Proof of The Gaussian Free-Energy Pair is Wasserstein-Closed

Each result cited is universally quantified over the data in its own statement.

We work in the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation; elementary order and arithmetic of real numbers is carried by The Real Numbers: Standing Notation and Background §background. The quadruple is a penalty pair by The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §pair, so Wasserstein-Closed Penalty Pairs §w2-closed applies to it. Since the letter $c$ denotes the variance vector here, we write $\ell\in\mathbb{R}$ for the level called $c$ in that definition; fix such an $\ell$. By The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair, $\mathcal{D}$ is the set of $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ of finite relative entropy with respect to $\gamma_{c}$, and $\mathcal{E}(\mu)=a\,H(\mu\,|\,\gamma_{c})$; the measure $\gamma_{c}$ is a probability measure on $\mathbb{R}^{d}$ by Diagonal Gaussian Measures on Euclidean Space §measure.

Step 1 (closed sublevel sets). Let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{D}$ with $\mathcal{E}(\mu_{n})\le\ell$ for every $n$, converging in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ to some $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By Convergent Sequence in a Metric Space, for every positive $\varepsilon$ there is $N\in\mathbb{N}$ with $W_{2}(\mu_{n},\mu)<\varepsilon$ for all $n\ge N$; as $W_{2}(\mu_{n},\mu)$ is nonnegative by The Quadratic Wasserstein Distance on Euclidean Space §distance, $|W_{2}(\mu_{n},\mu)-0|=W_{2}(\mu_{n},\mu)<\varepsilon$ for those $n$, so $\lim_{n\to\infty}W_{2}(\mu_{n},\mu)=0$ in the sense of Limit of a Sequence of Real Numbers. Hence Convergence in the Wasserstein Distance Implies Weak Convergence and Convergence of Integrals of Continuous Functions of Quadratic Growth §weak, read with $m=d$, gives $\mu_{n}\Rightarrow\mu$.

Step 2. Each $\mu_{n}$ has finite relative entropy with respect to $\gamma_{c}$, and since $a$ is positive, $a\,H(\mu_{n}\,|\,\gamma_{c})=\mathcal{E}(\mu_{n})\le\ell$ gives $H(\mu_{n}\,|\,\gamma_{c})\le\tfrac{\ell}{a}$. Apply Relative Entropy on Euclidean Space: the Gibbs Inequality, a Variational Criterion for Finite Relative Entropy, and Closed Sublevel Sets under Weak Convergence §closed with $m=d$, $\gamma=\gamma_{c}$, level $\tfrac{\ell}{a}$, the sequence $(\mu_{n})$ in $\mathcal{P}(\mathbb{R}^{d})$ and the limit $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})\subseteq\mathcal{P}(\mathbb{R}^{d})$; the weak convergence there is the same weak convergence for the Euclidean distance as in Step 1. It gives that $\mu$ has finite relative entropy with respect to $\gamma_{c}$ and $H(\mu\,|\,\gamma_{c})\le\tfrac{\ell}{a}$. As $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, this means $\mu\in\mathcal{D}$ by The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair, and multiplying by the positive number $a$ gives $\mathcal{E}(\mu)=a\,H(\mu\,|\,\gamma_{c})\le\ell$. This is condition Wasserstein-Closed Penalty Pairs §closed at level $\ell$.

Step 3 (bounded sublevel sets). Let $c_{\max}$ be the greatest variance, which is positive by The Diagonal Gaussian Density on Euclidean Space and Its Notation §variances, so $\tfrac{4c_{\max}}{a}$ is positive. Put $B=\tfrac{4c_{\max}}{a}\,\ell+2\sum_{i=1}^{d}c_{i}$. For $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le\ell$, The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §moment gives

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

This is condition Wasserstein-Closed Penalty Pairs §bounded at level $\ell$.

As $\ell$ was arbitrary, both conditions hold at every level, and the pair is Wasserstein-closed by Wasserstein-Closed Penalty Pairs §w2-closed. $\blacksquare$