= Solution
Assume again that all boundary terms vanish. Multiplying the <Fokker-Planck equation> by $x_k$ and integrating by parts gives
$$
\begin{aligned}
\frac d{dt}\langle x_k\rangle
&=-\int x_k\,\partial_i(A_iP)\,dV
+\frac12\int x_k\,\partial_i\partial_j(B_{ij}P)\,dV\\
&=\int\delta_{ik}A_iP\,dV
=\boxed{\langle A_k\rangle}.
\end{aligned}
$$
The diffusion term vanishes after the second integration by parts.
Similarly, $\partial_i(x_kx_l)=\delta_{ik}x_l+\delta_{il}x_k$ and
$$
\partial_i\partial_j(x_kx_l)
=\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}.
$$
Because $B$ is symmetric,
$$
\boxed{
\frac d{dt}\langle x_kx_l\rangle
=\langle x_lA_k+x_kA_l+B_{kl}\rangle.}
$$
These are the <First two moments of a multivariate Fokker-Planck equation>.
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