= Solution
For a smooth test function $f$, integration by parts in the <Fokker-Planck equation> gives
$$
\frac d{dt}\mathbb E f(x)=\mathbb E\left[A_i\partial_i f+\tfrac12B_{ij}\partial_i\partial_j f\right].
$$
We use the decay and absence of boundary flux intended in the question, sufficiently strong to remove the moment-weighted boundary terms. For $f=x_k$, only $\partial_i f=\delta_{ik}$ survives, giving $d\langle x_k\rangle/dt=\langle A_k\rangle$. For $f=x_kx_l$, the first derivative is $\delta_{ik}x_l+\delta_{il}x_k$ and the second is $\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}$. The <Fokker-Planck diffusion matrix> is symmetric, so
$$
\boxed{\frac d{dt}\langle x_kx_l\rangle=\langle x_lA_k+x_kA_l+B_{kl}\rangle.}
$$
The factor one-half cancels the two equal diffusion contributions, including when $k=l$.
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