2025/09/14 by Grange, Pascal, Yuan, Linglong
#FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.2509.11308
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. The Laplace transforms of these equations induce evolution equations for the exit probabilities and the 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 tend to infinity 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.