Locally Strictly Proper Second-Order Equation Operator on the Wasserstein Space

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $F$ be a second-order equation operator over $Q$, with value $F(\nu,r,q,Y)$ at $(\nu,q)\in\mathcal{V}(Q)$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$. In this definition the letters $r$ and $s$ denote real numbers; the dimension written $r$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation is not used.

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

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

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

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