Noise-Optimal Maps and Uniquely Noise-Mapped Pairs

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mu,\nu$ belong to the set $\mathcal{P}^{a}_{\rho}$ of The Measures Noise-Connected to the Reference Measure §space, and let $\Pi^{a}(\mu,\nu)$ be the set of couplings of finite noise cost of $\mu$ and $\nu$. The ordered pair $(\mu,\nu)$ is noise-connected 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, so that noise-optimal couplings of $\mu$ and $\nu$, those of Noise-Optimal Couplings §optimal, are defined. Let $\mathrm{id}$ be the identity map of $X$ and let $n_{a}:X\to\mathbb{R}$ be the Borel function of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel. For a Borel map $T:X\to X$ the map $(\mathrm{id},T)$ 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; hence, by claims 3 and 4 of Borel Measurability and Bounded Integration on a Metric Space, the map $x\mapsto T(x)-x$ from $X$ to $X$, the composite of $(\mathrm{id},T)$ with the continuous map $z\mapsto y-x$, is Borel, and the function $x\mapsto n_{a}(T(x)-x)$, the composite of $(\mathrm{id},T)$ with the Borel function $c_{a}$ of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs, is Borel and nonnegative, so that its integral against $\mu$ is defined in $[0,\infty]$.

1. (Noise-optimal map)¶ A Borel map $T:X\to X$ is a noise-optimal map from $\mu$ to $\nu$ if $T_{\#}\mu=\nu$, $T(x)-x\in X^{a}$ for every $x\in X$,

$$\int_{X}n_{a}\bigl(T(x)-x\bigr)\,\mu(dx)<\infty,$$

and the coupling $(\mathrm{id},T)_{\#}\mu$, which belongs to $\Pi^{a}(\mu,\nu)$ by Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §displacement, is noise-optimal. For such $T$ the map $T-\mathrm{id}:X\to X^{a}$, $x\mapsto T(x)-x$, is measurable by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable and square-integrable with respect to $\mu$ in the sense of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §space, since $|T(x)-x|_{a}^{2}=n_{a}(T(x)-x)$ for every $x\in X$; its class in $L^{2}(\mu;X^{a})$ is again written $T-\mathrm{id}$.

2. (Uniquely noise-mapped pair)¶ The ordered pair $(\mu,\nu)$ is uniquely noise-mapped if there is a noise-optimal map $T$ from $\mu$ to $\nu$ such that every noise-optimal coupling of $\mu$ and $\nu$ equals $(\mathrm{id},T)_{\#}\mu$.