Dimension reduction via timescale separation in stochastic dynamical systems

We review the development and applications of a systematic framework for dimension reduction in stochastic dynamical systems that exhibit a separation of timescales. When a multidimensional stochastic dynamical system possesses quantities that are approximately conserved on short timescales, it is common to observe on long timescales that trajectories remain close to some lower-dimensional subspace. Here, we derive explicit and general formulae for a reduced-dimension description of such a process that is exact in the limit of small noise and well-separated slow and fast dynamics. The Michaelis-Menten law of enzyme-catalyzed reactions is explored as a worked example. Extensions of the method are presented for infinite dimensional systems and processes coupled to general noise sources.