procedure let us consider an interaction potential of the form $V_{int}= { \lambda \over 4} \phi(x)^4$. In momentum space this potential generates a term in the Fokker-Planck equation for $\phi_k$ of the form $$\begin{aligned}
{\delta V_{int} \over \delta \phi_k} = \lambda \int {d^D p \over (2 \pi)^D} 
{d^D q \over (2 \pi)^D}
~\phi_p \phi_q \phi_{k-p-q}.\end{aligned}$$ A mean-field approximation to this term can be obtained by taking $p=-q$ and the noise average such that $$\begin{aligned}
{\delta V_{MF} \over \delta \phi_k} = 3 \lambda \left[ \int {d^D p \over (2 \pi)^D} 
\langle \phi_p \phi_{-p} \rangle \right] \phi_{k} \equiv 
3 \lambda G(t) \phi_k.\end{aligned}$$ Under this approximation the evolution equation can be written in terms of a mean-field Fokker-Planck operator where $V_{MF}$ substitutes the potential $V$, with $$\begin{aligned}
V_{MF} = V_{0} +  {3 \lambda \over 4} G(t)^2,\end{aligned}$$ where $V_{0}$ corresponds to the purely harmonic case, together with the self-consistency condition $$\begin{aligned}
G(t) = \int {d^D p \over (2 \pi)^D}  \int d \phi_p~ \phi_p \phi_{-p} P[\pi_q,\phi_p,t].\end{aligned}$$ It is then clear that the under this Gaussian approximation the theory remains harmonic, but with a time-dependent (self-consistently determined) mass.

Unfortunately the time dependence of the mean field potential destroys the separability of the solution into a function 