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

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

Throughout, the notation and background of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation are 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, which is stated in that setting, applies to $n$, $f$, $\varphi=f\circ p_{n}$ and $U_{\varphi}$. By The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain and The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain we have $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ and $\mathcal{D}\subseteq\mathcal{P}_{2}(X)$.

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 $\mu\in\mathcal{D}$, which lies in $\mathcal{P}^{a}_{\rho}$, 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 $L^{2}(\mu;X^{a})$.

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 $\Sigma(\mu)=\beta\,Z^{a}_{\mu}$ with $Z^{a}_{\mu}$ the noise score field of $\mu$, by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair. By The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain the measure $\mu$ has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, and $\mu\in\mathcal{P}_{2}(X)$ since $\mathcal{D}\subseteq\mathcal{P}_{2}(X)$; so 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 for $\mu$, 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, applied with $n$ and with the function there called $g$ taken to be $f\in C^{2}_{b}(\mathbb{R}^{n})$, gives

$$\langle Z^{a}_{\mu},\nabla_{a}\varphi\rangle_{\mu}=L^{a}_{\mu}(f).$$

The space $L^{2}(\mu;X^{a})$ is a real Hilbert space by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, so its inner product is bilinear, and therefore

$$\langle\Sigma(\mu),\nabla U_{\varphi}(\mu)\rangle_{\mu}=\langle\beta\,Z^{a}_{\mu},\nabla_{a}\varphi\rangle_{\mu}=\beta\,\langle Z^{a}_{\mu},\nabla_{a}\varphi\rangle_{\mu}=\beta\,L^{a}_{\mu}(f).$$

This is the first equality of claim 2. The second equality is the definition of $L^{a}_{\mu}(f)$ in The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional, applied with $\mu\in\mathcal{P}_{2}(X)$, $n$ and the function there called $g$ taken to be $f$, multiplied by $\beta$.

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 $Q=\mathcal{D}_{\Sigma}$; this is the bundle on which $F$ is defined, by The Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space §operator. That clause, applied with $\nu=\mu$, the real number $r$ and $q=\nabla_{a}\varphi$, gives

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

and $\beta\,\langle Z^{a}_{\mu},\nabla_{a}\varphi\rangle_{\mu}=\beta\,L^{a}_{\mu}(f)$ by the first equality of claim 2. It remains to compute the norm.

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 $\varphi\in\mathcal{F}C^{1}_{b}(X)$. 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, the $k$-th partial derivative of $\varphi$ is $\partial_{k}\varphi=(\partial_{k}f)\circ p_{n}$ for $k\in[n]$. Hence the formula for the noise gradient in The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, computed with the representation $(n,f)$, gives for every $x\in X$

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

By Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, the norm $\lVert\cdot\rVert_{\mu}$ is that of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations with $E=X^{a}$, so $\lVert\nabla_{a}\varphi\rVert_{\mu}^{2}=\int_{X}|\nabla_{a}\varphi|_{a}^{2}\,d\mu$, 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 $p=2$, and its integral with respect to $\mu$ is finite, $\partial_{k}\varphi$ 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. The additivity and the positive homogeneity of the integral of nonnegative measurable functions in Linearity and Monotonicity of the Lebesgue Integral §nonnegative, applied $n-1$ times and $n$ times respectively with the nonnegative coefficients $a_{k}$, give

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

a finite sum of real numbers. Substituting this and $\beta\,\langle Z^{a}_{\mu},\nabla_{a}\varphi\rangle_{\mu}=\beta\,L^{a}_{\mu}(f)$ into the expression for $F\bigl(\mu,r,\nabla U_{\varphi}(\mu)\bigr)$ gives the display of claim 3.