Moderate deviations for the Langevin equations: Strong damping and fast Markovian switching

In this paper, we obtain a moderate deviations principle (MDP) for a class of Langevin dynamic systems with a strong damping and fast Markovian switching. To facilitate our study, first, analysis of systems with bounded drifts is dealt with. To obtain the desired moderate deviations, the exponential tightness of the solution of the Langevin equation is proved. Then, the solution of its first-order approximation using local MDPs is examined. Finally, the MDPs are established. To enable the treatment of unbounded drifts, a reduction technique is presented near the end of the paper, which shows that Lipschitz continuous drifts can be dealt with.