Talagrand's Inequality in the Noise Norm for a Diagonal Gaussian Reference Measure on a Hilbert Space

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 $\mu\in\mathcal{P}(X)$ have finite relative entropy $H(\mu\,|\,\gamma_{c})$ with respect to $\gamma_{c}$. Let $\mathcal{P}^{a}_{\rho}$ be the set of measures noise-connected to $\rho$ and $W_{a}$ the noise Wasserstein distance.

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

2. (Talagrand's inequality)¶ The distance $W_{a}(\mu,\gamma_{c})$, defined by claim 1, satisfies

$$W_{a}(\mu,\gamma_{c})^{2}\le2\kappa\,H(\mu\,|\,\gamma_{c}).$$