The Ornstein-Uhlenbeck Operator with Noise Weights on Bounded $C^2$ Cylindrical Functions

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the noise weights $a_{k}$, the variances $c_{k}$, the set $\mathcal{F}C^{2}_{b}(X)$ of bounded $C^{2}$ cylindrical functions, and the partial derivatives $\partial_{k}$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cylindrical. 1. (Second partial derivatives)¶ Let $F\in\mathcal{F}C^{2}_{b}(X)$, with $F=g\circ p_{n}$ for some $n\in\mathbb{N}$ and $g\in C^{2}_{b}(\mathbb{R}^{n})$, the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded with $q=n$. By clause 2 of C^k Maps on a Euclidean Open Set, $g$ and each $\partial_{k}g$ ($k\in[n]$) are of class $C^{1}$, and they are bounded with bounded partial derivatives by Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded; so $g$ and each $\partial_{k}g$ belong to the set $C^{1}_{b}(\mathbb{R}^{n})$ of Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded, and $F\in\mathcal{F}C^{1}_{b}(X)$. By Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial, for $k\in\mathbb{N}$ the partial derivative $\partial_{k}F$ exists, $\partial_{k}F=(\partial_{k}g)\circ p_{n}$ if $k\le n$ and $\partial_{k}F=0$ if $k>n$. In both cases $\partial_{k}F\in\mathcal{F}C^{1}_{b}(X)$, the constant $0$ being the composite with $p_{n}$ of the constant function $0$ on $\mathbb{R}^{n}$; so the second partial derivative $\partial_{k}\partial_{k}F=\partial_{k}(\partial_{k}F)$ exists, and it is $0$ for $k>n$.

2. (The Ornstein-Uhlenbeck operator)¶ The Ornstein-Uhlenbeck operator with noise weights $a$ maps $F\in\mathcal{F}C^{2}_{b}(X)$, with $(n,g)$ as in clause 1, to the function $L^{a}F:X\to\mathbb{R}$,

$$L^{a}F(x)=\sum_{k=1}^{n}a_{k}\Bigl(\partial_{k}\partial_{k}F(x)-\frac{x_{k}}{c_{k}}\,\partial_{k}F(x)\Bigr)\qquad(x\in X),$$

which does not depend on the choice of $n$ and $g$: if also $F=g'\circ p_{n'}$, we may assume $n\le n'$ by symmetry, and the terms with $n<k\le n'$ vanish, since $\partial_{k}F=0$ and $\partial_{k}\partial_{k}F=0$ for $k>n$ by clause 1 applied with $(n,g)$.