  The recent experimental progresses in handling microscopic systems have
allowed to probe them at levels where fluctuations are prominent, calling for
stochastic modeling in a large number of physical, chemical and biological
phenomena. This has provided fruitful applications for established stochastic
methods and motivated further developments. These systems often involve
processes taking place on widely separated time scales. For an efficient
modeling one usually focuses on the slower degrees of freedom and it is of
great importance to accurately eliminate the fast variables in a controlled
fashion, carefully accounting for their net effect on the slower dynamics. This
procedure in general requires to perform two different operations: decimation
and coarse-graining. We introduce the asymptotic methods that form the basis of
this procedure and discuss their application to a series of physical,
biological and chemical examples. We then turn our attention to functionals of
the stochastic trajectories such as residence times, counting statistics,
fluxes, entropy production, etc. which have been increasingly studied in recent
years. For such functionals, the elimination of the fast degrees of freedom can
present additional difficulties and naive procedures can lead to blatantly
inconsistent results. Homogenization techniques for functionals are less
covered in the literature and we will pedagogically present them here, as
natural extensions of the ones employed for the trajectories. We will also
discuss recent applications of these techniques to the thermodynamics of small
systems and their interpretation in terms of information-theoretic concepts.
