Finite Free Fisher Information, the Free Score and the Free Fisher Information of a Probability Measure on the Real Line

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 $\mu\in\mathcal{P}_{2}(\mathbb{R})$, with $L^{2}(\mu;\mathbb{R})$, test functions and their gradient maps and tangent space $T_{\mu}$ as fixed there, let $\mu\boxtimes\mu\in\mathcal{P}(\mathbb{R}^{2})$ be the product measure of $\mu$ with itself, and for $\psi\in C_{c}^{\infty}(\mathbb{R})$ let $\psi'=\nabla\psi$ be its derivative and $F_{\psi}:\mathbb{R}^{2}\to\mathbb{R}$ the difference quotient of its derivative, which is integrable with respect to $\mu\boxtimes\mu$ by that clause. The symbol $\Phi^{*}$ always denotes the free Fisher information, never the cumulative distribution function $\Phi$ of Standard Normal Distribution.

1. (Finite free Fisher information)(i) The measure $\mu$ has finite free Fisher information if there is a real number $C\ge0$ such that

$$\Bigl|\int_{\mathbb{R}^{2}}F_{\psi}\,d(\mu\boxtimes\mu)\Bigr|\le C\,\lVert\nabla\psi\rVert_{\mu}\qquad\text{for every }\psi\in C_{c}^{\infty}(\mathbb{R}).$$

The set of all $\mu\in\mathcal{P}_{2}(\mathbb{R})$ with finite free Fisher information is denoted $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$.

2. (Free score)(ii) Let $\mu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$. The free score of $\mu$ is the unique $\Xi_{\mu}\in T_{\mu}$ such that

$$\langle\Xi_{\mu},\nabla\psi\rangle_{\mu}=\int_{\mathbb{R}^{2}}F_{\psi}\,d(\mu\boxtimes\mu)\qquad\text{for every }\psi\in C_{c}^{\infty}(\mathbb{R}),$$

that is, by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative (ii), $\int_{\mathbb{R}}\Xi_{\mu}\,\psi'\,d\mu=\int_{\mathbb{R}^{2}}F_{\psi}\,d(\mu\boxtimes\mu)$ for every $\psi$. It exists and is unique 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 (iv), applied to $\ell(\psi)=\int_{\mathbb{R}^{2}}F_{\psi}\,d(\mu\boxtimes\mu)$: this $\ell$ is linear in $\psi$ by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative (iv), and $|\ell(\psi)|\le C\lVert\nabla\psi\rVert_{\mu}$ for any $C$ as in clause 1.

3. (Free Fisher information)(iii) For $\mu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$, the free Fisher information of $\mu$ is the nonnegative real number

$$\Phi^{*}(\mu)=\lVert\Xi_{\mu}\rVert_{\mu}^{2}.$$