I was wondering if there is a way to derive the Fokker-Planck equation for Langevin equation with a general colored noise:
$$m \ddot{x} = -\frac{\partial V(x)}{\partial x} - \gamma \dot{x} + F(t)$$
with $\langle F(t) F(t') \rangle=\chi(t-t')$, some arbitrary (well-behaved) function.
I found this post Relation between Langevin and Fokker-Planck for exponentially correlated noise, which was incredibly useful. However, the trick of writing the evolution of the exponentially correlated noise appears to only work when the correlation is exponential (the OU process governing the dynamics of the noise generates exponential correlations). I figure that it may be possible to use a similar technique but using a different stochastic process rather than an OU process to generate the desired correlation. However, this seems like it would probably bring me back to the original problem. Is there a know alternative (non-perturbative) approach or a way to find such an appropriate auxiliary stochastic process for the dynamics of the noise?
