  We recently showed that the dynamics of coarse-grained observables in systems
out of thermal equilibrium are governed by the non-stationary generalized
Langevin equation [J. Chem. Phys. 147, 214110 (2017), J. Chem. Phys. 150,
174118 (2019)]. The derivation we presented in these two articles was based on
the assumption that the dynamics of the microscopic degrees of freedom was
deterministic. Here we extend the discussion to stochastic microscopic
dynamics. The fact that the non-stationary Generalized Langevin Equation also
holds for stochastic processes implies that methods designed to estimate the
memory kernel, drift term and fluctuating force term of this equation as well
as methods designed to propagate it numerically, can be applied to data
obtained in molecular dynamics simulations that employ a stochastic thermostat
or barostat.
