Test Data for an Intrinsic Second-Order Equation Operator on the Wasserstein Space and the Admissible Sets

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair for each positive $\delta\in\mathbb{R}$, defined on $\mathcal{V}(\mathcal{D}_{\Sigma})\times\mathbb{R}\times\mathcal{S}(d)$ and written $F^{\mp}_{\delta}(\nu,r,q,Y)$ at the element with $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$. Products of sets are Cartesian products, $|s|$ is the absolute value of $s\in\mathbb{R}$, and $\sqrt{t}$ for nonnegative real $t$ is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i). In this definition the letter $r$ denotes a real number; the dimension written $r$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation is not used.

1. (Test data)(i) The set of test data for $F$ is the product

$$\mathcal{W}=\mathcal{V}(\mathcal{D}_{\Sigma})\times\mathbb{R}\times\mathcal{S}(d),$$

which is the domain of $F^{-}_{\delta}$ and of $F^{+}_{\delta}$ for every positive $\delta\in\mathbb{R}$. Its elements are written $\xi=(\nu,r,q,Y)$, standing for the element $((\nu,q),r,Y)$ of that product, and for such a $\xi$ and a positive $\delta\in\mathbb{R}$ we write $F^{-}_{\delta}(\xi)=F^{-}_{\delta}(\nu,r,q,Y)$ and $F^{+}_{\delta}(\xi)=F^{+}_{\delta}(\nu,r,q,Y)$.

2. ($R$-bounded test data)(ii) Let $R\in\mathbb{R}$ be positive. A test datum $\xi=(\nu,r,q,Y)\in\mathcal{W}$ is $R$-bounded if

$$\sqrt{M_{2}(\nu)}<R,\qquad\bigl|\mathcal{E}(\nu)\bigr|<R,\qquad|r|<R,\qquad\lVert q\rVert_{\nu}<R,\qquad\lVert Y\rVert<R;$$

here $\mathcal{E}(\nu)$ is defined because $\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i), and $M_{2}(\nu)$ is a nonnegative real number because $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

3. (Admissible test data)(iii) Let $\delta,R\in\mathbb{R}$ be positive. The set $S^{-}_{\delta,R}$ consists of those $\xi\in\mathcal{W}$ that are $R$-bounded and for which there exists an $R$-bounded $\eta\in\mathcal{W}$ with

$$F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\eta)<R .$$

The set $S^{+}_{\delta,R}$ consists of those $\eta\in\mathcal{W}$ that are $R$-bounded and for which there exists an $R$-bounded $\xi\in\mathcal{W}$ satisfying the same inequality. Both sets depend on $F$, on the penalty pair, on $\delta$ and on $R$; when several operators are in play they are written $S^{-}_{\delta,R}(F)$ and $S^{+}_{\delta,R}(F)$.