First-Passage-Time Asymmetry for Biased Run-and-Tumble Processes

Abstract: We explore first-passage phenomenology for biased active processes with a renewal-type structure, focusing in particular on paradigmatic run-and-tumble models in both discrete and continuous state spaces. In general, we show there is no symmetry between distributions of first-passage times to symmetric barriers positioned in and against the bias direction; however, we give conditions for such a duality to be restored asymptotically (in the limit of a large barrier distance) and highlight connections to the Gallavotti-Cohen fluctuation relation and the method of images. Our general trajectory arguments are supported by exact analytical calculations of first-passage-time distributions for asymmetric run-and-tumble processes escaping from an interval of arbitrary width, and these calculations are confirmed with high accuracy via extensive numerics. Furthermore, we quantify the degree of violation of first-passage duality using Kullback-Leibler divergence and signal-to-noise ratios associated with the first-passage times to the two barriers. We reveal an intriguing dependence of such measures of first-passage asymmetry on the underlying tumbling dynamics which may inspire inference techniques based on first-passage-time statistics in active systems.