The Heat Gauge on the Probability Measures on Euclidean Space

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, with Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation in force, fix a dimension $q$, and let $Q$ be the heat gauge form on $\mathcal{P}(\mathbb{R}^{q})$, built from the multiscale Gaussian kernel; $Q(\mu,\nu)$ is a nonnegative real number by The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential (ii), and $\sqrt{t}$ is the nonnegative square root of a nonnegative real number $t$.

(The heat gauge)(i) The heat gauge on $\mathcal{P}(\mathbb{R}^{q})$ is the function

$$\varrho:\mathcal{P}(\mathbb{R}^{q})\times\mathcal{P}(\mathbb{R}^{q})\to\mathbb{R},\qquad\varrho(\mu,\nu)=\sqrt{Q(\mu,\nu)} .$$