Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, with the space $L^{2}(\mu;\mathbb{R}^{d})$, the tangent space $T_{\mu}$ and the test functions $\psi\in C^{\infty}_{c}(\mathbb{R}^{d})$ as fixed there. The gradient $Df(x)\in\mathbb{R}^{d}$ of a function $f$ of class $C^{1}$ on $\mathbb{R}^{d}$ is that of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii), $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space Is Open in Itself, and $\Delta f$ is the Laplacian of a function of class $C^{2}$ on $\mathbb{R}^{d}$. For $a\in\mathbb{R}^{d}$ the constant map $\mathbb{R}^{d}\to\mathbb{R}^{d}$ with value $a$ is Borel and $\int\lVert a\rVert^{2}\,d\mu=\lVert a\rVert^{2}<\infty$, so its class in $L^{2}(\mu;\mathbb{R}^{d})$ is defined and is again written $a$; the translations $\tau_{a}$ and push-forwards (Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv)) are as fixed there. Let $M$ be a nonnegative real number.

1. (Tangency)(i) Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{d}$ with $|\partial_{i}f(x)|\le M$ for all $x\in\mathbb{R}^{d}$ and $i\in[d]$. Then $|f(x)|\le|f(0_{\mathbb{R}^{d}})|+\sqrt{d}\,M\lVert x\rVert$ for every $x$; the gradient map $x\mapsto Df(x)$ is Borel with $\int_{\mathbb{R}^{d}}\lVert Df\rVert^{2}\,d\mu\le dM^{2}$, its class in $L^{2}(\mu;\mathbb{R}^{d})$ is again written $Df$, and

$$Df\in T_{\mu}.$$

2. (Constants)(ii) $a\in T_{\mu}$ for every $a\in\mathbb{R}^{d}$.

3. (The score identity)(iii) Suppose that $\mu$ has finite Fisher information, with score $\xi_{\mu}$, and let $f$ be as in claim 1 and moreover of class $C^{2}$ on $\mathbb{R}^{d}$ with $|\partial_{j}\partial_{i}f(x)|\le M$ for all $x\in\mathbb{R}^{d}$ and $i,j\in[d]$. Then $\Delta f$ is bounded and Borel, and

$$\langle\xi_{\mu},Df\rangle_{\mu}=-\int_{\mathbb{R}^{d}}\Delta f\,d\mu .$$

4. (Mean zero)(iv) If $\mu$ has finite Fisher information, with score $\xi_{\mu}$, then $\langle\xi_{\mu},a\rangle_{\mu}=0$ for every $a\in\mathbb{R}^{d}$.

5. (Translation)(v) Let $a\in\mathbb{R}^{d}$. Then $(\tau_{a})_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Let $\eta\in T_{\mu}$. Then for every representative of $\eta$ the map $x\mapsto\eta(x-a)$ is Borel, its class in $L^{2}((\tau_{a})_{\#}\mu;\mathbb{R}^{d})$ does not depend on the representative, has the same norm as $\eta$, and belongs to $T_{(\tau_{a})_{\#}\mu}$.