The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation. Let $V$ be a confining potential on $\mathbb{R}^{d}$, with gradient map $\nabla V$, Hessian matrices $D^{2}V(x)\in\mathcal{S}(d)$ and Laplacian $\Delta V$, let $\sigma\in\mathbb{R}$ be positive, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Langevin free-energy pair with potential $V$ and noise intensity $\sigma$. Penalty pairs, the translation Hessian $H_{\mathcal{E}}$, being Wasserstein-coercive and the map property are those of the setting, and $\mathrm{tr}$ is the trace; lower semicontinuity and continuity of real functions on subsets of $\mathcal{D}$ are taken relative to those subsets in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, and $\mathbb{R}$ carries the metric of The Absolute Value Metric on the Real Line. The letter $\sigma$ denotes the noise intensity; the swap map written $\sigma$ in Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §swap is not used.

1. (Penalty pair)¶ $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$. For every $\mu\in\mathcal{D}$ each entry of $D^{2}V$ and the function $\Delta V$ are integrable with respect to $\mu$, each entry of $H_{\mathcal{E}}(\mu)$ is the $\mu$-integral of the corresponding entry of $D^{2}V$, and $\mathrm{tr}\,H_{\mathcal{E}}(\mu)=\int_{\mathbb{R}^{d}}\Delta V\,d\mu$.

2. (Coercivity and the map property)¶ The pair is Wasserstein-coercive, and $\mathcal{D}$ has the map property.

3. (Semicontinuity and growth)¶ $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$, and there is $C\in\mathbb{R}$ with

4. (Continuity of the trace of the translation Hessian at bounded energy)¶ For every positive $R\in\mathbb{R}$ the restriction of $\mu\mapsto\mathrm{tr}\,H_{\mathcal{E}}(\mu)$ to $\{\mu\in\mathcal{D}:|\mathcal{E}(\mu)|\le R\}$ is continuous.

5. (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, with gradient map $\nabla\psi$ and Laplacian $\Delta\psi$; the maps $\mathrm{id}+t\,\nabla\psi$ and differentiability at $0$ are as in Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §variation. Then the function $\nabla V\cdot\nabla\psi$ is integrable with respect to $\mu$, and 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