A simple hypocoercivity analysis for the effective Mori-Zwanzig equation

We provide a simple hypocoercivity analysis for the effective Mori-Zwanzig equation governing the time evolution of noise-averaged observables in a stochastic dynamical system. Under the hypocoercivity framework mainly developed by Dolbeault, Mouhot and Schmeiser and further extended by Grothaus and Stilgenbauer, we prove that under the same conditions which lead to the geometric ergodicity of the Markov semigroup $e^{tK}$, the Mori-Zwanzig orthogonal semigroup $e^{tQKQ}$ is also geometrically ergodic, provided that $P=I-Q$ is a finite-rank, orthogonal projection operator in a certain Hilbert space. The result is applied to the widely used Mori-type effective Mori-Zwanzig equations in the coarse-grained modeling of molecular dynamics and leads to exponentially decaying estimates for the memory kernel and the fluctuation force.