The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, the Wasserstein space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, the space $L^{2}(\Omega;\mathbb{R}^{d})$ with the law $\mathcal{L}(X)$ of a class, and the constant classes $c_{a}$ and translations $\tau_{a}$ are as fixed there; push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv); $M_{2}$ is the second moment; points of $\mathbb{R}^{d}$ are read as $d$-tuples by Euclidean Points as Tuples of Real Numbers, the $i$th coordinate of $x$ being $x_{i}$, and the coordinates $X_{i}$ of a random vector $X$ are those of Random Vector and Its Law (ii); for a Borel map $\xi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\int\lVert\xi\rVert^{2}\,d\mu<\infty$ its class in $L^{2}(\mu;\mathbb{R}^{d})$ is again written $\xi$, as in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii). Test functions on $\mathcal{P}_{2}(\mathbb{R}^{d})$, their intrinsic gradients $\nabla\varphi(\mu)$ and translation Hessians $H_{\varphi}(\mu)$ are those of that definition, which requires $(\Omega,\mathcal{F},P)$ to be rich; the gradient $D\phi(x)$ and Hessian matrix $D^{2}\phi(x)$ of a function $\phi$ of class $C^{2}$ on $\mathbb{R}^{d}$ 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 $0_{d}$ denotes the $d\times d$ matrix all of whose entries are $0$, an element of the set $\mathcal{S}(d)$ of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i). Then the following hold.

1. (The mean)(i) Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. For every $i\in[d]$ the coordinate map $x\mapsto x_{i}$ is Borel and integrable with respect to $\mu$; the mean of $\mu$ is the point $m(\mu)\in\mathbb{R}^{d}$ whose $i$th coordinate is $\int_{\mathbb{R}^{d}}x_{i}\,\mu(dx)$. For every $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$ and every representative of $X$, each coordinate $X_{i}$ is integrable with respect to $P$ and $m(\mu)_{i}=\mathbb{E}[X_{i}]$. Moreover $\lVert m(\mu)\rVert^{2}\le M_{2}(\mu)$; $m((\tau_{a})_{\#}\mu)=m(\mu)+a$ for every $a\in\mathbb{R}^{d}$; and $\lVert m(\mu)-m(\nu)\rVert\le W_{2}(\mu,\nu)$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

2. (Centring)(ii) Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. The centred measure $\bar{\mu}=(\tau_{-m(\mu)})_{\#}\mu$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$ and satisfies $m(\bar{\mu})=0_{\mathbb{R}^{d}}$, $M_{2}(\bar{\mu})=M_{2}(\mu)-\lVert m(\mu)\rVert^{2}$, $\mu=(\tau_{m(\mu)})_{\#}\bar{\mu}$, and $\overline{(\tau_{a})_{\#}\mu}=\bar{\mu}$ for every $a\in\mathbb{R}^{d}$; for every $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$ one has $\mathcal{L}(X-c_{m(\mu)})=\bar{\mu}$; and for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$,

$$W_{2}(\bar{\mu},\bar{\nu})^{2}+\lVert m(\mu)-m(\nu)\rVert^{2}\le W_{2}(\mu,\nu)^{2},\qquad\text{in particular}\qquad W_{2}(\bar{\mu},\bar{\nu})\le W_{2}(\mu,\nu).$$

3. (Functions of the mean)(iii) Assume that $(\Omega,\mathcal{F},P)$ is rich, and let $\phi:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{d}$. Then $\mu\mapsto\phi(m(\mu))$ is a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$; at $\mu$ its intrinsic gradient is the class of the constant map $x\mapsto D\phi(m(\mu))$ and its translation Hessian is $D^{2}\phi(m(\mu))$.

4. (Centring of a test function)(iv) Assume that $(\Omega,\mathcal{F},P)$ is rich, and let $\varphi$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$. Then $\varphi^{\circ}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, $\varphi^{\circ}(\mu)=\varphi(\bar{\mu})$, is a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with $\varphi^{\circ}((\tau_{a})_{\#}\mu)=\varphi^{\circ}(\mu)$ for all $\mu$ and $a$, and its translation Hessian is $0_{d}$ at every $\mu$. Its intrinsic gradient at $\mu$ is obtained as follows: for every representative $\eta$ of $\nabla\varphi(\bar{\mu})\in L^{2}(\bar{\mu};\mathbb{R}^{d})$, each coordinate of $\eta$ is integrable with respect to $\bar{\mu}$, the point $\int\eta\,d\bar{\mu}\in\mathbb{R}^{d}$ with coordinates $\int\eta_{i}\,d\bar{\mu}$ and the class in $L^{2}(\mu;\mathbb{R}^{d})$ of the Borel map $x\mapsto\eta(x-m(\mu))$ do not depend on the representative, and

$$\nabla\varphi^{\circ}(\mu)\ \text{is the class of}\ x\mapsto\eta\bigl(x-m(\mu)\bigr)-\int_{\mathbb{R}^{d}}\eta\,d\bar{\mu}.$$