Well-Posedness of the Dyson Hamilton-Jacobi Equation in the Weyl Chamber below the Collision Threshold

In the setting of Second-Order Equations on Euclidean Open Sets, let $N\ge2$ be a natural number, let $\beta,\lambda\in\mathbb{R}$ be positive, and let $\theta,\kappa\in\mathbb{R}$ satisfy $0\le\theta$ and $0\le\kappa<2\beta$. Let $V_{1}:\mathbb{R}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}$, with $V_{1}''\ge0$, with $a_{0}|t|-b_{0}\le V_{1}(t)$ for some $a_{0},b_{0}\in\mathbb{R}$ with $0<a_{0}$ and every $t\in\mathbb{R}$, and of regular growth. Let $W_{N}$ and $P$ be as in The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality, so that $P$ is a penalty on $W_{N}$ by The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §penalty. Let $g:W_{N}\to\mathbb{R}$ be continuous into the real line, let $M\in\mathbb{R}$ satisfy $|g(x)|\le M$ for every $x\in W_{N}$, and let $F$ be the Dyson Hamilton-Jacobi operator with strength $\beta$, confinement $V_{1}$, discount $\lambda$, control cost $\theta$, noise intensity $\kappa$ and running cost $g$.

Then the following hold.

1. (Comparison)¶ If $u:W_{N}\to\mathbb{R}$ is a viscosity subsolution of $F$ on $W_{N}$ with $P$-subordinate growth from above and $v:W_{N}\to\mathbb{R}$ is a viscosity supersolution of $F$ on $W_{N}$ with $P$-subordinate growth from below, then $u(x)\le v(x)$ for every $x\in W_{N}$.

2. (Existence and uniqueness)¶ There is exactly one function $u:W_{N}\to\mathbb{R}$ that is both a viscosity subsolution and a viscosity supersolution of $F$ on $W_{N}$ and has $P$-subordinate growth from above and from below. It is continuous and satisfies $-\lambda^{-1}M\le u(x)\le\lambda^{-1}M$ for every $x\in W_{N}$.