Built upon the hypoelliptic analysis of the effective Mori-Zwanzig (EMZ) equation for observables of stochastic dynamical systems, we show that the obtained semigroup estimates for the EMZ equation can be used to drive prior estimates of the observable statistics for system in the equilibrium and non-equilibrium state. In addition, we introduce both first-principle and data-driven methods to approximate the EMZ memory kernel, and prove the convergence of the data-driven parametrization schemes using the regularity estimate of the memory kernel. The analysis results are validated numerically via the Monte-Carlo simulation of the Langevin dynamics for a Fermi-Pasta-Ulam chain model. With the same example, we also show the effectiveness of the proposed memory kernel approximation methods.

翻译：在对有效的Mori-Zwanzig(EMZ)等式进行观测随机动态系统观测结果的虚微分析的基础上,我们发现,获得的EMZ等式半组估计值可用于推动平衡和非平衡状态系统可观测统计数据的先前估计值。此外,我们引入了第一原则和数据驱动方法,以接近EMZ内核,并用记忆内核的规律性估计来证明数据驱动的平衡方案趋同。分析结果通过对Fermi-Pasta-Ulam链模型的Langevin动态模型的蒙特卡洛模拟进行数字验证。我们用同样的例子还展示了拟议的记忆内核近光方法的有效性。