  An analytical expression is derived for the transition path time distribution
for a one-dimensional particle crossing of a parabolic barrier. Two cases are
analyzed: (i) A non-Markovian process described by a generalized Langevin
equation with a power-law memory kernel and (ii) a Markovian process with a
noise violating the fluctuation-dissipation theorem, modeling the stochastic
dynamics generated by active forces. In the case (i) we show that the anomalous
dynamics strongly affecting the short time behavior of the distributions, but
this happens only for very rare events not influencing the overall statistics.
At long times the decay is always exponential, in disagreement with a recent
study suggesting a stretched exponential decay. In the case (ii) the active
forces do not substantially modify the short time behavior of the distribution,
but lead to an overall decrease of the average transition path time. These
findings offer some novel insights, useful for the analysis of experiments of
transition path times in (bio)molecular systems.
