This article is concerned with the averaging principle and its extensions
for stochastic dynamical systems with fast and slow degrees of
freedom. It is demonstrated how the \conventional" averaging principle
results from asymptotic multiscale analysis, how one can construct
an indicator for its (in-)appropriateness, and how, if inappropriate, it
may be extended into an improved approximation. The conventional
scheme contains averages over the entire accessible state space of the
fast degrees of freedom and may thus fail if these fast degrees of freedom
exhibit long-term (auto-)correlations. In contrast, the improved
scheme combines several conditional averages with a Markov jump process
that is designed to represent the
ipping process between the
conditional averages and thus incorporates the important long-term
correlations. All important steps of the derivation are illustrated by
numerical experiments. Application to problems from molecular dynamics
is discussed.
The stochastic dynamics of a well-stirred mixture of molecular species
interacting through different biochemical reactions can be
accurately modelled by the chemical master equation (CME). Research in
the biology and scientific computing community has
concentrated mostly on the development of numerical techniques to
approximate the solution of the CME via many realizations of the associated
Markov jump process. The domain of exact and/or efficient methods for
directly solving the CME is still widely open, which is due to its
large dimension that grows exponentially with the number of molecular
species involved. In this article, we present an exact solution
formula of the CME for arbitrary initial conditions in the case where
the underlying system is governed by monomolecular reactions. The
solution can be expressed in terms of the convolution of multinomial
and product Poisson distributions with time-dependent parameters
evolving according to the traditional reaction-rate equations. This
very structured representation allows to deduce any property of the
solution. The model class includes many interesting examples and may
also be used as the starting point for the design of new numerical
methods for the CME of more complex reaction systems.