ be a natural number, regarded as a real number through the canonical map, so that $N-1$ is positive. Let $\beta,\lambda\in\mathbb{R}$ be positive, let $\sigma\in\mathbb{R}$, let $V:\mathbb{R}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}=\mathbb{R}^{1}$ with derivative $V'=\partial_{1}V$, and let $g:\mathcal{P}_{2}(\mathbb{R})\to\mathbb{R}$. For $x\in\mathbb{R}^{N}$, $\mu^{N}_{x}\in\mathcal{P}_{2}(\mathbb{R})$ is the empirical measure of $x$. The Weyl chamber $W_{N}=\{x\in\mathbb{R}^{N}:x_{1}>x_{2}>\dots>x_{N}\}$ and the numbers $a_{kj}(x)$, equal to $(x_{k}-x_{j})^{-1}$ for $k\ne j$ and to $0$ for $k=j$, are those of The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity; $W_{N}$ is open by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §open. Put

1. (The operator)¶ The $N$-particle Dyson Hamilton-Jacobi operator with interaction $\beta$, confinement $V$, discount $\lambda$, noise $\sigma$ and mean-field cost $g$ is the Dyson Hamilton-Jacobi operator on $W_{N}$ with strength $b_{N}$, confinement $V$, discount $\lambda$, control cost $1$, noise intensity $\kappa_{N}$ and running cost $x\mapsto N\,g(\mu^{N}_{x})$; by the formula of that clause, for $x\in W_{N}$, $r\in\mathbb{R}$, $p\in\mathbb{R}^{N}$ and $X\in\mathcal{S}(N)$,

2. (The equation)¶ The $N$-particle Dyson Hamilton-Jacobi equation is

with $\Delta v=\operatorname{tr}(D^{2}v)$, that is, the Dyson Hamilton-Jacobi equation for the data of clause 1; its viscosity subsolutions and supersolutions are those of Viscosity Subsolution and Supersolution of a Second-Order Equation for the operator $F_{N}$ on $W_{N}$.