Mean-Field Limit of the N-Particle Dyson Game: the Value per Particle Converges to the Solution of the Dyson Hamilton-Jacobi Equation on the Wasserstein Space

In the setting of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations: $\beta,\sigma,\lambda$ are positive with $\sigma^{2}<\beta$, $V$ is a confining potential and $g$ a bounded uniformly continuous function on $\mathcal{P}_{2}(\mathbb{R})$. For each $N\ge2$, $v_{N}$ is the viscosity solution on the Weyl chamber $W_{N}=\{x_{1}>\dots>x_{N}\}$ of the $N$-particle Dyson Hamilton-Jacobi equation

and $u:\mathcal{D}\to\mathbb{R}$ is the bounded viscosity solution of the Dyson Hamilton-Jacobi equation on the Wasserstein space,

understood relative to the confined logarithmic-energy pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$, whose energy is $\mathcal{E}(\mu)=\frac{\beta}{4}\mathcal{E}_{\log}(\mu)+\int V\,d\mu$ and whose score is $\Sigma(\mu)=V'-\frac{\beta}{4}\Xi_{\mu}$, with $\Xi_{\mu}$ the free score. The $N$-particle potential is $P_{N}(x)=-\frac{\beta}{2(N-1)}\sum_{i<j}\log(x_{i}-x_{j})+\sum_{k}V(x_{k})$.

(Mean-field limit)¶ Let $\mu\in\mathcal{D}$, let $x^{N}\in W_{N}$ for every natural number $N\ge2$, and suppose that $\bigl(W_{2}(\mu^{N}_{x^{N}},\mu)\bigr)_{N}$ converges to $0$ and that there is $c\in\mathbb{R}$ with $P_{N}(x^{N})\le cN$ for every $N$. Then