Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle

The connection between absorbing boundary conditions and hard walls is well established in the mathematical literature for a variety of stochastic models, including for instance the Brownian motion. In this paper we explore this duality for a different type of process which is of particular interest in physics and biology, namely the run-tumble-particle, a toy model of active particle. For a one-dimensional run-and-tumble particle subjected to an arbitrary external force, we provide a duality relation between the exit probability, i.e. the probability that the particle exits an interval from a given boundary before a certain time $t$, and the cumulative distribution of its position in the presence of hard walls at the same time $t$. We show this relation for a run-and-tumble particle in the stationary state by explicitly computing both quantities. At finite time, we provide a derivation using the Fokker-Planck equation. All the results are confirmed by numerical simulations.