Proof of Upper Convergence: Mean-Field Solutions with the Tensor-Averaged Mollified Costs Are Asymptotically below the Solution with the Local Cost

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. The field axioms and the rules for adding inequalities, for multiplying them by nonnegative or positive real numbers, for taking inverses of positive reals, and for handling absolute values and maxima, 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). For nonnegative reals $s,t,a$, $s\le t$ implies $\sqrt{s}\le\sqrt{t}$, and $a\sqrt{t}=\sqrt{a^{2}t}$; both follow from Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and the uniqueness in Existence and Uniqueness of the Nonnegative Square Root, and are used without further mention. The sequences below are indexed by $N\in\mathbb{N}$, which plays the role of the index written $n$ in Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects.

Step 0 (Fixed constants). Let $b$ be a bound for $f$ (Bounded Real-Valued Function on a Set), and write $g_{0}(\nu)=\int_{\mathbb{R}^{d}}f\,d\nu$ for $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$; by Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, read with $m=d$, $g_{0}$ is uniformly continuous for $W_{2}$ and $|g_{0}|\le b$. By The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §growth there is $C_{g}\in\mathbb{R}$ with $M_{2}(\nu)\le C_{g}(1+|\mathcal{E}(\nu)|)$ for every $\nu\in\mathcal{D}$; put $C_{+}=\max\{C_{g},0\}$, so that $M_{2}(\nu)\le C_{+}(1+|\mathcal{E}(\nu)|)$ as $0<1+|\mathcal{E}(\nu)|$. Let $S$ be a positive real number with $\eta(z)\le S$ for every $z\in\mathbb{R}^{d}$, which exists by Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost §kernel, and let $\kappa_{d}$ be the normalising constant of closed balls in $\mathbb{R}^{d}$, which is positive by that clause.

Step 1 (The pair and the reference operator). By The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §pair and The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §coercive, $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a Wasserstein-coercive penalty pair whose penalty domain $\mathcal{D}$ has the map property, and by The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §closed it has closed score along couplings. Let $F$ be the Langevin Hamilton-Jacobi operator with common noise and density cost with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma$, control cost $\theta$, running cost $g_{0}$ and integrand $\Phi$, with $\delta$-shifts relative to the pair; it is a second-order equation operator over $\mathcal{D}_{\Sigma}$ by that clause. Since $g_{0}$ is bounded and uniformly continuous, hypothesis The Langevin Hamilton-Jacobi Operator with a Density Cost Satisfies the Hypotheses of the Comparison Principle and of Perron's Method §cost holds, and The Langevin Hamilton-Jacobi Operator with a Density Cost Satisfies the Hypotheses of the Comparison Principle and of Perron's Method §conclusion shows that $F$ is locally strictly proper and satisfies the shift-coercivity condition, the shift-semicontinuity condition and the second-order structure condition at uniquely mapped pairs. So $F$ is admissible as the reference operator of Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects.

Let $\nu\in\mathcal{D}_{\Sigma}$. Then $\nu\in\mathcal{D}$ has finite entropy by The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space §pair, hence is absolutely continuous by Basic Properties of the Entropy on the Wasserstein Space: Comparison with the Gaussian Relative Entropy, Lower Bound, Translation Invariance, Absolute Continuity, Closed Sublevel Sets and Lower Semicontinuity §absolutely-continuous, so $\nu\in\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$ and $0\le\mathcal{G}_{\Phi}(\nu)\le L$ for the density cost $\mathcal{G}_{\Phi}$. For $(\nu,q)$ in the bundle $\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$ write

so that $F(\nu,r,q,Y)=F_{0}(\nu,r,q,Y)-g_{0}(\nu)-\mathcal{G}_{\Phi}(\nu)$ by The Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost on the Wasserstein Space §operator.

Step 2 (The operators $F_{N}$, $F'_{N}$ and the defects). For $N\in\mathbb{N}$ let $F_{N}$ be the Langevin Hamilton-Jacobi operator with common noise with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma$, control cost $\theta$ and running cost $\tilde{c}_{N}$; by that clause it is a second-order equation operator over $\mathcal{D}_{\Sigma}$ with $F_{N}(\nu,r,q,Y)=F_{0}(\nu,r,q,Y)-\tilde{c}_{N}(\nu)$. Let $F'_{N}=F$. For $\nu\in\mathcal{D}_{\Sigma}$ put

By The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost §identity, applied with this $N$ and $\varepsilon=\varepsilon_{N}$, for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$

the integrand being Borel with values in $[0,L]$ and $\nu^{\otimes N}$ a probability measure (claim 6 of Borel Measurability and Bounded Integration on a Metric Space and claim 2 of Linearity and Monotonicity of the Lebesgue Integral). Hence $k_{N}(\nu)\le L-\mathcal{G}_{\Phi}(\nu)\le L$ for $\nu\in\mathcal{D}_{\Sigma}$, and $0\le h_{N}(\nu)\le L$ and $0\le h'_{N}(\nu)\le L$; we take $H=L$. For $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r$ and $Y$, since $k_{N}(\nu)\le h_{N}(\nu)$,

and $F'_{N}(\nu,r,q,Y)=F(\nu,r,q,Y)+h'_{N}(\nu)$.

Step 3 (The defects vanish locally uniformly). Let $R,C,\zeta\in\mathbb{R}$ be positive, $\zeta$ playing the role of the tolerance written $\varepsilon$ in Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects. The constants are chosen in this order: first

which depend only on $L$, $C_{+}$, $R$ and $\zeta$; then

which depend on $A$; and finally $N_{0}\in\mathbb{N}$ given by the hypothesis on $(\varepsilon_{N})$ for this $M$, so that $M\le N\varepsilon_{N}^{d}$ for $N\ge N_{0}$. Here $0\le K$, $1<A$, $0<B$ (claim 5 of Properties of Natural Number Powers in a Field and claim 4 there) and $0<M$. Let $N\ge N_{0}$ and let $\nu\in\mathcal{D}_{\Sigma}$ with $|\mathcal{E}(\nu)|\le R$ (the bound $\lVert\Sigma(\nu)\rVert_{\nu}\le C$ is not needed). Since $0<\varepsilon_{N}\le1$, $\nu\in\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$ (Step 1) and $1<A$, The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost §defect, applied with this $N$, $\varepsilon=\varepsilon_{N}$, $\mu=\nu$ and $A$ in place of its radius $R$, gives

For the second term: $M_{2}(\nu)\le C_{+}(1+R)$ by Step 0, so $2L\,M_{2}(\nu)\le K$; and $A-1=1+2K\zeta^{-1}\ge1$, so $(A-1)^{2}\ge A-1\ge2K\zeta^{-1}$ and $K\le\tfrac{\zeta}{2}(A-1)^{2}$; hence the second term is at most $\tfrac{\zeta}{2}$. For the first term: by claims 3 and 2 of Properties of Natural Number Powers in a Field, $(\varepsilon_{N}^{-1})^{d}\,\varepsilon_{N}^{d}=(\varepsilon_{N}^{-1}\varepsilon_{N})^{d}=1^{d}=1$, so $(\varepsilon_{N}^{-1})^{d}N^{-1}=(N\varepsilon_{N}^{d})^{-1}\le M^{-1}$, using $0<M\le N\varepsilon_{N}^{d}$. Since $4L^{2}B<\zeta^{2}M$, it follows that $L^{2}B\,(\varepsilon_{N}^{-1})^{d}N^{-1}\le L^{2}B\,M^{-1}<\tfrac{\zeta^{2}}{4}$, and so $L\sqrt{B\,(\varepsilon_{N}^{-1})^{d}N^{-1}}=\sqrt{L^{2}B\,(\varepsilon_{N}^{-1})^{d}N^{-1}}\le\sqrt{\zeta^{2}/4}=\tfrac{\zeta}{2}$. Therefore $k_{N}(\nu)\le\zeta$, and since $0<\zeta$, $h_{N}(\nu)+h'_{N}(\nu)=\max\{0,k_{N}(\nu)\}\le\zeta$. Thus $N_{0}$ serves as the index written $n_{0}$ in the hypothesis of Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects.

Step 4 (Uniform bounds and viscosity properties). By (1) and $|g_{0}|\le b$, $|\tilde{c}_{N}(\nu)|\le b+L$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and every $N$. So Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution §existence, applied with running cost $\tilde{c}_{N}$ and the bound $b+L$, gives a viscosity solution $w_{N}$ with $-\lambda_{0}^{-1}(b+L)\le w_{N}\le\lambda_{0}^{-1}(b+L)$; being bounded, it equals $\bar{u}_{N}$ by Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness. Hence $\bar{u}_{N}(\mu)\le\lambda_{0}^{-1}(b+L)$ for every $N$ and every $\mu\in\mathcal{D}$. As $u$ is bounded, there is a real $M_{u}$ with $-M_{u}\le u(\mu)$ for every $\mu\in\mathcal{D}$. By The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space §equation and The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space §equation, whose operator is $F_{N}$ by The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space §operator, $\bar{u}_{N}$ is a viscosity solution of $F_{N}$ relative to the pair, hence a viscosity subsolution of $F_{N}$ by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution. By The Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost on the Wasserstein Space §equation, $u$ is a viscosity solution of $F$ relative to the pair, hence a viscosity supersolution of $F=F'_{N}$ by the same clause.

Step 5 (Conclusion). All hypotheses of Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects hold for the pair of Step 1, the reference operator $F$, $H=L$, the operators $F_{N}$, $F'_{N}$ and defects $h_{N}$, $h'_{N}$ of Steps 2 and 3, the bounds $\lambda_{0}^{-1}(b+L)$ and $-M_{u}$ in place of its $b$ and $b'$, and the functions $\bar{u}_{N}$ in place of its $u_{n}$ and $u$ in place of its $v_{n}$ (Step 4). Let $R,\vartheta\in\mathbb{R}$ be positive. By Stability of Comparison on the Wasserstein Space under Locally Vanishing Cost Defects §stability, with $\vartheta$ in place of its tolerance $\theta$, there is $N_{1}\in\mathbb{N}$ with $\bar{u}_{N}(\mu)-u(\mu)\le\vartheta$ for every $N\ge N_{1}$ and every $\mu\in\mathcal{D}$ with $|\mathcal{E}(\mu)|\le R$, which is the assertion.