The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $V$ be an admissible cylindrical potential with head dimension $d$, with the functions $\partial_{k}V$ of Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §gradient, let $\beta\in\mathbb{R}$ be positive, and let $\gamma^{V}_{\beta}$ be the Gibbs measure of $V$ at temperature $\beta$. $a$ is the noise weight sequence, a weight sequence by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian. For $\mu\in\mathcal{P}_{2}(X)$, the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, such that $\partial_{k}V$ is integrable with respect to $\mu$ for every $k\in[d]$, the relative score of $\mu$ with respect to $\gamma^{V}_{\beta}$ and its components are those of that definition, with $L^{2}(\mu)$ and its norm $\lVert\cdot\rVert_{L^{2}(\mu)}$ as there.

(Weighted Fisher information relative to the Gibbs measure)¶ Let $\mu\in\mathcal{P}_{2}(X)$ be such that $\partial_{k}V$ is integrable with respect to $\mu$ for every $k\in[d]$, and let $\mu$ have a relative score $(\zeta^{V}_{k})_{k\in\mathbb{N}}$ with respect to $\gamma^{V}_{\beta}$. The measure $\mu$ has finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$ if the series $\sum_{k=1}^{\infty}a_{k}\lVert\zeta^{V}_{k}\rVert_{L^{2}(\mu)}^{2}$ converges, and its Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$ is then the sum

$$\mathcal{I}_{a}(\mu\,|\,\gamma^{V}_{\beta})=\sum_{k=1}^{\infty}a_{k}\lVert\zeta^{V}_{k}\rVert_{L^{2}(\mu)}^{2},$$

a nonnegative real number by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, the terms being nonnegative.