is taken to zero.

The time dependent expectation value over the stochastic fields $\xi$ of any quantity $\rho[\phi,\pi]$ is then given by $$\begin{aligned}
\langle \rho \rangle(t) = {\cal N} \int D\pi D\phi  P_{FP} [\phi,\pi,t] 
\rho[\phi,\pi].
\label{e8}\end{aligned}$$

In order to be useful this picture requires the explicit knowledge of $P_{FP} [\phi,\pi,t]$. The Langevin equation generates a time-dependent (Fokker-Planck) probability distribution $P_{FP}[\pi,\phi,t]$, which we can write formally $$\begin{aligned}
&& P_{FP}[\pi,\phi,t] = \nonumber \\ 
&& \qquad \langle  
\int d^D x  \delta \left[ \hat \pi(x,t) - \pi(x) \right] 
\delta \left[ \hat \phi (x,t) - \phi(x) \right] \rangle,
\label{e9}\end{aligned}$$ where the brackets denote, as before, average over the stochastic fields. The static fields are the arguments of $P_{FP}[\pi,\phi,t]$, whereas $\hat \pi(x,t), \ \hat \phi(x,t)$ obey the Langevin Eqs. (\[e4a\]- \[e4b\]).

The probability $P_{FP}[\pi,\phi,t]$, obeys a functional Fokker-Planck evolution equation that can be computed directly by differentiating Eq. (\[e9\]). Following Zinn-Justin [@Zinn] $$\begin{aligned}
&& \partial_t P_{FP} = \langle \left[ \int d^D x~ 
\partial_t  \hat \phi (x,t) {\delta \over \delta \hat \phi(x,t)} + 
\partial_t \hat \pi(x,t)   {\delta \over \delta \hat \pi(x,t)} 
\right] \nonumber \\
&&   \qquad \quad \times  \delta \left[ \hat \pi(x,t) - \pi(x) \right] 
\delta \left[ \hat \phi (x,t) - \phi(x) \right] \rangle. 
\label{e10}\end{aligned}$$ The properties 