The Centred Heat Gauge on the Wasserstein Space

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $m(\mu)$ be the mean and $\bar{\mu}$ the centred measure of $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, an element of $\mathcal{P}_{2}(\mathbb{R}^{d})$ by that clause, and let $\varrho$ be the heat gauge on $\mathcal{P}(\mathbb{R}^{d})$, read in dimension $q=d$; $\sqrt{t}$ is the nonnegative square root of a nonnegative real number $t$.

(The centred heat gauge)(i) The centred heat gauge on $\mathcal{P}_{2}(\mathbb{R}^{d})$ is the function

$$\rho:\mathcal{P}_{2}(\mathbb{R}^{d})\times\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R},\qquad \rho(\mu,\nu)=\sqrt{\lVert m(\mu)-m(\nu)\rVert^{2}+\varrho(\bar{\mu},\bar{\nu})^{2}}\ ,$$

the radicand being a nonnegative real number.