Large deviations for multi-scale Markovian switching systems with a small diffusion

This work develops asymptotic properties of randomly switching and time inhomogeneous dynamic systems under Brownian perturbation with a small diffusion. The switching process is modeled by a continuous-time Markov chain, which portraits discrete events that cannot be modeled by a diffusion process. In the model, there are two small parameters. One of them is ε associated with the generator of the continuoustime, inhomogeneous Markov chain, and the other is δ = δε signifies the small intensity of the diffusion. Assume ε → 0 and δε → 0 as ε → 0. This paper focuses on large deviations type of estimates for such Markovian switching systems with small diffusions. The ratio ε/δε can be a nonzero constant, or equal to 0, or ∞. These three different cases yield three different outcomes. This paper analyzes the three cases and present the corresponding asymptotic properties.