Noise Intrinsic Test Functions Pulled Back along a Mode Restriction, and the Squared Coarse Noise Wasserstein Distance as a Fine Test Function

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 $j$. Suppose that $p_{\#}\rho_{1}\in\mathcal{P}^{a}_{\rho_{2}}$, so that $p_{\#}\mu\in\mathcal{P}^{a}_{\rho_{2}}$ for every $\mu\in\mathcal{P}^{a}_{\rho_{1}}$ 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 §space; for $\varphi:\mathcal{P}^{a}_{\rho_{2}}\to\mathbb{R}$, $\varphi\circ p_{\#}:\mathcal{P}^{a}_{\rho_{1}}\to\mathbb{R}$ is the function $\mu\mapsto\varphi(p_{\#}\mu)$. Let $\mathcal{Q}_{1}\subseteq\mathcal{P}^{a}_{\rho_{1}}$ and $\mathcal{Q}_{2}\subseteq\mathcal{P}^{a}_{\rho_{2}}$ satisfy $p_{\#}\mu\in\mathcal{Q}_{2}$ for every $\mu\in\mathcal{Q}_{1}$. Noise intrinsic test functions on $\mathcal{Q}_{i}$ and their gradients along noise couplings, written $\nabla_{i}$, are taken at level $i$; for $\mu\in\mathcal{P}(X_{1})$ and $\eta\in L^{2}(p_{\#}\mu;X_{2}^{a})$, $j\circ\eta\circ p\in L^{2}(\mu;X_{1}^{a})$ is the class 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. The noise map property, noise-optimal maps $S$ and their displacements $S-\mathrm{id}_{2}\in L^{2}(\sigma;X_{2}^{a})$ for $\sigma\in\mathcal{P}^{a}_{\rho_{2}}$ are taken at level 2, and $\mathrm{id}_{2}-S$ denotes $-(S-\mathrm{id}_{2})$. Then the following hold.

1. (Pull-back of test functions)¶ Let $\varphi$ be a noise intrinsic test function on $\mathcal{Q}_{2}$ at level 2. Then $\varphi\circ p_{\#}$ is a noise intrinsic test function on $\mathcal{Q}_{1}$ at level 1, and

$$\nabla_{1}(\varphi\circ p_{\#})(\mu)=j\circ\nabla_{2}\varphi(p_{\#}\mu)\circ p\qquad\text{for every }\mu\in\mathcal{Q}_{1}.$$

2. (The squared coarse distance)¶ Suppose that $\mathcal{Q}_{2}$ has the noise map property at level 2, and let $\nu_{0}\in\mathcal{P}^{a}_{\rho_{2}}$. Let $\psi:\mathcal{P}^{a}_{\rho_{1}}\to\mathbb{R}$ be the function

$$\psi(\mu)=W_{a,2}(p_{\#}\mu,\nu_{0})^{2},$$

which is defined because $p_{\#}\mu,\nu_{0}\in\mathcal{P}^{a}_{\rho_{2}}$ and any two members of $\mathcal{P}^{a}_{\rho_{2}}$ are noise-connected at level 2 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. ¶Then $\psi$ is a noise intrinsic test function on $\mathcal{Q}_{1}$ at level 1. For $\mu\in\mathcal{Q}_{1}$ a noise-optimal map $S$ from $p_{\#}\mu$ to $\nu_{0}$ at level 2 exists by the noise map property of $\mathcal{Q}_{2}$, since $p_{\#}\mu\in\mathcal{Q}_{2}$, and the class $\mathrm{id}_{2}-S$ does not depend on its choice by The Noise Map Property of a Set of Probability Measures §map-property, taken at level 2. ¶For every such $\mu$ and $S$,

$$\nabla_{1}\psi(\mu)=2\,j\circ(\mathrm{id}_{2}-S)\circ p.$$

¶For every $\mu\in\mathcal{Q}_{1}$,

$$\lVert\nabla_{1}\psi(\mu)\rVert_{\mu,1}=2\,W_{a,2}(p_{\#}\mu,\nu_{0}).$$