Fluctuation relations and coarse-graining

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.