The Torus Heat Kernel

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 $s$, let $\lambda_{d}$ be Lebesgue measure, and let $\delta_{0}\in\mathcal{P}(\mathbb{T}^{d})$ be the Dirac measure at the origin, as in Existence and Uniqueness of a Continuous Periodic Density for the Torus Heat Semigroup Started at the Origin.

¶The torus heat kernel at time $s$ is the unique $\Theta_{s}\in C_{\mathrm{per}}$ such that $\mathbf{1}_{Q}\Theta_{s}$ is a density of $S_{s}\delta_{0}$ with respect to $\lambda_{d}$; it exists and is unique by Existence and Uniqueness of a Continuous Periodic Density for the Torus Heat Semigroup Started at the Origin §kernel.