Optimal FPE for non-linear 1d-SDE. I: Additive Gaussian colored noise

Many complex phenomena occurring in physics, chemistry, biology, finance, etc can be reduced, by some projection process, to a 1-d stochastic Differential equation (SDE) for the variable of interest. Typically, this SDE is both non-linear and non-Markovian, so a Fokker Planck equation (FPE), for the probability density function (PDF), is generally not obtainable. However, a FPE is desirable because it is the main tool to obtain relevant analytical statistical information such as stationary PDF and First Passage Time. This problem has been addressed by many authors in the past, but due to an incorrect use of the interaction picture (the standard tool to obtain a reduced FPE) previous theoretical results were incorrect, as confirmed by direct numerical simulation of the SDE. The pitfall lies in the rapid diverging behavior of the backward evolution of the trajectories for strong dissipative flows. We will show, in general, how to address this problem and we will derive the correct best FPE from a cumulant-perturbation approach. The specific perturbation method followed gives general validity to the results obtained, beyond the simple case of exponentially correlated Gaussian driving used here as an example: it can be applied even to non Gaussian drivings with a generic time correlation.