Proof of Cross-Level Inequalities between the Shifted Lifted N-Particle and Mean-Field Langevin Operators at Tensor Powers and through One-Particle Marginals

Each result cited is universally quantified over the data in its own statement.

Throughout, $N$ is read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background §numbers; it is positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and $N^{-1}$ exists. Write $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ for the particle-level and $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ for the configuration-level Langevin pair, both penalty pairs 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 latter applied at the configuration level of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level), so that $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$ and $\mathcal{D}_{N,\Sigma}\subseteq\mathcal{D}_{N}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair. By The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space §pair, $\Sigma(\nu)=\nabla V+\tfrac{\sigma^{2}}{2}\xi_{\nu}$ for $\nu\in\mathcal{D}_{\Sigma}$ and $\Sigma_{N}(P)=\nabla V_{N}+\tfrac{\sigma^{2}}{2}\xi_{P}$ for $P\in\mathcal{D}_{N,\Sigma}$. Each space $L^{2}(\rho;\mathbb{R}^{r})$ below is that of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields; by The Space of Square-Integrable Random Vectors §inner-product it is a real inner product space whose norm is $\lVert\cdot\rVert_{\rho}$, so Elementary Identities in a Real Inner Product Space applies to it. The rows $\gamma_{j}\in\mathbb{R}^{d}$ ($j\in[p]$) of $\Gamma$ and the matrix $\Gamma_{N}$ are those of The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space §common-noise.

Step 1 (The shifted operators). By The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted and The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space §operator, with $\langle\nabla V+\tfrac{\sigma^{2}}{2}\xi_{\nu},\cdot\rangle_{\nu}=\langle\Sigma(\nu),\cdot\rangle_{\nu}$, for $\nu\in\mathcal{D}_{\Sigma}$, $r\in\mathbb{R}$, $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $Y\in\mathcal{S}(d)$,

Likewise, by the same definition read at the configuration level and The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §operator, for $P\in\mathcal{D}_{N,\Sigma}$, $r\in\mathbb{R}$, $G\in L^{2}(P;\mathbb{R}^{dN})$ and $X\in\mathcal{S}(dN)$,

For a real inner product space $E$ with inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$, and $x,D\in E$, write

so that the third and fourth terms of $F^{+}_{\delta}(\nu,r,q,Y)$ and of $F^{+}_{N,\delta}(P,r,G,X)$ are $K_{L^{2}(\nu;\mathbb{R}^{d})}(q,\Sigma(\nu))$ and $K_{L^{2}(P;\mathbb{R}^{dN})}(G,\Sigma_{N}(P))$.

Step 2 (The trace terms agree). Let $P\in\mathcal{D}_{N}$; then $P^{[1]}\in\mathcal{D}$ by The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §marginal. We show

First, for $n\in\mathbb{N}$, real $n\times n$ matrices $A,B$ and $z\in\mathbb{R}^{n}$: by Matrix-Vector Product, Difference of Real Matrices, Scalar Multiple of a Real Matrix and claims 2 and 3 of Properties of Finite Sums, each coordinate of $(A-\delta B)z$ equals the corresponding coordinate of $Az-\delta(Bz)$, so the two points agree, and claim 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ gives

The matrices $X_{N}=\mathbb{Y}_{N}-\delta H_{\mathcal{E}_{N}}(P)$ and $X_{1}=\mathbb{Y}-\delta H_{\mathcal{E}}(P^{[1]})$ lie in $\mathcal{S}(dN)$ and $\mathcal{S}(d)$, as recorded in The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted. For $j\in[p]$, (2.2), the hypothesis on $\mathbb{Y}_{N}$ and $\mathbb{Y}$ with $a=\gamma_{j}$, and The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §hessian with $a=\gamma_{j}$ give

the last step by (2.2) again. Summing over $j\in[p]$, using the displayed formula of The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space §common-noise with $X=X_{N}$, claim 3 of Properties of Finite Sums, and The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound §rows applied to $A=\Gamma$ (with $p$ rows $\gamma_{j}$ and $d$ columns) and $X=X_{1}$,

which is (2.1). Consequently the trace terms satisfy $-\tfrac{1}{2}\mathrm{tr}(\Gamma_{N}^{\top}\Gamma_{N}X_{N})=N\bigl(-\tfrac{1}{2}\mathrm{tr}(\Gamma^{\top}\Gamma X_{1})\bigr)$.

Step 3 (At tensor powers). Suppose $g=\tilde{c}$, and let $\mu\in\mathcal{D}_{\Sigma}$, $r\in\mathbb{R}$ and $G\in L^{2}(\mu^{\otimes N};\mathbb{R}^{dN})$. Put $Q=\mu^{\otimes N}$ and $S=\Sigma(\mu)\in T_{\mu}$. By The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §tensor, $Q\in\mathcal{D}_{N,\Sigma}$, $\mathcal{E}_{N}(Q)=N\mathcal{E}(\mu)$ and $\Sigma_{N}(Q)=S^{\oplus}$. By Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor, $Q^{[1]}=\mu$, so $\Pi_{Q}$ maps $L^{2}(Q;\mathbb{R}^{dN})$ into $T_{\mu}\subseteq L^{2}(\mu;\mathbb{R}^{d})$; put $q_{1}=\Pi_{Q}(G)$, so that $(\mu,q_{1})$ and $(Q,G)$ lie in the bundles over $\mathcal{D}_{\Sigma}$ and $\mathcal{D}_{N,\Sigma}$ and both sides of the asserted inequality are defined. We compare the terms of $F^{+}_{N,\delta}(Q,Nr,G,\mathbb{Y}_{N})$ with $N$ times those of $F^{+}_{\delta}(\mu,r,q_{1},\mathbb{Y})$, as written in Step 1.

(3a) Discount terms: $\lambda_{0}(Nr-\delta\mathcal{E}_{N}(Q))=\lambda_{0}(Nr-\delta N\mathcal{E}(\mu))=N\lambda_{0}(r-\delta\mathcal{E}(\mu))$ by the field axioms.

(3b) Trace terms: since $Q\in\mathcal{D}_{N}$ and $Q^{[1]}=\mu$, Step 2 gives equality after multiplication by $N$.

(3c) Pairing terms: by The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §contraction $\Pi_{Q}$ is linear, and $\Pi_{Q}(S^{\oplus})=S$ by The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §product since $S\in T_{\mu}$; hence $\Pi_{Q}(G-\delta S^{\oplus})=q_{1}-\delta S$. By symmetry of the inner product (condition (a) of Real Inner Product Space §inner-product) and The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §pairing with $D=G-\delta S^{\oplus}$ and $g=S$,

(3d) Quadratic terms: The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §contraction with $D=G-\delta S^{\oplus}$ gives $N\lVert q_{1}-\delta S\rVert_{\mu}^{2}\le\lVert G-\delta S^{\oplus}\rVert_{Q}^{2}$. Since $0<\tfrac{\theta}{2}$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, claim 5 of Elementary Arithmetic in an Ordered Field gives $N\tfrac{\theta}{2}\lVert q_{1}-\delta S\rVert_{\mu}^{2}\le\tfrac{\theta}{2}\lVert G-\delta S^{\oplus}\rVert_{Q}^{2}$.

(3e) Running costs: by The Tensor-Averaged Running Cost of a Configuration-Space Cost §cost, $\tilde{c}(\mu)=N^{-1}\int c\,dQ$, so $\int_{\mathbb{R}^{dN}}c\,dQ=N\tilde{c}(\mu)=N g(\mu)$.

By (3a)--(3e) and the field axioms, $F^{+}_{N,\delta}(Q,Nr,G,\mathbb{Y}_{N})-N\,F^{+}_{\delta}(\mu,r,q_{1},\mathbb{Y})=\tfrac{\theta}{2}\lVert G-\delta S^{\oplus}\rVert_{Q}^{2}-N\tfrac{\theta}{2}\lVert q_{1}-\delta S\rVert_{\mu}^{2}$, which is nonnegative by (3d) and claim 3 of Elementary Arithmetic in an Ordered Field; the same claim yields $N\,F^{+}_{\delta}(\mu,r,\Pi_{\mu^{\otimes N}}(G),\mathbb{Y})\le F^{+}_{N,\delta}(\mu^{\otimes N},Nr,G,\mathbb{Y}_{N})$, which is Cross-Level Inequalities between the Shifted Lifted N-Particle and Mean-Field Langevin Operators at Tensor Powers and through One-Particle Marginals §tensor.

Step 4 (A negative coefficient). Put $\kappa=\tfrac{\theta}{2}\delta\delta-\delta$. We show $\kappa<0$. By claim 8 of Elementary Order Arithmetic in an Ordered Field, $0<\tfrac{\theta}{2}$ and $\tfrac{\theta}{2}<\theta$; as $\theta\le1$, claim 2 there gives $\tfrac{\theta}{2}<1$. Since $\delta<1$ and $0<\tfrac{\theta}{2}$, claim 10 there gives $\tfrac{\theta}{2}\delta<\tfrac{\theta}{2}$, so $\tfrac{\theta}{2}\delta<1$ by claim 2 there, and $\tfrac{\theta}{2}\delta-1<0$ by claim 1 there (adding $-1$). Since $0<\delta$, claim 10 there gives $\delta(\tfrac{\theta}{2}\delta-1)<\delta\cdot0=0$, and $\delta(\tfrac{\theta}{2}\delta-1)=\kappa$ by the field axioms. Hence $\kappa<0$ and $0\le-\kappa$ by claim 4 there.

Step 5 (Expansion of $K_{E}$). Let $E$ be a real inner product space and $x,D\in E$. By claims 5, 1 and 4 of Elementary Identities in a Real Inner Product Space, $|x-\delta D|^{2}=|x|^{2}-2\delta\langle x,D\rangle+\delta\delta|D|^{2}$, using $|\delta|=\delta$ (Absolute Value in an Ordered Field, as $0\le\delta$); by claim 1 there and Real Inner Product Space §norm, $\langle D,x-\delta D\rangle=\langle D,x\rangle-\delta|D|^{2}$; and $\langle x,D\rangle=\langle D,x\rangle$ by symmetry. With $\tfrac{\theta}{2}\cdot2=\theta$ and the field axioms,

Step 6 (Through one-particle marginals). Let $P\in\mathcal{D}_{N,\Sigma}$ satisfy $N g(P^{[1]})\le\int c\,dP$, let $r\in\mathbb{R}$ and $h\in T_{P^{[1]}}$. Write $\mu_{1}=P^{[1]}$, $D=\Sigma_{N}(P)$ and $D_{1}=\Sigma(\mu_{1})$. By The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §marginal, $\mu_{1}\in\mathcal{D}_{\Sigma}$, $N\mathcal{E}(\mu_{1})\le\mathcal{E}_{N}(P)$ and $D_{1}=\Pi_{P}(D)$; the product field $h^{\oplus}$ lies in $L^{2}(P;\mathbb{R}^{dN})$ by Product Fields and the Projection onto One-Particle Tangent Fields §product-field, so both sides of the asserted inequality are defined. We compare the terms of $F^{+}_{N,\delta}(P,Nr,h^{\oplus},\mathbb{Y}_{N})$ with $N$ times those of $F^{+}_{\delta}(\mu_{1},r,h,\mathbb{Y})$.

(6a) Discount terms: from $N\mathcal{E}(\mu_{1})\le\mathcal{E}_{N}(P)$ and $0\le\delta$, claim 5 of Elementary Arithmetic in an Ordered Field gives $\delta N\mathcal{E}(\mu_{1})\le\delta\mathcal{E}_{N}(P)$, so by claim 3 there $Nr-\delta\mathcal{E}_{N}(P)\le Nr-\delta N\mathcal{E}(\mu_{1})$ (the difference of the right and left sides being $\delta\mathcal{E}_{N}(P)-\delta N\mathcal{E}(\mu_{1})$), and multiplying by $0\le\lambda_{0}$ (claim 5 there) $\lambda_{0}(Nr-\delta\mathcal{E}_{N}(P))\le N\lambda_{0}(r-\delta\mathcal{E}(\mu_{1}))$.

(6b) Trace terms: since $P\in\mathcal{D}_{N}$, Step 2 gives equality after multiplication by $N$.

(6c) Quadratic and pairing terms: by Product Fields and the Projection onto One-Particle Tangent Fields §product-field, $\lVert h^{\oplus}\rVert_{P}^{2}=N\lVert h\rVert_{\mu_{1}}^{2}$; by The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §pairing with $g=h\in T_{\mu_{1}}$, $\langle D,h^{\oplus}\rangle_{P}=N\langle\Pi_{P}(D),h\rangle_{\mu_{1}}=N\langle D_{1},h\rangle_{\mu_{1}}$; and by The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §contraction, $N\lVert D_{1}\rVert_{\mu_{1}}^{2}\le\lVert D\rVert_{P}^{2}$, so claim 5 of Elementary Arithmetic in an Ordered Field with $0\le-\kappa$ (Step 4) and claim 4 of Elementary Order Arithmetic in an Ordered Field give $\kappa\lVert D\rVert_{P}^{2}\le\kappa N\lVert D_{1}\rVert_{\mu_{1}}^{2}$. Applying (5.1) in $L^{2}(P;\mathbb{R}^{dN})$ with $x=h^{\oplus}$ and in $L^{2}(\mu_{1};\mathbb{R}^{d})$ with $x=h$ and $D_{1}$ in place of $D$,

the inequality by claim 3 of Elementary Arithmetic in an Ordered Field, the difference of the two sides being $\kappa N\lVert D_{1}\rVert_{\mu_{1}}^{2}-\kappa\lVert D\rVert_{P}^{2}$.

(6d) Running costs: from $Ng(\mu_{1})\le\int c\,dP$, claim 4 of Elementary Order Arithmetic in an Ordered Field gives $-\int c\,dP\le-Ng(\mu_{1})$.

By Step 1 and the field axioms, $N\,F^{+}_{\delta}(\mu_{1},r,h,\mathbb{Y})-F^{+}_{N,\delta}(P,Nr,h^{\oplus},\mathbb{Y}_{N})$ is the sum of the differences of corresponding terms in (6a)--(6d), each nonnegative by claim 3 of Elementary Arithmetic in an Ordered Field (the trace difference being $0$); by claim 2 there the sum is nonnegative, and claim 3 there yields $F^{+}_{N,\delta}(P,Nr,h^{\oplus},\mathbb{Y}_{N})\le N\,F^{+}_{\delta}(P^{[1]},r,h,\mathbb{Y})$, which is Cross-Level Inequalities between the Shifted Lifted N-Particle and Mean-Field Langevin Operators at Tensor Powers and through One-Particle Marginals §marginal.