  It is a well established result that, in classical dynamical systems with
sufficient time-scale separation, the fast chaotic degrees of freedom are well
modeled by (Gaussian) white noise. In this paper, we present the stochastic
dynamical description for intermediate time-scale motions with insufficient
time-scale separation from the slow dynamical system. First, we analyze how the
fast deterministic dynamics can be viewed as stochastic dynamics under
experimental observation by intrinsic errors of measurement. Then, we present
how the stochastic dynamical description should be modified if intermediate
time-scale motions exist: the time correlation of the noise \xi is modified to
<\xi(t)\xi(t')> = C(x,p)\delta(t-t'), where C(x,p) is a smooth function of the
slow coordinate (x,p), and generally the cumulants of \xi except its average
vary as a smooth function of the slow coordinates (x,p). The analysis given in
this work actually shows that, regardless of the sufficiency of time-scale
separation, any complex (chaotic and ergodic) dynamical system can be well
described using Markov process, if we perfectly construct the deterministic
part of (extended) stochastic dynamics.
