Coercive Penalty Pairs on the Wasserstein Space

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\rho$ be the centred heat gauge, a metric on $\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Centred Heat Gauge: Metric Properties, Comparison with the Wasserstein Distance, Behaviour Under Translations, the Squared Gauge as a Test Function, and the Polarisation Inequality (i), and let $M_{2}$ be the second moment. The penalty domain $\mathcal{D}$ is a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i), so that for $c\in\mathbb{R}$ the set $\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$ is a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$ as well.

(Coercive penalty pair)(i) The penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is coercive if the following two conditions hold for every $c\in\mathbb{R}$.

1. (Sequentially compact sublevel sets)(ii) The set $\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$ is sequentially compact in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$.

2. (Bounded second moments on sublevel sets)(iii) There is $R\in\mathbb{R}$ such that $M_{2}(\mu)\le R$ for every $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le c$.