Tangency of Noise-Optimal Displacements out of a Measure with a Density Relative to a Diagonal Gaussian Measure, and the Noise Map Property

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 $\gamma_{c}$, let $\mathcal{P}^{a}_{\rho}$ be the set of The Measures Noise-Connected to the Reference Measure §space, and let $\mathrm{id}$ be the identity map of $X$. Noise-optimal maps $T$ and their displacements $T-\mathrm{id}\in L^{2}(\mu;X^{a})$, for $\mu\in\mathcal{P}^{a}_{\rho}$, are those of Noise-Optimal Maps and Uniquely Noise-Mapped Pairs, for $\mu\in\mathcal{P}^{a}_{\rho}$ the set $T^{a}_{\mu}\subseteq L^{2}(\mu;X^{a})$ is the noise tangent space at $\mu$, and the noise map property is that of The Noise Map Property of a Set of Probability Measures. Let $\mathcal{Q}_{c}$ be the set of the $\mu\in\mathcal{P}^{a}_{\rho}$ that have a density with respect to $\gamma_{c}$.

1. (Tangency)¶ Let $\mu\in\mathcal{Q}_{c}$ and $\nu\in\mathcal{P}^{a}_{\rho}$, and let $T$ be a noise-optimal map from $\mu$ to $\nu$. Then $T-\mathrm{id}\in T^{a}_{\mu}$.

2. (The noise map property)¶ The set $\mathcal{Q}_{c}$ has the noise map property.