Noise-Optimal Couplings

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, the sets $\Pi^{a}(\mu,\nu)$ of couplings of finite noise cost, the noise cost $I^{a}$ and noise-connectedness are those of Couplings of Finite Noise Cost and Their Noise Cost, and $W_{a}$ is the noise Wasserstein distance.

1. (Noise-optimal couplings)¶ Let $\mu,\nu\in\mathcal{P}(X)$ be such that $(\mu,\nu)$ is noise-connected, so that $W_{a}(\mu,\nu)$ is defined by The Noise Wasserstein Distance §distance. A coupling $\pi\in\Pi^{a}(\mu,\nu)$ is noise-optimal if $I^{a}(\pi)=W_{a}(\mu,\nu)^{2}$.