A Toolkit for Penalised Comparison on the Noise Wasserstein Space: Constant Test Functions and Linear Combinations of Test Functions, the Identity as a Unique Noise-Optimal Map, and Discrepancies Along the Push-Forward Under (id, id) and Along Glued Couplings

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $\mathrm{id}$ be the identity map of $X$ (First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §maps). For $\nu\in\mathcal{P}(X)$, the zero element $0_{\nu}$ of the real Hilbert space $L^{2}(\nu;X^{a})$ of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields is the class of the constant map with value $0_{X}$, by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert, $0_{X}$ being the zero vector of $X^{a}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert; the map $(\mathrm{id},\mathrm{id}):X\to X\times X$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing, and $(\mathrm{id},\mathrm{id})_{\#}\nu$ is its push-forward of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward. For $\nu,\mu\in\mathcal{P}(X)$, $\pi\in\Pi(\nu,\mu)$, $q\in L^{2}(\nu;X^{a})$ and $\eta\in L^{2}(\mu;X^{a})$, the discrepancy $\int_{X\times X}|q(x)-\eta(y)|_{a}^{2}\,\pi(dz)$ is the nonnegative real number of that clause, and square roots are the nonnegative square roots of Existence and Uniqueness of the Nonnegative Square Root. Noise intrinsic test functions, noise-optimal maps with their displacements, noise-optimal couplings and uniquely noise-mapped pairs are those of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §gradients and First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §maps. In claim 5, $X_{(3)}$, the maps $q_{1},q_{2},q_{3}$ on it, gluings and the composite coupling are those of Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs; the letters $q_{1},q_{2},q_{3}$ always denote these maps, and the letter $q$ without index a noise field. Then the following hold.

1. (Constant test functions)¶ Let $Q\subseteq\mathcal{P}^{a}_{\rho}$, let $c\in\mathbb{R}$, and let $\varphi_{c}:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ be the constant function with value $c$. Then $\varphi_{c}$ is a noise intrinsic test function on $Q$, and $\nabla\varphi_{c}(\mu)=0_{\mu}$ for every $\mu\in Q$.

2. (Linear combinations)¶ Let $Q\subseteq\mathcal{P}^{a}_{\rho}$, let $\varphi,\psi:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ be noise intrinsic test functions on $Q$, let $s,t\in\mathbb{R}$, and let $s\varphi+t\psi:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ be the function with value $s\varphi(\nu)+t\psi(\nu)$ at $\nu$. Then $s\varphi+t\psi$ is a noise intrinsic test function on $Q$, and

$$\nabla(s\varphi+t\psi)(\mu)=s\,\nabla\varphi(\mu)+t\,\nabla\psi(\mu)\qquad\text{for every }\mu\in Q,$$

the right-hand side being formed in the real vector space $L^{2}(\mu;X^{a})$.

3. (The identity)¶ Let $\mu\in\mathcal{P}^{a}_{\rho}$. Then $\mathrm{id}$ is a noise-optimal map from $\mu$ to $\mu$, and its displacement $\mathrm{id}-\mathrm{id}$ is $0_{\mu}$; the coupling $(\mathrm{id},\mathrm{id})_{\#}\mu$ is the only noise-optimal coupling of $\mu$ and $\mu$; consequently the ordered pair $(\mu,\mu)$ is uniquely noise-mapped; and $W_{a}(\mu,\mu)=0$.

4. (Discrepancies along $(\mathrm{id},\mathrm{id})_{\#}\nu$)¶ Let $\nu\in\mathcal{P}(X)$, let $q,\eta\in L^{2}(\nu;X^{a})$, and let $\Delta=(\mathrm{id},\mathrm{id})_{\#}\nu$. Then $\Delta\in\Pi^{a}(\nu,\nu)\subseteq\Pi(\nu,\nu)$, $I^{a}(\Delta)=0$, and

$$\int_{X\times X}|q(x)-\eta(y)|_{a}^{2}\,\Delta(dz)=\lVert q-\eta\rVert_{\nu}^{2}.$$

5. (Discrepancies along a glued composite coupling)¶ Let $\nu,\lambda,\mu\in\mathcal{P}_{2}(X)$, the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space; let $\pi_{12}\in\Pi(\nu,\lambda)$ and $\pi_{23}\in\Pi(\lambda,\mu)$, let $\sigma$ be a gluing of $\pi_{12}$ and $\pi_{23}$, and let $\pi_{13}=(q_{1},q_{3})_{\#}\sigma$, which belongs to $\Pi(\nu,\mu)$ by Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §composite, both applied with $\nu,\lambda,\mu$ in place of its $\mu,\lambda,\nu$. Let $q\in L^{2}(\nu;X^{a})$, $\eta\in L^{2}(\lambda;X^{a})$ and $\theta\in L^{2}(\mu;X^{a})$. Then

$$\sqrt{\int_{X\times X}|q(x)-\theta(y)|_{a}^{2}\,\pi_{13}(dz)}\ \le\ \sqrt{\int_{X\times X}|q(x)-\eta(y)|_{a}^{2}\,\pi_{12}(dz)}+\sqrt{\int_{X\times X}|\eta(x)-\theta(y)|_{a}^{2}\,\pi_{23}(dz)} .$$