Effective Mori-Zwanzig equation for the reduced-order modeling of stochastic systems.

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.