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

Each result cited is universally quantified over the data in its own statement.

Preliminaries. 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, $p_{\#}\mu\in\mathcal{P}^{a}_{\rho_{2}}$ for every $\mu\in\mathcal{P}^{a}_{\rho_{1}}$, and 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 §lipschitz, $W_{a,2}(p_{\#}\mu,p_{\#}\mu')\le W_{a,1}(\mu,\mu')$ for all $\mu,\mu'\in\mathcal{P}^{a}_{\rho_{1}}$. We also record an identity of displacement pairings, called (J). Let $\mu,\nu\in\mathcal{P}(X_{1})$, $\pi\in\Pi^{a,1}(\mu,\nu)$ and $\pi'=(p\times p)_{\#}\pi$, which belongs to $\Pi^{a,2}(p_{\#}\mu,p_{\#}\nu)$ with $I^{a,2}(\pi')\le I^{a,1}(\pi)$ 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 §couplings, and let $\eta\in L^{2}(p_{\#}\mu;X^{a}_{2})$. Then (J) states

$$\mathcal{J}^{a,1}(j\circ\eta\circ p,\pi)=\mathcal{J}^{a,2}(\eta,\pi'),$$

where $\mathcal{J}^{a,i}$ is the noise displacement pairing of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §pairing taken at level $i$. To prove (J), fix a representative $\eta$; then $j\circ\eta\circ p$ is a representative of its class 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 §fields. Let $\delta_{1}$ and $\delta_{2}$ be the displacement fields of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §displacement-field for $\pi$ at level 1 and for $\pi'$ at level 2, with the sets $D_{a,i}$ of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs at level $i$. Let $z=(x,y)\in D_{a,1}$. Then $y-x\in X^{a}_{1}$, and $p(y)-p(x)=p(y-x)\in X^{a}_{2}$ 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 §linear and 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 §noise; so $(p\times p)(z)\in D_{a,2}$ and $\delta_{2}((p\times p)(z))=p(\delta_{1}(z))$. Since $\eta(p(x))\in X^{a}_{2}$ and $\delta_{1}(z)\in X^{a}_{1}$, 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 §adjoint gives

$$\bigl\langle j(\eta(p(x))),\delta_{1}(z)\bigr\rangle_{a,1}=\bigl\langle\eta(p(x)),p(\delta_{1}(z))\bigr\rangle_{a,2}=\bigl\langle\eta(\pi_{1}((p\times p)(z))),\delta_{2}((p\times p)(z))\bigr\rangle_{a,2}.$$

Let $h_{1}:X_{1}\times X_{1}\to\mathbb{R}$ and $h_{2}:X_{2}\times X_{2}\to\mathbb{R}$ be the integrands of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §pairing for $j\circ\eta\circ p$ along $\pi$ and for $\eta$ along $\pi'$; they are Borel, and $h_{2}$ is integrable with respect to $\pi'$, by that claim. The display says $h_{1}=h_{2}\circ(p\times p)$ on $D_{a,1}$, whose complement is $\pi$-null because $\pi(D_{a,1})=1$ by Couplings of Finite Noise Cost and Their Noise Cost §finite and $\pi$ is a probability measure. The map $p\times p$ is Borel 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 §borel, so $h_{2}\circ(p\times p)$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and by claim 2 of Image Measures, Measures with Densities, and Change of Variables it is integrable with respect to $\pi$ with $\int h_{2}\circ(p\times p)\,d\pi=\int h_{2}\,d\pi'=\mathcal{J}^{a,2}(\eta,\pi')$. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, $h_{1}$ is integrable with the same integral, which is (J).

Claim (pullback). Let $\varphi$ be a noise intrinsic test function on $\mathcal{Q}_{2}$ at level 2 and write $\Phi=\varphi\circ p_{\#}$. We verify the three properties of Noise Intrinsic Test Functions on the Noise Wasserstein Space §test at level 1.

Property (a). Let $\mu\in\mathcal{P}^{a}_{\rho_{1}}$ and $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$. Since $\varphi$ is continuous at $p_{\#}\mu$ by Noise Intrinsic Test Functions on the Noise Wasserstein Space §continuity at level 2, Continuous Map Between Metric Spaces provides $\delta>0$ such that $|\varphi(\sigma)-\varphi(p_{\#}\mu)|<\varepsilon$ for every $\sigma\in\mathcal{P}^{a}_{\rho_{2}}$ with $W_{a,2}(p_{\#}\mu,\sigma)<\delta$. If $\mu'\in\mathcal{P}^{a}_{\rho_{1}}$ and $W_{a,1}(\mu,\mu')<\delta$, then $p_{\#}\mu'\in\mathcal{P}^{a}_{\rho_{2}}$ and $W_{a,2}(p_{\#}\mu,p_{\#}\mu')<\delta$ by the preliminaries, so $|\Phi(\mu')-\Phi(\mu)|<\varepsilon$. Thus $\Phi$ is continuous on $\mathcal{P}^{a}_{\rho_{1}}$.

Property (b). Let $\mu\in\mathcal{Q}_{1}$, so that $p_{\#}\mu\in\mathcal{Q}_{2}$. By Noise Intrinsic Test Functions on the Noise Wasserstein Space §differentiability at level 2, $\varphi$ is differentiable along noise couplings at $p_{\#}\mu$, and $\eta=\nabla_{2}\varphi(p_{\#}\mu)$ lies in $T^{a,2}_{p_{\#}\mu}$. Put $\xi=j\circ\eta\circ p\in L^{2}(\mu;X^{a}_{1})$. Let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$. By Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §differentiable together with Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §gradient, both at level 2 and applied to the gradient $\eta=\nabla_{2}\varphi(p_{\#}\mu)$, choose $\theta>0$ such that $|\varphi(\sigma)-\varphi(p_{\#}\mu)-\mathcal{J}^{a,2}(\eta,\pi'')|\le\varepsilon\sqrt{I^{a,2}(\pi'')}$ for every $\sigma\in\mathcal{P}^{a}_{\rho_{2}}$ and every $\pi''\in\Pi^{a,2}(p_{\#}\mu,\sigma)$ with $I^{a,2}(\pi'')<\theta^{2}$. Let $\nu\in\mathcal{P}^{a}_{\rho_{1}}$ and $\pi\in\Pi^{a,1}(\mu,\nu)$ with $I^{a,1}(\pi)<\theta^{2}$, and let $\pi'=(p\times p)_{\#}\pi$. Then $p_{\#}\nu\in\mathcal{P}^{a}_{\rho_{2}}$, $\pi'\in\Pi^{a,2}(p_{\#}\mu,p_{\#}\nu)$ and $I^{a,2}(\pi')\le I^{a,1}(\pi)<\theta^{2}$ by the preliminaries. Using (J), the choice of $\theta$, and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, applied to the nonnegative square roots of $I^{a,2}(\pi')\le I^{a,1}(\pi)$, and multiplying by $\varepsilon>0$,

$$\bigl|\Phi(\nu)-\Phi(\mu)-\mathcal{J}^{a,1}(\xi,\pi)\bigr|=\bigl|\varphi(p_{\#}\nu)-\varphi(p_{\#}\mu)-\mathcal{J}^{a,2}(\eta,\pi')\bigr|\le\varepsilon\sqrt{I^{a,2}(\pi')}\le\varepsilon\sqrt{I^{a,1}(\pi)} .$$

So $\Phi$ is differentiable along noise couplings at $\mu$ with gradient $\xi$, and by the uniqueness in Differentiability of a Function on the Noise-Connected Measures Along Noise Couplings, and Its Gradient §gradient, $\nabla_{1}\Phi(\mu)=\xi=j\circ\nabla_{2}\varphi(p_{\#}\mu)\circ p$, which is the asserted formula. Moreover $\xi\in T^{a,1}_{\mu}$ 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, since $\eta\in T^{a,2}_{p_{\#}\mu}$.

Property (c). Let $\mu\in\mathcal{Q}_{1}$, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{Q}_{1}$, and let $(\pi_{n})_{n\in\mathbb{N}}$ be a sequence of couplings of vanishing noise cost from $(\mu_{n})$ to $\mu$ at level 1 (Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §couplings). Put $\pi'_{n}=(p\times p)_{\#}\pi_{n}$. By the preliminaries, $\pi'_{n}\in\Pi^{a,2}(p_{\#}\mu_{n},p_{\#}\mu)$ and $0\le I^{a,2}(\pi'_{n})\le I^{a,1}(\pi_{n})$, and $p_{\#}\mu_{n},p_{\#}\mu\in\mathcal{Q}_{2}$. Given $\varepsilon>0$, choose $n_{0}$ with $|I^{a,1}(\pi_{n})-0|<\varepsilon$ for $n\ge n_{0}$ (Limit of a Sequence of Real Numbers), so that $I^{a,1}(\pi_{n})<\varepsilon$, as $I^{a,1}(\pi_{n})\ge0$ by Couplings of Finite Noise Cost and Their Noise Cost §cost; then $|I^{a,2}(\pi'_{n})-0|<\varepsilon$ for $n\ge n_{0}$. So $(\pi'_{n})$ is a sequence of couplings of vanishing noise cost from $(p_{\#}\mu_{n})$ to $p_{\#}\mu$ at level 2. Let $\eta_{n}=\nabla_{2}\varphi(p_{\#}\mu_{n})$ and $\eta=\nabla_{2}\varphi(p_{\#}\mu)$. By Noise Intrinsic Test Functions on the Noise Wasserstein Space §gradient-continuity at level 2 and Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §strong, the discrepancies $\int|\eta_{n}(x')-\eta(y')|_{a,2}^{2}\,\pi'_{n}(dw)$, the integration variable being $w=(x',y')\in X_{2}\times X_{2}$, tend to $0$. By property (b), $\nabla_{1}\Phi(\mu_{n})=j\circ\eta_{n}\circ p$ and $\nabla_{1}\Phi(\mu)=j\circ\eta\circ p$. For representatives and every $z=(x,y)\in X_{1}\times X_{1}$, 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 §linear and 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 §noise give

$$\bigl|j(\eta_{n}(p(x)))-j(\eta(p(y)))\bigr|_{a,1}^{2}=\bigl|j\bigl(\eta_{n}(p(x))-\eta(p(y))\bigr)\bigr|_{a,1}^{2}=\bigl|\eta_{n}(p(x))-\eta(p(y))\bigr|_{a,2}^{2},$$

so the integrand of the level-1 discrepancy along $\pi_{n}$ is the integrand of the level-2 discrepancy along $\pi'_{n}$ composed with $p\times p$. Both integrands are Borel and nonnegative by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy, at the respective levels, so claim 2 of Image Measures, Measures with Densities, and Change of Variables shows that the two discrepancies are equal for every $n$. Hence the level-1 discrepancies tend to $0$, that is, $(\nabla_{1}\Phi(\mu_{n}))$ converges strongly to $\nabla_{1}\Phi(\mu)$ along $(\pi_{n})$ (Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost §strong). Therefore $\Phi$ is a noise intrinsic test function on $\mathcal{Q}_{1}$ at level 1.

Claim (distance). Let $\varphi:\mathcal{P}^{a}_{\rho_{2}}\to\mathbb{R}$ be $\varphi(\sigma)=W_{a,2}(\sigma,\nu_{0})^{2}$, so that $\psi=\varphi\circ p_{\#}$. Since $\mathcal{Q}_{2}$ has the noise map property at level 2, Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §distance, taken at level 2, shows that $\varphi$ is a noise intrinsic test function on $\mathcal{Q}_{2}$ and that $\nabla_{2}\varphi(\sigma)=2(\mathrm{id}_{2}-S_{\sigma})$ for every $\sigma\in\mathcal{Q}_{2}$ and every noise-optimal map $S_{\sigma}$ from $\sigma$ to $\nu_{0}$.

Claim (distance-test). By claim 1, applied to $\varphi$, $\psi$ is a noise intrinsic test function on $\mathcal{Q}_{1}$ at level 1.

Claim (distance-gradient). Let $\mu\in\mathcal{Q}_{1}$ and let $S$ be a noise-optimal map from $p_{\#}\mu$ to $\nu_{0}$ at level 2. Since $p_{\#}\mu\in\mathcal{Q}_{2}$, claim 1 and the formula above give $\nabla_{1}\psi(\mu)=j\circ\bigl(2(\mathrm{id}_{2}-S)\bigr)\circ p$, which equals $2\,j\circ(\mathrm{id}_{2}-S)\circ p$ 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 §fields-linear.

Claim (distance-norm). Let $\mu\in\mathcal{Q}_{1}$, put $\sigma=p_{\#}\mu\in\mathcal{Q}_{2}$, and let $S$ be a noise-optimal map from $\sigma$ to $\nu_{0}$, which exists by The Noise Map Property of a Set of Probability Measures §map-property. By the gradient formula and 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-isometry, $\lVert\nabla_{1}\psi(\mu)\rVert_{\mu,1}=\lVert2(\mathrm{id}_{2}-S)\rVert_{\sigma,2}$. For $x\in X_{2}$, $S(x)-x\in X^{a}_{2}$ by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, and, $X^{a}_{2}$ being a real inner product space under $\langle\cdot,\cdot\rangle_{a,2}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert, the identity $x-S(x)=-(S(x)-x)$ and the homogeneity and symmetry of its inner product (Real Inner Product Space §inner-product) give $|2(x-S(x))|_{a,2}^{2}=4\,|S(x)-x|_{a,2}^{2}=4\,n_{a,2}(S(x)-x)$, with $n_{a,2}$ the function of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel at level 2. The function $x\mapsto n_{a,2}(S(x)-x)$ is $c_{a,2}\circ(\mathrm{id}_{2},S)$, with $c_{a,2}$ the Borel function of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs and $(\mathrm{id}_{2},S)$ Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing, both of its components being Borel: $S$ by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, and $\mathrm{id}_{2}$ because it is continuous, by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. By The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations, Linearity and Monotonicity of the Lebesgue Integral §nonnegative and claim 2 of Image Measures, Measures with Densities, and Change of Variables,

$$\lVert2(\mathrm{id}_{2}-S)\rVert_{\sigma,2}^{2}=\int_{X_{2}}4\,n_{a,2}(S(x)-x)\,\sigma(dx)=4\int_{X_{2}\times X_{2}}c_{a,2}\,d\bigl((\mathrm{id}_{2},S)_{\#}\sigma\bigr)=4\,I^{a,2}\bigl((\mathrm{id}_{2},S)_{\#}\sigma\bigr)=4\,W_{a,2}(\sigma,\nu_{0})^{2};$$

here $(\mathrm{id}_{2},S)_{\#}\sigma\in\Pi^{a,2}(\sigma,\nu_{0})$ and is noise-optimal by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, so the third equality is Couplings of Finite Noise Cost and Their Noise Cost §cost and the fourth is Noise-Optimal Couplings §optimal. Thus $\lVert\nabla_{1}\psi(\mu)\rVert_{\mu,1}^{2}=(2\,W_{a,2}(p_{\#}\mu,\nu_{0}))^{2}$, and both numbers being nonnegative, claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert\nabla_{1}\psi(\mu)\rVert_{\mu,1}=2\,W_{a,2}(p_{\#}\mu,\nu_{0})$.