Survival probability of a run-and-tumble particle in the presence of a drift

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 behaviours 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.

Run-and-tumble motion in a harmonic potential: field theory and entropy production

“…The formalism presented in section 2 can be generalised to anisotropic self-propulsion and asymmetric tumbling rates, which has applications in the context of gene regulation [29][30][31][32]. First, we consider the case where the self-propulsion velocities are w + Ω for right-moving particles and −w + Ω for left-moving particles, similar to a persistent random walk [56,57]. The coupled Fokker-Planck equations in (3) then read…”

Condensation transition in the late-time position of a Run-and-Tumble particle

“…This method has been used to compute several observables, e.g. the survival probability, of a fixed-T RTP [45,46,52,55]. For the sake of simplicity, we will henceforth focus on the model where the speed v 0 of the particle is kept fixed.…”

“…Note that in the canonical and perhaps the most well studied RTP model, the speed of the particle is a constant v 0 > 0 and does not vary from one run to another, corresponding to the choice W (v) = δ(v − v 0 ). Nevertheless, RTP models with generic W (v) have also been studied [41,45,46,55].…”

Generalized run-and-tumble model in 1D geometry for an arbitrary distribution of drift velocities

“…Continuous distributions P (v) naturally arise when particles, in dimension d > 1, are confined between two walls. Similar versions of the RTP model to a continuum of states have already been considered in the past in the study of survival probabilities [35,38].…”