Proof of Lower Convergence: The Solution with the Local Cost Is Asymptotically below the Mean-Field Solutions with the Mollified Costs

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, 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. For nonnegative reals $s\le t$, $\sqrt{s}\le\sqrt{t}$ by Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and Existence and Uniqueness of the Nonnegative Square Root, and $\sqrt{t}$ is nonnegative. 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 $|\mathrm{tr}\,H_{\mathcal{E}}(\nu)|\le C_{g}(1+|\mathcal{E}(\nu)|)$ for every $\nu\in\mathcal{D}$, where $H_{\mathcal{E}}$ is the translation Hessian and $\mathrm{tr}$ the trace named there; put $C_{+}=\max\{C_{g},0\}$, so that $\mathrm{tr}\,H_{\mathcal{E}}(\nu)\le C_{+}(1+|\mathcal{E}(\nu)|)$.

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, 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 and The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §closed, $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a Wasserstein-coercive penalty pair with closed score along couplings whose penalty domain has the map property. 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, a second-order equation operator over $\mathcal{D}_{\Sigma}$ by that clause. Since $g_{0}$ is bounded and uniformly continuous, 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. Let $\nu\in\mathcal{D}_{\Sigma}$. By The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space §pair, $\nu\in\mathcal{D}\cap\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ has finite entropy and finite Fisher information, and $\Sigma(\nu)=\nabla V+\tfrac{\sigma^{2}}{2}\xi_{\nu}$; 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 it is absolutely continuous, so the density cost $\mathcal{G}_{\Phi}(\nu)$ is defined, with $0\le\mathcal{G}_{\Phi}(\nu)\le L$. 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 and the defects). For $N\in\mathbb{N}$ let $F_{N}=F$ and $h_{N}=0$ on $\mathcal{D}_{\Sigma}$, and 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 $g_{\varepsilon_{N}}$, a second-order equation operator over $\mathcal{D}_{\Sigma}$ by that clause; since $g_{\varepsilon_{N}}=g_{0}+\mathcal{G}_{\Phi,\varepsilon_{N}}$ by The Mollified Mean-Field Cost and the Mollified N-Particle Cost of a Local Coupling §mean-field, where $\mathcal{G}_{\Phi,\varepsilon_{N}}$ is the mollified density cost with integrand $\Phi$, kernel $\eta$ and scale $\varepsilon_{N}$, that clause gives $F'_{N}(\nu,r,q,Y)=F_{0}(\nu,r,q,Y)-g_{0}(\nu)-\mathcal{G}_{\Phi,\varepsilon_{N}}(\nu)$. For $\nu\in\mathcal{D}_{\Sigma}$ put $h'_{N}(\nu)=\mathcal{G}_{\Phi}(\nu)-\mathcal{G}_{\Phi,\varepsilon_{N}}(\nu)$. Then $F(\nu,r,q,Y)-h_{N}(\nu)=F_{N}(\nu,r,q,Y)$ and

By Translation and Mollification Estimates in the Integrable Norm for a Density of Finite Fisher Information, and the Mollified Density Cost §cost, applied to the absolutely continuous $\nu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ with a density of $\nu$ (which exists as recorded there), the kernel $\eta$ and $\varepsilon=\varepsilon_{N}$,

with $\mathcal{I}(\nu)$ the Fisher information; and $h'_{N}(\nu)\le\mathcal{G}_{\Phi}(\nu)\le L$ because $0\le\mathcal{G}_{\Phi,\varepsilon_{N}}(\nu)$ by The Mollified Density Cost is Bounded, Lipschitz for the Wasserstein Distance, and Below the Density Cost §bound. So $0\le h_{N},h'_{N}\le H$ with $H=L$.

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 depends only on $\sigma$, $C$, $C_{+}$ and $R$ and is nonnegative; then $\tau=\zeta\,\bigl(L\sqrt{J}+1\bigr)^{-1}$, which is positive; and finally $N_{0}\in\mathbb{N}$ given by the hypothesis on $(\varepsilon_{N})$ for this $\tau$, so that $\varepsilon_{N}\le\tau$ for $N\ge N_{0}$. Let $\nu\in\mathcal{D}_{\Sigma}$ with $|\mathcal{E}(\nu)|\le R$ and $\lVert\Sigma(\nu)\rVert_{\nu}\le C$. By The Potential Gradient Paired with the Score, and a Fisher Information Bound on the Domain of the Langevin Free-Energy Pair §fisher, whose field $\Sigma(\nu)=\nabla V+\tfrac{\sigma^{2}}{2}\xi_{\nu}$ is the one of the pair (Step 1), and by Step 0,

the first term being bounded because $0\le\lVert\Sigma(\nu)\rVert_{\nu}\le C$; hence $\mathcal{I}(\nu)\le J$. For $N\ge N_{0}$, (1) gives

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 The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost §regularity, $|g_{\varepsilon_{N}}(\nu)|\le b+L$ 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 $g_{\varepsilon_{N}}$ and the bound $b+L$, gives a viscosity solution $w_{N}$ with $-\lambda_{0}^{-1}(b+L)\le w_{N}\le\lambda_{0}^{-1}(b+L)$, which equals $u'_{N}$ 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 $-\lambda_{0}^{-1}(b+L)\le u'_{N}(\mu)$ for every $N$ and every $\mu\in\mathcal{D}$. As $u$ is bounded, there is a real $M_{u}$ with $u(\mu)\le M_{u}$ for every $\mu\in\mathcal{D}$. By The Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost on the Wasserstein Space §equation, $u$ is a viscosity solution of $F=F_{N}$ 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 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, $u'_{N}$ is a viscosity solution, hence a viscosity supersolution, of $F'_{N}$ relative to the pair.

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