The Gibbs Entropy Pair of a K-Semiconvex Potential is Lambda-Displacement Convex with Lambda the Temperature over the Variance-to-Noise Bound Minus K

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 $K\in\mathbb{R}$ be nonnegative and let $V$ be an admissible cylindrical potential with semiconvexity constant $K$, 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 Gibbs entropy pair with potential $V$ and temperature $\beta$, whose hypothesis holds with this $\kappa$; it is a noise penalty pair by The Gibbs 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. ($\lambda$-displacement convexity)¶ The pair is $\bigl(\frac{\beta}{\kappa}-K\bigr)$-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,

$$\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)+\frac{1}{2}\Bigl(\frac{\beta}{\kappa}-K\Bigr)I^{a}(\pi)\le\mathcal{E}(\nu).$$

2. (Displacement convexity)¶ If $K\le\beta/\kappa$, the pair is displacement convex.