Penalty Pairs with Closed Score Along Couplings

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$. For $\nu\in\mathcal{D}_{\Sigma}$ the score $\Sigma(\nu)$ lies in $T_{\nu}$, hence in $L^{2}(\nu;\mathbb{R}^{d})$, by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i). Sequences of couplings of vanishing cost and weak convergence along them are those of that definition.

(Closed score along couplings)(i) The penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ has closed score along couplings if the following holds for every nonnegative $R\in\mathbb{R}$. Let $(\nu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{D}_{\Sigma}$ with

$$\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le R\qquad(n\in\mathbb{N}),$$

let $\nu\in\mathcal{D}$, and let $(\pi_{n})_{n\in\mathbb{N}}$ be a sequence of couplings of vanishing cost from $(\nu_{n})_{n\in\mathbb{N}}$ to $\nu$. Then $\nu\in\mathcal{D}_{\Sigma}$, and the sequence $(\Sigma(\nu_{n}))_{n\in\mathbb{N}}$ converges weakly to $\Sigma(\nu)$ along $(\pi_{n})_{n\in\mathbb{N}}$.