Proof of 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

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. Throughout, $d=1$ with the identifications of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative, as in The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations; the pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$, the operator $F$ and the function $u$ are those of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §limit-equation. Natural numbers begin at $1$ by Natural Numbers and are read as real numbers through the canonical map; by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, every $n\in\mathbb{N}$ is positive and $n^{-1}$ exists and is positive. Since $0\le|g(\nu)|\le b_{g}$ for every $\nu$, $b_{g}\ge0$.

We fix once and for all the data of Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy: $\varepsilon_{0}=1-\sigma^{2}/\beta$, positive because $\sigma^{2}<\beta$; a natural number $N_{0}\ge2$ with $N_{0}\varepsilon_{0}\ge1$, which exists because claim 2 of The Archimedean Property of the Real Numbers gives $m\in\mathbb{N}$ with $1<m\varepsilon_{0}$, and $N_{0}=m+1$ serves; and $C_{0}$ and $e_{*}$ as fixed there. For positive $\tau$, $\bar{u}_{\tau}$ and $\underline{u}_{\tau}$ are the functions of that lemma for these data, and $h_{\tau}$ and the operators $F^{h}$ ($h\in\mathbb{R}$) are those of 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, stated for the same functions.

Step 1 (the reference operator). By The Dyson Hamilton-Jacobi Equation with Common Noise and a Confining Potential on the Wasserstein Space of the Real Line §operator, $F$ is the Hamilton-Jacobi operator with common noise and penalty drift of the pair with discount $\lambda$, common-noise intensity $0$, control cost $1$ and running cost $g$, a second-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts relative to the pair. By The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score and Regular Penalised Maxima §pair the pair is a penalty pair and $H_{\mathcal{E}}(\mu)=\int_{\mathbb{R}}V''\,d\mu$ for $\mu\in\mathcal{D}$; by The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score and Regular Penalised Maxima §coercive it is Wasserstein-coercive and $\mathcal{D}$ has the map property; by The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score and Regular Penalised Maxima §closed it has closed score along couplings.

The operator $F$ is defined, through The Dyson Hamilton-Jacobi Equation with Common Noise and a Confining Potential on the Wasserstein Space of the Real Line §operator, as an instance of the scalar-intensity operator of The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space §operator; the hypotheses lemma stated for exactly that operator is the version cited below, which is therefore the type-correct one here (its successor is stated for the matrix-noise form of the equation). As in the published proof of Well-Posedness of the Dyson Hamilton-Jacobi Equation with Common Noise and a Confining Potential: Existence and Uniqueness of a Bounded Viscosity Solution, we apply The Hamilton-Jacobi Operator with Common Noise and Penalty Drift Satisfies the Hypotheses of the Comparison Principle for a Displacement Convex Pair to the pair, with $\lambda_{0}=\lambda$, $\theta=1$ (so $0<\theta\le1$), $\kappa=0$, running cost $g$ and the operator $F$. For $\mu\in\mathcal{D}$, $H_{\mathcal{E}}(\mu)$ is a $1\times1$ matrix, and by the definition of the trace with $p=1$ its trace is its sole entry, the number $H_{\mathcal{E}}(\mu)$. Hence (Convexity) is The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score and Regular Penalised Maxima §convex; (Semicontinuity) and (Growth) are The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score and Regular Penalised Maxima §growth; (Hessian continuity) is The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score and Regular Penalised Maxima §hessian; and for (Running cost), $|g(\nu)|\le b_{g}$ for every $\nu$, so $g$ is bounded, and $g$ is uniformly continuous by The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §data. By The Hamilton-Jacobi Operator with Common Noise and Penalty Drift Satisfies the Hypotheses of the Comparison Principle for a Displacement Convex Pair §conclusion, $F$ is locally strictly proper and satisfies the shift-coercivity condition, the second-order structure condition at uniquely mapped pairs, and the shift-semicontinuity condition in the sense of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity. The statement of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space coincides word for word with that of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space: the same test data and $R$-bounded test data, the same convergence of test data in clause The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §converging, the same two implications in clause The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §level, and the same quantification over $0<\delta<1$ and $0<R$ in clause The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity. So $F$ satisfies the shift-semicontinuity condition of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity. The pair and $F$ therefore satisfy every hypothesis that Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects imposes on the pair and on the reference operator.

For $h\in\mathbb{R}$, $F^{h}$ is the operator of The Dyson Hamilton-Jacobi Equation with Common Noise and a Confining Potential on the Wasserstein Space of the Real Line §operator with running cost $\nu\mapsto g(\nu)+h$, hence a second-order equation operator over $\mathcal{D}_{\Sigma}$ with $\delta$-shifts relative to the pair; and comparing the formula of 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 with that of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §limit-equation,

Step 2 ($u$ is a solution in the notion of the stability theorem). By The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §limit-equation, $u$ is a viscosity solution of the equation of The Dyson Hamilton-Jacobi Equation with Common Noise and a Confining Potential on the Wasserstein Space of the Real Line §equation for $V$, $\lambda$, intensity $0$ and $g$, and $|u(\mu)|\le\lambda^{-1}b_{g}$ for every $\mu\in\mathcal{D}$. By The Dyson Hamilton-Jacobi Equation with Common Noise and a Confining Potential on the Wasserstein Space of the Real Line §equation and The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space §equation, this means that $u$ is a viscosity solution of $F$ relative to the pair in the sense of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution: it has penalty-subordinate growth from above and from below, and it is a viscosity subsolution and a viscosity supersolution of $F$ in the sense of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution and Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution. Clauses Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution and Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution impose the same growth hypotheses and require the same conclusions, but only for the $\delta$ with $0<\delta<1$, which are among the positive $\delta$ for which the clauses of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space require them. Hence $u$ is a viscosity subsolution and a viscosity supersolution of $F$ relative to the pair in the sense of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space.

Step 3 (upper asymptotic comparison). For $n\in\mathbb{N}$ put $\tau_{n}=n^{-1}$, which is positive. We apply Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects to the pair and the reference operator $F$ (Step 1), with $H=2b_{g}$, which is nonnegative; for every $n$, $F_{n}=F^{h_{\tau_{n}}}$, $h_{n}$ the constant function $h_{\tau_{n}}$ on $\mathcal{D}_{\Sigma}$, $F'_{n}=F$ and $h'_{n}=0$; $b=\lambda^{-1}b_{g}$ and $b'=-\lambda^{-1}b_{g}$; and $u_{n}=\bar{u}_{\tau_{n}}$, $v_{n}=u$. Its hypotheses hold. $F_{n}$ and $F'_{n}$ are second-order equation operators over $\mathcal{D}_{\Sigma}$ by Step 1. $0\le h_{n}\le H$ by 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 §defect, and $0\le h'_{n}\le H$. By (1), $F(\nu,r,q,Y)-h_{n}(\nu)=F_{n}(\nu,r,q,Y)$ and $F'_{n}(\nu,r,q,Y)=F(\nu,r,q,Y)+h'_{n}(\nu)$. For the vanishing of the defects, let $R,C,\varepsilon$ be positive; 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 §defect gives a positive $\tau_{0}$ with $h_{\tau}\le\varepsilon$ whenever $0<\tau\le\tau_{0}$, and claim 3 of The Archimedean Property of the Real Numbers gives $n_{0}\in\mathbb{N}$ with $0<n_{0}^{-1}<\tau_{0}$; for $n\ge n_{0}$ we have $0<\tau_{n}\le n_{0}^{-1}<\tau_{0}$, so $h_{n}(\nu)+h'_{n}(\nu)=h_{\tau_{n}}\le\varepsilon$ for every $\nu\in\mathcal{D}_{\Sigma}$. Next, $u_{n}(\mu)\le b$ by Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §bounds and $b'\le v_{n}(\mu)$ by Step 2, for every $\mu\in\mathcal{D}$. Finally, $u_{n}$ is a viscosity subsolution of $F_{n}$ by 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 §subsolution with $\tau=\tau_{n}$, and $v_{n}=u$ is a viscosity supersolution of $F'_{n}=F$ by Step 2, both relative to the pair in the sense of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space. By Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects §stability, for all positive $R,\theta$ there is $n_{1}\in\mathbb{N}$ with

Step 4 (lower asymptotic comparison). We apply Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects again, with the same pair, $F$, $H$, $b$, $b'$ and $\tau_{n}$, but now $F_{n}=F$, $h_{n}=0$, $F'_{n}=F^{-h_{\tau_{n}}}$, $h'_{n}$ the constant function $h_{\tau_{n}}$, $u_{n}=u$ and $v_{n}=\underline{u}_{\tau_{n}}$. By (1) with $h=-h_{\tau_{n}}$, $F'_{n}(\nu,r,q,Y)=F(\nu,r,q,Y)+h'_{n}(\nu)$, and trivially $F-h_{n}=F_{n}$; the bounds on $h_{n},h'_{n}$ and the vanishing of the defects hold exactly as in Step 3. $u_{n}(\mu)\le b$ by Step 2 and $b'\le v_{n}(\mu)$ by Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §bounds; $u$ is a viscosity subsolution of $F$ by Step 2, and $\underline{u}_{\tau_{n}}$ is a viscosity supersolution of $F^{-h_{\tau_{n}}}$ by 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 §supersolution. By Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects §stability, for all positive $R,\theta$ there is $n_{2}\in\mathbb{N}$ with

Step 5 (conclusion). Let $\mu$, $(x^{N})_{N\ge2}$ and $c$ be as in the statement, and let $\varepsilon$ be positive. Put $R=|\mathcal{E}(\mu)|+1$ and $\theta=\varepsilon/4$, let $n_{1}$ and $n_{2}$ be given by (2) and (3) for these $R,\theta$, let $n$ be the larger of $n_{1},n_{2}$, and put $\tau=\tau_{n}$. Since $|\mathcal{E}(\mu)|\le R$,

For $k\in\mathbb{N}$ let $N_{k}=N_{0}+k-1$ (so $N_{1}=N_{0}$ and $N_{k+1}=N_{k}+1$) and $y^{k}=x^{N_{k}}\in W_{N_{k}}$ (the points written $x^{k}$ in Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §sequences). The sequence $(N_{k})_{k}$ is strictly increasing with $N_{1}\ge N_{0}$, and $N_{k}\ge k$ because $N_{0}\ge2$. Given a positive $\eta$, the convergence of $(W_{2}(\mu^{N}_{x^{N}},\mu))_{N}$ to $0$ (Limit of a Sequence of Real Numbers) gives $M$ with $W_{2}(\mu^{N}_{x^{N}},\mu)<\eta$ for $N\ge M$, hence $W_{2}(\mu^{N_{k}}_{y^{k}},\mu)<\eta$ for $k\ge M$; so $(W_{2}(\mu^{N_{k}}_{y^{k}},\mu))_{k}$ converges to $0$. Moreover $P_{N_{k}}(y^{k})\le cN_{k}$ for every $k$. By Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §sequences, applied with this $\tau$, these $N_{k}$, $y^{k}$, $\mu$, $c'=c$ and $\varepsilon/4$, there is $k_{0}$ with

Let $N\ge N_{0}+k_{0}-1$. Then $k=N-N_{0}+1\in\mathbb{N}$ satisfies $k\ge k_{0}$, $N_{k}=N$ and $y^{k}=x^{N}$, and with (4),

so $\bigl|v_{N}(x^{N})/N-u(\mu)\bigr|\le\varepsilon/2<\varepsilon$. As $\varepsilon$ was arbitrary, $\bigl(v_{N}(x^{N})/N\bigr)_{N}$ converges to $u(\mu)$ in the sense of Limit of a Sequence of Real Numbers.