This talk provides a survey of recent approaches to the spectral theory of Markov processes, with application to model reduction and simulation. The lecture will focus on diffusion models on finite-dimensional Euclidian space, with hypo-elliptic generator.
Real eigenfunctions provide a decomposition of the state space into 'almost'-absorbing subsets. It is shown that the process mixes rapidly in each of these subsets prior to exiting, and that the conditional distributions of exit times are approximately exponential. These results may be viewed as an extension of the perturbation theory of Wentzell and Freidlin.
References:
I. Kontoyiannis and S.P. Meyn, Spectral Theory and Limit Theory for Geometrically Ergodic Markov Processes, Annals of Applied Prob., Volume 13, pp. 304-362, 2003.
I. Kontoyiannis and S.P. Meyn, Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes, Submitted for publication, 2003.
W. Huisinga, S. P. Meyn, and C. Schuette. Phase Transitions & Metastability in Markovian and Molecular Systems. Annals of Applied Probability, Volume 14, No. 1, pp. 419-458, 2004