The Confined Logarithmic-Energy Pair on the Wasserstein Space of 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 identifications of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative; for $\mu\in\mathcal{P}_{2}(\mathbb{R})$, $L^{2}(\mu;\mathbb{R})$ is the space of square-integrable vector fields and $T_{\mu}$ the tangent space, which equals $L^{2}(\mu;\mathbb{R})$ by On the Real Line the Tangent Space is the Whole Space of Square-Integrable Vector Fields (i). The set $\mathcal{D}_{\log}$ and the logarithmic energy $\mathcal{E}_{\log}$, and the set $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ of measures of finite free Fisher information with the free score $\Xi_{\mu}\in T_{\mu}$, are those of those definitions; integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii). Let $V$ be a confining potential, with derivative $V'$ (both Borel by A Confining Potential and Its Derivative are Continuous and Borel (ii)), and let $\beta\in\mathbb{R}$ be positive; $\tfrac{\beta}{4}$ is the product of $\beta$ with the multiplicative inverse of $4=(1+1)+(1+1)$, and is positive by claims 8, 3, 7 and 5 of Elementary Order Arithmetic in an Ordered Field.

(The confined logarithmic-energy pair)(i) The confined logarithmic-energy pair with potential $V$ and inverse temperature $\beta$ is the quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ given as follows. The set $\mathcal{D}$ consists of the $\mu\in\mathcal{D}_{\log}$ for which $V$ is $\mu$-integrable, and

$$\mathcal{E}:\mathcal{D}\to\mathbb{R},\qquad\mathcal{E}(\mu)=\tfrac{\beta}{4}\,\mathcal{E}_{\log}(\mu)+\int_{\mathbb{R}}V\,d\mu .$$

The set $\mathcal{D}_{\Sigma}$ consists of the $\mu\in\mathcal{D}\cap\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ with $\int_{\mathbb{R}}(V')^{2}\,d\mu<\infty$; for such $\mu$ the function $V'$ has a class in $L^{2}(\mu;\mathbb{R})$, again written $V'$, and

$$\Sigma(\mu)=V'-\tfrac{\beta}{4}\,\Xi_{\mu}\in L^{2}(\mu;\mathbb{R})=T_{\mu},$$

the combination being formed in the vector space $L^{2}(\mu;\mathbb{R})$.