Heat Regularisation of an Intrinsic Test Function: a Laplacian Test Function whose Laplacian is Minus the Pairing of the Gradient with the Score

In the setting of Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation, let $\varphi$ be an intrinsic test function on $\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$. For $\mu\in\mathcal{P}(\mathbb{T}^{d})$ and every $s$ the measure $S_{s}\mu$ is absolutely continuous by Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation §measures, so $\nabla\varphi(S_{s}\mu)\in T_{S_{s}\mu}$ is defined, and $A^{\mu}_{s}\nabla\varphi(S_{s}\mu)$ is its heat average. Then the following hold.

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

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

$$\mathcal{L}(\varphi\circ S_{s})(\mu)=-\bigl\langle\nabla\varphi(S_{s}\mu),\xi_{S_{s}\mu}\bigr\rangle_{S_{s}\mu}.$$

3. (Convergence as the time vanishes)¶ Let $\mu\in\mathcal{P}^{\mathrm{ac}}(\mathbb{T}^{d})$, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}(\mathbb{T}^{d})$ that converges to $\mu$ in $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$, and let $(s_{n})_{n\in\mathbb{N}}$ be a sequence of real numbers with $0<s_{n}\le\tfrac12$ that converges to $0$. Then both sequences

$$\bigl((S_{s_{n}}\mu_{n},\nabla\varphi(S_{s_{n}}\mu_{n}))\bigr)_{n\in\mathbb{N}}\qquad\text{and}\qquad\bigl((\mu_{n},A^{\mu_{n}}_{s_{n}}\nabla\varphi(S_{s_{n}}\mu_{n}))\bigr)_{n\in\mathbb{N}}$$

converge along couplings to $(\mu,\nabla\varphi(\mu))$.