The Dyson Hamilton-Jacobi Equation in Singular-Cost Form on the Weyl Chamber

In the setting of Second-Order Equations on Euclidean Open Sets, let $N\ge2$ be a natural number, let $W_{N}$ be the Weyl chamber, open by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §open, and let $S:W_{N}\to\mathbb{R}$ be the sum of the inverse squared gaps of The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity, $S(x)=\sum_{k=1}^{N}\sum_{j=1}^{N}a_{kj}(x)^{2}$ with $a_{kj}(x)=(x_{k}-x_{j})^{-1}$ for $k\ne j$, which exists since $x_{k}\ne x_{j}$ by the definition of $W_{N}$, and $a_{kk}(x)=0$. Let $\lambda,\theta\in\mathbb{R}$ be positive, let $\kappa,\omega\in\mathbb{R}$ be nonnegative, let $c\in\mathbb{R}$ 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 singular-cost Dyson Hamilton-Jacobi operator with pair-cost coefficient $c$, confinement coefficient $\omega$, discount $\lambda$, control cost $\theta$, noise intensity $\kappa$ and running cost $g$ is the second-order equation operator on $W_{N}$

2. (The equation)¶ The singular-cost Dyson Hamilton-Jacobi equation is

that is, $F(x,u(x),Du(x),D^{2}u(x))=0$ for $x\in W_{N}$. Its classical sub- and supersolutions are those of Classical Subsolution and Supersolution of a Second-Order Equation and its viscosity sub- and supersolutions those of Viscosity Subsolution and Supersolution of a Second-Order Equation, for $F$ on $W_{N}$.