Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $s$ be a real number with $0<s\le\tfrac12$, let $S_{s}$ be the heat semigroup at time $s$, and let $\lambda_{d}$ be Lebesgue measure. Finite entropy and the entropy $\mathrm{Ent}$ are those of that definition; $\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$, the score $\xi$ and the torus Fisher information $\mathcal{I}$ are those of Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus §finite. Let $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$. Then the following hold.

1. (Distance to the identity)¶ $W_{\mathbb{T}}(S_{s}\mu,\mu)^{2}\le d\,s$.

2. (Contraction)¶ $W_{\mathbb{T}}(S_{s}\mu,S_{s}\nu)\le W_{\mathbb{T}}(\mu,\nu)$.

3. (Density)¶ There is a $\mathbb{Z}^{d}$-periodic function $p:\mathbb{R}^{d}\to\mathbb{R}$ of class $C^{2}$ on $\mathbb{R}^{d}$ with $p(x)>0$ for every $x\in\mathbb{R}^{d}$ such that $\mathbf{1}_{Q}p$ is a density of $S_{s}\mu$ with respect to $\lambda_{d}$.

4. (Entropy)¶ $S_{s}\mu$ has finite entropy; if $\mu$ has finite entropy, then $\mathrm{Ent}(S_{s}\mu)\le\mathrm{Ent}(\mu)$.

5. (Score)¶ With $p$ as in clause 3, $S_{s}\mu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ and $\xi_{S_{s}\mu}$ is the class of the map $x\mapsto p(x)^{-1}\nabla p(x)$.