 Solution

For a smooth test function $f$, integration by parts in the Fokker-Planck equation givesWe 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 $d⟨x_k⟩/dt=⟨A_k⟩$. For $f=x_k{x}_l$, 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, soThe factor one-half cancels the two equal diffusion contributions, including when $k=l$.