A $\lambda$-Displacement Convex Penalty Pair Has a $\lambda$-Monotone Score Along Optimal Couplings

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $\lambda\in\mathbb{R}$, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ that is $\lambda$-displacement convex, and let $\mu,\nu\in\mathcal{D}_{\Sigma}$, whose scores $\Sigma(\mu)$ and $\Sigma(\nu)$ lie in $L^{2}(\mu;\mathbb{R}^{d})$ and $L^{2}(\nu;\mathbb{R}^{d})$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i). Displacement pairings $\mathcal{J}$ are those of The Intrinsic Calculus on the Wasserstein Space: Standing Notation (i), and optimal couplings those of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i). For $\pi\in\Pi(\mu,\nu)$, $\pi^{\top}\in\Pi(\nu,\mu)$ denotes its image under the swap, which belongs to $\Pi(\nu,\mu)$ by that clause.

1. ($\lambda$-monotonicity along an optimal coupling)(i) For every optimal coupling $\pi\in\Pi(\mu,\nu)$,

$$\lambda\,W_{2}(\mu,\nu)^{2}\ \le\ -\mathcal{J}(\Sigma(\mu),\pi)-\mathcal{J}(\Sigma(\nu),\pi^{\top}).$$

2. ($\lambda$-monotonicity at uniquely mapped pairs)(ii) Suppose that both ordered pairs $(\mu,\nu)$ and $(\nu,\mu)$ are uniquely mapped, let $S$ be an optimal map from $\mu$ to $\nu$ and $S'$ an optimal map from $\nu$ to $\mu$, with classes $\mathrm{id}-S\in L^{2}(\mu;\mathbb{R}^{d})$ and $\mathrm{id}-S'\in L^{2}(\nu;\mathbb{R}^{d})$ as in The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (i). Then

$$\lambda\,W_{2}(\mu,\nu)^{2}\ \le\ \langle\Sigma(\mu),\mathrm{id}-S\rangle_{\mu}+\langle\Sigma(\nu),\mathrm{id}-S'\rangle_{\nu}.$$