The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation. Let $V$ be a confining potential on $\mathbb{R}^{d}$, let $\sigma\in\mathbb{R}$ be positive, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Langevin free-energy pair with potential $V$ and noise intensity $\sigma$, a penalty pair by The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §pair. Closed score along couplings, regular penalised maxima and displacement convexity are those of the definitions cited. The letter $\sigma$ denotes the noise intensity; the swap map written $\sigma$ in Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §swap is not used.

1. (Closed score)¶ The pair has closed score along couplings.

2. (Regular penalised maxima)¶ The pair has regular penalised maxima.

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