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

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $F$ be a second-order equation operator on the lift over $Q$, defined on the product of the preimage $Q^{\Lambda}$ of $Q$ under the law map, of $\mathbb{R}$, of the space $L^{2}(\Omega;\mathbb{R}^{d})$ and of the set $\mathcal{S}(d)$ of symmetric real $d\times d$ matrices. 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

$$\lambda\,(r-s)\ \le\ F(X,r,V,\mathbb{X})-F(X,s,V,\mathbb{X})$$

for every $X\in Q^{\Lambda}$, every $V\in L^{2}(\Omega;\mathbb{R}^{d})$, every $\mathbb{X}\in\mathcal{S}(d)$ 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 $R$.