  For one-dimensional Jump-Drift and Jump-Diffusion processes converging
towards some steady state, the large deviations of a long dynamical trajectory
are described from two perspectives. Firstly, the joint probability of the
empirical time-averaged density, of the empirical time-averaged current and of
the empirical time-averaged jump-flow are studied via the large deviations at
Level 2.5. Secondly, the joint probability of the empirical jumps and of the
empirical excursions between consecutive jumps are analyzed via the large
deviations at Level 2.5 for the alternate Markov chain that governs the series
of all the jump events of a long trajectory. These two general frameworks are
then applied to three examples of positive jump-drift processes without
diffusion, and to two examples of jump-diffusion processes, in order to
illustrate various simplifications that may occur in rate functions and in
contraction procedures.
