Compatible Noise Penalty Pairs at a Fine and a Coarse Level

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 $p$. 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 $i$. Then $\mathcal{D}_{\Sigma,1}\subseteq\mathcal{D}_{1}\subseteq\mathcal{P}^{a}_{\rho_{1}}\subseteq\mathcal{P}_{2}(X_{1})\subseteq\mathcal{P}(X_{1})$ by 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 §moments and The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, all taken at level 1 (here $\mathcal{P}_{2}(X_{1})$ is the set of measures with finite second moment, not the pair $\mathcal{P}_{2}$), so that $p_{\#}\mu\in\mathcal{P}(X_{2})$ for every $\mu\in\mathcal{D}_{1}$ 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 §pushforward-measure.

1. (Compatible penalty pairs)¶ Let $c_{0}\in\mathbb{R}$ be nonnegative. The pairs $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are compatible with constant $c_{0}$ if $p_{\#}\mu\in\mathcal{D}_{2}$ for every $\mu\in\mathcal{D}_{1}$, $p_{\#}\mu\in\mathcal{D}_{\Sigma,2}$ for every $\mu\in\mathcal{D}_{\Sigma,1}$, and, the left side being defined by the first of these conditions,

$$\mathcal{E}_{2}(p_{\#}\mu)\le\mathcal{E}_{1}(\mu)+c_{0}\qquad\text{for every }\mu\in\mathcal{D}_{1}.$$

They are compatible if they are compatible with some nonnegative constant.