The Density Cost Along Optimal Maps is Controlled by the Monotonicity of the Score

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, with the space $L^{2}(\mu;\mathbb{R}^{d})$ and its inner product $\langle\cdot,\cdot\rangle_{\mu}$, let finite Fisher information, the set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and the score $\xi_{\mu}$ be those of that definition, and let convex Lipschitz integrands, the set $\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$ and the density cost $\mathcal{G}_{\Phi}$ be those of that definition. Let $L\in\mathbb{R}$ be nonnegative and let $\Phi$ be a convex Lipschitz integrand with constant $L$. Let $\mu,\nu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ be absolutely continuous, so that $\mu,\nu\in\mathcal{P}_{2}^{\mathrm{ac}}(\mathbb{R}^{d})$, let $T$ be an optimal map from $\mu$ to $\nu$ and $T'$ an optimal map from $\nu$ to $\mu$, and let $\mathrm{id}-T\in L^{2}(\mu;\mathbb{R}^{d})$ and $\mathrm{id}-T'\in L^{2}(\nu;\mathbb{R}^{d})$ be the classes of The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable. The number $d$ is read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background §numbers.

1. (Displacement bound)¶ For every positive $A\in\mathbb{R}$,