Splitting a Bounded First Variation into a Square-Integrable Potential Force and a Finite Fisher Information

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, with the test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, their gradient maps $\nabla\psi$ and Laplacians $\Delta\psi$, and the norms $\lVert\cdot\rVert_{\mu}$ of the spaces $L^{2}(\mu;\mathbb{R}^{d})$; integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, and finite Fisher information and the set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ are those of that definition. Let $V$ be a confining potential on $\mathbb{R}^{d}$, with gradient $DV(x)$, gradient map $\nabla V:x\mapsto DV(x)$ and Laplacian $\Delta V$, let $\kappa$ be a positive real number, and let $\mu\in\mathcal{P}(\mathbb{R}^{d})$ be such that $V$ is integrable with respect to $\mu$.

1. (Integrable force)¶ $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, the function $\lVert\nabla V\rVert$ is integrable with respect to $\mu$, and for every $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ the function $\nabla V\cdot\nabla\psi$ is Borel and integrable with respect to $\mu$.

2. (Splitting)¶ Suppose that there is a nonnegative real number $C$ with

Then $\int_{\mathbb{R}^{d}}\lVert\nabla V\rVert^{2}\,d\mu<\infty$, so that $\nabla V$ has a class in $L^{2}(\mu;\mathbb{R}^{d})$, again written $\nabla V$, and $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$.