I have a term of the following form:
$$\mathbb{E}[X_iX_j \nabla_xv_i \cdot v_j]$$
For scalar random variables $X_i$. The vector valued functions $v = v(x,X_k)$ depend on a deterministic variable $x$ and the random variables $X_k$.
Can I say that this expression is equal to:
$M_{ij} \nabla_x w_i \cdot w_j$ ? For some deterministic matrix $M$ and vector valued functions $w = w_i(x)$? My hunch is that if everything was independent, or at the very least the $X_i$'s and $v_i$'s then clearly we can just set $M_{ij} = \mathbb{E}[X_i X_j]$ and $\mathbb{E}[v_i] = w_i$ and use the fact that $\nabla_x$ can be pulled out as a linear operator.
Furthermore, I feel like I can apply a sort of simple function boostrap argument, at least for the scalar case: suppose we have $f = f_a(x)\chi_a$
$$\mathbb{E}[X_iX_j\partial_xf^i g^i] = \mathbb{E}[X_iX_j\partial_xf^i_a(x)\chi_a f^j_b(x)\chi_b] = \mathbb{E}[X_iX_j\chi_a \chi_b]\partial_xf^i_a(x)f^j_b(x)$$
Then just call $M$ this huge jumble of characteristic functions and the random variables.
So if this works for simple functions then perhaps I can argue that this works in the limit? I might have to assume very strong properties for $f$ though since I know convergence of a function's derivatives is a bit iffy, but this is acceptable to me.