Is there a version of Stein's lemma for multivariate Gaussians, I am attempting to solve integrals of the form:
$$ \mathbb{E}(\mathbf{x} g(\mathbf{x})) =\int_{\mathbb{R}^p} \mathbf{x} g(\mathbf{x}) \mathcal{N}(\mathbf{x}| \mu, \Sigma) d\mathbf{x} $$
and
$$ \mathbb{E}(\mathbf{x}^T\mathbf{x} g(\mathbf{x})) $$
where $\mathbf{x} \in \mathbb{R}^p$ is a Gaussian vector with mean $\mu$, and covariance matrix $\Sigma$ and $g: \mathbb{R}^p \to\mathbb{R}$