Diffusion models arising in analysis of large biochemical models and other complex systems are typically far too complex for exact solution or even meaningful simulation. The purpose of this paper is to develop foundations for model reduction and new modeling techniques for diffusion models.
These foundations are all based upon the recent spectral theory of Markov processes. The main assumption imposed is $V$-uniform ergodicity of the process. This is equivalent to any common formulation of exponential ergodicity and is known to be far weaker than the Donsker--Varadahn conditions in large deviations theory. Under this assumption it is shown that the associated semigroup admits a spectral gap in a weighted $L_\infty$-norm and 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 represent a significant expansion of the classical Wentzell--Freidlin theory. In particular, the results require no special structure beyond geometric ergodicity; reversibility is not assumed and meaningful conclusions can be drawn even for models with significant variability.
"Phase transitions and metastability in Markovian and molecular systems." Ann. Appl. Probab. 14 (1) 419 - 458, February 2004. https://doi.org/10.1214/aoap/1075828057