Fluctuation of stochastic systems with average equilibrium point (Q2462098)

A dynamical system whose velocity field is perturbed by a rapidly varying continuous time Markov chain is considered. The velocity has a slow and a fast component. The latter, along with the aforementioned Markov chain dynamics, is parametrized by a small positive parameter. The velocity fireld averaged over the stationary distribution of the Markov chain is assumed to have a unique equilibrium at zero and a single time scale, i.e., the fast component averages out to zero. Under these conditions, as the small parameter tends to zero, the joint process defined by the a scaled time integral of the fast component of the velocity and the associated fluctuation process is shown to converge in law to a diffusion whose extended generator is characterized.