The Potential Gradient Paired with the Score, and a Fisher Information Bound on the Domain of the Langevin Free-Energy Pair

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, with the space $L^{2}(\nu;\mathbb{R}^{d})$, its inner product $\langle\cdot,\cdot\rangle_{\nu}$ and its norm $\lVert\cdot\rVert_{\nu}$; finite Fisher information, the set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, the score $\xi_{\nu}$ and the Fisher information $\mathcal{I}(\nu)=\lVert\xi_{\nu}\rVert_{\nu}^{2}$ are those of that definition, and integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Let $V$ be a confining potential on $\mathbb{R}^{d}$, with gradient map $\nabla V:x\mapsto DV(x)$ and Laplacian $\Delta V$; let $\sigma\in\mathbb{R}$ be positive, with $\sigma^{2}=\sigma\sigma$, $\tfrac{\sigma^{2}}{2}$ as in The Langevin Free-Energy Pair of a Confining Potential on the Wasserstein Space and $\tfrac{\sigma^{4}}{4}=\tfrac{\sigma^{2}}{2}\cdot\tfrac{\sigma^{2}}{2}$; and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Langevin free-energy pair with potential $V$ and noise intensity $\sigma$, a penalty pair by The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §pair, with translation Hessian $H_{\mathcal{E}}(\nu)\in\mathcal{S}(d)$ at $\nu\in\mathcal{D}$ and $\mathrm{tr}$ the trace. The letter $\sigma$ denotes the noise intensity; the swap map written $\sigma$ in Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §swap is not used. Let $\nu\in\mathcal{D}_{\Sigma}$.

1. (The potential gradient paired with the score)¶ The function $\lVert\nabla V\rVert^{2}$ is integrable with respect to $\nu$, so that the class of $\nabla V$ belongs to $L^{2}(\nu;\mathbb{R}^{d})$ and is again written $\nabla V$; the function $\Delta V$ is integrable with respect to $\nu$; and

2. (Fisher information bound)¶ With $\Sigma(\nu)=\nabla V+\tfrac{\sigma^{2}}{2}\xi_{\nu}\in L^{2}(\nu;\mathbb{R}^{d})$,