Wasserstein-Coercive Penalty Pairs

In the setting of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: 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 §pair, 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)¶ The penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ is Wasserstein-coercive if for every $c\in\mathbb{R}$ the set

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

This differs from Coercive Penalty Pairs on the Wasserstein Space §coercive in the metric: sequential compactness of the sublevel sets is required for the Wasserstein distance rather than for the centred heat gauge. The second clause of that definition, bounding the second moment on each sublevel set, is not imposed here.