Gaussian Integration by Parts for Functions of Finitely Many Coordinates Relative to a Diagonal Gaussian Measure on a Hilbert Space

In the settings of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, with the coordinates $x_{k}$, the coordinate maps $p_{n}$ and the Euclidean norm $\lVert\cdot\rVert$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, and integrals and integrability as in Measure Spaces and the Lebesgue Integral: Standing Notation, let $c$ be a variance sequence, $\gamma_{c}$ the diagonal Gaussian measure on $X$ with variances $c$, $H_{c}$ the Cameron-Martin space of $c$, and, for $h\in H_{c}$, $\ell_{h}$ the Paley-Wiener functional of $h$ relative to $c$.

Let $n\in\mathbb{N}$, let $M\in\mathbb{R}$ be nonnegative, and let $\psi:\mathbb{R}^{n}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{n}$, where the class $C^{1}$ and the partial derivatives $\partial_{i}\psi$ are understood as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives with $q=n$, the set $\mathbb{R}^{n}$ being open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and suppose that

$$|\partial_{i}\psi(y)|\le M\bigl(1+\lVert y\rVert\bigr)\qquad\text{for all }y\in\mathbb{R}^{n}\text{ and }i\in[n].$$

Let $\varphi=\psi\circ p_{n}:X\to\mathbb{R}$ and, for $i\in[n]$, $\varphi_{i}=(\partial_{i}\psi)\circ p_{n}:X\to\mathbb{R}$.

1. (Integrability)¶ The following functions are Borel and integrable with respect to $\gamma_{c}$:¶ the functions $\varphi$ and $\varphi_{i}$ ($i\in[n]$);¶ the functions $x\mapsto x_{k}\varphi(x)$ ($k\in\mathbb{N}$); and¶ the functions $\ell_{h}\varphi$ ($h\in H_{c}$).

2. (Coordinate directions)¶¶ For every $i\in[n]$,

$$\int_{X}\varphi_{i}\,d\gamma_{c}=\frac{1}{c_{i}}\int_{X}x_{i}\,\varphi(x)\,\gamma_{c}(dx).$$

¶For every $k\in\mathbb{N}$ with $k>n$,

$$\int_{X}x_{k}\,\varphi(x)\,\gamma_{c}(dx)=0.$$

3. (Cameron-Martin directions)¶ For every $h\in H_{c}$, with coordinates $h_{i}$,

$$\sum_{i=1}^{n}h_{i}\int_{X}\varphi_{i}\,d\gamma_{c}=\int_{X}\ell_{h}\,\varphi\,d\gamma_{c}.$$