Test Data for a Second-Order Equation Operator on the Lift of the Wasserstein Space and the Admissible Sets

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: 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 on the lift over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair for each positive $\delta\in\mathbb{R}$. The space $L^{2}(\Omega;\mathbb{R}^{d})$ with its norm $\lVert\cdot\rVert_{L^{2}}$ and the law $\mathcal{L}(X)$ of a class are those of that clause, $\mathcal{D}_{\Sigma}^{\Lambda}$ is the preimage of $\mathcal{D}_{\Sigma}$ under the law map, $\mathcal{S}(d)$ with its norm $\lVert\cdot\rVert$ is as in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iii), products of sets are Cartesian products, and $|s|$ is the absolute value of $s\in\mathbb{R}$. 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

$$\mathcal{W}=\mathcal{D}_{\Sigma}^{\Lambda}\times\mathbb{R}\times L^{2}(\Omega;\mathbb{R}^{d})\times\mathcal{S}(d),$$

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

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

$$\lVert X\rVert_{L^{2}}<R,\qquad\bigl|\mathcal{E}(\mathcal{L}(X))\bigr|<R,\qquad|r|<R,\qquad\lVert V\rVert_{L^{2}}<R,\qquad\lVert\mathbb{X}\rVert<R;$$

here $\mathcal{E}(\mathcal{L}(X))$ is defined because $\mathcal{L}(X)\in\mathcal{D}_{\Sigma}$ by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iv) and $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i).

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 $S^{+}_{\delta,R}(F)$.