Virtual Slow Manifolds: The Fast Stochastic Case

The persistently fast evolutionary behavior of certain differential systems may have intrinsically slow features. We consider fast stochastic systems whose solution trajectories are slowly changing distributions and assume that we do not have access to the equations of the system, only to a simulator or legacy code that performs step-wise time integration of the system. We characterize the set of all possible instantaneous solutions by points on a low-dimensional virtual slow manifold (VSM) and show how, when there is a sufficiently large gap between the time scales of the fast and slow behaviors, we can restrict the fast behavior observed numerically through simulation to the VSM and perform important computations there.