  We consider the fluctuations of generalized currents in stochastic Markovian
dynamics. The large deviations of current fluctuations are shown to obey a
Gallavotti-Cohen (GC) type symmetry in systems with a finite state space.
However, this symmetry is not guaranteed to hold in systems with an infinite
state space. A simple example of such a case is the Zero-Range Process (ZRP).
Here we discuss in more detail the already reported breakdown of the GC
symmetry in the context of the ZRP with open boundaries and we give a physical
interpretation of the phases that appear. Furthermore, the earlier analytical
results for the single-site case are extended to cover multiple-site systems.
We also use our exact results to test an efficient numerical algorithm of
Giardina, Kurchan and Peliti, which was developed to measure the current large
deviation function directly. We find that this method breaks down in some
phases which we associate with the gapless spectrum of an effective
Hamiltonian.
