Degenerate Elliptic Second-Order Equation Operators on the Wasserstein Space

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$, let $F$ be a second-order equation operator over $Q$, and let $\mathcal{V}(Q)$ be the bundle of vector fields over $Q$. The set $\mathcal{S}(d)$ of symmetric real $d\times d$ matrices and the order $\preceq$ on it are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i). In this definition the letter $q$ denotes a vector field; the dimension written $q$ in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i) and in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i) is not used.

(Degenerate ellipticity)(i) The operator $F$ is degenerate elliptic if

$$F(\nu,r,q,Y)\le F(\nu,r,q,X)$$

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