  We discuss applications of a recently developed method for model reduction
based on linear response theory of weakly coupled dynamical systems. We apply
the weak coupling method to simple stochastic differential equations with slow
and fast degrees of freedom. The weak coupling model reduction method results
in general in a non-Markovian system, we therefore discuss the Markovianization
of the system to allow for straightforward numerical integration. We compare
the applied method to the equations obtained through homogenization in the
limit of large time scale separation between slow and fast degrees of freedom.
We look at ensemble spread from a fixed initial condition, correlation
functions and exit times from domain. The weak coupling method gives better
results in all test cases, albeit with a higher numerical cost.
