Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $s$ be a real number with $0<s\le\tfrac12$, let $\Theta_{s}$ be the torus heat kernel and $S_{s}$ the heat semigroup at time $s$, and let $\lambda_{d}$ be Lebesgue measure. Let $T_{\mu}$ be the tangent space at $\mu\in\mathcal{P}(\mathbb{T}^{d})$, let $\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ and the score $\xi$ be as in Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §finite, and let $\mathcal{J}_{\mathbb{T}}$ be the torus displacement pairing. Let $g_{s}$ be the Gaussian kernel of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails, read with $q=d$, and let $N_{s}\in\mathcal{P}(\mathbb{R}^{d})$ be the measure with density $g_{s}$ with respect to $\lambda_{d}$, which exists by claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder and Image Measures, Measures with Densities, and Change of Variables.

Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and let $F:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Borel with $\int\lVert F\rVert^{2}\,d(S_{s}\mu)<\infty$. Then the following hold.

1. (Heat average)¶ For every $x\in\mathbb{R}^{d}$ the map $y\mapsto\mathbf{1}_{Q}(y)\,\Theta_{s}(y-x)\,F(y)$ is integrable with respect to $\lambda_{d}$, component by component, and the map $a:\mathbb{R}^{d}\to\mathbb{R}^{d}$,

$$a(x)=\int_{Q}F(y)\,\Theta_{s}(y-x)\,\lambda_{d}(dy),$$

has the following properties: $a(x)=\int F\bigl(\pi(x+z)\bigr)\,N_{s}(dz)$ for every $x\in\mathbb{R}^{d}$; each component $a_{j}$ belongs to $C^{1}_{\mathrm{per}}$, with $\partial_{i}a_{j}(x)=-\int_{Q}F_{j}(y)\,\partial_{i}\Theta_{s}(y-x)\,\lambda_{d}(dy)$ for all $x$ and $i,j\in[d]$; $a$ is unchanged when $F$ is replaced by a Borel map that agrees with it $S_{s}\mu$-almost everywhere; and $\int\lVert a\rVert^{2}\,d\mu\le\int\lVert F\rVert^{2}\,d(S_{s}\mu)$.

2. (Tangency)¶ If the class of $F$ in $L^{2}(S_{s}\mu;\mathbb{R}^{d})$ belongs to $T_{S_{s}\mu}$, then the class of $a$ in $L^{2}(\mu;\mathbb{R}^{d})$ belongs to $T_{\mu}$.

3. (Divergence identity)¶ With $S_{s}\mu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ by The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure §density,

$$\int\sum_{i=1}^{d}\partial_{i}a_{i}\,d\mu=-\langle F,\xi_{S_{s}\mu}\rangle_{S_{s}\mu}.$$

4. (Synchronous couplings)¶ Let $\nu\in\mathcal{P}(\mathbb{T}^{d})$ and $\gamma\in\Pi(\mu,\nu)$, let $\gamma\boxtimes N_{s}\in\mathcal{P}(\mathbb{R}^{d+d+d})$ be the product measure, read with $q=d+d$ and $p=d$, and let $\gamma^{s}$ be its image under the map $\tau:\mathbb{R}^{d+d+d}\to\mathbb{R}^{d+d}$ with $\tau(\iota^{d+d,d}(w,z))=\iota^{d,d}\bigl(\pi(\mathrm{pr}_{1}(w)+z),\pi(\mathrm{pr}_{2}(w)+z)\bigr)$ for $w\in\mathbb{R}^{d+d}$ and $z\in\mathbb{R}^{d}$, where $\iota^{q,p}$ are the concatenation maps of that lemma. The map $\tau$ is Borel: it is the pairing of the two maps $\pi\circ(\mathrm{pr}_{j}\circ\mathrm{pr}^{d+d,d}_{1}+\mathrm{pr}^{d+d,d}_{2})$, $j=1,2$, each a composite of the Borel coordinate projections, a sum of Borel maps (componentwise, claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and the Borel wrapping map $\pi$ of The Half-Open Unit Cell Tiles Euclidean Space §wrap. Then $\gamma^{s}\in\Pi(S_{s}\mu,S_{s}\nu)$, $I_{\mathbb{T}}(\gamma^{s})=I_{\mathbb{T}}(\gamma)$, and $\mathcal{J}_{\mathbb{T}}(F,\gamma^{s})=\mathcal{J}_{\mathbb{T}}(a,\gamma)$, the pairings being taken with the classes of $F$ in $L^{2}(S_{s}\mu;\mathbb{R}^{d})$ and of $a$ in $L^{2}(\mu;\mathbb{R}^{d})$.