  The transition mechanism of jump processes between two different subsets in
state space reveals important dynamical information of the processes and
therefore has attracted considerable attention in the past years. In this
paper, we study the first passage path ensemble of both discrete-time and
continuous-time jump processes on a finite state space. The main approach is to
divide each first passage path into nonreactive and reactive segments and to
study them separately. The analysis can be applied to jump processes which are
non-ergodic, as well as continuous-time jump processes where the waiting time
distributions are non-exponential. In the particular case that the jump
processes are both Markovian and ergodic, our analysis elucidates the relations
between the study of the first passage paths and the study of the transition
paths in transition path theory. We provide algorithms to numerically compute
statistics of the first passage path ensemble. The computational complexity of
these algorithms scales with the complexity of solving a linear system, for
which efficient methods are available. Several examples demonstrate the wide
applicability of the derived results across research areas.
