Gaussian Integration by Parts Relative to a Diagonal Gaussian Measure for Functions with a Bounded Hessian

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector, with scaling map $S_{c}$, and let $\gamma_{c}$ be the diagonal Gaussian measure with variances $c$. Being of class $C^{2}$ on $\mathbb{R}^{d}$, and the gradient $Df(x)$ and the partial derivatives of such a function, are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous; $\Delta f$ is the Laplacian and $u\cdot v$ the dot product; integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Let $M\in\mathbb{R}$ be nonnegative and let $f:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{d}$ with $|\partial_{j}\partial_{i}f(x)|\le M$ for all $x\in\mathbb{R}^{d}$ and $i,j\in[d]$. For $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the class $Df\in L^{2}(\mu;\mathbb{R}^{d})$ of the gradient map is that of Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §integrable, and the class $S_{c}\in L^{2}(\mu;\mathbb{R}^{d})$ is that of Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information.

1. (The Ornstein-Uhlenbeck generator)¶ The function

$$L_{c}f:\mathbb{R}^{d}\to\mathbb{R},\qquad L_{c}f(x)=\Delta f(x)-S_{c}(x)\cdot Df(x),$$

is Borel and integrable with respect to every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

2. (Integration by parts)¶ Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ have finite Fisher information relative to $\gamma_{c}$, with relative score $\zeta^{c}_{\mu}$. Then

$$\langle\zeta^{c}_{\mu},Df\rangle_{\mu}=-\int_{\mathbb{R}^{d}}L_{c}f\,d\mu .$$