Locally Strictly Proper First-Order Equation Operators 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 $Q\subseteq\mathcal{P}^{a}_{\rho}$ and let $F$ be a first-order equation operator over $Q$, with value $F(\nu,r,q)$ at $(\nu,q)\in\mathcal{V}^{a}(Q)$ and $r\in\mathbb{R}$.

1. (Properness constant at a level)¶ Let $R,\lambda\in\mathbb{R}$ be positive. We say that $\lambda$ is a properness constant for $F$ at $R$ if

$$\lambda\,(r-s)\ \le\ F(\nu,r,q)-F(\nu,s,q)$$

for every $(\nu,q)\in\mathcal{V}^{a}(Q)$ and all $r,s\in\mathbb{R}$ with $-R\le s\le r\le R$.

2. (Local strict properness)¶ The operator $F$ is locally strictly proper if for every positive $R\in\mathbb{R}$ there is a properness constant for $F$ at $R$.