Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy

In the setting of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations, with the data $\beta,\sigma,\lambda,V,g,b_{g}$ of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §data, the chambers $W_{N}$, potentials $P_{N}$ and numbers $\kappa_{N}$ of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §particles, the solutions $v_{N}$ of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §particle-equation, and $\mathcal{D}$ as in The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §limit-equation. Let $\varepsilon_{0}=1-\sigma^{2}/\beta$ and fix $C_{0}\ge0$ as in The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §dissipation, and $e_{*}$ as in The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §bounds. Let $N_{0}\ge2$ be a natural number with $N_{0}\varepsilon_{0}\ge1$. Let $\omega_{g}(s)$, for $s\ge0$, be the least upper bound of $\{|g(\mu)-g(\nu)|:\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}),\ W_{2}(\mu,\nu)\le s\}$, a nonempty set bounded above by $2b_{g}$; thus $\omega_{g}$ is nondecreasing and $|g(\mu)-g(\nu)|\le\omega_{g}(W_{2}(\mu,\nu))$.

Regularisation. Let $\tau\in\mathbb{R}$ be positive. For $N\ge N_{0}$, let $\overline{w}^{0}_{N,\tau}$ and $\underline{w}^{0}_{N,\tau}$ be the functions $\overline{w}$ of Part A and $\underline{w}$ of Part B of Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty, applied with $n=N$, $D=W_{N}$, $P=P_{N}$, discount $\lambda$, $\theta=1$, $\kappa=\kappa_{N}$, $\varepsilon=\varepsilon_{0}$, $C=C_{0}N$, running cost $x\mapsto N\,g(\mu^{N}_{x})$ with $\rho(s)=N\,\omega_{g}(s/\sqrt{N})$, $M=\lambda^{-1}Nb_{g}$, $\eta=\frac{1}{N}$, this $\tau$, and $u=v=v_{N}$; the hypotheses of that lemma hold (Step 1 of the proof). With $L_{N}=N(\lambda^{-1}b_{g}+1)$ put

Let $p_{0,N}$ be the least value of $P_{N}$ on $W_{N}$ (it exists by that lemma), $r_{N}(x)\ge0$ the square root of $2\tau\bigl(2\lambda^{-1}Nb_{g}+(P_{N}(x)-p_{0,N})/N\bigr)$, $\theta_{1,N}=1-\frac{1}{2N\varepsilon_{0}}$, $\theta_{2,N}=1+\frac{1}{N\varepsilon_{0}}$, and $g^{\pm}_{N}(x)=N\,g(\mu^{N}_{x})\pm\bigl(C_{0}+N\,\omega_{g}(r_{N}(x)/\sqrt{N})\bigr)$. Let $\bar{u}_{\tau}:\mathcal{D}\to\mathbb{R}$ be the upper half-relaxed limit of $(\overline{w}_{N,\tau})_{N\ge N_{0}}$ and $\underline{u}_{\tau}:\mathcal{D}\to\mathbb{R}$ the lower half-relaxed limit of $(\underline{w}_{N,\tau})_{N\ge N_{0}}$; the growth conditions of Upper and Lower Half-Relaxed Limits along Empirical Measures of Functions on the Weyl Chambers hold with $c=\lambda^{-1}b_{g}+|e_{*}|+1$ by clause 1 below. Put $R_{\tau}=\sqrt{4\tau\lambda^{-1}b_{g}}$.

1. (The regularised solutions)¶ For every natural number $N\ge N_{0}$ and all $x,y\in W_{N}$: (i)¶ $-L_{N}\le\overline{w}_{N,\tau}(x)\le\lambda^{-1}Nb_{g}+|e_{*}|$, $v_{N}(x)-\frac{1}{N}P_{N}(x)\le\overline{w}_{N,\tau}(x)$, and $-\lambda^{-1}Nb_{g}-|e_{*}|\le\underline{w}_{N,\tau}(x)\le L_{N}$, $\underline{w}_{N,\tau}(x)\le v_{N}(x)+\frac{1}{N}P_{N}(x)$; (ii)¶ $\overline{w}_{N,\tau}$ and $-\underline{w}_{N,\tau}$ are semiconvex on $W_{N}$ with constant $\tau^{-1}$; (iii)¶ at every point $x$ at which $\overline{w}_{N,\tau}$, respectively $\underline{w}_{N,\tau}$, is differentiable, its gradient has norm at most $\tau^{-1}r_{N}(x)$; (iv)¶ $\overline{w}_{N,\tau}(x)\le\overline{w}_{N,\tau}(y)+\frac{1}{2\tau}\lVert x-y\rVert\bigl(3\lVert x-y\rVert+2r_{N}(y)\bigr)$ and $\underline{w}_{N,\tau}(x)\ge\underline{w}_{N,\tau}(y)-\frac{1}{2\tau}\lVert x-y\rVert\bigl(3\lVert x-y\rVert+2r_{N}(y)\bigr)$; (v)¶ $\overline{w}_{N,\tau}$ is a viscosity subsolution on $W_{N}$ of the penalty-drift Hamilton-Jacobi operator with potential $P_{N}$, discount $\lambda$, control cost $\theta_{1,N}$, noise intensity $\kappa_{N}$ and running cost $g^{+}_{N}$, and $\underline{w}_{N,\tau}$ is a viscosity supersolution on $W_{N}$ of the one with control cost $\theta_{2,N}$ and running cost $g^{-}_{N}$; the functions $g^{\pm}_{N}$ are bounded, and Borel on $W_{N}$.

2. (Bounds)¶ $-\lambda^{-1}b_{g}\le\underline{u}_{\tau}(\mu)\le\lambda^{-1}b_{g}$ and $-\lambda^{-1}b_{g}\le\bar{u}_{\tau}(\mu)\le\lambda^{-1}b_{g}$ for every $\mu\in\mathcal{D}$.

3. (Modulus)¶ For all $\mu,\mu'\in\mathcal{D}$, with $W=W_{2}(\mu,\mu')$,

4. (Bounds along convergent configurations)¶ Let $(N_{k})_{k\in\mathbb{N}}$ be a strictly increasing sequence of natural numbers with $N_{1}\ge N_{0}$, let $x^{k}\in W_{N_{k}}$ and $\mu\in\mathcal{D}$ be such that $(W_{2}(\mu^{N_{k}}_{x^{k}},\mu))_{k}$ converges to $0$, and let $\varepsilon\in\mathbb{R}$ be positive. Then there is $k_{0}$ with

If moreover $P_{N_{k}}(x^{k})\le c'N_{k}$ for every $k$, for some $c'\in\mathbb{R}$, then $k_{0}$ can be chosen so that also

5. (Realising levels)¶ Let $\mu\in\mathcal{D}$. There is a strictly increasing sequence $(N_{k})_{k\in\mathbb{N}}$ of natural numbers with $N_{1}\ge N_{0}$ such that, for every sequence of points $x^{k}\in W_{N_{k}}$ with $(W_{2}(\mu^{N_{k}}_{x^{k}},\mu))_{k}$ converging to $0$ and $P_{N_{k}}(x^{k})\le c'N_{k}$ for every $k$ and some $c'\in\mathbb{R}$, the sequence $\bigl(\overline{w}_{N_{k},\tau}(x^{k})/N_{k}\bigr)_{k}$ converges to $\bar{u}_{\tau}(\mu)$. Likewise there is a strictly increasing sequence $(N'_{k})_{k}$ with $N'_{1}\ge N_{0}$ along which, under the same conditions with $N'_{k}$ in place of $N_{k}$, $\bigl(\underline{w}_{N'_{k},\tau}(x^{k})/N'_{k}\bigr)_{k}$ converges to $\underline{u}_{\tau}(\mu)$.