Test Data for a First-Order Equation Operator on the Noise Wasserstein Space and the Admissible Sets

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ and let $F$ be a first-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 $\mathcal{V}^{a}(\mathcal{D}_{\Sigma})\times\mathbb{R}$. Products of sets are Cartesian products and $|s|$ is the absolute value of $s\in\mathbb{R}$.

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

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

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

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

$$W_{a}(\nu,\rho)<R,\qquad\bigl|\mathcal{E}(\nu)\bigr|<R,\qquad|r|<R,\qquad\lVert q\rVert_{\nu}<R;$$

here $\mathcal{E}(\nu)$ is defined because $\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$, and $W_{a}(\nu,\rho)$ is a nonnegative real number because $\nu,\rho\in\mathcal{P}^{a}_{\rho}$ (Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §reference), so that the pair is noise-connected by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected.

3. (Admissible test data)¶ 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 noise penalty pair, on $\delta$ and on $R$.