随机动力系统的一个重要研究目标是考察不确定性如何随时间演化和发展，而这通常可以通过随机微分方程解过程的概率密度，即Fokker-Planck方程的解来研究。由于大部分随机系统相应的Fokker-Planck方程的闭形式解难以得到，因此通常需要做近似逼近。目前，关于高斯过程驱动系统方面的研究已经有很多结果，但非高斯过程方面还存在很多不足。首先，本项目拟考察Marcus意义下乘性Lévy过程驱动的Fokker-Planck方程，分析解的性质并设计数值方法，分析算法的稳定性和收敛性。然后，将一维情形推广到二维。最后，利用Fokker-Planck方程的近似解得到似然函数的闭形式，来研究稳定Lévy过程驱动下多尺度系统的投影积分法。本项目需要建立新的方法和技巧来处理非局部奇异积分、高维问题中的‘维数灾难’、极大似然估计等问题。

An important objective of stochastic dynamical systems is to quantify how uncertainty propagates and evolves. It can be achieved by the probability density of the solution for the stochastic differential equations, i.e. the solution of Fokker-Planck equation. The closed-form solution of Fokker-Planck equation exists only for a handful of cases, so we need to consider the numerical approximation. The Fokker-Planck equation driven by Gaussian processes obtained a number of achievements in recent years, but the research of the non-Gaussian cases is still in its infancy. Firstly, in this project we hope to consider the Fokker-Planck equation for Marcus stochastic differential equations with stable Lévy processes, analyze the properties of the solution, develop numerical schemes and investigate the stability and convergence. Then, we extend it to the two-dimensional case. Finally, using the numerical solution of Fokker-Planck equation we expect to get the closed-form of the likelihood, and consider the projective integration of multiscale systems driven by stable Lévy processes. To overcome the difficulties of nonlocal integral, the 'curse of dimensionality' of high dimensional systems, the maximum likelihood estimation and so on, we need to provide new methods and techniques.