Forward-backward correspondence between stationary structure and splitting probabilities in active matter

Abstract

Active particles confined by hard walls accumulate at boundaries and may become dynamically adsorbed due to directional persistence. In this work, we show that the same persistence mechanism also gives rise to a finite wall splitting probability, meaning that a particle initialized at a wall can reach the opposite boundary before returning to its starting point. By comparing forward and backward evolution equations directly in position--velocity phase space, we derive exact relations linking stationary distributions and splitting probabilities for run-and-tumble, active Brownian, and active Ornstein--Uhlenbeck particles. In particular, we show that the stationary density is generated by the spatial derivative of the splitting probability, while the distribution of dynamically adsorbed particles at the walls is encoded in wall splitting probabilities. The correspondence is valid in arbitrary spatial dimension and establishes an exact bridge between stationary and first-passage descriptions of confined active matter, revealing them as complementary representations of the same persistence-driven dynamics.