We consider a one-dimensional run-and-tumble particle, or persistent random walk, in the presence of an absorbing boundary located at the origin. After each tumbling event, which occurs at a constant rate γ, the (new) velocity of the particle is drawn randomly from a distribution W(v). We study the survival probability S(x, t) of a particle starting from x ⩾ 0 up to time t and obtain an explicit expression for its double Laplace transform (with respect to both x and t) for an arbitrary velocity distribution W(v), not necessarily symmetric. This result is obtained as a consequence of Spitzer’s formula, which is well known in the theory of random walks and can be viewed as a generalization of the Sparre Andersen theorem. We then apply this general result to the specific case of a two-state particle with velocity v ₀, the so-called persistent random walk (PRW), and in the presence of a constant drift μ and obtain an explicit expression for S(x, t), for which we present more detailed results. Depending on the drift μ, we find a rich variety of behaviors for S(x, t), leading to three distinct cases: (i) subcritical drift −v ₀ <μ < v ₀, (ii) supercritical drift μ < −v ₀ and (iii) critical drift μ = −v ₀. In these three cases, we obtain exact analytical expressions for the survival probability S(x, t) and establish connections with existing formulae in the mathematics literature. Finally, we discuss some applications of these results to record statistics and to the statistics of last-passage times.

我们考虑在原点存在吸收边界的情况下的一维奔跑和翻滚粒子或持续随机游走。在以恒定速率γ发生的每个翻滚事件之后，粒子的（新）速度从分布W ( v ) 中随机抽取。我们研究从x ⩾ 0 到时间t的粒子的生存概率S ( x , t )并获得其双拉普拉斯变换（关于x和t）的显式表达式，用于任意速度分布W ( v)，不一定是对称的。这个结果是作为斯皮策公式的结果获得的，该公式在随机游走理论中是众所周知的，可以看作是对 Sparre Andersen 定理的推广。然后，我们将这个一般结果应用于速度为v ₀的二态粒子的特定情况，即所谓的持久随机游走 (PRW)，并且在存在恒定漂移μ的情况下，并获得S ( x , t )，我们提供了更详细的结果。根据漂移μ，我们发现S ( x , t ) 的多种行为，导致三种不同的情况：(i)亚临界漂移 - v ₀ < μ < v ₀，(ii)超临界漂移μ < - v ₀和 (iii)临界漂移μ = - v ₀。在这三种情况下，我们获得了生存概率S ( x , t ) 的精确解析表达式，并与数学文献中的现有公式建立了联系。最后，我们讨论了这些结果在记录统计数据和最后一次通道时间统计方面的一些应用。