the particular case of the harmonic potential. These solutions are fairly simple but to the best of my knowledge have not been discussed in the literature of stochastic field theories. A mean-field formulation of the dynamical problem is given in section \[MF\] and it is found that its solution is similar in effort to solving the associated Langevin mean-field equation, but with the stochastic averages already taken into account. In section \[Dyn\] I discuss applications of the solutions of section \[Sol\] to the understanding of the full non-linear dynamics of the system. Although qualitative it will be argued that these solutions capture the essentials of the full dynamics in the critical domain and permit an understanding of the relevant thermalization times. A brief discussion of spinodal instabilities is also included. Finally in section \[Conc\] I summarize the conclusions of this paper.

The Langevin and Fokker-Planck equations for scalar fields {#FP}
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We start by formulating the problem. The second order Langevin equation for a scalar fields $\phi(x,t)$ with an arbitrary interaction potential $V(\phi)$ is given by $$\begin{aligned}
\left( \partial_t^2  -\nabla^2 \right) \phi(x) +
 {\delta V(\phi) \over \delta \phi(x)} + \eta \dot \phi(x)= \xi({\bf x},t).
\label{e1}\end{aligned}$$ The stochastic fields $\xi(x,t)$ are 