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Many complex phenomena occurring in physics,chemistry, biology, finance, etc.can be reduced, by some projection process, to a 1-d stochastic DifferentialEquation (SDE) for the variable of interest. Typically, this SDE is bothnon-linear and non-markovian, so a Fokker Planck equation (FPE), for theprobability density function (PDF), is generally not obtainable. However, a FPEis desirable because it is the main tool to obtain relevant analyticalstatistical information such as stationary PDF and First Passage Time. Thisproblem has been addressed by many authors in the past, but due to an incorrectuse of the interaction picture (the standard tool to obtain a reduced FPE)previous theoretical results were incorrect, as confirmed by direct numericalsimulation of the SDE. We will show, in general, how to address the problem andwe will derived the correct best FPE from a perturbation approach. The methodfollowed and the results obtained have a general validity beyond the simplecase of exponentially correlated Gaussian driving used here as an example; theycan be applied even to non Gaussian drivings with a generic time correlation.

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