The Noise-Optimal Map: Transport Cost, Stability Along Nearly Optimal Couplings and Stability Under Perturbation of the Source

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, so that 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, with the noise Wasserstein distance $W_{a}$. Let $\Pi^{a}(\mu,\nu)$ be the set of couplings of finite noise cost of $\mu$ and $\nu$, and let $I^{a}$ be the noise cost, so that $I^{a}(\pi)=\int_{X\times X}c_{a}\,d\pi$ with $c_{a}$ the Borel function of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs. Noise-optimal maps $T$ from $\mu$ to $\nu$, their displacements $T-\mathrm{id}\in L^{2}(\mu;X^{a})$, and uniquely noise-mapped pairs are those of that definition; $n_{a}$ is the Borel function of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §borel; sequences and limits of sequences of real numbers are those of those definitions. Then the following hold.

1. (Transport cost)¶ If $T$ is a noise-optimal map from $\mu$ to $\nu$, then

$$\lVert T-\mathrm{id}\rVert_{\mu}^{2}=W_{a}(\mu,\nu)^{2}.$$

2. (Stability along nearly optimal couplings)¶ Suppose that the ordered pair $(\mu,\nu)$ is uniquely noise-mapped, let $T$ be a noise-optimal map from $\mu$ to $\nu$, and let $(\gamma_{n})_{n\in\mathbb{N}}$ be a sequence in $\Pi^{a}(\mu,\nu)$ such that the sequence $(I^{a}(\gamma_{n}))_{n\in\mathbb{N}}$ converges to $W_{a}(\mu,\nu)^{2}$. For every $n\in\mathbb{N}$, $y-x\in X^{a}$ for $\gamma_{n}$-almost every $z$ by Couplings of Finite Noise Cost and Their Noise Cost §finite, and $T(x)-x\in X^{a}$ for every $z$, so $T(x)-y\in X^{a}$ for $\gamma_{n}$-almost every $z$, $X^{a}$ being a linear subspace of $X$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert. The function $z\mapsto n_{a}(T(x)-y)$ on $X\times X$ is Borel and nonnegative by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §pairing and claims 3 and 4 of Borel Measurability and Bounded Integration on a Metric Space, and it is $\gamma_{n}$-integrable: for $\gamma_{n}$-almost every $z$ it is dominated by $2n_{a}(T(x)-x)+2c_{a}(z)$, by the inequality $|u-v|_{a}^{2}\le2|u|_{a}^{2}+2|v|_{a}^{2}$ in the real Hilbert space $X^{a}$ of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert, and the integral of this bound against $\gamma_{n}$ is $2\int_{X}n_{a}(T(x)-x)\,\mu(dx)+2I^{a}(\gamma_{n})<\infty$ by the change-of-variables formula with $(\pi_{1})_{\#}\gamma_{n}=\mu$, so that The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison applies. Then

$$\lim_{n\to\infty}\int_{X\times X}n_{a}\bigl(T(x)-y\bigr)\,\gamma_{n}(dz)=0.$$

3. (Stability under perturbation of the source)¶ Let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}^{a}_{\rho}$ and let $(\gamma_{n})_{n\in\mathbb{N}}$ be a sequence of couplings of vanishing noise cost from $(\mu_{n})_{n\in\mathbb{N}}$ to $\mu$. Suppose that the ordered pair $(\mu,\nu)$ and every ordered pair $(\mu_{n},\nu)$ are uniquely noise-mapped, and let $S$ be a noise-optimal map from $\mu$ to $\nu$ and, for every $n\in\mathbb{N}$, $S_{n}$ a noise-optimal map from $\mu_{n}$ to $\nu$. Then the sequence $(S_{n}-\mathrm{id})_{n\in\mathbb{N}}$, with $S_{n}-\mathrm{id}\in L^{2}(\mu_{n};X^{a})$, converges strongly to $S-\mathrm{id}\in L^{2}(\mu;X^{a})$ along $(\gamma_{n})_{n\in\mathbb{N}}$.