# Statistical Analysis of the First Passage Path Ensemble of Jump Processes

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## Abstract

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.

## Keywords

Jump process Non-ergodic process Non-exponential distribution First passage path Transition path theory## Notes

### Acknowledgements

This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through Grant CRC 1114, by the Einstein Foundation Berlin through Project CH4 of Einstein Center for Mathematics (ECMath) and though the BMBF, Grant Number 031A307.

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