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.

通过某些投影过程，可以将物理，化学，生物学，金融等领域中发生的许多复杂现象简化为目标变量的一维随机微分方程（SDE）。通常，此SDE既是非线性的又是非马尔可夫的，因此通常无法获得概率密度函数（PDF）的Fokker Planck方程（FPE）。但是，FPE是可取的，因为它是获取相关分析统计信息（例如固定PDF和首次通过时间）的主要工具。过去，许多作者已经解决了这个问题，但是由于不正确使用交互作用图片（获得降低的FPE的标准工具），先前的理论结果是不正确的，这已通过SDE的直接数值模拟得到了证实。陷阱在于强耗散流的轨迹向后演化的快速发散行为。通常，我们将展示如何解决该问题，并且将从累积量扰动方法中得出正确的最佳FPE。除了这里以指数相关的高斯驱动为例的简单情况之外，遵循的特定摄动方法还对获得的结果提供了总体有效性：它甚至可以应用于具有一般时间相关性的非高斯驱动。