The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations

This setting fixes the standing data of results on the mean-field limit of the $N$-particle Dyson game. It is layered on N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level, read with particle dimension $d=1$ and with the identifications of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative, and on Second-Order Equations on Euclidean Open Sets; it introduces no concept and asserts nothing beyond what its references provide. Natural numbers are regarded as real numbers through the canonical map. The potentials $P_{N}$ below always carry their index, so that they do not clash with the letter $P$ reserved for laws on the configuration space by N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles.

1. (Data)¶ $\beta,\sigma,\lambda\in\mathbb{R}$ are positive with $\sigma^{2}<\beta$, where $\sigma^{2}=\sigma\sigma$; $V:\mathbb{R}\to\mathbb{R}$ is a confining potential with derivative $V'$; $g:\mathcal{P}_{2}(\mathbb{R})\to\mathbb{R}$ is uniformly continuous for $W_{2}$ and the metric of The Absolute Value Metric on the Real Line, and $b_{g}\in\mathbb{R}$ satisfies $|g(\nu)|\le b_{g}$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R})$.

2. (The particle level)¶ For every natural number $N\ge2$, the Weyl chamber $W_{N}\subseteq\mathbb{R}^{N}$, the numbers

and the potential $P_{N}(x)=H_{b_{N}}(x)+\sum_{k=1}^{N}V(x_{k})$ on $W_{N}$ are those of The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability, which is in force for the data of clause 1. For $x\in\mathbb{R}^{N}$, $\mu^{N}_{x}\in\mathcal{P}_{2}(\mathbb{R})$ is the empirical measure of $x$.

3. (The $N$-particle equation)¶ $F_{N}$ is the $N$-particle Dyson Hamilton-Jacobi operator with interaction $\beta$, confinement $V$, discount $\lambda$, noise $\sigma$ and mean-field cost $g$, so that the $N$-particle equation reads

$F_{N}$ is the Dyson Hamilton-Jacobi operator with strength $b_{N}$, confinement $V$, discount $\lambda$, control cost $1$, noise intensity $\kappa_{N}$ and running cost $x\mapsto N\,g(\mu^{N}_{x})$, and Well-Posedness of the Dyson Hamilton-Jacobi Equation in the Weyl Chamber below the Collision Threshold §well-posed applies to it: its hypotheses on $V$ and $0\le\kappa_{N}<2b_{N}$ hold by The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §hypotheses and The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §penalty, and the running cost is continuous on $W_{N}$ with $|N\,g(\mu^{N}_{x})|\le Nb_{g}$ by Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §empirical (read with $d=1$). $v_{N}:W_{N}\to\mathbb{R}$ denotes the unique viscosity solution of $F_{N}$ on $W_{N}$ with $P_{N}$-subordinate growth from above and from below provided by that clause; it is continuous and $|v_{N}(x)|\le\lambda^{-1}Nb_{g}$ for every $x\in W_{N}$.

4. (The limit equation)¶ $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is the confined logarithmic-energy pair with potential $V$ and inverse temperature $\beta$, and $F$ is the Dyson Hamilton-Jacobi operator with confining potential $V$, discount $\lambda$, common-noise intensity $0$ and running cost $g$, so that for $(\nu,q)$ in the bundle $\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(1)$

with $\Xi_{\nu}$ the free score of $\nu$. $u:\mathcal{D}\to\mathbb{R}$ denotes the unique bounded viscosity solution of the Dyson Hamilton-Jacobi equation for these data provided by Well-Posedness of the Dyson Hamilton-Jacobi Equation with Common Noise and a Confining Potential: Existence and Uniqueness of a Bounded Viscosity Solution §existence and Well-Posedness of the Dyson Hamilton-Jacobi Equation with Common Noise and a Confining Potential: Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness; it satisfies $|u(\mu)|\le\lambda^{-1}b_{g}$ for every $\mu\in\mathcal{D}$.