The Relative Free Energy and the Relative Score of a Probability Measure for a Potential on an Open Set

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ let $L^{2}(\mu;\mathbb{R}^{d})$ be the space of square-integrable vector fields, with norm $\lVert\cdot\rVert_{\mu}$. Finite entropy, the entropy $\mathrm{Ent}$ and the set $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ are those of that definition; finite Fisher information and the set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ are those of that definition, and the score of $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ is $\xi_{\mu}\in L^{2}(\mu;\mathbb{R}^{d})$; integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Let $D\subseteq\mathbb{R}^{d}$ be open, let $U:D\to\mathbb{R}$ be of class $C^{2}$ on $D$, with gradient $DU(x)$ at $x\in D$, and let $a\in\mathbb{R}$ be positive. Let $\bar{U}:\mathbb{R}^{d}\to\mathbb{R}$ and $\nabla U:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the maps equal to $U$ and to $DU$ on $D$ and to $0$ and $0_{\mathbb{R}^{d}}$ off $D$; they are Borel. Indeed, for $t\in\mathbb{R}$ the set $\{x\in\mathbb{R}^{d}:\bar{U}(x)>t\}$ is the union of $\{x\in D:U(x)>t\}$, which is open: if $x\in D$ and $U(x)>t$, the continuity of $U$ at $x$ (claim 1 of C^k Maps on a Euclidean Open Set) gives a positive $r$ with $|U(y)-U(x)|<U(x)-t$ for every $y\in D$ with $\lVert y-x\rVert<r$, and as $D$ is open we may shrink $r$ so that every $y\in\mathbb{R}^{d}$ with $\lVert y-x\rVert<r$ lies in $D$, so that $U(y)>t$ for all such $y$; this set is united with $\mathbb{R}^{d}\setminus D$ when $t<0$ and with nothing otherwise; both pieces are Borel by claim 4 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, so $\bar{U}$ is Borel by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line. The same argument applies to each component of $\nabla U$, the partial derivatives of $U$ being continuous on $D$, and $\nabla U$ is Borel by claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets.

1. (Relative free energy)¶ The set $\mathcal{D}_{U,a}$ consists of the $\mu\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{d})$ with $\mu(D)=1$ for which $\bar{U}$ is integrable with respect to $\mu$. The relative free energy with potential $U$ and temperature $a$ is

2. (Relative score)¶ The set $\mathcal{D}^{\Sigma}_{U,a}$ consists of the $\mu\in\mathcal{D}_{U,a}\cap\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ with $\int_{\mathbb{R}^{d}}\lVert\nabla U\rVert^{2}\,d\mu<\infty$. For such $\mu$ the class of $\nabla U$ in $L^{2}(\mu;\mathbb{R}^{d})$ is again written $\nabla U$, and the relative score of $\mu$ is

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