The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c=(c_{1},\dots,c_{d})\in\mathbb{R}^{d}$ with $c_{i}>0$ for every $i\in[d]$, let $\gamma_{c}$ be the diagonal Gaussian measure with variances $c$, and let $a\in\mathbb{R}$ be positive (the temperature). Finite relative entropy with respect to $\gamma_{c}$ and $H(\cdot\,|\,\gamma_{c})$ are those of that definition on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$; finite Fisher information relative to $\gamma_{c}$ and the relative score $\zeta^{c}_{\mu}\in T_{\mu}$ are those of that definition, $T_{\mu}$ being the tangent space, a linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$ 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 §closed.

(The Gaussian free-energy pair)¶ The Gaussian free-energy pair with variances $c$ and temperature $a$ is the quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ given as follows. The set $\mathcal{D}$ consists of the $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ of finite relative entropy with respect to $\gamma_{c}$, and

$$\mathcal{E}:\mathcal{D}\to\mathbb{R},\qquad\mathcal{E}(\mu)=a\,H(\mu\,|\,\gamma_{c}).$$

The set $\mathcal{D}_{\Sigma}$ consists of the $\mu\in\mathcal{D}$ of finite Fisher information relative to $\gamma_{c}$, and

$$\Sigma(\mu)=a\,\zeta^{c}_{\mu}\in T_{\mu}\qquad(\mu\in\mathcal{D}_{\Sigma}).$$