  We consider the application of fluctuation relations to the dynamics of
coarse-grained systems, as might arise in a hypothetical experiment in which a
system is monitored with a low-resolution measuring apparatus. We analyze a
stochastic, Markovian jump process with a specific structure that lends itself
naturally to coarse-graining. A perturbative analysis yields a reduced
stochastic jump process that approximates the coarse-grained dynamics of the
original system. This leads to a non-trivial fluctuation relation that is
approximately satisfied by the coarse-grained dynamics. We illustrate our
results by computing the large deviations of a particular stochastic jump
process. Our results highlight the possibility that observed deviations from
fluctuation relations might be due to the presence of unobserved degrees of
freedom.
