A Displacement Convex Penalty Pair Has a Monotone Score Along Optimal Couplings

In the setting of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ that is displacement convex, let $\mu,\nu\in\mathcal{D}_{\Sigma}$ and let $\pi$ be an optimal coupling of $\mu$ and $\nu$. The coordinate projections $\mathrm{pr}_{1},\mathrm{pr}_{2}$ of $\mathbb{R}^{d+d}$ are those of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation §plans; in this statement we write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$ for the two components of a point $z\in\mathbb{R}^{d+d}$, both of them positions here, so that the letter $p$, reserved there for a velocity, is not used.

(Monotonicity of the score along an optimal coupling)¶ The function

on $\mathbb{R}^{d+d}$ is integrable with respect to $\pi$, and