Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $X\in L^{2}(\Omega;\mathbb{R}^{d})$ and let $\mu=\mathcal{L}(X)$ be its law, which belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (ii). Let $L^{2}(\mu;\mathbb{R}^{d})$ be the space of square-integrable vector fields with respect to $\mu$, let $\mathrm{id}$ be the identity map of $\mathbb{R}^{d}$, an element of $L^{2}(\mu;\mathbb{R}^{d})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (ii), and let finite Fisher information, the set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, the score $\xi_{\mu}$ and the Fisher information $\mathcal{I}(\mu)$ be as defined there. The natural number $d$ is read as a real number where a real number is required, and $\mathbb{E}$ is the expectation on $(\Omega,\mathcal{F},P)$.

1. (Composition)(i) For $\xi\in L^{2}(\mu;\mathbb{R}^{d})$, the composition $\xi\circ X:\Omega\to\mathbb{R}^{d}$ of a representative of $\xi$ with a representative of $X$ is a square-integrable random vector whose class in $L^{2}(\Omega;\mathbb{R}^{d})$ does not depend on the two representatives chosen; this class is denoted $\xi\circ X$. The map $\xi\mapsto\xi\circ X$ from $L^{2}(\mu;\mathbb{R}^{d})$ to $L^{2}(\Omega;\mathbb{R}^{d})$ is linear and preserves inner products,

$$\langle\xi\circ X,\eta\circ X\rangle_{L^{2}}=\langle\xi,\eta\rangle_{\mu},\qquad \lVert\xi\circ X\rVert_{L^{2}}=\lVert\xi\rVert_{\mu}\qquad(\xi,\eta\in L^{2}(\mu;\mathbb{R}^{d})),$$

and $\mathrm{id}\circ X=X$.

2. (The lifted score)(ii) If $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, then $\lVert\xi_{\mu}\circ X\rVert_{L^{2}}^{2}=\mathcal{I}(\mu)$ and, for every test function $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$,

$$\langle\xi_{\mu}\circ X,\nabla\psi\circ X\rangle_{L^{2}}=-\int_{\mathbb{R}^{d}}\Delta\psi\,d\mu=-\mathbb{E}[\Delta\psi\circ X].$$

3. (Second-moment identity)(iii) For every $\nu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, whether or not it is the law of an element of $L^{2}(\Omega;\mathbb{R}^{d})$, $\langle\xi_{\nu},\mathrm{id}\rangle_{\nu}=-d$, the identity map lying in $L^{2}(\nu;\mathbb{R}^{d})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (ii). In particular, if $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, then

$$\langle\xi_{\mu}\circ X,X\rangle_{L^{2}}=-d .$$