The Score Integrates by Parts Against Every Compactly Supported Continuously Differentiable Vector Field

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, with the space $L^{2}(\mu;\mathbb{R}^{d})$ and its inner product $\langle\cdot,\cdot\rangle_{\mu}$; finite Fisher information, the set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and the score $\xi_{\mu}$ are those of that definition. Compact support of a function on $\mathbb{R}^{d}$ refers to the topology of the open subsets of $\mathbb{R}^{d}$ (Metric Open Sets Form a Topology and Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$), a function is bounded as defined there, a map into $\mathbb{R}^{d}$ is bounded when its Euclidean norm is, and integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Let $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$, and let $\eta:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be a map whose components $\eta_{i}$ ($i\in[d]$) are of class $C^{1}$ on $\mathbb{R}^{d}$, in the sense of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, and compactly supported; its divergence is the function $\operatorname{div}\eta=\sum_{i=1}^{d}\partial_{i}\eta_{i}$.

1. (Integration by parts)¶ The map $\eta$ is Borel and bounded, so that its class belongs to $L^{2}(\mu;\mathbb{R}^{d})$ and is again written $\eta$; the function $\operatorname{div}\eta$ is Borel and bounded, hence integrable with respect to $\mu$; and