McCann's Tangent Inequality on the Torus, and Monotonicity of the Score Along Optimal Maps

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ and $\mathrm{Ent}$ be as in The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets §finite-entropy-set, and let $\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ and the score $\xi_{\mu}$ be as in Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §finite. Members of $\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ are absolutely continuous by The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets §nonnegative, so optimal maps out of them exist by McCann's Theorem on the Flat Torus: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map with a Periodic Potential §map; the displacement field of a Borel map is a bounded Borel map by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, so its class lies in $L^{2}(\mu;\mathbb{R}^{d})$ for every $\mu\in\mathcal{P}(\mathbb{T}^{d})$.

1. (Tangent inequality)¶ Let $\mu,\nu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ with $\mu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$, and let $T$ be an optimal map from $\mu$ to $\nu$. Then

$$\mathrm{Ent}(\nu)\ge\mathrm{Ent}(\mu)+\langle\xi_{\mu},v_{T}\rangle_{\mu}.$$

2. (Monotone score)¶ Let $\mu,\nu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})\cap\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$, let $T$ be an optimal map from $\mu$ to $\nu$ and $S$ one from $\nu$ to $\mu$. Then

$$\langle\xi_{\mu},v_{T}\rangle_{\mu}+\langle\xi_{\nu},v_{S}\rangle_{\nu}\le0 .$$