  Systems composed of large numbers of interacting agents often admit an
effective coarse-grained description in terms of a multidimensional stochastic
dynamical system, driven by small-amplitude intrinsic noise. In applications to
biological, ecological, chemical and social dynamics it is common for these
models to posses quantities that are approximately conserved on short
timescales, in which case system trajectories are observed to remain close to
some lower-dimensional subspace. Here, we derive explicit and general formulae
for a reduced-dimension description of such processes that is exact in the
limit of small noise and well-separated slow and fast dynamics. The
Michaelis-Menten law of enzyme-catalyzed reactions, and the link between the
Lotka-Voltera and Wright-Fisher processes are explored as a simple worked
examples. Extensions of the method are presented for infinite dimensional
systems and processes coupled to non-Gaussian noise sources.
