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

Each result cited is universally quantified over the data in its own statement. We apply Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality with $D=W_{N}$, which is open and nonempty by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §open, and with the penalty $P$ (The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §penalty, whose hypothesis is the linear lower bound on $V_{1}$). By The Dyson Hamilton-Jacobi Equation for N Controlled Particles in the Weyl Chamber §operator, $F$ is the penalty-drift Hamilton-Jacobi operator on $W_{N}$ with potential $P$ and the given coefficients.

One-sided drift. Since $V_{1}''\ge0$, The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §monotone gives $0\le(DP(x)-DP(y))\cdot(x-y)$ for all $x,y\in W_{N}$. For every $R\in\mathbb{R}$ we may therefore take $c_{R}=0$ in hypothesis Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §one-sided, since $-0\cdot\lVert x-y\rVert^{2}=0$ (claim 1 of Zero Products and Elementary Identities in a Field).

Dissipation. Since $0<\lambda$, $0\le\kappa<2\beta$, $V_{1}''\ge0$, $V_{1}$ has the linear lower bound and regular growth, The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §dissipation gives $\varepsilon,C$ with $0<\varepsilon\le1$ for which hypothesis Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §dissipation holds.

Hence all hypotheses of Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality hold, and its claims Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §comparison and Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §well-posed are claims 1 and 2.