function to obtain $$\begin{aligned}
{\delta \hat \pi(x,t) \over \delta \xi(x,t)} = {1\over 2}.\end{aligned}$$ Bringing together all the terms results finally in $$\begin{aligned}
\partial_t P_{FP}[\pi,\phi,t]=-{\cal H}_{\rm FP} P_{FP}[\pi,\phi,t].
\label{e11}\end{aligned}$$ where $$\begin{aligned}
{\cal H}_{\rm FP} &=& - {\Omega \over 2} 
{\delta^2 \over \delta \pi^2 } + \pi {\delta \over \delta 
\phi} \nonumber \\
&& \qquad \qquad - {\delta \over \delta \pi} 
\left( \eta \pi - \nabla^2 \phi + 
{\delta V(\phi) \over \delta \phi}\right) 
\label{e12}\end{aligned}$$

It is useful to think in terms of the functional operator ${\cal H}_{\rm FP}$ as the generator of infinitesimal time displacements of the probability functional $P_{\rm FP}$. If, as in most applications, the potential $V(\phi)$ is explicitly time independent we can invoke a separation ansatz for $P_{FP}$ such that $$\begin{aligned}
P_{FP} [\pi,\phi,t] = {\cal P} [\pi,\phi] T(t)
\label{e13}\end{aligned}$$ Thus we can regard Eq. (\[e11\]) as an analog of a functional Schrödinger equation, in imaginary time. Then we can write the time independent and dependent equations $$\begin{aligned}
{\cal H}_{\rm FP} {\cal P}_n = E_n {\cal P}_n, \quad \partial_t T(t) = 
- E_n T(t).
\label{e14}\end{aligned}$$ The functional dependence on the fields is now limited to the static probability eigenfunctionals ${\cal P}_n$. The time evolution of the Fokker-Planck distribution is completely 