by the evolution of Eq (\[e1\]-\[e2\]). With the choice Eq. (\[e2\]) this average is a Gaussian functional integral of the form $$\begin{aligned}
\int D\xi \rho[\pi_{\xi},\phi_{\xi}] e^{- {1 \over 2 \Omega }
\int d^4 x \xi^2}
\label{e6}\end{aligned}$$ where the $\xi$ subscripts in the field and its momentum denote their functional dependence on the stochastic fields $\xi(x,t)$, through the evolution of Eq. (\[e4a\]-\[e4b\]). Relation (\[e6\]) and (\[e2\]) can be generalized to fields $\xi$ with more complicated Gaussian distributions.

In the presence of the stochastic fields $\xi(x,t)$ the initial conditions for the fields $\pi(x), \ \phi(x)$ may only be know statistically themselves. They can then be expressed in terms of a functional probability distribution $P[\pi,\phi]$, at the initial time.

Throughout the evolution we assume that a time dependent probability distribution $P_{\rm FP} [\phi(x),\pi(x),t]$, a functional of the time independent fields and a function of time, exists. This is the Fokker-Planck probability distribution, which we require, as usual, to be positive definite and normalizable in the sense: $$\begin{aligned}
{\cal N} \int D\pi D\phi  P_{FP} [\phi,\pi,t] = 1.
\label{e7}\end{aligned}$$ Note that in general $\cal N$ as well as other expectation values of the fields may be formally infinite as the ultraviolet cutoff of the theory 