  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.
