The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure

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 $s$, let $\lambda_{d}$ be Lebesgue measure, and 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. Then the following hold.

1. (Regularity and bounds)¶ $\Theta_{s}\in C^{2}_{\mathrm{per}}$, $\Theta_{s}(-v)=\Theta_{s}(v)$ for every $v\in\mathbb{R}^{d}$, and $\int_{Q}\Theta_{s}\,d\lambda_{d}=1$. There are real numbers $c_{s}$ and $C_{s}$ with $0<c_{s}\le C_{s}$ such that $c_{s}\le\Theta_{s}(v)\le C_{s}$, $|\partial_{i}\Theta_{s}(v)|\le C_{s}$ and $|\partial_{j}\partial_{i}\Theta_{s}(v)|\le C_{s}$ for all $v\in\mathbb{R}^{d}$ and $i,j\in[d]$.

2. (Density of a heat-smoothed measure)¶ Let $\mu\in\mathcal{P}(\mathbb{T}^{d})$. For every $y\in\mathbb{R}^{d}$ the function $x\mapsto\Theta_{s}(y-x)$ is continuous and bounded by clause 1, so $p_{\mu}(y)=\int\Theta_{s}(y-x)\,\mu(dx)$ is a real number. The function $p_{\mu}$ belongs to $C^{2}_{\mathrm{per}}$, with $\partial_{i}p_{\mu}(y)=\int\partial_{i}\Theta_{s}(y-x)\,\mu(dx)$ and $c_{s}\le p_{\mu}(y)\le C_{s}$ for all $y\in\mathbb{R}^{d}$ and $i\in[d]$; the function $\mathbf{1}_{Q}p_{\mu}$ is a density of $S_{s}\mu$ with respect to $\lambda_{d}$; and $S_{s}\mu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$, with $\xi_{S_{s}\mu}$ the class of the map $y\mapsto p_{\mu}(y)^{-1}\nabla p_{\mu}(y)$.

3. (Lipschitz dependence on the measure)¶ There is a real number $L_{s}\ge0$ such that, with $p_{\mu}$ and $p_{\nu}$ as in clause 2,

$$|p_{\mu}(y)-p_{\nu}(y)|\le L_{s}\,W_{\mathbb{T}}(\mu,\nu)\qquad\text{and}\qquad\lVert\nabla p_{\mu}(y)-\nabla p_{\nu}(y)\rVert\le L_{s}\,W_{\mathbb{T}}(\mu,\nu)$$

for all $\mu,\nu\in\mathcal{P}(\mathbb{T}^{d})$ and $y\in\mathbb{R}^{d}$.