The Score Paired with the Identity, and the Laplacian of a Convex Potential Bounded by its Pairing with the Score

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, with the space $L^{2}(\mu;\mathbb{R}^{d})$ and its inner product $\langle\cdot,\cdot\rangle_{\mu}$, let finite Fisher information, the set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and the score $\xi_{\mu}$ be those of that definition, and let integrable be as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Let $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ be absolutely continuous. The identity map $\mathrm{id}$ of $\mathbb{R}^{d}$ is Borel, being continuous, and $\int_{\mathbb{R}^{d}}\lVert x\rVert^{2}\,\mu(dx)$ is the second moment of $\mu$, which is finite; its class in $L^{2}(\mu;\mathbb{R}^{d})$ is again written $\mathrm{id}$.

1. (The score paired with the identity)¶ $\langle\xi_{\mu},\mathrm{id}\rangle_{\mu}=-d$, with $d$ read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background §numbers.

2. (Laplacian comparison)¶ Let $G\subseteq\mathbb{R}^{d}$ be open and convex, so that $G\in\mathcal{B}(\mathbb{R}^{d})$ by Euclidean Space and Lebesgue Measure: Standing Notation §borel, with $\mu(G)=1$, let $\varphi:G\to\mathbb{R}$ be convex on $G$, and let $A\in\mathcal{B}(\mathbb{R}^{d})$ satisfy $A\subseteq G$ and $\mu(A)=1$ and be such that $\varphi$ is twice differentiable at every point of $A$, with first-order coefficient $D\varphi(y)$ and Hessian $D^{2}\varphi(y)$ at $y\in A$. Let $g:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and $\Delta:\mathbb{R}^{d}\to\mathbb{R}$ be the maps with $g(y)=D\varphi(y)$ and $\Delta(y)=\mathrm{tr}\,D^{2}\varphi(y)$, the trace, for $y\in A$, and $g(y)=0_{\mathbb{R}^{d}}$ and $\Delta(y)=0$ for $y\in\mathbb{R}^{d}\setminus A$; they are Borel by The Points of Twice Differentiability of a Convex Function: a Borel Set of Full Measure, and Borel Measurability of the Gradient and Hessian on It §borel, read with $d$ in place of $n$, and claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and $0\le\Delta$ by The Points of Twice Differentiability of a Convex Function: a Borel Set of Full Measure, and Borel Measurability of the Gradient and Hessian on It §nonnegative. Suppose that $\int_{\mathbb{R}^{d}}\lVert g\rVert^{2}\,d\mu<\infty$, and write $g$ also for the class of $g$ in $L^{2}(\mu;\mathbb{R}^{d})$. Then $\Delta$ is integrable with respect to $\mu$, and