The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector, with greatest variance $c_{\max}$, scaling map $S_{c}$ and weighted square $|x|_{c}^{2}$ ($x\in\mathbb{R}^{d}$), let $a\in\mathbb{R}$ be positive, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian free-energy pair with variances $c$ and temperature $a$. Penalty pairs and the translation Hessian $H_{\mathcal{E}}$ are those of the setting, the translations $\tau_{h}:x\mapsto x+h$ ($h\in\mathbb{R}^{d}$) are those of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants, the vector written $a$ there being written $h$ here, the mean $m(\mu)\in\mathbb{R}^{d}$ is that of The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean, and $M_{2}$ is the second moment; the map $\ell^{c}_{\mu}$ is the Ornstein-Uhlenbeck functional; lower semicontinuity on $\mathcal{D}$ is taken relative to $\mathcal{D}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, and the map property is that of the setting.

1. (Penalty pair)¶ $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$.

2. (Translations and the translation Hessian)¶ For every $\mu\in\mathcal{D}$ and $h\in\mathbb{R}^{d}$, $(\tau_{h})_{\#}\mu\in\mathcal{D}$ and

$$\mathcal{E}\bigl((\tau_{h})_{\#}\mu\bigr)=\mathcal{E}(\mu)+a\,S_{c}(h)\cdot m(\mu)+\tfrac{a}{2}\,|h|_{c}^{2};$$

consequently $H_{\mathcal{E}}(\mu)$ is the diagonal matrix whose $i$th diagonal entry is $a/c_{i}$, for every $i\in[d]$.

3. (First variation on the penalty domain)¶ Let $\mu\in\mathcal{D}$ and let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ be a test function, the maps $\mathrm{id}+t\,\nabla\psi$ and differentiability at $0$ being as in Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §variation. There is a positive $t_{0}\in\mathbb{R}$ such that $(\mathrm{id}+t\,\nabla\psi)_{\#}\mu\in\mathcal{D}$ for every $t\in(-t_{0},t_{0})$ and the function $t\mapsto\mathcal{E}\bigl((\mathrm{id}+t\,\nabla\psi)_{\#}\mu\bigr)$ on $(-t_{0},t_{0})$ is differentiable at $0$ with derivative $-a\,\ell^{c}_{\mu}(\psi)$.

4. (Lower bound)¶ $0\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$.

5. (Moment bound)¶ For every $\mu\in\mathcal{D}$,

$$M_{2}(\mu)\le\frac{4c_{\max}}{a}\,\mathcal{E}(\mu)+2\sum_{i=1}^{d}c_{i}.$$

6. (Lower semicontinuity)¶ $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$.

7. (The map property)¶ $\mathcal{D}$ has the map property.