Uphill and downhill first passage of an active Brownian particle: Asymmetry and exact path reweighting
First passage processes in active systems combine stochastic transport with self-propulsion and orientational persistence, making motion along and against an external bias sensitive to the internal active dynamics. For passive biased diffusion, opposite exits can have different splitting probabilities while their conditional first passage time distributions remain identical. We study how this relation changes for an active Brownian particle driven by a constant external force between two absorbing boundaries. Self-propulsion breaks the equality of the uphill and downhill first passage time distributions and modifies the splitting probabilities. Nevertheless, the two directional path ensembles remain exactly related by spatial reflection, which pairs downhill and uphill first passage paths of the same duration while preserving their orientational history. The log-ratio of the probabilities of a path and its reflected partner defines a path-dependent asymmetry functional and provides an exact reweighting between the two ensembles. In the symmetric half-weighted representation, the uphill and downhill first passage time distributions coincide for arbitrary orientational persistence. The same path relation also allows rare uphill statistics to be reconstructed from the more frequently sampled downhill trajectories. Numerical simulations confirm the weighted equality across the explored bias and persistence regimes, while perturbative and asymptotic analyses clarify how orientational persistence produces the directional asymmetry of the unweighted statistics. The exact relation between the two directional path ensembles suggests that similar symmetry-based reconstruction protocols may be found in other nonequilibrium first-passage problems.