The Gaussian Entropy Pair is Uniformly Displacement Convex, with Modulus the Temperature over the Variance-to-Noise Bound

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, so that the reference measure is $\rho=\gamma_{c}$, let $\beta$ and $\kappa$ be positive real numbers with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $\beta$, whose hypothesis holds with this $\kappa$; it is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ by The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair. $\lambda$-displacement convexity and displacement convexity are those of Lambda-Displacement Convexity of a Noise Penalty Pair §convex and Lambda-Displacement Convexity of a Noise Penalty Pair §displacement-convex.

1. (Uniform displacement convexity)¶ The pair is $\frac{\beta}{\kappa}$-displacement convex: for every $\mu\in\mathcal{D}_{\Sigma}$, every $\nu\in\mathcal{D}$ and every noise-optimal coupling $\pi\in\Pi^{a}(\mu,\nu)$, with $\mathcal{J}^{a}$ the noise displacement pairing and $I^{a}$ the noise cost,

$$\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)+\frac{\beta}{2\kappa}\,I^{a}(\pi)\le\mathcal{E}(\nu).$$

2. (Displacement convexity)¶ The pair is displacement convex.