The Lifted Free Score: Norm, Weak Identity and the Second-Moment Identity

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, in dimension $d=1$, with the real line identified with $\mathbb{R}^{1}$ and $\mathbb{R}^{2}=\mathbb{R}^{1+1}$ as in One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative. Let $X\in L^{2}(\Omega;\mathbb{R})$ and let $\mu=\mathcal{L}(X)$ be its law, which belongs to $\mathcal{P}_{2}(\mathbb{R})$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (ii). Let $\mathrm{id}$ be the identity map of $\mathbb{R}$, an element of $L^{2}(\nu;\mathbb{R})$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R})$ 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); for $\xi\in L^{2}(\mu;\mathbb{R})$ let $\xi\circ X\in L^{2}(\Omega;\mathbb{R})$ be the composition; for $\psi\in C_{c}^{\infty}(\mathbb{R})$ let $\psi'=\nabla\psi$ be its derivative and $F_{\psi}$ the difference quotient of its derivative; and let finite free Fisher information, the set $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$, the free score $\Xi_{\nu}$ and the free Fisher information $\Phi^{*}(\nu)$ of a $\nu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ be as defined there.

1. (The lifted free score)(i) If $\mu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$, then $\lVert\Xi_{\mu}\circ X\rVert_{L^{2}}^{2}=\Phi^{*}(\mu)$ and, for every $\psi\in C_{c}^{\infty}(\mathbb{R})$,

$$\langle\Xi_{\mu}\circ X,\psi'\circ X\rangle_{L^{2}}=\int_{\mathbb{R}^{2}}F_{\psi}\,d(\mu\boxtimes\mu).$$

2. (Second-moment identity)(ii) For every $\nu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$, whether or not it is the law of an element of $L^{2}(\Omega;\mathbb{R})$, $\langle\Xi_{\nu},\mathrm{id}\rangle_{\nu}=1$. In particular, if $\mu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$, then

$$\langle\Xi_{\mu}\circ X,X\rangle_{L^{2}}=1 .$$