On complex spectra and metastability of Markov models

The purpose of this paper is to develop methods for model reduction for diffusion processes that exhibit cyclic behavior. For this purpose we extend techniques based on the spectral theory of Markov processes to the case of complex spectra. The main idea is to augment the state process for the diffusion with a clock process. For each complex eigenvalue for the original diffusion there exists a real eigenvalue for the augmented process. Results concerning metastability (or quasi-stationarity) are then applied to the augmented process. The results are illustrated through a linear diffusion, and an empirical model of combustion dynamics.