A Bound on the Gibbs Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Fisher Information Relative to the Gibbs Measure

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 noise gradient $\nabla_{a}V$, let $\beta\in\mathbb{R}$ be positive, let $\gamma^{V}_{\beta}$ be the Gibbs measure of $V$ at temperature $\beta$, let $\mu\in\mathcal{P}_{2}(X)$, the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, be such that $V$ is integrable with respect to $\mu$, and let $R\in\mathbb{R}$ be nonnegative. By Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §integrable, $|\nabla_{a}V|_{a}$ and each $\partial_{k}V$ are integrable with respect to $\mu$, so The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential and The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential apply to $\mu$. For $n\in\mathbb{N}$, $C^{2}_{b}(\mathbb{R}^{n})$ is the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded with $q=n$; for $g\in C^{2}_{b}(\mathbb{R}^{n})$, $L^{a,V}_{\mu}(g)$ is the Gibbs Ornstein-Uhlenbeck functional of $\mu$ at $g$ with potential $V$ and temperature $\beta$, which applies since $\mu\in\mathcal{P}_{2}(X)$ and $V$ is integrable with respect to $\mu$; $g\circ p_{n}$ is a bounded $C^{2}$ cylindrical function, and its noise gradient $\nabla_{a}(g\circ p_{n})$ is bounded in $|\cdot|_{a}$ by that clause. $\lVert\cdot\rVert_{\mu}$ is the norm of $L^{2}(\mu;X^{a})$, the space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, and the class of $\nabla_{a}(g\circ p_{n})$ in it is again written $\nabla_{a}(g\circ p_{n})$. Suppose that

$$\bigl|L^{a,V}_{\mu}(g)\bigr|\le R\,\lVert\nabla_{a}(g\circ p_{n})\rVert_{\mu}\qquad\text{for every }n\in\mathbb{N}\text{ and every }g\in C^{2}_{b}(\mathbb{R}^{n}).$$

(Finite Fisher information relative to the Gibbs measure)¶ Then $\mu$ has a relative score with respect to $\gamma^{V}_{\beta}$ and finite Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$, and $\mathcal{I}_{a}(\mu\,|\,\gamma^{V}_{\beta})\le R^{2}$.