(\[solcomb\]) remains real as required.

The overdamped limit is recovered as $\eta^2 >> k^2 +m^2$. Then $$\begin{aligned}
&& E_N^- = N {k^2 +m^2 \over \eta}, \label{Over} \\
&& a_-^2= {\beta \over 2} {k^2 + m^2 \over \eta^2} <<  
b_-^2 = {\beta \over 2} (k^2 + m^2).\end{aligned}$$ The eigenfunctionals associated with $E_N^+ \simeq N \eta 
[1 + (k^2 +m^2)/\eta^2]$ are, in contrast, damped away rapidly.

The characteristic thermalization times of the system are thus set by $t_N = 1/E_N$. For the long wave length modes ($k\simeq0$) we have that the equilibration time $t_{eq} \sim \eta/(k^2+ m^2)$. In the converse limit, of short wavelengths, the decay of all states is controlled at leading order by $t_{eq} \sim 2/ \eta$, which is scale independent. In this sense the evolution of the short wavelength modes is quantitatively different from the overdamped Langevin system.

A mean-field approximation to the non-linear Fokker-Planck dynamics {#MF}
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It would no doubt be desirable to be able to solve the Fokker-Planck equations for more general (interacting) potentials. At the next level of complexity it is possible to attempt a solution of the Fokker-Planck equation in a mean-field approximation, which we discuss in this section.

In order to illustrate the 