The Dyson Hamilton-Jacobi Equation for N Controlled Particles in the Weyl Chamber

In the setting of Second-Order Equations on Euclidean Open Sets, let $N\ge2$ be a natural number, let $\beta\in\mathbb{R}$ be positive, let $V_{1}:\mathbb{R}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}$, and let $W_{N}$, $a_{kj}$ and $P=H_{\beta}+\sum_{k}V_{1}(x_{k})$ be as in The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality; $W_{N}$ is open by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §open and $P$ is of class $C^{2}$ on $W_{N}$ by The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §regularity. Let $\lambda\in\mathbb{R}$ be positive, let $\theta,\kappa\in\mathbb{R}$ be nonnegative and let $g:W_{N}\to\mathbb{R}$. Traces, squared norms and halves are written as in The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space.

1. (The operator)¶ The Dyson Hamilton-Jacobi operator with strength $\beta$, confinement $V_{1}$, discount $\lambda$, control cost $\theta$, noise intensity $\kappa$ and running cost $g$ is the penalty-drift Hamilton-Jacobi operator on $D=W_{N}$ with potential $P$ and these coefficients; by The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §regularity,

2. (The equation)¶ The Dyson Hamilton-Jacobi equation is the penalty-drift Hamilton-Jacobi equation for these data on $W_{N}$.