taken to be Gaussian and white, characterized by $$\begin{aligned}
\langle \xi({\bf x}, t) \rangle = 0,   \quad  
\langle \xi ({\bf x},t) \xi({\bf x'},t') \rangle = \Omega \delta^3 
({\bf x}-{\bf x'})\delta (t-t').
\label{e2}\end{aligned}$$ Although we have written this equation for a single scalar field $\phi(x,t)$ the generalization of Eq. (\[e1\]-\[e2\]) and of what follows, to a $O(N)$ symmetric theory, with N the number of flavors, is straightforward.

We can proceed to reduce the order of this differential system, by introducing field generalized momenta conjugate to $\phi$: $$\begin{aligned}
\partial_t \phi(x) &=& \pi(x), \label{e4a} \\
\partial_t \pi(x) &=& -\eta \pi(x)  + \nabla^2 \phi(x) 
- {\delta V(\phi) 
\over \delta \phi(x)} + \xi(x).
\label{e4b}\end{aligned}$$

An interesting limit of Eqs. (\[e4a\]-\[e4b\]) arises when $\eta$ is large, or more precisely when $\partial_t \pi(x) << \eta \pi(x)$. Then we can write $$\begin{aligned}
\eta \partial_t \phi(x) &=&  \nabla^2 \phi(x) - {\delta V(\phi) \over 
\delta \phi(x)} + \xi(x).
\label{e5}\end{aligned}$$ Although this is the more conventional form of the Langevin field equation [@Zinn; @Parisi] I will refer to it as the overdamped limit of Eqs. (\[e4a\]-\[e4b\]).

The objective of the Langevin analysis is to measure the expectation value (over the stochastic fields) of any functional $\rho$ of the fields $\pi(x),\phi(x)$, generated 