The Half-Relaxed Limits of the Regularised N-Particle Dyson Solutions are Viscosity Sub- and Supersolutions of the Limit Dyson Equation up to a Cost Defect

In the setting of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations, with the confined logarithmic-energy pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §limit-equation, and with $\omega_{g}$, $R_{\tau}$, $\bar{u}_{\tau}$ and $\underline{u}_{\tau}$ as in Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy for positive $\tau\in\mathbb{R}$. For $h\in\mathbb{R}$ let $F^{h}$ be the Dyson Hamilton-Jacobi operator with confining potential $V$, discount $\lambda$, common-noise intensity $0$ and running cost $g+h$, that is

so that $F^{0}$ is the limit operator $F$ of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §limit-equation. For positive $\tau$ put $h_{\tau}=\omega_{g}\bigl(\sqrt{2}\,R_{\tau}\bigr)$. The functions $\bar{u}_{\tau}$ and $\underline{u}_{\tau}$ are bounded by Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §bounds, so they have penalty-subordinate growth from above and from below by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth, the pair being Wasserstein-coercive by The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score and Regular Penalised Maxima §coercive.

1. (The cost defect)¶ $0\le h_{\tau}\le2b_{g}$ for every positive $\tau$, and for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\tau_{0}$ with $h_{\tau}\le\varepsilon$ whenever $0<\tau\le\tau_{0}$.

2. (Subsolution)¶ For every positive $\tau$, $\bar{u}_{\tau}$ is a viscosity subsolution of $F^{h_{\tau}}$ relative to the pair.

3. (Supersolution)¶ For every positive $\tau$, $\underline{u}_{\tau}$ is a viscosity supersolution of $F^{-h_{\tau}}$ relative to the pair.