Finite Torus Fisher Information, the Torus Score and the Torus Fisher Information of a Probability Measure on the Torus

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\mu\in\mathcal{P}(\mathbb{T}^{d})$, with $\langle\cdot,\cdot\rangle_{\mu}$ and $\lVert\cdot\rVert_{\mu}$ as in Optimal Transport on the Flat Torus: Standing Notation §fields, gradients and Laplacians of periodic functions as in Optimal Transport on the Flat Torus: Standing Notation §calculus, and the tangent space $T_{\mu}$. The symbol $\mathcal{I}$ denotes the Fisher information below, never a cost.

1. (Finite torus Fisher information)¶ The measure $\mu$ has finite torus Fisher information if there is a real number $C\ge0$ with

$$\Bigl|\int\Delta f\,d\mu\Bigr|\le C\,\lVert\nabla f\rVert_{\mu}\qquad\text{for every }f\in C^{\infty}_{\mathrm{per}}.$$

The set of such $\mu$ is denoted $\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$.

2. (Score)¶ For $\mu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ the score of $\mu$ is the unique $\xi_{\mu}\in T_{\mu}$ with

$$\langle\xi_{\mu},\nabla f\rangle_{\mu}=-\int\Delta f\,d\mu\qquad\text{for every }f\in C^{\infty}_{\mathrm{per}} .$$

It exists and is unique by The Torus Tangent Space: Linearity of the Periodic Calculus, Closed Subspace, and Representation of Bounded Functionals on Gradients §representation, applied to $\ell(f)=-\int\Delta f\,d\mu$ and to a constant $C$ as in clause 1: $\ell$ is linear by The Torus Tangent Space: Linearity of the Periodic Calculus, Closed Subspace, and Representation of Bounded Functionals on Gradients §linear and the linearity of the integral (claim 2 of Linearity and Monotonicity of the Lebesgue Integral), and $|\ell(f)|=|\int\Delta f\,d\mu|$ by claim 2 of Properties of the Absolute Value in an Ordered Field.

3. (Fisher information)¶ For $\mu\in\mathcal{P}^{\mathcal{I}}(\mathbb{T}^{d})$ the torus Fisher information of $\mu$ is $\mathcal{I}(\mu)=\lVert\xi_{\mu}\rVert_{\mu}^{2}$.