The Heat Semigroup on the Probability Measures on the Torus

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $s$ be a real number with $0<s\le\tfrac12$, let $\lambda_{d}$ be Lebesgue measure, and let $\mu\in\mathcal{P}(\mathbb{T}^{d})$. Let $g_{s}*\mu$ be the Gaussian smoothing of $\mu$ at scale $s$, that lemma read with $q=d$; it is nonnegative, Borel and of integral $1$ with respect to $\lambda_{d}$ by Gaussian Smoothing of a Probability Measure on Euclidean Space: Regularity, Mass, Duality, Approximation of a Bounded Function with a Modulus of Continuity, and the Pairing Identity §duality, so the measure $\mu_{s}$ with density $g_{s}*\mu$ with respect to $\lambda_{d}$ belongs to $\mathcal{P}(\mathbb{R}^{d})$.

(Heat semigroup)¶ The heat semigroup at time $s$ sends $\mu$ to $S_{s}\mu=\pi_{\#}\mu_{s}$, which belongs to $\mathcal{P}(\mathbb{T}^{d})$ by The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §wrap.