Survival probability of a run-and-tumble particle in the presence of a drift
Benjamin De Bruyne, Satya N Majumdar, Grégory Schehr
Abstract
Abstract 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.