The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, intrinsic test functions on a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$, their gradients along couplings and their translation Hessians are those of that definition, and $m(\mu)$ is the mean of $\mu$. Being of class $C^{2}$ on $\mathbb{R}^{d}$, the gradient $D\phi(a)$ and the Hessian matrix $D^{2}\phi(a)$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii), $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space Is Open in Itself, and $I_{d}$ is the identity matrix of Euclidean Space and Real Matrices: Standing Notation (ii). For $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $a\in\mathbb{R}^{d}$, the class in $L^{2}(\mu;\mathbb{R}^{d})$ of the constant map with value $a$ is again written $a$, as in Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields.

1. (The squared distance to a fixed measure)(i) Let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ have the map property, let $\nu_{0}\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and let $\psi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be the function $\psi(\mu)=W_{2}(\mu,\nu_{0})^{2}$. Then $\psi$ is an intrinsic test function on $Q$. For $\mu\in Q$ and any optimal map $S_{\mu}$ from $\mu$ to $\nu_{0}$,

$$\nabla\psi(\mu)=2\,(\mathrm{id}-S_{\mu}),$$

and $H_{\psi}(\mu)=2I_{d}$ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

2. (Functions of the mean)(ii) Let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\phi:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{d}$, and let $\chi:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be the function $\chi(\mu)=\phi(m(\mu))$. Then $\chi$ is an intrinsic test function on $Q$, with

$$\nabla\chi(\mu)=D\phi\bigl(m(\mu)\bigr)\quad(\mu\in Q),\qquad H_{\chi}(\mu)=D^{2}\phi\bigl(m(\mu)\bigr)\quad\bigl(\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})\bigr).$$