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Built upon the hypoelliptic analysis of the effective Mori-Zwanzig (EMZ)equation for observables of stochastic dynamical systems, we show that theobtained semigroup estimates for the EMZ equation can be used to drive priorestimates of the observable statistics for system in the equilibrium andnon-equilibrium state. In addition, we introduce both first-principle anddata-driven methods to approximate the EMZ memory kernel, and prove theconvergence of the data-driven parametrization schemes using the regularityestimate of the memory kernel. The analysis results are validated numericallyvia the Monte-Carlo simulation of the Langevin dynamics for a Fermi-Pasta-Ulamchain model. With the same example, we also show the effectiveness of theproposed memory kernel approximation methods.

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