Finite Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space as Componentwise Integration by Parts against Bounded $C^1$ Functions

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector and $\gamma_{c}$ the diagonal Gaussian measure with variances $c$. Finite Fisher information relative to $\gamma_{c}$, the relative score $\zeta^{c}_{\nu}$ and the relative Fisher information $\mathcal{I}(\nu\,|\,\gamma_{c})$ are those of that definition, and $L^{2}(\nu;\mathbb{R}^{d})$ is the space of square-integrable vector fields. $C^{1}_{b}(\mathbb{R}^{d})$ is the set of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded with $q=d$, with the partial derivatives $\partial_{i}\psi$ of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives. Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and let $L^{2}(\nu)=L^{2}(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\nu)$ be the real Hilbert space of The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §hilbert, with inner product $\langle\cdot,\cdot\rangle_{L^{2}(\nu)}$ and norm $\lVert\cdot\rVert_{L^{2}(\nu)}$. For $\xi\in L^{2}(\nu;\mathbb{R}^{d})$ and $i\in[d]$, $\xi_{i}\in L^{2}(\nu)$ denotes the class of the $i$-th component of a representative of $\xi$; that component is square-integrable by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §coordinates, applied on the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\nu)$, and its class does not depend on the representative, since two representatives agree outside a $\nu$-null set and so do their $i$-th components.

1. (Integrability)¶ For every $\psi\in C^{1}_{b}(\mathbb{R}^{d})$ and $i\in[d]$, the functions $\psi$ and $\partial_{i}\psi$ are Borel, bounded and $2$-integrable with respect to $\nu$, and the function $y\mapsto y_{i}\psi(y)/c_{i}-\partial_{i}\psi(y)$ is integrable with respect to $\nu$; the class of $\psi$ in $L^{2}(\nu)$ is again written $\psi$.

2. (Componentwise integration by parts)¶ The measure $\nu$ has finite Fisher information relative to $\gamma_{c}$ if and only if for every $i\in[d]$ there is $g_{i}\in L^{2}(\nu)$ with

$$\langle g_{i},\psi\rangle_{L^{2}(\nu)}=\int_{\mathbb{R}^{d}}\Bigl(\frac{y_{i}}{c_{i}}\,\psi(y)-\partial_{i}\psi(y)\Bigr)\,\nu(dy)\qquad\text{for every }\psi\in C^{1}_{b}(\mathbb{R}^{d}).$$

3. (Agreement)¶ Let $\nu$ have finite Fisher information relative to $\gamma_{c}$ and let $i\in[d]$. Then exactly one $g_{i}\in L^{2}(\nu)$ satisfies the identity of claim 2 for every $\psi\in C^{1}_{b}(\mathbb{R}^{d})$, namely $g_{i}=(\zeta^{c}_{\nu})_{i}$; moreover

$$\mathcal{I}(\nu\,|\,\gamma_{c})=\sum_{i=1}^{d}\bigl\lVert(\zeta^{c}_{\nu})_{i}\bigr\rVert_{L^{2}(\nu)}^{2}.$$