The Squared Wasserstein Distance to a Fixed Measure: a One-Sided Bound Along Couplings, Differentiability at a Uniquely Mapped Source, and the Translation and Centring Identities

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The mean $m(\rho)\in\mathbb{R}^{d}$ and the centred measure $\bar{\rho}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ of $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ are those of those clauses, and $(\tau_{a})_{\#}\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ for $a\in\mathbb{R}^{d}$ by The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class (i), the translations $\tau_{a}$ being those fixed there. Then the following hold.

1. (A one-sided bound along couplings)(i) Let $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be an optimal map from $\mu$ to $\nu$, with $\mathrm{id}-T\in L^{2}(\mu;\mathbb{R}^{d})$. Then for every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and every $\pi\in\Pi(\mu,\rho)$,

$$W_{2}(\rho,\nu)^{2}\ \le\ W_{2}(\mu,\nu)^{2}+2\,\mathcal{J}(\mathrm{id}-T,\pi)+I(\pi).$$

2. (Differentiability at a uniquely mapped source)(ii) Suppose that the ordered pair $(\mu,\nu)$ is uniquely mapped, and let $T$ be an optimal map from $\mu$ to $\nu$. Then the function

$$\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R},\qquad\rho\mapsto W_{2}(\rho,\nu)^{2},$$

is differentiable along couplings at $\mu$, with gradient along couplings $2\,(\mathrm{id}-T)$.

3. (Translations)(iii) For all $a,b\in\mathbb{R}^{d}$,

$$W_{2}\bigl((\tau_{a})_{\#}\mu,(\tau_{b})_{\#}\nu\bigr)^{2}=W_{2}(\mu,\nu)^{2}+2\,(a-b)\cdot\bigl(m(\mu)-m(\nu)\bigr)+\lVert a-b\rVert^{2}.$$

4. (Centring)(iv)

$$W_{2}(\mu,\nu)^{2}=W_{2}(\bar{\mu},\bar{\nu})^{2}+\lVert m(\mu)-m(\nu)\rVert^{2}.$$