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.