Noise Penalty Pairs with Closed Score Along Noise Couplings

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho}$. For $\nu\in\mathcal{D}_{\Sigma}$ the score $\Sigma(\nu)$ lies in $T^{a}_{\nu}$, hence in $L^{2}(\nu;X^{a})$, by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair. Sequences of couplings of vanishing noise cost and weak convergence along them are those of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §couplings, the measures involved lying in $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$.

(Closed score along noise couplings)¶ The noise penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ has closed score along noise 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 noise cost from $(\nu_{n})_{n\in\mathbb{N}}$ to $\nu$. Then $\nu\in\mathcal{D}_{\Sigma}$, so that $\Sigma(\nu)\in T^{a}_{\nu}\subseteq L^{2}(\nu;X^{a})$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, and the sequence $(\Sigma(\nu_{n}))_{n\in\mathbb{N}}$ converges weakly to $\Sigma(\nu)$ along $(\pi_{n})_{n\in\mathbb{N}}$.