The Centred Heat Gauge: Metric Properties, Comparison with the Wasserstein Distance, Behaviour Under Translations, the Squared Gauge as a Test Function, and the Polarisation Inequality

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\rho$ be the centred heat gauge on $\mathcal{P}_{2}(\mathbb{R}^{d})$, $m(\mu)$ the mean and $\bar{\mu}$ the centred measure of $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and $\varrho$ the heat gauge; the multiscale Gaussian kernel $K$, the potentials $K*\mu$ and the heat gauge form $Q$ are read in dimension $q=d$, so that $\varrho(\mu,\nu)^{2}=Q(\mu,\nu)$, and $C_{Q}$ and $N_{0},\dots,N_{3}$ are nonnegative real numbers as in 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 (vi) and 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 (vii). The translations $\tau_{a}$, push-forwards (Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv)), test functions on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with their intrinsic gradients and translation Hessians, the class in $L^{2}(\mu;\mathbb{R}^{d})$ of a Borel map $\xi$ with $\int\lVert\xi\rVert^{2}d\mu<\infty$ (again written $\xi$), the gradient $D\psi(x)$ of a function of class $C^{1}$ on $\mathbb{R}^{d}$ (Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii)) and the matrix $0_{d}$ are as in The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions; $I_{d}$ is the identity matrix. For $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ put

$$\psi_{\mu,\nu}=K*\bar{\mu}-K*\bar{\nu},$$

a function of class $C^{3}$ on $\mathbb{R}^{d}$ 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 (vii), and let $E_{\mu,\nu}\in\mathbb{R}^{d}$ be the point with $i$th coordinate $\int_{\mathbb{R}^{d}}\partial_{i}\psi_{\mu,\nu}\,d\bar{\mu}$, each $\partial_{i}\psi_{\mu,\nu}$ being bounded and Borel. Then the following hold.

1. (Metric and comparison with the Wasserstein distance)(i) $\rho$ is a metric on $\mathcal{P}_{2}(\mathbb{R}^{d})$, and for all $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$,

$$\lVert m(\mu)-m(\nu)\rVert\le\rho(\mu,\nu),\qquad\varrho(\bar{\mu},\bar{\nu})\le\rho(\mu,\nu),\qquad\rho(\mu,\nu)\le C_{\rho}\,W_{2}(\mu,\nu),$$

where $C_{\rho}=\sqrt{1+C_{Q}}$.

2. (Translations)(ii) For all $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $a,b\in\mathbb{R}^{d}$,

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

3. (The potential of the pair)(iii) For all $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $x\in\mathbb{R}^{d}$ and $i,j,l\in[d]$,

$$|\psi_{\mu,\nu}(x)|\le N_{0}\,\rho(\mu,\nu),\qquad|\partial_{i}\psi_{\mu,\nu}(x)|\le N_{1}\,\rho(\mu,\nu),\qquad|\partial_{j}\partial_{i}\psi_{\mu,\nu}(x)|\le N_{2}\,\rho(\mu,\nu),\qquad|\partial_{l}\partial_{j}\partial_{i}\psi_{\mu,\nu}(x)|\le N_{3}\,\rho(\mu,\nu),$$

and $\lVert E_{\mu,\nu}\rVert\le\sqrt{d}\,N_{1}\,\rho(\mu,\nu)$; moreover $\psi_{\nu,\mu}=-\psi_{\mu,\nu}$, the point with coordinates $\int\partial_{i}\psi_{\mu,\nu}\,d\bar{\nu}$ equals $E_{\mu,\nu}$, and $E_{\nu,\mu}=-E_{\mu,\nu}$.

4. (The squared gauge as a test function)(iv) Assume that $(\Omega,\mathcal{F},P)$ is rich, let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and let $\Theta_{\nu},\Xi_{\nu}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be given by $\Theta_{\nu}(\sigma)=\varrho(\bar{\sigma},\bar{\nu})^{2}$ and $\Xi_{\nu}(\sigma)=\rho(\sigma,\nu)^{2}$. Then $\Theta_{\nu}$ is a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with $\Theta_{\nu}((\tau_{a})_{\#}\sigma)=\Theta_{\nu}(\sigma)$ for all $\sigma$ and $a$, and at every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ its translation Hessian is $0_{d}$ and its intrinsic gradient is the class of

$$x\mapsto2\bigl(D\psi_{\mu,\nu}(x-m(\mu))-E_{\mu,\nu}\bigr).$$

Likewise $\Xi_{\nu}$ is a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, and at every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ its translation Hessian is $2I_{d}$ and its intrinsic gradient is the class of

$$x\mapsto2\bigl(m(\mu)-m(\nu)-E_{\mu,\nu}+D\psi_{\mu,\nu}(x-m(\mu))\bigr).$$

Since $\rho$ is symmetric by claim 1, the same holds for $\sigma\mapsto\rho(\mu,\sigma)^{2}$ with $\mu$ fixed, with the roles of $\mu$ and $\nu$ interchanged.

5. (Polarisation inequality)(v) Let $\mu,\nu,\hat{\mu},\hat{\nu}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and put $\psi=\psi_{\hat{\mu},\hat{\nu}}$. Then

$$\varrho(\bar{\mu},\bar{\nu})^{2}\le\Bigl(2\varrho(\bar{\mu},\bar{\hat{\mu}})^{2}+2\int\psi\,d\bar{\mu}-2\int\psi\,d\bar{\hat{\mu}}\Bigr)+\Bigl(2\varrho(\bar{\nu},\bar{\hat{\nu}})^{2}-2\int\psi\,d\bar{\nu}+2\int\psi\,d\bar{\hat{\nu}}\Bigr)+\varrho(\bar{\hat{\mu}},\bar{\hat{\nu}})^{2},$$

with equality when $\mu=\hat{\mu}$ and $\nu=\hat{\nu}$; here $\psi$ is bounded and Borel, so each integral is a real number.