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

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 wall before returning to its starting point. By comparing forward and backward evolution equations 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 equals the spatial derivative of the splitting probability, while the fraction of adsorbed particles is encoded in the splitting probabilities evaluated at the walls. The correspondence is valid in arbitrary spatial dimensions and establishes an exact bridge between stationary and first-passage descriptions of confined active matter, revealing stationary and first-passage observables as complementary representations of the same persistence-driven dynamics.