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

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 $p:X_{1}\to X_{2}$ and mode embedding $j:X_{2}\to X_{1}$. Borel sets, Borel maps and push-forwards are those of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures and Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward, taken at the levels concerned, and $p\times p:X_{1}\times X_{1}\to X_{2}\times X_{2}$ is the map $(x,y)\mapsto(p(x),p(y))$. Noise-connectedness of an ordered pair of measures, Lipschitz maps and continuity are those of the definitions cited. For $\mu\in\mathcal{P}(X_{i})$, measurability of maps into $X_{i}^{a}$ and square-integrability with respect to $\mu$ are those of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields taken at level $i$, and the class in $L^{2}(\mu;X_{i}^{a})$ of such a map is that of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §classes. For a map $g:X_{2}\to X_{2}^{a}$, $j\circ g\circ p$ is the map $x\mapsto j(g(p(x)))$ on $X_{1}$. Then the following hold.

1. (Linearity)¶ The maps $p$ and $j$ are linear.

2. (Left inverse)¶ $p(j(y))=y$ for every $y\in X_{2}$; that is, $p\circ j=\mathrm{id}_{2}$.

3. (Norms)¶ $|p(x)|_{2}\le|x|_{1}$ for every $x\in X_{1}$, and $|j(y)|_{1}=|y|_{2}$ for every $y\in X_{2}$.

4. (Noise norms)¶ For every $x\in X_{1}^{a}$ one has $p(x)\in X_{2}^{a}$ and $|p(x)|_{a,2}\le|x|_{a,1}$. For every $y\in X_{2}^{a}$ one has $j(y)\in X_{1}^{a}$ and $|j(y)|_{a,1}=|y|_{a,2}$.

5. (Adjointness for the noise pairings)¶ For all $y\in X_{2}^{a}$ and $x\in X_{1}^{a}$,

$$\langle j(y),x\rangle_{a,1}=\langle y,p(x)\rangle_{a,2}.$$

6. (Borel measurability and moments)¶ ¶$p$ is Lipschitz with constant $1$ from $(X_{1},d_{1})$ to $(X_{2},d_{2})$, hence continuous, and Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space; likewise $p\times p$ is Lipschitz with constant $1$, for the distances of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs taken at the levels concerned, hence continuous and Borel. ¶For every $\mu\in\mathcal{P}(X_{1})$ one has $p_{\#}\mu\in\mathcal{P}(X_{2})$. ¶If $\mu\in\mathcal{P}_{2}(X_{1})$, then $p_{\#}\mu\in\mathcal{P}_{2}(X_{2})$.

7. (Contraction of noise costs and distances)¶ Let $\mu,\nu\in\mathcal{P}(X_{1})$. For every $\pi\in\Pi^{a,1}(\mu,\nu)$ one has $(p\times p)_{\#}\pi\in\Pi^{a,2}(p_{\#}\mu,p_{\#}\nu)$ and

$$I^{a,2}\bigl((p\times p)_{\#}\pi\bigr)\le I^{a,1}(\pi).$$

¶Consequently, if the ordered pair $(\mu,\nu)$ is noise-connected at level 1, then the ordered pair $(p_{\#}\mu,p_{\#}\nu)$ is noise-connected at level 2 and

$$W_{a,2}(p_{\#}\mu,p_{\#}\nu)\le W_{a,1}(\mu,\nu).$$

This applies to all $\mu,\nu\in\mathcal{P}^{a}_{\rho_{1}}$, any two of which are noise-connected at level 1 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.

8. (Push-forward of the noise-connected measures)¶ Here $p_{\#}\rho_{1}\in\mathcal{P}(X_{2})$ by clause 6. Suppose that $p_{\#}\rho_{1}\in\mathcal{P}^{a}_{\rho_{2}}$. Then $p_{\#}\mu\in\mathcal{P}^{a}_{\rho_{2}}$ for every $\mu\in\mathcal{P}^{a}_{\rho_{1}}$, and

$$W_{a,2}(p_{\#}\mu,\rho_{2})\le W_{a,1}(\mu,\rho_{1})+W_{a,2}(p_{\#}\rho_{1},\rho_{2}).$$

9. (Pulled-back noise fields)¶ Let $\mu\in\mathcal{P}(X_{1})$, so that $p_{\#}\mu\in\mathcal{P}(X_{2})$ by clause 6, and let $g:X_{2}\to X_{2}^{a}$ be measurable and square-integrable with respect to $p_{\#}\mu$. Then $j\circ g\circ p$ takes values in $X_{1}^{a}$, is measurable as a map into $X_{1}^{a}$ and square-integrable with respect to $\mu$, and

$$\int_{X_{1}}\bigl|j(g(p(x)))\bigr|_{a,1}^{2}\,\mu(dx)=\int_{X_{2}}|g(y)|_{a,2}^{2}\,(p_{\#}\mu)(dy).$$

¶Its class in $L^{2}(\mu;X_{1}^{a})$ depends only on the class of $g$ in $L^{2}(p_{\#}\mu;X_{2}^{a})$; for a class $g$ in $L^{2}(p_{\#}\mu;X_{2}^{a})$ it is again written $j\circ g\circ p$. ¶For every $g\in L^{2}(p_{\#}\mu;X_{2}^{a})$ one has $\lVert j\circ g\circ p\rVert_{\mu,1}=\lVert g\rVert_{p_{\#}\mu,2}$. ¶The map $L^{2}(p_{\#}\mu;X_{2}^{a})\to L^{2}(\mu;X_{1}^{a})$, $g\mapsto j\circ g\circ p$, is linear.

10. (Cylindrical functions)¶ Let $\psi\in\mathcal{F}C^{1}_{b}(X_{2})$. Then $\psi\circ p\in\mathcal{F}C^{1}_{b}(X_{1})$, and

$$\nabla_{a,1}(\psi\circ p)(x)=j\bigl(\nabla_{a,2}\psi(p(x))\bigr)\qquad\text{for every }x\in X_{1}.$$

11. (Tangent fields)¶ Let $\mu\in\mathcal{P}(X_{1})$ and $g\in T^{a,2}_{p_{\#}\mu}$. Then the class $j\circ g\circ p$ of clause 9 belongs to $T^{a,1}_{\mu}$.