Let $\mu$ be a centered Gaussian measure on $\mathbb{R}^N$ with the covariance matrix $\sigma$.
Let $F : \mathbb{R}^N \to \mathbb{R}^N$ be some smooth, bounded mapping.
Then, I wonder if it is possible to expand the integral \begin{equation} \int_{\mathbb{R}^N} F_l(y)F_k(y) d\mu(y) \end{equation} in terms of the entries of $\sigma$? Here, $l,k \in \{1, \cdots, N\}$ denote the components.
Since many even moment of a Gaussian measure is expressible by products of the entries of $\sigma$, I guess there must be a formula. But I cannot find a relevant reference or finish the calculations myself..
Could anyone help me?