Noise-Optimal Maps out of the Entropy Domain of a Diagonal Gaussian Reference Measure

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $c$ be a variance sequence and suppose that the reference measure is $\rho=\gamma_{c}$, the diagonal Gaussian measure on $X$ with variances $c$, which is admissible since $\gamma_{c}\in\mathcal{P}_{2}(X)$ by Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §moment. Let $\kappa$ be a positive real number with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, where $a$ is the sequence of noise weights. Let $\mathcal{P}^{a}_{\rho}$ be the set of The Measures Noise-Connected to the Reference Measure §space, and let uniquely noise-mapped pairs be those of Noise-Optimal Maps and Uniquely Noise-Mapped Pairs. Let $\mu\in\mathcal{P}(X)$ have finite relative entropy with respect to $\gamma_{c}$.

1. (Noise-connected)¶ $\mu\in\mathcal{P}^{a}_{\rho}$.

2. (Noise-optimal maps out of the entropy domain)¶ For every $\nu\in\mathcal{P}^{a}_{\rho}$ the ordered pair $(\mu,\nu)$, whose first member lies in $\mathcal{P}^{a}_{\rho}$ by claim 1, is uniquely noise-mapped.