Proof of On Integrals of Cylindrical Functions the Gibbs Score Drift is the Gibbs Ornstein-Uhlenbeck Functional, and the Gibbs Score-Drift Hamilton-Jacobi Operator Takes Its Pointwise Form

Each result cited is universally quantified over the data in its own statement. Elementary real-number arithmetic and the manipulation of finite sums are carried by The Real Numbers: Standing Notation and Background §background.

Throughout, the notation of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation is in force by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background, so Integrals of Bounded C^2 Cylindrical Functions are Noise Intrinsic Test Functions with the Noise Gradient as Gradient applies to $n$, $f$, $\varphi=f\circ p_{n}$ and $U_{\varphi}$. Let $d$ be the head dimension of $V$ (Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §admissible), so that $\partial_{k}V=0$ for $k>d$ by Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient. By The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §penalty-domain and The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain, $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$, and every $\mu\in\mathcal{D}$ lies in $\mathcal{P}_{2}(X)$ with $V$ integrable with respect to $\mu$ (Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §entropy, Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound §domain).

Claim 1. Since $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$, Integrals of Bounded C^2 Cylindrical Functions are Noise Intrinsic Test Functions with the Noise Gradient as Gradient §test, applied with $Q=\mathcal{D}$, shows that $U_{\varphi}$ is a noise intrinsic test function on $\mathcal{D}$, and for $\mu\in\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$, Integrals of Bounded C^2 Cylindrical Functions are Noise Intrinsic Test Functions with the Noise Gradient as Gradient §gradient gives $\nabla U_{\varphi}(\mu)=\nabla_{a}\varphi$ in $L^{2}(\mu;X^{a})$.

Preparation: coordinates of $\nabla_{a}\varphi$. Since $f\in C^{2}_{b}(\mathbb{R}^{n})$, also $f\in C^{1}_{b}(\mathbb{R}^{n})$, as recorded in The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §pairing-functional; so $(n,f)$ is a representation of $\varphi\in\mathcal{F}C^{1}_{b}(X)$, and by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial $\partial_{k}\varphi=(\partial_{k}f)\circ p_{n}$ for $k\in[n]$. By The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, computed with $(n,f)$, the $k$-th coordinate of $\nabla_{a}\varphi(x)\in X^{a}$ is $a_{k}\partial_{k}f(p_{n}(x))$ for $k\le n$ and $0$ for $k>n$, and

$$|\nabla_{a}\varphi(x)|_{a}^{2}=\sum_{k=1}^{n}a_{k}\,\partial_{k}f(p_{n}(x))^{2}\qquad(x\in X).\tag{1}$$

Claim 2. Let $\mu\in\mathcal{D}_{\Sigma}$. Then $\mu\in\mathcal{D}$, so $\nabla U_{\varphi}(\mu)=\nabla_{a}\varphi$ by claim 1, and $\Sigma(\mu)=\beta Z^{a}_{\mu}+\nabla_{a}V$ by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §pair, where, by The Gibbs Entropy Pair on the Noise Wasserstein Space: Relative Entropy to the Gibbs Measure and the Gibbs Score Field §score-domain, $\mu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, $\int_{X}|\nabla_{a}V|_{a}^{2}\,d\mu<\infty$, and the class of $\nabla_{a}V$ lies in $T^{a}_{\mu}\subseteq L^{2}(\mu;X^{a})$. The inner product of the real Hilbert space $L^{2}(\mu;X^{a})$ (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields) is bilinear, so

$$\langle\Sigma(\mu),\nabla_{a}\varphi\rangle_{\mu}=\beta\,\langle Z^{a}_{\mu},\nabla_{a}\varphi\rangle_{\mu}+\langle\nabla_{a}V,\nabla_{a}\varphi\rangle_{\mu}.\tag{2}$$

The Gaussian term. As $\mu\in\mathcal{P}_{2}(X)$, the hypotheses of The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency hold, and The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §pairing-functional (with the function there called $g$ taken to be $f$) gives $\langle Z^{a}_{\mu},\nabla_{a}\varphi\rangle_{\mu}=L^{a}_{\mu}(f)$, the noise Ornstein-Uhlenbeck functional.

The potential term. By The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations (with $E=X^{a}$, as fixed in Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields), $\langle\nabla_{a}V,\nabla_{a}\varphi\rangle_{\mu}=\int_{X}\langle\nabla_{a}V(x),\nabla_{a}\varphi(x)\rangle_{a}\,\mu(dx)$. For $x\in X$, Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, applied with $h=\nabla_{a}\varphi(x)\in X^{a}$ and the coordinates of the preparation, gives

$$\langle\nabla_{a}V(x),\nabla_{a}\varphi(x)\rangle_{a}=\sum_{k=1}^{d}\partial_{k}V(x)\,h_{k}=\sum_{k=1}^{n}a_{k}\,\partial_{k}V(x)\,\partial_{k}f(p_{n}(x)),$$

since both sums equal the sum over $k\le\min(d,n)$: in the first, $h_{k}=0$ for $k>n$; in the second, $\partial_{k}V(x)=0$ for $k>d$. For $k\in[n]$ the function $x\mapsto\partial_{k}V(x)\partial_{k}f(p_{n}(x))$ is integrable with respect to $\mu$, as recorded in the preamble of The Gibbs Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates (which applies since $\mu\in\mathcal{P}_{2}(X)$ and $V$ is integrable with respect to $\mu$). By Linearity and Monotonicity of the Lebesgue Integral §integrable,

$$\langle\nabla_{a}V,\nabla_{a}\varphi\rangle_{\mu}=\sum_{k=1}^{n}a_{k}\int_{X}\partial_{k}V(x)\,\partial_{k}f(p_{n}(x))\,\mu(dx).$$

Assembly. In The Gibbs Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional, read with $g=f$, the $k$-th integrand is the sum of $\frac{x_{k}}{c_{k}}\partial_{k}f(p_{n}(x))-\partial_{k}\partial_{k}f(p_{n}(x))$ and $\beta^{-1}\partial_{k}V(x)\partial_{k}f(p_{n}(x))$, each integrable (as recorded there and in The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates); by Linearity and Monotonicity of the Lebesgue Integral §integrable and The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional,

$$L^{a,V}_{\mu}(f)=L^{a}_{\mu}(f)+\frac{1}{\beta}\sum_{k=1}^{n}a_{k}\int_{X}\partial_{k}V(x)\,\partial_{k}f(p_{n}(x))\,\mu(dx).$$

Multiplying by $\beta$ and comparing with (2) and the two terms computed above gives $\langle\Sigma(\mu),\nabla U_{\varphi}(\mu)\rangle_{\mu}=\beta\,L^{a,V}_{\mu}(f)$.

Claim 3. Let $\mu\in\mathcal{D}_{\Sigma}$ and $r\in\mathbb{R}$. By claim 1, $\nabla U_{\varphi}(\mu)=\nabla_{a}\varphi$ is an element of $L^{2}(\mu;X^{a})$, and $\mu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{P}^{a}_{\rho}$, so $(\mu,\nabla U_{\varphi}(\mu))$ belongs to the bundle $\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ of The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §bundle, applied with $Q=\mathcal{D}_{\Sigma}$. Then The Hamilton-Jacobi Equation with Gibbs Score Drift on a Hilbert Space §operator, applied with $\nu=\mu$, the real number $r$ and $q=\nabla_{a}\varphi$, gives

$$F\bigl(\mu,r,\nabla U_{\varphi}(\mu)\bigr)=\lambda_{0}\,r+\frac{\theta}{2}\,\lVert\nabla_{a}\varphi\rVert_{\mu}^{2}+\beta\,\langle Z^{a}_{\mu},\nabla_{a}\varphi\rangle_{\mu}+\langle\nabla_{a}V,\nabla_{a}\varphi\rangle_{\mu}-g(\mu),$$

and by (2) and claim 2 the sum of the third and fourth terms is $\langle\Sigma(\mu),\nabla_{a}\varphi\rangle_{\mu}=\beta\,L^{a,V}_{\mu}(f)$. It remains to compute the norm. By The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, $\lVert\nabla_{a}\varphi\rVert_{\mu}^{2}=\int_{X}|\nabla_{a}\varphi|_{a}^{2}\,d\mu$, the square of the nonnegative square root of a nonnegative real number being that number. For $k\in[n]$ the function $\partial_{k}\varphi$ is Borel 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 $(\partial_{k}\varphi)^{2}=|\partial_{k}\varphi|^{2}$ is measurable and nonnegative by Power-Integrable Functions and the p-Seminorm §measurable-power with $p=2$, with finite integral, $\partial_{k}\varphi$ being $2$-integrable with respect to $\mu$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable in the sense of Power-Integrable Functions and the p-Seminorm §space. By (1) and the additivity and positive homogeneity of the integral of nonnegative measurable functions in Linearity and Monotonicity of the Lebesgue Integral §nonnegative,

$$\lVert\nabla_{a}\varphi\rVert_{\mu}^{2}=\int_{X}\sum_{k=1}^{n}a_{k}\,\partial_{k}f(p_{n}(x))^{2}\,\mu(dx)=\sum_{k=1}^{n}a_{k}\int_{X}\partial_{k}f(p_{n}(x))^{2}\,\mu(dx),$$

a finite sum of real numbers. Substituting gives the display of claim 3.