  We search for steady states in a class of fluctuating and driven physical
systems that exhibit sustained currents. We find that the physical concept of a
steady state, well known for systems at equilibrium, must be generalised to
describe such systems. In these, the generalisation of a steady state is
associated with a stationary probability density of micro-states and a
deterministic dynamical system whose trajectories the system follows on
average. These trajectories are a manifestation of non-stationary macroscopic
currents observed in these systems. We determine precise conditions for the
steady state to exist as well as the requirements for it to be stable. We
illustrate this with some examples.
