solving the corresponding Fokker-Planck equation. Unfortunately the task of solving for these non-linear functional equations in general is quite monumental. The main difficulty is connected with the non-local form of the equations: in x-space this arises from the Laplacian term, while in k space the difficulty is transfered to the non-linear terms.

Below I give closed form solutions to the equations in the fields and their conjugate momenta for the particular case of the harmonic potential. These are still interesting since they offer an alternative to the direct solutions of the Langevin equation, which, clearly, can also be found in the linear case. In the former picture, however, the averages over the stochastic fields are already taken into account.

It is simpler to build some intuition for the solution of the overdamped Fokker-Planck equation first to which I now turn.

The overdamped Fokker-Plank equation and its solutions {#sOFP}
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The advantage of the harmonic case is that in Fourier space different modes decouple in the Langevin equation: $$\eta \partial_t  \phi_k(t) = (k^2 +m^2) \phi_k(t) + \xi_k (t)
\label{Lang_k}$$ where the stochastic fields obey $$\begin{aligned}
\langle \xi_k(t) \rangle = 0, \quad 
\langle \xi_k(t)\xi_{-k}(t') \rangle = \Omega \delta(t-t')\end{aligned}$$ and $$\begin{aligned}
\phi(x,t) = 