Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mu\in\mathcal{P}(X)$. For $\nu\in\mathcal{P}(X)$, $\Pi(\nu,\mu)$ is the set of couplings of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background, $\Pi^{a}(\mu,\nu)$ is the set of Couplings of Finite Noise Cost and Their Noise Cost §couplings, and for $\pi\in\Pi^{a}(\mu,\nu)$ the noise cost $I^{a}(\pi)=\int_{X\times X}c_{a}\,d\pi$ is a nonnegative real number by Couplings of Finite Noise Cost and Their Noise Cost §cost. $\mathcal{P}^{a}_{\rho}$ is the set of The Measures Noise-Connected to the Reference Measure §space. The set $D_{a}$ and the function $c_{a}$ are those of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §noise-space; the spaces $L^{2}(\mu;X^{a})$, $L^{2}(\nu;X^{a})$ and $L^{2}(\pi;X^{a})$, with inner products $\langle\cdot,\cdot\rangle_{\mu}$, $\langle\cdot,\cdot\rangle_{\nu}$, $\langle\cdot,\cdot\rangle_{\pi}$ and norms $\lVert\cdot\rVert_{\mu}$, $\lVert\cdot\rVert_{\nu}$, $\lVert\cdot\rVert_{\pi}$, and measurability of maps into $X^{a}$ are those of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields; $\mathrm{id}$ is the identity map of $X$; and square roots are those of Existence and Uniqueness of the Nonnegative Square Root. Then the following hold.

1. (The displacement field)¶ Let $\nu\in\mathcal{P}(X)$ and $\pi\in\Pi^{a}(\mu,\nu)$. The map $\delta:X\times X\to X^{a}$ with $\delta(z)=y-x$ for $z\in D_{a}$ and $\delta(z)=0_{X}$ for $z\in(X\times X)\setminus D_{a}$ is measurable into $X^{a}$ and square-integrable with respect to $\pi$, and its class in $L^{2}(\pi;X^{a})$, again written $\delta$, satisfies

$$\lVert\delta\rVert_{\pi}^{2}=I^{a}(\pi).$$

2. (The noise displacement pairing)¶ Let $\nu\in\mathcal{P}(X)$, $\pi\in\Pi^{a}(\mu,\nu)$ and $\eta\in L^{2}(\mu;X^{a})$, and let $\delta$ be the displacement field of claim 1 for $\pi$. For every representative of $\eta$ the function $X\times X\to\mathbb{R}$ with value $\langle\eta(x),\delta(z)\rangle_{a}$ at $z$ is Borel and integrable with respect to $\pi$, and its integral does not depend on the representative chosen. That integral,

$$\mathcal{J}^{a}(\eta,\pi)=\int_{X\times X}\langle\eta(x),\delta(z)\rangle_{a}\,\pi(dz)\in\mathbb{R},$$

whose integrand equals $\langle\eta(x),y-x\rangle_{a}$ at every $z\in D_{a}$, is called the noise displacement pairing of $\eta$ along $\pi$.

3. (Bound)¶ For $\nu\in\mathcal{P}(X)$, $\pi\in\Pi^{a}(\mu,\nu)$ and $\eta\in L^{2}(\mu;X^{a})$,

$$\bigl|\mathcal{J}^{a}(\eta,\pi)\bigr|\le\lVert\eta\rVert_{\mu}\,\sqrt{I^{a}(\pi)} .$$

4. (Linearity in the field)¶ For $\nu\in\mathcal{P}(X)$, $\pi\in\Pi^{a}(\mu,\nu)$, $\eta,\xi\in L^{2}(\mu;X^{a})$ and $s,t\in\mathbb{R}$,

$$\mathcal{J}^{a}(s\,\eta+t\,\xi,\pi)=s\,\mathcal{J}^{a}(\eta,\pi)+t\,\mathcal{J}^{a}(\xi,\pi).$$

5. (Displacement couplings)¶ Let $h\in L^{2}(\mu;X^{a})$, let $h:X\to X^{a}$ also denote any representative of it, and let $t\in\mathbb{R}$; the following conclusions hold for every such representative. Then the map $S_{t}:X\to X$, $S_{t}(x)=x+t\,h(x)$, is Borel, the coupling $\pi_{t}=(\mathrm{id},S_{t})_{\#}\mu$ belongs to $\Pi^{a}(\mu,(S_{t})_{\#}\mu)$, and

$$I^{a}(\pi_{t})=t^{2}\,\lVert h\rVert_{\mu}^{2},\qquad\mathcal{J}^{a}(\eta,\pi_{t})=t\,\langle\eta,h\rangle_{\mu}\quad\text{for every }\eta\in L^{2}(\mu;X^{a}).$$

¶ If moreover $\mu\in\mathcal{P}^{a}_{\rho}$, then $(S_{t})_{\#}\mu\in\mathcal{P}^{a}_{\rho}$.

6. (The cross pairing)¶ Let $\nu\in\mathcal{P}(X)$, $\pi\in\Pi(\nu,\mu)$, $q\in L^{2}(\nu;X^{a})$ and $\eta\in L^{2}(\mu;X^{a})$. For all representatives of $q$ and of $\eta$ the function $X\times X\to\mathbb{R}$ with value $\langle q(x),\eta(y)\rangle_{a}$ at $z$ is Borel and integrable with respect to $\pi$, and its integral does not depend on the representatives chosen. That integral,

$$\mathcal{K}^{a}(q,\eta,\pi)=\int_{X\times X}\langle q(x),\eta(y)\rangle_{a}\,\pi(dz)\in\mathbb{R},$$

is called the cross pairing of $q$ and $\eta$ along $\pi$. ¶ It satisfies $|\mathcal{K}^{a}(q,\eta,\pi)|\le\lVert q\rVert_{\nu}\,\lVert\eta\rVert_{\mu}$. ¶ Moreover, for all representatives the function $X\times X\to\mathbb{R}$ with value $|q(x)-\eta(y)|_{a}^{2}$ at $z$ is Borel, nonnegative and integrable with respect to $\pi$, its integral, called the discrepancy of $q$ and $\eta$ along $\pi$, does not depend on the representatives chosen, and

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

7. (Vanishing)¶ Suppose moreover $\mu\in\mathcal{P}^{a}_{\rho}$, let $\eta\in L^{2}(\mu;X^{a})$, and suppose that for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $\theta\in\mathbb{R}$ with $0<\theta$ such that

$$\bigl|\mathcal{J}^{a}(\eta,\pi)\bigr|\le\varepsilon\,\sqrt{I^{a}(\pi)}$$

for every $\nu\in\mathcal{P}^{a}_{\rho}$ and every $\pi\in\Pi^{a}(\mu,\nu)$ with $I^{a}(\pi)<\theta^{2}$. Then $\eta$ is the zero element of $L^{2}(\mu;X^{a})$.