  We develop a thorough mathematical analysis of the effective Mori-Zwanzig
(EMZ) equation governing the dynamics of noise-averaged observables in
stochastic differential equations driven by multiplicative Gaussian white
noise. Building upon recent work on hypoelliptic operators, we prove that the
EMZ memory kernel and fluctuation terms converge exponentially fast in time to
a unique equilibrium state which admits an explicit representation. We apply
the new theoretical results to the Langevin dynamics of a high-dimensional
particle system with smooth interaction potential.
