Brenier's Theorem in the Noise Norm: Noise-Optimal Couplings out of a Measure with a Density Relative to a Diagonal Gaussian Measure are Induced by a Unique Map

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $c$ be a variance sequence with diagonal Gaussian measure $\gamma_{c}$, and 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 and noise-optimal couplings of $\mu$ and $\nu$ are defined. Noise-optimal maps and uniquely noise-mapped pairs are those of Noise-Optimal Maps and Uniquely Noise-Mapped Pairs, and $\mathrm{id}$ is the identity map of $X$. Suppose that $\mu$ has a density with respect to $\gamma_{c}$.

1. (Optimal couplings are induced by maps)¶ For every noise-optimal coupling $\pi$ of $\mu$ and $\nu$ there is a noise-optimal map $T$ from $\mu$ to $\nu$ with $\pi=(\mathrm{id},T)_{\#}\mu$.

2. (Uniqueness)¶ The ordered pair $(\mu,\nu)$ is uniquely noise-mapped.