Wasserstein-Coercive Penalty Pairs

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ be the Wasserstein space with its distance. 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.

(Wasserstein-coercive penalty pair)(i) The penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is Wasserstein-coercive if for every $c\in\mathbb{R}$ the set

$$\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$$

is sequentially compact in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$.