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We study classical stochastic systems with discrete states, coupled toswitching external environments. For fast environmental processes we derivereduced dynamics for the system itself, focusing on corrections to theadiabatic limit of infinite time scale separation. In some cases, this leads tomaster equations with negative transition `rates' or bursting events. We devisea simulation algorithm in discrete time to unravel these master equations intosample paths, and provide an interpretation of bursting events. Focusing onstochastic population dynamics coupled to external environments, we discuss aseries of approximation schemes combining expansions in the inverse switchingrate of the environment, and a Kramers--Moyal expansion in the inverse size ofthe population. This places the different approximations in relation toexisting work on piecewise-deterministic and piecewise-diffusive Markovprocesses. We apply the model reduction methods to different examples includingsystems in biology and a model of crack propagation.

physical review. e

Model reduction methods for classical stochastic systems with  fast-switching environments: reduced master equations, stochastic  differential equations, and applications