Splitting probabilities and mean first passage times across multiple thresholds of jump-and-drift transition paths

We apply stochastic-trajectory analysis to derive exact expressions for the mean first-passage times of jump-and-drift transition paths across two or more consecutive thresholds. We perform the analysis of the crossing statistics in terms of dimensionless quantities and show that, for particles starting between two thresholds, such statistics are directly related to the probability of not crossing one threshold and to the splitting probability of crossing the second one. We additionally derive a relationship for the mean first-passage time of the transition paths crossing two consecutive thresholds for particles starting outside them. The results are relevant to several physical and engineering applications including the case of flow discharge in fluvial environments, which is shown.