First passage of a run-and-tumble particle with exponentially-distributed tumble duration in the presence of a drift

Abstract We consider a run-and-tumble particle on a finite interval [ a , b ] with two absorbing end points. The particle has an internal velocity state that switches between three values v , 0 , − v at exponential times, thus incorporating positive tumble times. Moreover, a constant drift is added to the run-and-tumble motion at all times. The combination of these two features constitutes the main novelty of our model. The densities of the first-passage time through a (given the initial position and velocity states) satisfy certain forward Fokker–Planck equations, whose Laplace transforms induce evolution equations for the exit probabilities and mean first-passage times of the particle. We solve these equations explicitly for all possible initial states. We consider the limiting regimes of instantaneous tumble and/or the limit of large b to confirm consistency with existing results in the literature. In particular, in the limit of a half-line (large b ), the mean first-passage time conditioned on the exit through a is an affine function of the initial position if the drift is positive, as in the case of instantaneous tumble.