of the $\delta$ allow us to trade the functional derivatives of $\hat \pi(x,t), \hat \phi(x,t)$ for derivatives relative to $\pi(x), \phi(x)$. These can be taken out of the stochastic average $\langle ... \rangle$. The only remaining term inside this average results from the appearance of the stochastic fields $\xi$ in the equation of motion for $\hat \pi (x,t)$. This term is of the form $$\begin{aligned}
\langle \int d^D x \xi (x,t)  \delta \left[ \hat \pi(x,t) - \pi(x) \right] \rangle.\end{aligned}$$ The Gaussianity of the stochastic fields enforces the identity [@Zinn] $$\begin{aligned}
&& \langle \xi(x,t)  \delta \left[ \hat \pi(x,t) - \pi(x) \right] \rangle
= \Omega \langle {\delta  \over \delta \xi (x,t)} \delta \left[ \hat \pi(x,t) 
- \pi(x) \right] \rangle \nonumber \\
&& = \Omega \langle {\delta \hat \pi(x,t) \over 
\delta \xi(x,t)}
{\delta \over  \hat \pi(x,t)} \delta \left[ \hat \pi(x,t) 
- \pi(x) \right] \rangle.\end{aligned}$$ Once again we can trade the functional derivative relative to $\hat \pi (x,t)$ for another relative to $\pi (x)$. The expectation value of ${\delta \hat \pi(x,t) \over 
\delta \xi(x,t)}$ can in turn be obtained from the formal integration of the equation of motion by using a regularized delta function in time. We must effectively take half of this delta 