Prove $E\left[\mathbf g(\mathbf x)(\mathbf x- \mathbf m)^\intercal\right]=E\left[\mathbf G_{\mathbf x}(\mathbf x)\right]\mathbf P$ for Gaussian x

Question

$\mathbf x \in \mathbb R^n$ is a Gaussian random variable with mean $\mathbf m$ and covariance matrix $\mathbf P$, i.e.

$$\mathbf x \sim \mathcal N(\mathbf m, \mathbf P)$$

I want prove the following property:

$$E\left[\mathbf g(\mathbf x)(\mathbf x- \mathbf m)^\intercal\right]=E\left[\mathbf G_{\mathbf x}(\mathbf x)\right]\mathbf P$$

where $\mathbf G_{\mathbf x}$ is the Jacobian matrix of $\mathbf g(\mathbf x)$ (assumed to be a differentiable function) and $E[\cdot]$ is the expected value w.r.t. $\mathbf x$.

Hints/references are welcome.

Answer 1

Assume wlog that $\mathbf m= \mathbf 0$. First apply the product rule to see that $$\nabla \left(\mathbf g(\mathbf x)N(\mathbf x)\right) = \mathbf G_{\mathbf x}(\mathbf x)N(\mathbf x)-\mathbf g(\mathbf x) \; \mathbf x^\intercal \mathbf P^{-1} N(\mathbf x)$$

where $N$ denotes the normal density with covariance $\mathbf P$.

Then observe that the integral of the left-hand side vanishes assuming that $gN$ decays sufficiently quickly.