Out of an Absolutely Continuous Measure the Optimal Map and the Optimal Displacement are Tangent

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and suppose that $\mu$ is absolutely continuous. Let $T_{\mu}$ be the tangent space at $\mu$ and let $\mathrm{id}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the identity map, whose class belongs to $L^{2}(\mu;\mathbb{R}^{d})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (ii).

1. (The pair is uniquely mapped)(i) The ordered pair $(\mu,\nu)$ is uniquely mapped; in particular there is an optimal map from $\mu$ to $\nu$.

2. (Tangency of the optimal map and of the optimal displacement)(ii) Let $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be an optimal map from $\mu$ to $\nu$. Then $\int_{\mathbb{R}^{d}}\lVert T\rVert^{2}\,d\mu=M_{2}(\nu)<\infty$, so the class of $T$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$ and is again written $T$, and

$$T\in T_{\mu},\qquad \mathrm{id}-T\in T_{\mu},$$

where $M_{2}(\nu)$ is the second moment of $\nu$.