Cross-Level Data and Eta-Consistency of a Fine and a Coarse Level at (delta, R)

In the setting of The Real Numbers: Standing Notation and Background and Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation, let levels 1 and 2 be as in Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §levels, with the level-indexed notation of Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §notation, and let $(\kappa,p,j)$ be a mode-restriction link from level 1 to level 2, with mode restriction $p$ and mode embedding $j$. For $i\in\{1,2\}$ let $\mathcal{P}_{i}=(\mathcal{D}_{i},\mathcal{D}_{\Sigma,i},\mathcal{E}_{i},\Sigma_{i})$ be a noise penalty pair on $\mathcal{P}^{a}_{\rho_{i}}$ at level $i$, and suppose that $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are compatible. For $i\in\{1,2\}$ let $F_{i}$ be a first-order equation operator over $\mathcal{D}_{\Sigma,i}$ at level $i$, with $\delta$-shifts $F^{-}_{i,\delta}$ and $F^{+}_{i,\delta}$ relative to $\mathcal{P}_{i}$ for positive $\delta\in\mathbb{R}$, functions on $\mathcal{V}^{a,i}(\mathcal{D}_{\Sigma,i})\times\mathbb{R}$ whose value at $((\nu,q),r)$ is written $F^{\mp}_{i,\delta}(\nu,r,q)$. For $\mu\in\mathcal{D}_{1}$ the measures $\mu$ and $\rho_{1}$ lie in $\mathcal{P}^{a}_{\rho_{1}}$ (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 $W_{a,1}(\mu,\rho_{1})$ is a nonnegative real number; $|s|$ is the absolute value of $s\in\mathbb{R}$. In this item the letter $\eta$ denotes a nonnegative real number.

1. (Cross-level data)¶ A cross-level datum is a triple $(\mu,r,h)$ with $\mu\in\mathcal{D}_{\Sigma,1}$, $r\in\mathbb{R}$ and $h\in T^{a,2}_{p_{\#}\mu}$. For such a triple, $p_{\#}\mu\in\mathcal{D}_{\Sigma,2}$ by Compatible Noise Penalty Pairs at a Fine and a Coarse Level §compatible and $T^{a,2}_{p_{\#}\mu}\subseteq L^{2}(p_{\#}\mu;X_{2}^{a})$ by The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §tangent, taken at level 2, so that $(p_{\#}\mu,h)\in\mathcal{V}^{a,2}(\mathcal{D}_{\Sigma,2})$; and the class $j\circ h\circ p$ of Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields §fields lies in $T^{a,1}_{\mu}$ by Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields §tangent, hence in $L^{2}(\mu;X_{1}^{a})$, so that $(\mu,j\circ h\circ p)\in\mathcal{V}^{a,1}(\mathcal{D}_{\Sigma,1})$. The score $\Sigma_{1}(\mu)$ lies in $T^{a,1}_{\mu}$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair. Let $R\in\mathbb{R}$ be positive. The cross-level datum is $R$-bounded if

$$W_{a,1}(\mu,\rho_{1})<R,\qquad|\mathcal{E}_{1}(\mu)|<R,\qquad\lVert\Sigma_{1}(\mu)\rVert_{\mu,1}<R,\qquad|r|<R,\qquad\lVert h\rVert_{p_{\#}\mu,2}<R.$$

2. (Sub-consistency)¶ Let $\delta,R\in\mathbb{R}$ be positive. The fine level with $F_{1}$ and $\mathcal{P}_{1}$ and the coarse level with $F_{2}$ and $\mathcal{P}_{2}$ are $\eta$-sub-consistent at $(\delta,R)$ if

$$F^{-}_{2,\delta}(p_{\#}\mu,r,h)\le F^{-}_{1,\delta}(\mu,r,j\circ h\circ p)+\eta$$

for every $R$-bounded cross-level datum $(\mu,r,h)$.

3. (Super-consistency)¶ With $\delta$ and $R$ as in clause 2, the two levels are $\eta$-super-consistent at $(\delta,R)$ if

$$F^{+}_{1,\delta}(\mu,r,j\circ h\circ p)\le F^{+}_{2,\delta}(p_{\#}\mu,r,h)+\eta$$

for every $R$-bounded cross-level datum $(\mu,r,h)$.

4. (Consistency)¶ With $\delta$ and $R$ as in clause 2, the two levels are $\eta$-consistent at $(\delta,R)$ if they are both $\eta$-sub-consistent and $\eta$-super-consistent at $(\delta,R)$.