The Shift-Coercivity Condition for a First-Order Equation Operator on the Noise Wasserstein Space

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 $\mathcal{P}^{a}_{\rho}$ and let $F$ be a first-order equation operator over $\mathcal{D}_{\Sigma}$, with the admissible sets $S^{-}_{\delta,R}$ and $S^{+}_{\delta,R}$ of test data, taken relative to this operator and this pair. 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.

1. (Score bound at a level)¶ Let $\delta,R\in\mathbb{R}$ satisfy $0<\delta<1$ and $0<R$, and let $C\in\mathbb{R}$ be nonnegative. We say that $C$ is a score bound for $F$ at $(\delta,R)$ if every test datum $(\nu,r,q)$ belonging to $S^{-}_{\delta,R}$ or to $S^{+}_{\delta,R}$ satisfies

$$\lVert\Sigma(\nu)\rVert_{\nu}\ \le\ C .$$

2. (The shift-coercivity condition)¶ The operator $F$ satisfies the shift-coercivity condition if for all $\delta,R\in\mathbb{R}$ with $0<\delta<1$ and $0<R$ there is a score bound for $F$ at $(\delta,R)$.