The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score and Regular Penalised Maxima

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, in dimension $d=1$, with the identifications of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative, so that $\mathcal{S}(1)$ consists of the $1\times1$ matrices, each read as its sole entry. Let $V$ be a confining potential, with second derivative $V''$, let $\beta\in\mathbb{R}$ be positive, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the confined logarithmic-energy pair with potential $V$ and inverse temperature $\beta$. Penalty pairs and the translation Hessian $H_{\mathcal{E}}$, being Wasserstein-coercive, the map property, closed score along couplings, regular penalised maxima and displacement convexity are those of the definitions cited; lower semicontinuity and continuity of real functions on subsets of $\mathcal{D}$ are taken relative to those subsets in $(\mathcal{P}_{2}(\mathbb{R}),W_{2})$, and $\mathbb{R}$ carries the metric of The Absolute Value Metric on the Real Line.

1. (Penalty pair)¶ $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is a penalty pair on $\mathcal{P}_{2}(\mathbb{R})$. For every $\mu\in\mathcal{D}$ the function $V''$ is $\mu$-integrable and $H_{\mathcal{E}}(\mu)=\int_{\mathbb{R}}V''\,d\mu$.

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

3. (Closed score)¶ The pair has closed score along couplings.

4. (Regular penalised maxima)¶ The pair has regular penalised maxima.

5. (Displacement convexity)¶ The pair is displacement convex.

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

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