Splitting a Bounded First Variation into a Square-Integrable Potential Force and a Finite Free Fisher Information

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. The set $\mathcal{D}_{\log}$ is that of The Logarithmic Energy of a Probability Measure on the Real Line §energy, and the set $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ of measures of finite free Fisher information is that of that definition. Let $V$ be a confining potential, with derivative $V'$, let $a\in\mathbb{R}$ be positive, and let $\mu\in\mathcal{D}_{\log}$ be such that $V$ is $\mu$-integrable.

1. (Integrable force)¶ $V'$ is $\mu$-integrable, so that for every $\psi\in C_{c}^{\infty}(\mathbb{R})$ the function $V'\psi'$ is $\mu$-integrable, $\psi'$ being bounded by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives.

2. (Splitting)¶ Suppose that there is a nonnegative $C\in\mathbb{R}$ with

Then $\int_{\mathbb{R}}(V')^{2}\,d\mu<\infty$, so that $V'$ has a class in $L^{2}(\mu;\mathbb{R})$, and $\mu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$.