Existence and Uniqueness of a Continuous Periodic Density for the Torus Heat Semigroup Started at the Origin

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. The Dirac measure $\delta_{0}$ at the origin of $\mathbb{R}^{d}$ is a probability measure on $\mathbb{R}^{d}$ by Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §measure, and $\delta_{0}(Q)=1$ because $0\in Q$, the half-open cell of The Half-Open Unit Cell Tiles Euclidean Space consisting of the points all of whose coordinates $x_{i}$ satisfy $0\le x_{i}<1$; so $\delta_{0}\in\mathcal{P}(\mathbb{T}^{d})$.

¶There is exactly one $\Theta\in C_{\mathrm{per}}$ such that $\mathbf{1}_{Q}\Theta$ is a density of $S_{s}\delta_{0}$ with respect to $\lambda_{d}$.