The First Variation of the Logarithmic Energy on the Real Line along Gradient Perturbations of the 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 identifications, the test functions $\psi\in C_{c}^{\infty}(\mathbb{R})$, their derivatives $\psi'=\nabla\psi$ and the difference quotients $F_{\psi}$ of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative; $F_{\psi}$ is $\mu\boxtimes\mu$-integrable for every $\mu\in\mathcal{P}(\mathbb{R})$ by that clause. The set $\mathcal{D}_{\log}$ and the logarithmic energy $\mathcal{E}_{\log}$ are those of that definition, and the set $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ of measures of finite free Fisher information and the free score $\Xi_{\mu}$ are those of that definition. For $\psi\in C_{c}^{\infty}(\mathbb{R})$ and $t\in\mathbb{R}$, $\mathrm{id}+t\psi':\mathbb{R}\to\mathbb{R}$ is the map $x\mapsto x+t\psi'(x)$, and $(\mathrm{id}+t\psi')_{\#}\mu\in\mathcal{P}_{2}(\mathbb{R})$ for $\mu\in\mathcal{P}_{2}(\mathbb{R})$ by The Push-Forward of a Probability Measure with Finite Second Moment by the Identity Perturbed along the Gradient of a Test Function: Borel, Finite Second Moment, the Diagonal Coupling and the Wasserstein Bound §borel. Open intervals and differentiability at an interior point are those of Derivative at an Interior Point.

1. (First variation)¶ Let $\mu\in\mathcal{D}_{\log}$ and $\psi\in C_{c}^{\infty}(\mathbb{R})$. There is a positive $t_{0}\in\mathbb{R}$ such that $(\mathrm{id}+t\psi')_{\#}\mu\in\mathcal{D}_{\log}$ for every $t$ in the open interval $(-t_{0},t_{0})$, and the function $(-t_{0},t_{0})\to\mathbb{R}$, $t\mapsto\mathcal{E}_{\log}\bigl((\mathrm{id}+t\psi')_{\#}\mu\bigr)$, is differentiable at $0$ with derivative

2. (Finite free Fisher information)¶ If moreover $\mu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$, that derivative equals $-\langle\Xi_{\mu},\nabla\psi\rangle_{\mu}$.