Proof of The Mean-Field Limit of the Lifted N-Particle Hamilton-Jacobi Equation with a Mollified Local Coupling

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here; results adopting N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level are applied, for each $N\in\mathbb{N}$, in that setting with this $N$. The field axioms and the rules for adding inequalities, for multiplying them by positive real numbers and for handling absolute values, from Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field and Properties of the Absolute Value in an Ordered Field, are used without further mention; so is $0<N$ (claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field). Let $b$ be a bound for $f$ (Bounded Real-Valued Function on a Set). For each $N$ let $g_{\varepsilon_{N}}$ be the mollified mean-field cost with data $f$, $\Phi$, $\eta$ and $\varepsilon_{N}$; by The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost §regularity, $g_{\varepsilon_{N}}$ is uniformly continuous for $W_{2}$ with $|g_{\varepsilon_{N}}|\le b+L$, and $c_{N,\varepsilon_{N}}$ is uniformly continuous with $|c_{N,\varepsilon_{N}}|\le N(b+L)$. Put $b_{N}=N(b+L)+(b+L)$; since $0\le b+L$ and $0<N$, both $|c_{N,\varepsilon_{N}}|\le b_{N}$ and $|g_{\varepsilon_{N}}|\le b_{N}$. The hypotheses of Upper Convergence: Mean-Field Solutions with the Tensor-Averaged Mollified Costs Are Asymptotically below the Solution with the Local Cost and of Lower Convergence: The Solution with the Local Cost Is Asymptotically below the Mean-Field Solutions with the Mollified Costs hold for the present data, the sequence $(\varepsilon_{N})$ satisfying both of the conditions stated there. Let $\bar{u}_{N}$ be the function so named in Upper Convergence: Mean-Field Solutions with the Tensor-Averaged Mollified Costs Are Asymptotically below the Solution with the Local Cost, the unique bounded viscosity solution of the Langevin Hamilton-Jacobi equation with common noise with running cost the tensor-averaged cost of $c_{N,\varepsilon_{N}}$, and $u'_{N}$ the function so named in Lower Convergence: The Solution with the Local Cost Is Asymptotically below the Mean-Field Solutions with the Mollified Costs, the unique bounded viscosity solution of that equation with running cost $g_{\varepsilon_{N}}$ (the remaining coefficients being $V$, $\sigma$, $\lambda_{0}$, $\Gamma$ and $\theta$ in both cases). The function $u$ is the same in both corollaries and in the present statement.

Step 1 (Upper bound at tensor powers). Fix $N$. By Cross-Level Comparison at Tensor Powers: a Lifted N-Particle Subsolution Lies below N Times a Mean-Field Supersolution with the Tensor-Averaged Cost §solutions, applied with $V$, $\lambda_{0}$, $\sigma$, $\theta$, $p$, $\Gamma$, the cost $c=c_{N,\varepsilon_{N}}$ and the bound $b_{N}$: the unique bounded viscosity solution of its $N$-particle equation, which is the lifted $N$-particle equation with running cost $c_{N,\varepsilon_{N}}$ and so has $U_{N}$ as its unique bounded viscosity solution, and the unique bounded viscosity solution of its mean-field equation, which is $\bar{u}_{N}$, satisfy

Let $R,\vartheta\in\mathbb{R}$ be positive, and let $N_{A}\in\mathbb{N}$ be given by Upper Convergence: Mean-Field Solutions with the Tensor-Averaged Mollified Costs Are Asymptotically below the Solution with the Local Cost §convergence for $R$ and $\vartheta$. For $N\ge N_{A}$ and $\mu\in\mathcal{D}$ with $|\mathcal{E}(\mu)|\le R$, (1) and that clause give

Step 2 (Lower bound at one-particle marginals). Fix $N$. Every $P\in\mathcal{D}_{N}$ belongs to $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{dN})\subseteq\mathcal{P}_{2}(\mathbb{R}^{dN})$, by The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space §pair and The Entropy of a Probability Measure on Euclidean Space §entropy read at the configuration level (N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level); so The Mollified N-Particle Cost Dominates N Times the Mollified Mean-Field Cost of the One-Particle Marginal §domination, applied with $\varepsilon=\varepsilon_{N}$, gives the cost domination

which is the hypothesis Cross-Level Comparison through One-Particle Marginals: N Times a Mean-Field Subsolution Lies below a Lifted N-Particle Supersolution under Cost Domination §domination for $c=c_{N,\varepsilon_{N}}$ and $g=g_{\varepsilon_{N}}$. By Cross-Level Comparison through One-Particle Marginals: N Times a Mean-Field Subsolution Lies below a Lifted N-Particle Supersolution under Cost Domination §solutions, applied with $V$, $\lambda_{0}$, $\sigma$, $\theta$, $p$, $\Gamma$, these $c$ and $g$ and the bound $b_{N}$: the unique bounded viscosity solution of its mean-field equation, which is $u'_{N}$, and that of its $N$-particle equation, which is $U_{N}$, satisfy

Let $R,\vartheta\in\mathbb{R}$ be positive, and let $N_{B}\in\mathbb{N}$ be given by Lower Convergence: The Solution with the Local Cost Is Asymptotically below the Mean-Field Solutions with the Mollified Costs §convergence for $R$ and $\vartheta$. Let $N\ge N_{B}$ and $P\in\mathcal{D}_{N}$ with $|\mathcal{E}(P^{[1]})|\le R$; since $P^{[1]}\in\mathcal{D}$ (The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §marginal), that clause applied to $\mu=P^{[1]}$ and (3) give

This proves claim 2 with $N_{1}=N_{B}$.

Step 3 (Convergence at tensor powers). Let $R,\vartheta\in\mathbb{R}$ be positive, let $N_{A}$ and $N_{B}$ be as in Steps 1 and 2 for these $R$ and $\vartheta$, and put $N_{1}=\max\{N_{A},N_{B}\}$. Let $N\ge N_{1}$ and $\mu\in\mathcal{D}$ with $|\mathcal{E}(\mu)|\le R$. Then $P=\mu^{\otimes N}\in\mathcal{D}_{N}$ by The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §tensor, and $P^{[1]}=\mu$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor, read with dimension parameter $d$ as in N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles; so $|\mathcal{E}(P^{[1]})|\le R$ and (4) gives $u(\mu)-\vartheta\le N^{-1}U_{N}(\mu^{\otimes N})$. Together with (2), $-\vartheta\le N^{-1}U_{N}(\mu^{\otimes N})-u(\mu)\le\vartheta$, that is, $\bigl|N^{-1}U_{N}(\mu^{\otimes N})-u(\mu)\bigr|\le\vartheta$. This proves claim 1.