Title: Mean first-passage time of a run-and-tumble particle with exponentially-distributed tumble duration in the presence of a drift Show Chinese title

Title: 有漂移情况下具有指数分布停顿时间的跑跳粒子的平均首次通过时间

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. 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. Show Chinese abstract

Abstract: 我们考虑一个在有限区间$[a,b]$上的跑动-翻转粒子，其两个端点为吸收点。 该粒子具有一个内部速度状态，在指数时间切换为三个值$v,0,-v$，从而包含正的翻转时间。 此外，在所有时间都向跑动-翻转运动中添加了一个恒定的漂移。 这两个特征的结合构成了我们模型的主要创新点。 首次通过时间通过$a$的密度（给定初始位置和速度状态）满足某些前向 Fokker--Planck 方程。 这些方程的拉普拉斯变换诱导了粒子的退出概率和平均首次通过时间的演化方程。 我们对所有可能的初始状态显式求解了这些方程。 我们考虑瞬时翻转的极限情况和/或大$b$趋于无穷大的极限，以确认与文献中现有结果的一致性。 特别是，在半直线的极限（大$b$）下，当漂移为正时，条件退出通过$a$的平均首次通过时间是初始位置的线性函数，如同瞬时翻转的情况一样。