The one-dimensional run-and-tumble particle (RTP) is one of the simplest examples of active matter. The persistent nature of an RTP can lead to novel phenomena such as accumulation at walls in the absence of attractive
particle interactions and motility-based phase separation. Most theoretical studies of RTPs are based on the analysis of the forward or backward Kolmogorov
equation, whose solution determines the probability distribution of sample paths. The effects of a confining wall are typically implemented by supplementing the
Kolmogorov equation with some form of non-sticky or sticky boundary condition. However, from a biological perspective, one would like to develop more bio physically motivated models of microorganism-boundary interactions. A starting point for such an approach is to develop a probabilistic theory that allows one to incorporate boundary conditions at the level of individual sample paths.We have previously shown how to achieve this for both non-sticky and sticky partially absorbing boundaries. In this paper we extend the theory to include
the effects of diffusion. One of the non-trivial consequences of combining drift-diffusion with tumbling is that the zero diffusion limit D → 0 is singular in the sense that the number of boundary conditions is doubled when D > 0. We use stochastic calculus to derive the forward Kolmogorov equation for two distinct boundary conditions that reduce, respectively, to non-sticky and sticky boundary conditions in the zero-diffusion limit. In the latter case, it is necessary to include a boundary layer in a neighbourhood of the wall and use singular perturbation theory. We also treat the wall as partially absorbing by assuming that the particle is absorbed when the amount of boundary-particle contact time (discrete or continuous local time) exceeds a quenched random threshold. Finally, we analyse the survival probability and corresponding first-passage time density for absorption by deriving the corresponding backward Kolmogorov equation.