Proof of The Gibbs Entropy Pair Has Closed Score Along Noise Couplings

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

Real order and arithmetic, square roots, the Archimedean property, limits of real sequences with their limit laws and order properties, and claim 2 of Uniqueness of Limits and Boundedness of Convergent Real Sequences (convergent real sequences are bounded) are used through The Real Numbers: Standing Notation and Background §background, without further mention.

Reduction. Let $d$ be the head dimension and $v$ the profile of $V=v\circ p_{d}$, with the noise gradient $\nabla_{a}V$ and the functions $\partial_{k}V$ of Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient, and write $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ for the Gibbs entropy pair of the statement. By Noise Penalty Pairs with Closed Score Along Noise Couplings §closed, with sequences of couplings of vanishing noise cost and weak convergence along them those of Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §couplings and Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §weak, as fixed by First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §couplings, we must show the following. Let $R\in\mathbb{R}$ be nonnegative, let $(\nu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{D}_{\Sigma}$ with $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le R$ for every $n\in\mathbb{N}$, let $\nu\in\mathcal{D}$, and let $(\pi_{n})_{n\in\mathbb{N}}$ satisfy $\pi_{n}\in\Pi^{a}(\nu_{n},\nu)$ for every $n$ and $\lim_{n\to\infty}I^{a}(\pi_{n})=0$. Then $\nu\in\mathcal{D}_{\Sigma}$, and $\lim_{n\to\infty}\mathcal{K}^{a}(\Sigma(\nu_{n}),\eta,\pi_{n})=\langle\Sigma(\nu),\eta\rangle_{\nu}$ for every $\eta\in L^{2}(\nu;X^{a})$. We fix $R$, $(\nu_{n})_{n\in\mathbb{N}}$, $\nu$ and $(\pi_{n})_{n\in\mathbb{N}}$. As in Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background, a point $z\in X\times X$ is written $z=(x,y)$ with $x=\pi_{1}(z)$ and $y=\pi_{2}(z)$; $\bar{a}$ is the real bound $a_{k}\le\bar{a}$ ($k\in\mathbb{N}$) on the noise weights fixed by the setting, and $(e_{k})_{k\in\mathbb{N}}$, $(f_{k})_{k\in\mathbb{N}}$ and the coordinates $x_{k}=\langle x,e_{k}\rangle$ are those of the setting. Limits are taken as $n\to\infty$.

Step 0 (membership). By The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain and The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §penalty-domain, $\nu$ and every $\nu_{n}$ lie in $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ and have finite relative entropy with respect to $\gamma^{V}_{\beta}$. By Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §entropy, each of them has finite relative entropy with respect to $\gamma_{c}$ and $V$ is integrable with respect to it; by Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §domain, each of them lies in $\mathcal{P}_{2}(X)$. By The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain, each $\nu_{n}$ has a relative score with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$, and also a relative score with respect to $\gamma_{c}$, finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\nu_{n}<\infty$. Let $Z_{n}=Z^{a}_{\nu_{n}}\in L^{2}(\nu_{n};X^{a})$ be the noise score field of $\nu_{n}$, so that $\lVert Z_{n}\rVert_{\nu_{n}}^{2}=\mathcal{I}_{a}(\nu_{n}\,|\,\gamma_{c})$ by The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field. By Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §tangent the class of $\nabla_{a}V$ lies in $L^{2}(\nu_{n};X^{a})$, and by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §pair, $\Sigma(\nu_{n})=\beta Z_{n}+\nabla_{a}V$.

Step 1 (integrals of functions of quadratic growth). We show: if $h:X\to\mathbb{R}$ is continuous and there is $A'\in\mathbb{R}$ with $|h(x)|\le A'(1+|x|^{2})$ for every $x\in X$, then $h$ is integrable with respect to $\nu$ and every $\nu_{n}$, and $\lim\int_{X}h\,d\nu_{n}=\int_{X}h\,d\nu$. Since $\pi_{n}\in\Pi^{a}(\nu_{n},\nu)$, the pair $(\nu_{n},\nu)$ is noise-connected by Couplings of Finite Noise Cost and Their Noise Cost §connected, and $W_{a}(\nu_{n},\nu)^{2}\le I^{a}(\pi_{n})$ by The Noise Wasserstein Distance §distance. As $\nu_{n},\nu\in\mathcal{P}^{a}_{\rho}$ by Step 0, The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §comparison gives $W_{2}(\nu_{n},\nu)\le\sqrt{\bar{a}}\,W_{a}(\nu_{n},\nu)$, hence $0\le W_{2}(\nu_{n},\nu)^{2}\le\bar{a}\,I^{a}(\pi_{n})$. Given a positive $\varepsilon\in\mathbb{R}$, $\bar{a}\,I^{a}(\pi_{n})<\varepsilon^{2}$ for all large $n$, and then $W_{2}(\nu_{n},\nu)<\varepsilon$; so $\lim W_{2}(\nu_{n},\nu)=0$, and since $\nu_{n},\nu\in\mathcal{P}_{2}(X)$ by Step 0, the claim is Wasserstein Convergence on a Hilbert Space: Weak Convergence, Integrals of Continuous Functions of Quadratic Growth, Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Compactness §quadratic.

We shall use repeatedly that finite sums and products of continuous real functions on $X$ are continuous: if $u_{1},u_{2}:X\to\mathbb{R}$ are continuous, every sequence converging to a point $x$ of $X$ is carried by $u_{1}$ and $u_{2}$ to sequences converging to $u_{1}(x)$ and $u_{2}(x)$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential, so by the limit laws the same holds for $u_{1}+u_{2}$ and $u_{1}u_{2}$, which are therefore continuous by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset; the same criterion shows that constant functions are continuous, as they carry every sequence to a constant sequence. Continuous real functions on $X$ are Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space.

Step 2 (uniform bounds). Let $n\in\mathbb{N}$. By Step 0, $\nu_{n}$ satisfies the hypotheses of Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §field, which gives $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}^{2}=\lVert\beta Z_{n}+\nabla_{a}V\rVert_{\nu_{n}}^{2}=\beta^{2}\,\mathcal{I}_{a}(\nu_{n}\,|\,\gamma^{V}_{\beta})$; hence $0\le\beta^{2}\,\mathcal{I}_{a}(\nu_{n}\,|\,\gamma^{V}_{\beta})\le R^{2}$.

Let $h_{0}:X\to\mathbb{R}$, $h_{0}(x)=\sum_{k=1}^{d}x_{k}^{2}$. Each coordinate function $x\mapsto x_{k}$ is Lipschitz by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, hence continuous by A Lipschitz Map is Uniformly Continuous and Continuous, so $h_{0}$ is continuous. By Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square, $h_{0}(x)=\lVert p_{d}(x)\rVert^{2}$; since all coordinates of $0_{X}$ vanish, $p_{d}(0_{X})$ is the origin of $\mathbb{R}^{d}$, and Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity gives $\lVert p_{d}(x)\rVert\le|x|$. So $0\le h_{0}(x)\le|x|^{2}\le1+|x|^{2}$, and by Step 1 the sequence of the real numbers $m_{n}=\int_{X}h_{0}\,d\nu_{n}$ converges to $\int_{X}h_{0}\,d\nu$. A convergent real sequence being bounded, there is a nonnegative $M\in\mathbb{R}$ with $m_{n}\le M$ for every $n\in\mathbb{N}$.

Let $C_{*}\in\mathbb{R}$ be the constant of Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §fisher-bound, which depends only on $V$, $\beta$, $a$ and $c$, and put $A=|C_{*}|\,(1+R^{2}+M)$, a nonnegative real number. By Step 0 each $\nu_{n}$ lies in $\mathcal{P}_{2}(X)$, $V$ is integrable with respect to it, and it has a relative score with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$; so that clause gives

$$\beta^{2}\,\mathcal{I}_{a}(\nu_{n}\,|\,\gamma_{c})+\int_{X}|\nabla_{a}V|_{a}^{2}\,d\nu_{n}\le C_{*}\bigl(1+\beta^{2}\,\mathcal{I}_{a}(\nu_{n}\,|\,\gamma^{V}_{\beta})+m_{n}\bigr)\le|C_{*}|\,(1+R^{2}+M)=A,$$

the bracket lying between $0$ and $1+R^{2}+M$ by the previous two paragraphs. Both summands on the left are nonnegative, the first being $\beta^{2}\lVert Z_{n}\rVert_{\nu_{n}}^{2}$ by Step 0 and the second the integral of a nonnegative function. Since $\lVert\cdot\rVert_{\nu_{n}}$ is the norm of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, we obtain for every $n\in\mathbb{N}$

$$\textup{(2a)}\quad\lVert\beta Z_{n}\rVert_{\nu_{n}}^{2}\le A,\qquad\qquad\textup{(2b)}\quad\lVert\nabla_{a}V\rVert_{\nu_{n}}^{2}=\int_{X}|\nabla_{a}V|_{a}^{2}\,d\nu_{n}\le A .$$

Step 3 (the Gaussian part). Let $(\mathcal{D}^{G},\mathcal{D}^{G}_{\Sigma},\mathcal{E}^{G},\Sigma^{G})$ be the Gaussian entropy pair with temperature $\beta$ of The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair, whose hypothesis The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §hypothesis holds with the $\kappa$ of the statement. By Step 0 and The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain, $\nu$ and every $\nu_{n}$ lie in $\mathcal{D}^{G}$; by Step 0 and The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain, every $\nu_{n}$ lies in $\mathcal{D}^{G}_{\Sigma}$; and by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair, $\Sigma^{G}(\nu_{n})=\beta Z_{n}$, so $\lVert\Sigma^{G}(\nu_{n})\rVert_{\nu_{n}}\le A^{1/2}$ by (2a). The sequence $(\pi_{n})_{n\in\mathbb{N}}$ is a sequence of couplings of vanishing noise cost from $(\nu_{n})_{n\in\mathbb{N}}$ to $\nu$ in the sense of Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §couplings. Hence The Gaussian Entropy Pair Has Closed Score Along Noise Couplings §closed, read through Noise Penalty Pairs with Closed Score Along Noise Couplings §closed with $A^{1/2}$ in place of its $R$, shows that $\nu\in\mathcal{D}^{G}_{\Sigma}$ and that $(\Sigma^{G}(\nu_{n}))_{n\in\mathbb{N}}$ converges weakly to $\Sigma^{G}(\nu)$ along $(\pi_{n})_{n\in\mathbb{N}}$. By The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain, $\nu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$; let $Z=Z^{a}_{\nu}$ be its noise score field, so that $\Sigma^{G}(\nu)=\beta Z$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair. By Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §weak,

$$\lim\mathcal{K}^{a}(\beta Z_{n},\eta,\pi_{n})=\langle\beta Z,\eta\rangle_{\nu}\qquad\text{for every }\eta\in L^{2}(\nu;X^{a}).$$

Step 4 ($\int_{X}|\nabla_{a}V|_{a}^{2}\,d\nu\le A$). Let $g_{0}=|\nabla_{a}V|_{a}^{2}:X\to\mathbb{R}$. By Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, $g_{0}(x)=\sum_{k=1}^{d}a_{k}\,\partial_{k}V(x)^{2}$ with each $\partial_{k}V$ continuous, so $g_{0}$ is continuous and nonnegative. For $M'\in\mathbb{N}$ let $u_{M'}=\min\{g_{0},M'\}$, the pointwise minimum with the constant function $M'$; it is continuous by claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space, hence Borel, and $0\le u_{M'}\le M'\le M'(1+|x|^{2})$ by claims 1 and 3 of Elementary Properties of the Minimum of Two Elements. By Step 1, $\int_{X}u_{M'}\,d\nu=\lim\int_{X}u_{M'}\,d\nu_{n}$; and by claim 6(c) of Borel Measurability and Bounded Integration on a Metric Space, for $\sigma=\nu$ and for each $\sigma=\nu_{n}$ the integral of $u_{M'}$ with respect to $\sigma$ is also its integral as a nonnegative function. For each $n$, $u_{M'}\le g_{0}$ (claim 1 of Elementary Properties of the Minimum of Two Elements), so the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative and (2b) give $\int_{X}u_{M'}\,d\nu_{n}\le A$, and hence $\int_{X}u_{M'}\,d\nu\le A$ by the order of limits. By claims 1 and 3 of Elementary Properties of the Minimum of Two Elements, $u_{M'}\le u_{M'+1}$, and $u_{M'}(x)=g_{0}(x)$ whenever $g_{0}(x)\le M'$; for each $x\in X$ there is such an $M'$ by the Archimedean property, so $\sup_{M'}u_{M'}(x)=g_{0}(x)$. By Monotone Convergence Theorem on $(X,\mathcal{B}(X),\nu)$,

$$\int_{X}|\nabla_{a}V|_{a}^{2}\,d\nu=\sup_{M'\in\mathbb{N}}\int_{X}u_{M'}\,d\nu\le A<\infty .$$

By Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §tangent the class of $\nabla_{a}V$ lies in $L^{2}(\nu;X^{a})$, and $\lVert\nabla_{a}V\rVert_{\nu}^{2}\le A$ by the norm of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations.

Step 5 ($\nu\in\mathcal{D}_{\Sigma}$). By Step 0, $\nu\in\mathcal{P}_{2}(X)$ and $V$ is integrable with respect to $\nu$; by Step 3, $\nu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$; by Step 4, $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\nu<\infty$. So the ``if'' direction of Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §splitting shows that $\nu$ has a relative score with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$. As $\nu\in\mathcal{D}$, The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain gives $\nu\in\mathcal{D}_{\Sigma}$, and by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §pair, $\Sigma(\nu)=\beta Z^{a}_{\nu}+\nabla_{a}V=\beta Z+\nabla_{a}V$, with $Z$ the noise score field of Step 3.

Step 6 (cylindrical test fields). Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$, the set of Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical, with a representation $(m,\psi)$ in the sense of Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §representation, and let $k\in\mathbb{N}$.

(a) Lipschitz bound. By Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, $\psi$ is of class $C^{1}$ on $\mathbb{R}^{m}$ and there are nonnegative reals $B,B_{1},\dots,B_{m}$ with $|\psi|\le B$ and $|\partial_{i}\psi|\le B_{i}$ on $\mathbb{R}^{m}$ for $i\in[m]$; put $L=\sum_{i=1}^{m}B_{i}$. Let $u,w\in\mathbb{R}^{m}$ and $h=u-w$. The set $\mathbb{R}^{m}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, so claims 2 and 3 of Derivatives Along a Segment for C^1 Functions on a Euclidean Open Set, applied with $\mathbb{R}^{m}$, $\psi$, the point $w$, the increment $h$ and the interval $[0,1]$, show that $F(\tau)=\psi(w+\tau h)$ is continuous on $[0,1]$ and differentiable at every $\tau\in(0,1)$ with $F'(\tau)=\sum_{i=1}^{m}\partial_{i}\psi(w+\tau h)\,h_{i}$. By Mean Value Theorem on a Closed Real Interval there is $\tau_{0}\in(0,1)$ with $\psi(u)-\psi(w)=F(1)-F(0)=F'(\tau_{0})$, so, by the coordinate bound $|h_{i}|\le\lVert h\rVert$ of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate, $|\psi(u)-\psi(w)|\le\sum_{i=1}^{m}B_{i}|h_{i}|\le L\,\lVert u-w\rVert$. Taking $u=p_{m}(x)$ and $w=p_{m}(y)$ and using $\lVert p_{m}(x)-p_{m}(y)\rVert\le|x-y|$ from Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, we get $|\varphi(x)-\varphi(y)|\le L\,|x-y|$ for all $x,y\in X$. Thus $\varphi$ is Lipschitz, hence continuous by A Lipschitz Map is Uniformly Continuous and Continuous, and $|\varphi|\le B$.

(b) The field $\varphi f_{k}$. The map $\varphi f_{k}:X\to X^{a}$, $y\mapsto\varphi(y)f_{k}$, takes values in $X^{a}$, a linear subspace of $X$ containing $f_{k}=a_{k}^{1/2}e_{k}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert and The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis. Its coordinate along $e_{i}$ is $\langle\varphi(y)f_{k},e_{i}\rangle=a_{k}^{1/2}\varphi(y)\langle e_{k},e_{i}\rangle$, which is $a_{k}^{1/2}\varphi(y)$ for $i=k$ and $0$ otherwise, a Borel function of $y$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel; so $\varphi f_{k}$ is measurable into $X^{a}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable. By orthonormality of $(f_{i})_{i\in\mathbb{N}}$ in $X^{a}$ (The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis), $\langle\varphi(y)f_{k},f_{i}\rangle_{a}$ is $\varphi(y)$ for $i=k$ and $0$ otherwise, and $|\varphi(y)f_{k}|_{a}^{2}=\varphi(y)^{2}\le B^{2}$. Hence, by the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative, $\varphi f_{k}$ is square-integrable with respect to every $\sigma\in\mathcal{P}(X)$; its class in $L^{2}(\sigma;X^{a})$, again written $\varphi f_{k}$, has coordinates $(\varphi f_{k})_{k}=\varphi$ and $(\varphi f_{k})_{i}=0$ for $i\ne k$, in the sense of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates.

(c) Discrepancy bound. Let $n\in\mathbb{N}$. Since $\pi_{n}\in\Pi^{a}(\nu_{n},\nu)\subseteq\Pi(\nu_{n},\nu)$, Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy, applied with $\nu_{n}$ and $\nu$ in place of its $\nu$ and $\mu$ and with $\varphi f_{k}$ in both slots, shows that the function $z\mapsto|\varphi(x)f_{k}-\varphi(y)f_{k}|_{a}^{2}=(\varphi(x)-\varphi(y))^{2}$ is Borel, nonnegative and integrable with respect to $\pi_{n}$. By (a) it is at most $L^{2}|x-y|^{2}$, so by the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative, the quadratic cost $I(\pi_{n})$ and the support bound $I(\pi_{n})\le\bar{a}\,I^{a}(\pi_{n})$ of Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §support-bound,

$$D_{n}(\varphi):=\int_{X\times X}(\varphi(x)-\varphi(y))^{2}\,\pi_{n}(dz)\le L^{2}\,I(\pi_{n})\le L^{2}\,\bar{a}\,I^{a}(\pi_{n}) .$$

Step 7 (the potential part on the fields $\varphi f_{k}$). Let $\varphi\in\mathcal{F}C^{1}_{b}(X)$ and $k\in\mathbb{N}$, with $B$ and $L$ as in Step 6(a). We show $\lim\mathcal{K}^{a}(\nabla_{a}V,\varphi f_{k},\pi_{n})=\langle\nabla_{a}V,\varphi f_{k}\rangle_{\nu}$, the class of $\nabla_{a}V$ being taken in $L^{2}(\nu_{n};X^{a})$ (Step 0) in the first slot and in $L^{2}(\nu;X^{a})$ (Step 4) in the second. Let $g:X\to\mathbb{R}$, $g(x)=\partial_{k}V(x)\,\varphi(x)$.

(a) A pointwise identity. By The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis, $f_{k}=a_{k}^{1/2}e_{k}\in X^{a}$, whose coordinates $\langle f_{k},e_{j}\rangle$ are $a_{k}^{1/2}$ for $j=k$ and $0$ otherwise. So Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity gives $\langle\nabla_{a}V(x),f_{k}\rangle_{a}=\sum_{j=1}^{d}\partial_{j}V(x)\langle f_{k},e_{j}\rangle$, which is $a_{k}^{1/2}\partial_{k}V(x)$ if $k\le d$, and $0=a_{k}^{1/2}\partial_{k}V(x)$ if $k>d$, as then $\partial_{k}V=0$ by Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient. By bilinearity of $\langle\cdot,\cdot\rangle_{a}$, for all $x,y\in X$,

$$\langle\nabla_{a}V(x),\varphi(y)f_{k}\rangle_{a}=a_{k}^{1/2}\,\partial_{k}V(x)\,\varphi(y).$$

(b) Bounds on $g$. By Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity and Step 6(a), $\partial_{k}V$ and $\varphi$ are continuous, so $g$ is continuous, hence Borel. If $k\le d$, $a_{k}\partial_{k}V(x)^{2}$ is one of the nonnegative summands of $g_{0}(x)$ in Step 4; if $k>d$ it is $0$; so $g(x)^{2}\le B^{2}\partial_{k}V(x)^{2}\le(B^{2}/a_{k})\,g_{0}(x)$, $a_{k}$ being positive. With (2b), Step 4 and the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative,

$$\int_{X}g^{2}\,d\sigma\le\frac{B^{2}A}{a_{k}}\qquad\text{for }\sigma=\nu\text{ and for }\sigma=\nu_{n},\ n\in\mathbb{N}.$$

Since $2|g|\le1+g^{2}$, the same monotonicity and claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space show that $\int_{X}|g|\,d\sigma<\infty$, so $g$ is integrable with respect to each such $\sigma$.

(c) Convergence of $\int_{X}g\,d\nu_{n}$. Let $\varepsilon\in\mathbb{R}$ be positive. First choose, by the Archimedean property, a positive $M''\in\mathbb{R}$ with $B^{2}A/(a_{k}M'')<\varepsilon/3$. Let $g_{M''}=\max\{-M'',\min\{g,M''\}\}$, pointwise with the constant functions $\pm M''$, with the maximum and minimum of Maximum of Two Elements of a Totally Ordered Set and Minimum of Two Elements of a Totally Ordered Set; it is continuous by claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space, applied twice, and $|g_{M''}|\le M''\le M''(1+|x|^{2})$. We claim $|g(x)-g_{M''}(x)|\le g(x)^{2}/M''$ for every $x\in X$: if $|g(x)|\le M''$, then $g_{M''}(x)=g(x)$; if $g(x)>M''$, then $g_{M''}(x)=M''$ and $0<g(x)-M''<g(x)<g(x)^{2}/M''$, since $g(x)/M''>1$; if $g(x)<-M''$, then $g_{M''}(x)=-M''$ and $0<-M''-g(x)<|g(x)|<g(x)^{2}/M''$. Hence, by Linearity and Monotonicity of the Lebesgue Integral §integrable (linearity, $|\int f|\le\int|f|$ and monotonicity) and (b), for $\sigma=\nu$ and $\sigma=\nu_{n}$,

$$\Bigl|\int_{X}g\,d\sigma-\int_{X}g_{M''}\,d\sigma\Bigr|\le\frac{1}{M''}\int_{X}g^{2}\,d\sigma\le\frac{B^{2}A}{a_{k}M''}<\frac{\varepsilon}{3}.$$

Then, for this $M''$, choose by Step 1, applied to $g_{M''}$ with $A'=M''$, an $n_{0}\in\mathbb{N}$ with $|\int_{X}g_{M''}\,d\nu_{n}-\int_{X}g_{M''}\,d\nu|<\varepsilon/3$ for every $n\ge n_{0}$. For $n\ge n_{0}$ the triangle inequality gives $|\int_{X}g\,d\nu_{n}-\int_{X}g\,d\nu|<\varepsilon$. So $\lim\int_{X}g\,d\nu_{n}=\int_{X}g\,d\nu$.

(d) The cross pairing. Fix $n$, and on $(X\times X,\mathcal{B}(X\times X),\pi_{n})$ consider the maps $\Gamma(z)=\nabla_{a}V(x)$ and $\Theta(z)=(\varphi(y)-\varphi(x))f_{k}$ into $X^{a}$. They are measurable: $\Gamma=\nabla_{a}V\circ\pi_{1}$, with $\nabla_{a}V$ measurable into $X^{a}$ by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity and $\pi_{1},\pi_{2}$ Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §product-sigma, so claim 4 of Borel Measurability and Bounded Integration on a Metric Space applies; and $\Theta=(\varphi f_{k})\circ\pi_{2}-(\varphi f_{k})\circ\pi_{1}$, measurable by Step 6(b), the same claim and Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §operations. Since $(\pi_{1})_{\#}\pi_{n}=\nu_{n}$ by Couplings of Two Borel Probability Measures on a Hilbert Space and Their Quadratic Cost §coupling, claim 2 of Image Measures, Measures with Densities, and Change of Variables and (2b) give $\int|\Gamma|_{a}^{2}\,d\pi_{n}=\int_{X}|\nabla_{a}V|_{a}^{2}\,d\nu_{n}\le A$, and, by (a) and (b), that claim also shows that $z\mapsto\langle\nabla_{a}V(x),\varphi(x)f_{k}\rangle_{a}=a_{k}^{1/2}g(x)$ is integrable with respect to $\pi_{n}$ with integral $a_{k}^{1/2}\int_{X}g\,d\nu_{n}$. Moreover $|\Theta(z)|_{a}^{2}=(\varphi(y)-\varphi(x))^{2}$ by Step 6(b), so $\int|\Theta|_{a}^{2}\,d\pi_{n}=D_{n}(\varphi)$. By Measurable Maps into a Hilbert Space with an Orthonormal Basis: Norms, Inner Products and Linear Combinations, Synthesis from Coordinates, Square-Integrability, and Almost-Everywhere Equality §square-integrable, $\langle\Gamma,\Theta\rangle_{a}$ is integrable with respect to $\pi_{n}$ and, by Step 6(c),

$$|\Delta_{n}|\le A^{1/2}D_{n}(\varphi)^{1/2}\le A^{1/2}L\,\bigl(\bar{a}\,I^{a}(\pi_{n})\bigr)^{1/2},\qquad\Delta_{n}:=\int_{X\times X}\langle\Gamma,\Theta\rangle_{a}\,d\pi_{n}.$$

Now $\langle\Gamma(z),\Theta(z)\rangle_{a}=\langle\nabla_{a}V(x),\varphi(y)f_{k}\rangle_{a}-\langle\nabla_{a}V(x),\varphi(x)f_{k}\rangle_{a}$, and the first term is integrable with respect to $\pi_{n}$ with integral $\mathcal{K}^{a}(\nabla_{a}V,\varphi f_{k},\pi_{n})$ by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §cross, applied with $\nu_{n}$ and $\nu$ in place of its $\nu$ and $\mu$, $\nabla_{a}V$ and $\varphi f_{k}$ being representatives of the classes involved. So by Linearity and Monotonicity of the Lebesgue Integral §integrable,

$$\mathcal{K}^{a}(\nabla_{a}V,\varphi f_{k},\pi_{n})=a_{k}^{1/2}\int_{X}g\,d\nu_{n}+\Delta_{n}.$$

(e) The limit. $\lim\Delta_{n}=0$, because $\Delta_{n}^{2}\le A\,L^{2}\,\bar{a}\,I^{a}(\pi_{n})$ and $\lim I^{a}(\pi_{n})=0$. With (c), $\lim\mathcal{K}^{a}(\nabla_{a}V,\varphi f_{k},\pi_{n})=a_{k}^{1/2}\int_{X}g\,d\nu$. On the other hand, the inner product of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations and (a) with $x=y$ give $\langle\nabla_{a}V,\varphi f_{k}\rangle_{\nu}=\int_{X}\langle\nabla_{a}V(y),\varphi(y)f_{k}\rangle_{a}\,\nu(dy)=a_{k}^{1/2}\int_{X}g\,d\nu$. This proves the claim of Step 7.

Step 8 (approximation by finite sums). Let $\mathcal{S}$ be the set of the elements $\xi=\sum_{i=1}^{N}\varphi_{i}f_{i}$ of $L^{2}(\nu;X^{a})$ with $N\in\mathbb{N}$ and $\varphi_{1},\dots,\varphi_{N}\in\mathcal{F}C^{1}_{b}(X)$, each $\varphi_{i}f_{i}$ being the class of Step 6(b). We show: for every $\eta\in L^{2}(\nu;X^{a})$ and every positive $\varepsilon\in\mathbb{R}$ there is $\xi\in\mathcal{S}$ with $\lVert\eta-\xi\rVert_{\nu}<\varepsilon$. By The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates the series $\sum_{i=1}^{\infty}\lVert\eta_{i}\rVert_{L^{2}(\nu)}^{2}$ converges with sum $\lVert\eta\rVert_{\nu}^{2}$; with $s_{M'}$ its partial sums, Series of Real Numbers §convergent gives $N\in\mathbb{N}$ with $\lVert\eta\rVert_{\nu}^{2}-s_{N}<\varepsilon^{2}/2$. For $i\in[N]$, the classes of the members of $\mathcal{F}C^{1}_{b}(X)$ are dense in $L^{2}(\nu)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §density, so by the meaning of density and claim 3 of Characterization of the Closure in a Metric Space by Open Balls there is $\varphi_{i}\in\mathcal{F}C^{1}_{b}(X)$ with $\lVert\eta_{i}-\varphi_{i}\rVert_{L^{2}(\nu)}^{2}<\varepsilon^{2}/(2N)$. Let $\xi=\sum_{i=1}^{N}\varphi_{i}f_{i}\in\mathcal{S}$. By Step 6(b) and the linearity of coordinates in The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, $(\eta-\xi)_{l}=\eta_{l}-\varphi_{l}$ for $l\le N$ and $(\eta-\xi)_{l}=\eta_{l}$ for $l>N$. So for $M'\ge N$ the $M'$-th partial sum of the series $\sum_{l}\lVert(\eta-\xi)_{l}\rVert_{L^{2}(\nu)}^{2}$ equals $\sum_{l=1}^{N}\lVert\eta_{l}-\varphi_{l}\rVert_{L^{2}(\nu)}^{2}+s_{M'}-s_{N}$, and letting $M'\to\infty$, The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates gives

$$\lVert\eta-\xi\rVert_{\nu}^{2}=\sum_{l=1}^{N}\lVert\eta_{l}-\varphi_{l}\rVert_{L^{2}(\nu)}^{2}+\lVert\eta\rVert_{\nu}^{2}-s_{N}<\frac{\varepsilon^{2}}{2}+\frac{\varepsilon^{2}}{2}=\varepsilon^{2}.$$

Step 9 (weak convergence of the potential part). For $n\in\mathbb{N}$ the cross pairing $\mathcal{K}^{a}(\cdot,\cdot,\pi_{n})$ is linear in each slot, on $L^{2}(\nu_{n};X^{a})$ and on $L^{2}(\nu;X^{a})$ respectively: representatives of sums and real multiples may be taken pointwise by The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, $\langle\cdot,\cdot\rangle_{a}$ is bilinear, all integrands are integrable by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §cross, and the integral is linear by Linearity and Monotonicity of the Lebesgue Integral §integrable. Hence for $\xi=\sum_{i=1}^{N}\varphi_{i}f_{i}\in\mathcal{S}$, Step 7, the limit laws and the linearity of $\langle\nabla_{a}V,\cdot\rangle_{\nu}$ give $\lim\mathcal{K}^{a}(\nabla_{a}V,\xi,\pi_{n})=\sum_{i=1}^{N}\langle\nabla_{a}V,\varphi_{i}f_{i}\rangle_{\nu}=\langle\nabla_{a}V,\xi\rangle_{\nu}$.

Now let $\eta\in L^{2}(\nu;X^{a})$ and let $\varepsilon\in\mathbb{R}$ be positive. First choose, by Step 8, $\xi\in\mathcal{S}$ with $\lVert\eta-\xi\rVert_{\nu}<\varepsilon/(3(A^{1/2}+1))$; then, for this $\xi$, choose $n_{1}\in\mathbb{N}$ with $|\mathcal{K}^{a}(\nabla_{a}V,\xi,\pi_{n})-\langle\nabla_{a}V,\xi\rangle_{\nu}|<\varepsilon/3$ for every $n\ge n_{1}$. For $n\ge n_{1}$, linearity in the second slot, the choice of $n_{1}$ for the middle term, the bound Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §cross-bound with $\lVert\nabla_{a}V\rVert_{\nu_{n}}\le A^{1/2}$ from (2b), and The Cauchy-Schwarz Inequality in a Real Inner Product Space in $L^{2}(\nu;X^{a})$ with $\lVert\nabla_{a}V\rVert_{\nu}\le A^{1/2}$ from Step 4 give

$$\begin{aligned} \bigl|\mathcal{K}^{a}(\nabla_{a}V,\eta,\pi_{n})-\langle\nabla_{a}V,\eta\rangle_{\nu}\bigr|&\le\bigl|\mathcal{K}^{a}(\nabla_{a}V,\eta-\xi,\pi_{n})\bigr|+\bigl|\mathcal{K}^{a}(\nabla_{a}V,\xi,\pi_{n})-\langle\nabla_{a}V,\xi\rangle_{\nu}\bigr|+\bigl|\langle\nabla_{a}V,\xi-\eta\rangle_{\nu}\bigr|\\ &\le\frac{A^{1/2}\,\varepsilon}{3(A^{1/2}+1)}+\frac{\varepsilon}{3}+\frac{A^{1/2}\,\varepsilon}{3(A^{1/2}+1)}<\varepsilon . \end{aligned}$$

Hence $\lim\mathcal{K}^{a}(\nabla_{a}V,\eta,\pi_{n})=\langle\nabla_{a}V,\eta\rangle_{\nu}$ for every $\eta\in L^{2}(\nu;X^{a})$.

Step 10 (conclusion). Let $\eta\in L^{2}(\nu;X^{a})$. By Step 0, $\Sigma(\nu_{n})=\beta Z_{n}+\nabla_{a}V$ in $L^{2}(\nu_{n};X^{a})$, so by the linearity of $\mathcal{K}^{a}(\cdot,\eta,\pi_{n})$ in the first slot (Step 9), Step 3, Step 9, the limit laws, the linearity of the inner product of $L^{2}(\nu;X^{a})$ and Step 5,

$$\lim\mathcal{K}^{a}(\Sigma(\nu_{n}),\eta,\pi_{n})=\lim\bigl(\mathcal{K}^{a}(\beta Z_{n},\eta,\pi_{n})+\mathcal{K}^{a}(\nabla_{a}V,\eta,\pi_{n})\bigr)=\langle\beta Z,\eta\rangle_{\nu}+\langle\nabla_{a}V,\eta\rangle_{\nu}=\langle\Sigma(\nu),\eta\rangle_{\nu}.$$

So, by Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §weak, $(\Sigma(\nu_{n}))_{n\in\mathbb{N}}$ converges weakly to $\Sigma(\nu)$ along $(\pi_{n})_{n\in\mathbb{N}}$. Together with $\nu\in\mathcal{D}_{\Sigma}$ from Step 5, this is the closed-score property of Noise Penalty Pairs with Closed Score Along Noise Couplings §closed.