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Systems composed of large numbers of interacting agents often admit aneffective coarse-grained description in terms of a multidimensional stochasticdynamical system, driven by small-amplitude intrinsic noise. In applications tobiological, ecological, chemical and social dynamics it is common for thesemodels to posses quantities that are approximately conserved on shorttimescales, in which case system trajectories are observed to remain close tosome lower-dimensional subspace. Here, we derive explicit and general formulaefor a reduced-dimension description of such processes that is exact in thelimit of small noise and well-separated slow and fast dynamics. TheMichaelis-Menten law of enzyme-catalyzed reactions, and the link between theLotka-Voltera and Wright-Fisher processes are explored as a simple workedexamples. Extensions of the method are presented for infinite dimensionalsystems and processes coupled to non-Gaussian noise sources.

Dimension reduction for stochastic dynamical systems forced onto a  manifold by large drift: a constructive approach with examples from  theoretical biology