On Random Dynamical Systems and Levels of Their Description

We consider dynamical systems in which a (typically vector-valued) dependent variable evolves according to autonomous dynamics switch-ing randomly according to Markovian laws that change with the value of the dependent variable. Such systems are known as “random evo-lutions ” or, in electrical engineering contexts, as “switching systems”. Systems of this type are encountered in applications from electrical engineering to cell biology (our paper was inspired by a recent model for genetic oscillators). We review the derivation of the forward Kol-mogorov master equations for the probabilities to find the system in a certain state at some time. In the limit of an infinite switching rate solutions of our system converge almost surely to solutions of an av-eraged problem. The classical tool of a Chapman-Enskog expansion, well-known from kinetic theory, provides diffusion approximations to first order in the scale parameter. Our work focusses on a few typical examples. The analysis is formal, but we expect that the results hold rigorously in analogy to earlier results from kinetic theory (see [11])