Heat Regularisation of the Entropy: a Laplacian Test Function whose Laplacian is Minus the Fisher Information

In the setting of Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation, for $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and every $s$ the measure $S_{s}\mu$ belongs to $\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})\cap\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ by Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation §measures, so $\mathrm{Ent}(S_{s}\mu)$ and $\xi_{S_{s}\mu}\in T_{S_{s}\mu}$ are defined, and $A^{\mu}_{s}\xi_{S_{s}\mu}$ is the heat average of the score. Then the following hold.

1. (Laplacian test function)¶ For every $s$, the map $\mathrm{Ent}\circ S_{s}$ is a Laplacian test function; its gradient field is $(\mu,x)\mapsto A^{\mu}_{s}\xi_{S_{s}\mu}(x)$, and $\nabla(\mathrm{Ent}\circ S_{s})(\mu)=A^{\mu}_{s}\xi_{S_{s}\mu}$ for every $\mu\in\mathcal{P}(\mathbb{T}^{d})$.

2. (Laplacian)¶ For every $s$ and every $\mu\in\mathcal{P}(\mathbb{T}^{d})$,

$$\mathcal{L}(\mathrm{Ent}\circ S_{s})(\mu)=-\mathcal{I}(S_{s}\mu).$$