The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $\mu\in\mathcal{P}_{2}(X)$, the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, have a relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$. The coordinate of an element of $L^{2}(\mu;X^{a})$ along $f_{k}$ is that of The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates; $\langle\cdot,\cdot\rangle_{\mu}$ and $\lVert\cdot\rVert_{\mu}$ are the inner product and norm of $L^{2}(\mu;X^{a})$; $\nabla_{a}$ is the noise gradient and $T^{a}_{\mu}$ the noise tangent space at $\mu$.

1. (The noise score field)¶ There is exactly one $Z^{a}_{\mu}\in L^{2}(\mu;X^{a})$ whose coordinate along $f_{k}$ is $a_{k}^{1/2}\zeta_{k}$ for every $k\in\mathbb{N}$, and $\lVert Z^{a}_{\mu}\rVert_{\mu}^{2}=\mathcal{I}_{a}(\mu\,|\,\gamma_{c})$.

2. (Pairing with noise gradients)¶ Let $n\in\mathbb{N}$ and let $g\in C^{1}_{b}(\mathbb{R}^{n})$, the set of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded with $q=n$, with partial derivatives $\partial_{k}g$ as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, so that $g\circ p_{n}$ is a bounded $C^{1}$ cylindrical function and, for $k\in[n]$, $(\partial_{k}g)\circ p_{n}$ is its $k$-th partial derivative 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, whose class lies in $L^{2}(\mu)$ 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. Then

$$\langle Z^{a}_{\mu},\nabla_{a}(g\circ p_{n})\rangle_{\mu}=\sum_{k=1}^{n}a_{k}\,\langle\zeta_{k},(\partial_{k}g)\circ p_{n}\rangle_{L^{2}(\mu)} .$$

3. (Pairing with the Ornstein-Uhlenbeck functional)¶ Let $n\in\mathbb{N}$ and let $g\in C^{2}_{b}(\mathbb{R}^{n})$, the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded with $q=n$; then $g\in C^{1}_{b}(\mathbb{R}^{n})$, a function of class $C^{2}$ being of class $C^{1}$ by clause 2 of C^k Maps on a Euclidean Open Set. With $L^{a}_{\mu}(g)$ the noise Ornstein-Uhlenbeck functional of $\mu$ at $g$,

$$\langle Z^{a}_{\mu},\nabla_{a}(g\circ p_{n})\rangle_{\mu}=L^{a}_{\mu}(g).$$

4. (Head approximation)¶ For $n\in\mathbb{N}$, $\tilde{\mu}_{n}$ has finite Fisher information relative to $\tilde{\gamma}_{n}$ by The Relative Score of the Rescaled Head: Projections of the Score, the Ornstein-Uhlenbeck Functional of the Head, and the Converse under Bounded Head Fisher Information §head-score, so its relative score $\zeta^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}$ lies in $L^{2}(\tilde{\mu}_{n};\mathbb{R}^{n})$ and has the lift $\Lambda_{n}\zeta^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}\in L^{2}(\mu;X^{a})$ of Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §lift. Then

$$\bigl\lVert Z^{a}_{\mu}-\Lambda_{n}\zeta^{\tilde{c}^{(n)}}_{\tilde{\mu}_{n}}\bigr\rVert_{\mu}^{2}=\mathcal{I}_{a}(\mu\,|\,\gamma_{c})-\mathcal{I}(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n}),$$

and this tends to $0$ as $n\to\infty$.

5. (Tangency)¶ $Z^{a}_{\mu}\in T^{a}_{\mu}$.