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

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 $q$. Let $K$ be the multiscale Gaussian kernel, with the scales $s_{k}$ and weights $w_{k}$ of that lemma, $K*\mu$ the potential of $\mu\in\mathcal{P}(\mathbb{R}^{q})$ and $\mathcal{K}$ the kernel pairing; let $g_{s}*\mu$ be the Gaussian smoothing of $\mu$ at scale $s$, with Lebesgue measure $\lambda_{q}$ and integrable as there; and let series of real numbers and their sums be those of Series of Real Numbers. The pointwise difference $f-g$ of two real functions on $\mathbb{R}^{q}$ is as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (iv). For $\mu,\nu,\mu',\nu'\in\mathcal{P}(\mathbb{R}^{q})$ put

$$B(\mu,\nu;\mu',\nu')=\mathcal{K}(\mu,\mu')-\mathcal{K}(\mu,\nu')-\mathcal{K}(\nu,\mu')+\mathcal{K}(\nu,\nu'),\qquad Q(\mu,\nu)=B(\mu,\nu;\mu,\nu);$$

$Q$ is called the heat gauge form. The set $\mathcal{P}_{2}(\mathbb{R}^{q})$, the couplings $\Pi(\mu,\nu)$ with their quadratic cost $I$, and the quadratic Wasserstein distance $W_{2}$ are those of these definitions, read in dimension $q$; for $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$ and $\pi\in\Pi(\mu,\nu)$ the cost $I(\pi)$ is a nonnegative real number by Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound (iii). Then the following hold, where $\mu,\nu,\sigma,\mu',\nu'$ denote arbitrary elements of $\mathcal{P}(\mathbb{R}^{q})$.

1. (Algebra)(i) $B(\mu,\nu;\mu',\nu')=B(\mu',\nu';\mu,\nu)$, $B(\nu,\mu;\mu',\nu')=-B(\mu,\nu;\mu',\nu')$, and

$$B(\mu,\sigma;\mu',\nu')=B(\mu,\nu;\mu',\nu')+B(\nu,\sigma;\mu',\nu');$$

consequently $Q(\mu,\nu)=Q(\nu,\mu)$, $Q(\mu,\mu)=0$, and

$$Q(\mu,\sigma)=Q(\mu,\nu)+2\,B(\mu,\nu;\nu,\sigma)+Q(\nu,\sigma).$$

2. (Representation and positivity)(ii) For every $k\in\mathbb{N}$ the product $(g_{s_{k}}*\mu-g_{s_{k}}*\nu)(g_{s_{k}}*\mu'-g_{s_{k}}*\nu')$ is integrable, the series below converges absolutely, and

$$B(\mu,\nu;\mu',\nu')=\sum_{k=1}^{\infty}w_{k}\int_{\mathbb{R}^{q}}\bigl(g_{s_{k}}*\mu-g_{s_{k}}*\nu\bigr)\bigl(g_{s_{k}}*\mu'-g_{s_{k}}*\nu'\bigr)\,d\lambda_{q};$$

in particular $Q(\mu,\nu)=\sum_{k=1}^{\infty}w_{k}\int_{\mathbb{R}^{q}}(g_{s_{k}}*\mu-g_{s_{k}}*\nu)^{2}\,d\lambda_{q}$, and $0\le Q(\mu,\nu)$.

3. (Cauchy-Schwarz)(iii) $|B(\mu,\nu;\mu',\nu')|\le\sqrt{Q(\mu,\nu)}\,\sqrt{Q(\mu',\nu')}$, with the nonnegative square roots.

4. (Separation)(iv) $Q(\mu,\nu)=0$ if and only if $\mu=\nu$.

5. (Triangle inequality)(v) $\sqrt{Q(\mu,\sigma)}\le\sqrt{Q(\mu,\nu)}+\sqrt{Q(\nu,\sigma)}$.

6. (Wasserstein bound)(vi) There is a nonnegative real number $C_{Q}$, depending only on $q$, such that for all $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$ and every $\pi\in\Pi(\mu,\nu)$ one has $Q(\mu,\nu)\le C_{Q}\,I(\pi)$, and consequently $\sqrt{Q(\mu,\nu)}\le\sqrt{C_{Q}}\,W_{2}(\mu,\nu)$.

7. (The potential of a difference)(vii) There are nonnegative real numbers $N_{0},N_{1},N_{2},N_{3}$, depending only on $q$, such that for all $\mu,\nu\in\mathcal{P}(\mathbb{R}^{q})$ the function $\psi=K*\mu-K*\nu$, which is of class $C^{3}$ on $\mathbb{R}^{q}$, satisfies

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

for all $x\in\mathbb{R}^{q}$ and $i,j,l\in[q]$. Moreover, for all $\mu,\nu,\mu',\nu'\in\mathcal{P}(\mathbb{R}^{q})$, the function $\psi=K*\mu-K*\nu$ is bounded and Borel and

$$B(\mu,\nu;\mu',\nu')=\int_{\mathbb{R}^{q}}\psi\,d\mu'-\int_{\mathbb{R}^{q}}\psi\,d\nu' .$$

8. (Metric)(viii) The function $(\mu,\nu)\mapsto\sqrt{Q(\mu,\nu)}$ is a metric on $\mathcal{P}(\mathbb{R}^{q})$.