# Survival probability and position distribution of a run and tumble particle in $U ( x ) = \alpha | x |$ potential with an absorbing boundary

Sujit Kumar Nath, Sanjib Sabhapandit

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Sujit Kumar Nath, Sanjib Sabhapandit. Survival probability and position distribution of a run and tumble particle in $U ( x ) = \alpha | x |$ potential with an absorbing boundary. Journal of Statistical Mechanics: Theory and Experiment, 2024, 2024 (9), pp.093205. 10.1088/1742-5468/ad6c2c. hal-04933342

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Survival probability and position distribution of a
run and tumble particle in $U(x) = α|x|$ potential with an
absorbing boundary

To cite this article: Sujit Kumar Nath and Sanjib Sabhapandit *J. Stat. Mech*. (2024) 093205

View the [article online](#) for updates and enhancements.

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# Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential with an absorbing boundary

Sujit Kumar Nath<sup>1,2,3,*</sup> and Sanjib Sabhapandit<sup>4,*</sup>

1 Aix Marseille Univ, Université de Toulon, CNRS, Turing Center for Living Systems, CPT (UMR 7332), Marseille, France

2 Division of Cardiovascular Sciences, Faculty of Biology, Medicine and Health, The University of Manchester, M13 9PT, Manchester, United Kingdom

3 School of Computing and Faculty of Biological Sciences, University of Leeds, LS2 9JT, Leeds, United Kingdom

4 Raman Research Institute, Bangalore 560080, India

E-mail: sujit-kumar.nath@univ-amu.fr and sanjib@rri.res.in

Received 2 June 2024

Accepted for publication 29 July 2024

Published 17 September 2024

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CrossMark

Online at stacks.iop.org/JSTAT/2024/093205

https://doi.org/10.1088/1742-5468/ad6c2c

**Abstract.** We study the late-time exponential decay of the survival probability $S_±(t, a|x_0) ∼ e^{-θ(a)t}$, of a one-dimensional run-and-tumble particle starting from $x_0 < a$ with an initial orientation $σ(0) = ±1$, under a confining potential $U(x) = α|x|$ with an absorbing boundary at $x = a > 0$. We find that the decay rate $θ(a)$ of the survival probability has a strong dependence on the location $a$ of the absorbing boundary, which undergoes a freezing transition at a critical value $a = a_c = (v_0 - α)\sqrt{v_0^2 - α^2/(2αγ)}$, where $v_0 > α$ is the self-propulsion speed and $γ$ is the tumbling rate of the particle. For $a > a_c$, the value of $θ(a)$ increases monotonically from zero, as $a$ decreases from infinity, until it attains the maximum value $θ(a_c)$ at $a = a_c$. For $0 < a < a_c$, the value of $θ(a)$ freezes to the value

<figure><img src="image_3.png" /></figure>

<footnote>* Authors to whom any correspondence should be addressed.</footnote>
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

$\theta(a) = \theta(a_c)$. We also obtain the propagator with the absorbing boundary condition at $x = a$. Our analytical results are supported by numerical simulations.

**Keywords:** driven diffusive systems, first passage, stochastic particle dynamics

## Contents

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>2. Run and tumble particle in U(x) = α|x| potential</td><td>4</td></tr><tr><td>2.1. Survival probability</td><td>6</td></tr><tr><td>2.2. Leading order decay of the survival probability</td><td>8</td></tr><tr><td>3. Comparison of the decay rate θ(a) and the inverse of the mean first passage time T<sub>±</sub>(a)</td><td>11</td></tr><tr><td>3.1. Estimation of 1/T<sub>±</sub>(a) and θ(a) for a ≫ a<sub>c</sub></td><td>13</td></tr><tr><td>4. The propagator</td><td>14</td></tr><tr><td>5. Conclusion</td><td>18</td></tr><tr><td>Acknowledgments</td><td>19</td></tr><tr><td>Appendix A. Details of calculations of the survival probability in section 2.1</td><td>19</td></tr><tr><td>Appendix B. Details of calculations of the propagator in section 4</td><td>20</td></tr><tr><td>Appendix C. Survival probability by integrating the propagator</td><td>22</td></tr><tr><td>Appendix D. Contour integration for evaluating the survival probability in (x, t) space</td><td>23</td></tr><tr><td>References</td><td>25</td></tr></table>

## 1. Introduction

The applications of target-search problem can be found across disciplines, such as physics [1–9], ecology [10, 11], cellular processes in biology [12–15], computer science [16] etc. In this context, the time at which the searcher reaches the target for the first time, known as the first passage time (FPT), is a relevant quantity. The probability density function (PDF) $F(t, a|x_0)$ of the FPT for a target located at $x = a$ and its cumulative distribution $S(t, a|x_0) = \int_t^\infty F(t', a|x_0)dt'$, known as the survival probability with an absorbing boundary at $x = a$, starting from $x_0$, are two of the most well-studied quantities in random processes. Animals, bacteria, and several other species forage for their
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

foods, mates or shelters in a random fashion [17]. Therefore, the natural questions of interest are ‘How long does a forager need to encounter its target?’, ‘How does this hitting time depend on the location of the target?’, etc. Also, it is quite naturally observed that a forager always searches for its target starting from a home-location and remains confined in a territory around it. Even for a few animal species, such as the desert ants, being confined in a territory around their nests, while searching for food, is so crucial that any violation of this condition may be life threatening for them [18]. Therefore, there is always a pull on the forager toward its home-territory, which influences the forager not to go too far from its home while executing the search process. One simple way to model such random movements executed by a forager, is to consider the run and tumble dynamics, where the forager chooses its direction of movement randomly and moves in that direction at a constant speed [19–22]. The pull toward its home can be modelled by a confining potential having its minimum at the location of the forager’s nest or starting location [23–29].

The study of first passage problems was started in the time of Schrödinger and Smoluchowski [30, 31]. Later, Kramers studied the escape rate of a Brownian particle over a potential barrier under the influence of thermal fluctuations by computing the flux across the barrier, and found that it is well approximated by the van’t Hoff–Arrhenius form at low temperature and high barrier height limit [32–34]. It has also been found that the other alternative approaches using FPT and survival probability can be useful to study such escape rates [35–39]. Since then, the study of FPT and survival probability has been extensively developed. Although the mean first passage time (MFPT) has been extensively studied in various contexts [40–42], the full distribution of the FPT, or equivalently, the survival probability carries much more information. However, the computation of the FPT distribution and the survival probability are much harder compared to the MFPT [38, 43–46].

It is evident that the distribution of the FPT or the survival probability must depend on the distance between the location of the absorbing barrier and the starting point. Therefore, understanding the dependence of the FPT and survival probability on the position of the absorbing barrier is of immense importance for a clear understanding of the random search problem. For a Brownian particle in a ‘sufficiently’ confining potential, generically, $S(t, a|0) \sim e^{-\theta(a)t}$, where $x=0$ being the starting point of the particle as well as the minima of the potential. Recently, it has been found that for potentials having the property $U(x) \sim |x|$ when $x \to -\infty$, the decay rate $\theta(a)$ undergoes a freezing transition depending on the position of the barrier $a$ [47]. Above a critical value $a = a_c$, $\theta(a)$ decreases with increasing value of $a$. However, $\theta(a)$ freezes to a constant value below a critical value $a_c$. Other than the decay rate, freezing transitions of other parameters such as the strength of the confining potential which minimizes the MFPT, depending on the position of the absorbing boundary, have also been studied for confined Brownian particles [48]. Similar transitions at various parameter values (such as resetting rate, potential exponent, etc) which minimize MFPT, have also been studied in models with stochastic resetting and reflecting boundaries [49–53]. Therefore, a natural question arises whether such freezing transitions exist for other random processes, such as for a run and tumble particle (RTP) which closely resembles the bacteria or animal movements.
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

The steady state, relaxation time, extreme values, and first passage properties to the origin (minima of the potential) from a position away from the origin have been previously studied for an RTP in various types of confining potential [40, 54–57]. In particular, for a confining potential of the form $U(x) = \alpha|x|^\nu$, without any absorbing barrier, the steady-state distribution of the position is supported within a finite interval $[-(v_0/(\alpha\nu))^{1/(\nu-1)}, (v_0/(\alpha\nu))^{1/(\nu-1)}]$ for $\nu > 1$ [54], where $v_0$ denotes the self-propulsion speed. Therefore in this case, an RTP starting at the origin cannot reach a target position outside this interval. On the other hand, for $\nu < 1$, the stationary distribution tends to a delta function at the origin, i.e. $P_{\text{st}}(x) = \delta(x)$ [54]. In the marginal case $\nu = 1$, with $v_0 > \alpha$, the stationary state is supported over the entire line $-\infty < x < \infty$ [54]. In this article we study the first passage properties to a location away from the origin for a one-dimensional RTP in a confining potential $U(x) = \alpha|x|$. This is the same potential where the freezing transition of the decay rate $\theta(a)$ was observed for a Brownian particle [47]. A natural question is whether such a freezing transition also occurs for an active particle in the same potential.

Here, we calculate the late time behavior of the survival probability for an RTP in the potential $U(x) = \alpha|x|$ for $-\infty < x < a$, with an absorbing boundary at $x = a > 0$. We find that starting at the origin with the orientation $\sigma(0) = \pm 1$, the survival probability decays exponentially, $S_{\pm}(t, a|0) \sim e^{-\theta(a)t}$, at long times. The location dependent decay exponent $\theta(a)$ indeed undergoes a freezing transition, similar to the one observed in the passive case [47]. There is a critical value of the location $a = a_c = (v_0 - \alpha)\sqrt{v_0^2 - \alpha^2/(2\alpha\gamma)}$ such that $\theta(a)$ is constant for $0 < a < a_c$ and decreases monotonically with increasing $a > a_c$. We also obtain the propagator in the presence of a potential with an absorbing boundary at $x = a$.

The paper is organized as follows. In section 2 we define the model and obtain the survival probability using a backward Fokker-Planck equation approach. In section 3 we present a comparison between the inverse of the MFPT and the decay rate $\theta(a)$. In section 4, we obtain the propagator for the position of the particle in the presence of the absorbing barrier. Finally, we summarize our findings in section 5. We present an alternative derivation of the survival probability of the propagator in appendix C. Some details of the calculations are relegated to appendix D.

## 2. Run and tumble particle in $U(x) = \alpha|x|$ potential

We consider an RTP in one dimension, subjected to an external potential $U(x) = \alpha|x|$. The equation of motion reads

$$ \frac{dx}{dt} = -\alpha \operatorname{sgn}(x) + v_0 \sigma(t), \quad (1) $$

where $\sigma(t)$ is a dichotomous telegraphic noise that alternates between $\pm 1$ intermittently with a constant Poissonian rate $\gamma$. The self-propulsion speed $v_0$ is a constant. There is an absorbing boundary at a position $x = a > 0$. Starting at a position $x_0 < a$, we want
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

<figure><img src="image_4.png" /><figcaption>**Figure 1.** The panel above is a cartoonist's bug-on-a-conveyor-belt representation (top view) of the model. The left and right conveyor belts are running with velocities $+\alpha$ and $-\alpha$ respectively, in the opposite directions, resembling the attractive potential $U(x) = \alpha|x|$. The bug on the conveyor belt is the run and tumble random walker running with a randomly switching velocity $v_0\sigma(t)$ without any slipping. The position of the flag on the ground is the location of the absorbing boundary. The panel below shows a sample path of the random walker.</figcaption></figure>

to obtain the PDF of finding the RTP at position $x$ at time $t$, and the probability that the particle survives (i.e. it does not cross the position $a$) up to time $t$. Note that for any $\alpha > v_0$, the net velocity is always toward the origin, and hence for any $x_0 < a$, the particle moves toward the origin and arrives the origin at a finite time, and subsequently sits at the origin forever. Here, we consider the more interesting case of $\alpha < v_0$. A cartoonist's representation and a sample path of the model are shown in figure 1. As our primary goal in this paper is to investigate the existence of a freezing transition of the survival probability depending on the position of the absorbing boundary, and we present a detailed study of the survival probability in the following sections, postponing the study of the position distribution in section 4.
## 2.1. Survival probability

The survival probability can be most conveniently derived using a backward Fokker-Planck approach. To this end, let $S_±(t, a|x_0)$ be the survival probability, i.e. the probability that an RTP starting from the position $x_0 < a$, with $\sigma(0) = \pm 1$ respectively, does not cross the location $a > 0$ (hence survives) up to time $t$.

Starting with (1), and analyzing the starting move within an initial infinitesimal time interval $[0, dt]$, we obtain the backward Fokker-Planck equation for $S_±(t, a|x_0)$ as [43]

$$ \frac{\partial S_+}{\partial t} = [v_0 - \alpha \operatorname{sgn}(x_0)] \frac{\partial S_+}{\partial x_0} - \gamma S_+ + \gamma S_-, \quad (2) $$

$$ \frac{\partial S_-}{\partial t} = -[v_0 + \alpha \operatorname{sgn}(x_0)] \frac{\partial S_-}{\partial x_0} - \gamma S_- + \gamma S_+, \quad (3) $$

with the initial condition $S_±(0, a|x_0) = 1$. A particle starting at the absorbing boundary with positive velocity, i.e. $\sigma(0) = 1$, crosses the boundary immediately, implying the boundary condition $S_+(t, a|a) = 0$. On the other hand, if the particle starts from the boundary with a negative velocity, i.e. $\sigma(0) = -1$, it does not get absorbed immediately. Therefore, the boundary condition $S_-(t, a|a)$ is not specified. A particle starting from $-\infty$, cannot cross the level $a$ at any finite time, and hence, always survives, yielding the boundary condition $S_±(t, a| -\infty) = 1$.

To proceed, it is convenient to take Laplace transform of equations (2) and (3), with respect to $t$, which gives

$$ [v_0 - \alpha \operatorname{sgn}(x_0)] \tilde{S}'_+ - (\gamma + s) \tilde{S}_+ + \gamma \tilde{S}_- = -1, \quad (4) $$

$$ [v_0 + \alpha \operatorname{sgn}(x_0)] \tilde{S}'_- + (\gamma + s) \tilde{S}_- - \gamma \tilde{S}_+ = 1, \quad (5) $$

where we have already used the initial condition. The boundary conditions become $\tilde{S}_±(s, a| -\infty) = 1/s$ and $\tilde{S}_+(s, a|a) = 0$. Solving (4) and (5) with these boundary conditions we get (see appendix A)

$$ \tilde{S}_{\pm}(s, a|x_0) = \frac{1}{s} \left[ 1 - \tilde{F}_{\pm}(s, a|x_0) \right], \quad (6) $$

where

$$ \tilde{F}_{\pm}(s, a|x_0) = \frac{p}{\mathbb{D}} e^{-[\alpha(\gamma+s)(a-|x_0|)+pa]/(v_0^2-\alpha^2)} \tilde{\psi}_{\pm}(s|x_0), \quad (7) $$

with $p$ and $\mathbb{D}$ are given by

$$ p = \sqrt{v_0^2 (\gamma + s)^2 - \gamma^2 (v_0^2 - \alpha^2)}, \quad (8) $$

$$ \mathbb{D} = v_0 [p - \alpha (\gamma + s)] - \alpha [p - v_0 (\gamma + s)] e^{-2pa/(v_0^2 - \alpha^2)}. \quad (9) $$
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

The functions $\tilde{\psi}_{\pm}(s|x_0)$ in (7) are given by,

(1) for $0 < x_0 \le a$:

$$ \tilde{\psi}_{+}(s|x_0) = \frac{1}{p} \left\{ v_0 [p - \alpha(\gamma + s)] e^{px_0/(v_0^2 - \alpha^2)} - \alpha [p - v_0(\gamma + s)] e^{-px_0/(v_0^2 - \alpha^2)} \right\}, \quad (10) $$

$$ \tilde{\psi}_{-}(s|x_0) = \left[ \frac{v_0(\gamma + s) - p}{p\gamma(v_0 + \alpha)} \right] \left\{ v_0 [p - \alpha(\gamma + s)] e^{px_0/(v_0^2 - \alpha^2)} + \alpha [p + v_0(\gamma + s)] e^{-px_0/(v_0^2 - \alpha^2)} \right\}, \quad (11) $$

(2) for $x_0 < 0$:

$$ \tilde{\psi}_{+}(s|x_0) = (v_0 - \alpha) e^{px_0/(v_0^2 - \alpha^2)} \quad (12) $$

$$ \tilde{\psi}_{-}(s|x_0) = \frac{1}{\gamma} [v_0(\gamma + s) - p] e^{px_0/(v_0^2 - \alpha^2)}. \quad (13) $$

Similar expressions for the Laplace transform of the survival probability, as in (6), have previously been presented in [58]. However, a detailed study of the survival probability in the time domain, including the freezing transition discussed in our paper, was not carried out. Evidently, $\tilde{S}_{\pm}(s, a|x_0)$ can also be obtained by integrating the final position of the propagator with an absorbing boundary condition at $x = a$, as shown in appendix C. We obtain the propagator with the absorbing boundary condition by solving the forward Fokker-Planck equation in section 4.

In the time domain, the survival probability is given by the Bromwich integral

$$ S_{\pm}(t, a|x_0) = \frac{1}{2\pi i} \int_{0^{+}-i\infty}^{0^{+}+i\infty} \tilde{S}_{\pm}(s, a|x_0) e^{st} ds, \quad (14) $$

where $\tilde{S}_{\pm}(s, a|x_0)$ are given by (6). For simplicity, now we consider the initial position $x_0 = 0$, for which

$$ \tilde{F}_{\pm}(s, a|0) = \frac{p}{\mathbb{D}} e^{-[p+\alpha(\gamma+s)]a/(v_0^2-\alpha^2)} \tilde{\psi}_{\pm}(s), \quad (15) $$

with

$$ \tilde{\psi}_{+}(s) = (v_0 - \alpha), \quad (16) $$

$$ \tilde{\psi}_{-}(s) = \frac{1}{\gamma} [v_0(\gamma + s) - p], \quad (17) $$

where $\mathbb{D}$ is given by (9) and $p$ is given by (8). For $a \to \infty$, we have $\tilde{F}_{\pm}(s, a|0) \to 0$. Therefore, $\tilde{S}_{\pm}(s, a|0) \to 1/s$ has a simple pole at $s = 0$, and consequently, $S_{\pm}(t, a|0) = 1$. On the other hand, for any finite $a$, we get $\tilde{F}_{\pm}(s \to 0, a|0) = 1 + O(s)$. Therefore, $s = 0$ is no longer a pole and the most dominant contribution comes from the singularity closer to $s = 0$. Note that $p = 0$ corresponds to a pair of branch-points at

$$ s_{b}^{\pm} = \gamma \left[ -1 \pm \sqrt{1 - (\alpha/v_{0})^{2}} \right]. \quad (18) $$
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

On the other hand, if $\mathbb{D}=0$ has a solution for real $s$ within $(s_b^+, 0)$, then it becomes the most dominant singularity (a simple pole). Expressing $\mathbb{D}$ completely in terms of $p$, we get

$$ \mathbb{D}(p) = v_0 \left[ p - (\alpha/v_0) \sqrt{p^2 + \gamma^2(v_0^2 - \alpha^2)} \right] - \alpha \left[ p - \sqrt{p^2 + \gamma^2(v_0^2 - \alpha^2)} \right] e^{-2pa/(v_0^2 - \alpha^2)}. \quad (19) $$

Note that although $\mathbb{D}(p=0) = 0$, it does not correspond to a pole as $\tilde{F}_{\pm}(s, a|0) \propto p/\mathbb{D}$. It is easy to check that $\mathbb{D}(p) = (v_0 - \alpha)p + \mathcal{O}(1/p)$ as $p \to \infty$. On the other hand, for small $p$, we get

$$ \mathbb{D}(p) = \frac{2\alpha\gamma(a_c - a)}{\sqrt{v_0^2 - \alpha^2}} p + \mathcal{O}(p^2), \quad (20) $$

with

$$ a_c = \frac{(v_0 - \alpha) \sqrt{v_0^2 - \alpha^2}}{2\alpha\gamma}. \quad (21) $$

Therefore, $\mathbb{D}(p)$ for positive $p$, stays always positive for any $a < a_c$, and consequently, the contribution comes from the branch-point $s_b$. On the other hand, for $a > a_c$, $\mathbb{D}(p) < 0$ for small $p$, then at some value $p = p^*$, $\mathbb{D}(p^*) = 0$, and $\mathbb{D}(p) > 0$ for $p > p^*$ (see figure 2). Therefore, for $a > a_c$, $\tilde{F}_{\pm}(s, a|0)$ has a pole at

$$ s^* = -\gamma + \frac{1}{v_0} \sqrt{p^{*2} + \gamma^2(v_0^2 - \alpha^2)}. \quad (22) $$

Note that $p^* \in (0, \alpha\gamma)$, with $p^* \to 0$, as $a \to a_c^+$ and $p^* \to \alpha\gamma$ as $a \to \infty$. Consequently, $s^* \in (s_b^+, 0)$ with $s^* \to s_b^+$ as $a \to a_c^+$ and $s^* \to 0$ as $a \to \infty$. Therefore, we have

$$ S_{\pm}(t, a|0) = \begin{cases} Q_{\pm}(t, a) e^{-|s_b^+|t}, & \text{for } a \le a_c \\ R_{\pm}(a) e^{-|s^+|t} + Q_{\pm}(t, a) e^{-|s_b^+|t} & \text{for } a > a_c. \end{cases} \quad (23) $$

where $R_{\pm}(a) = \lim_{s \to s^*} (s - s^*) \tilde{S}_{\pm}(s, a|0)$ arise from the residue at $s^*$ and are given by

$$ R_{\pm}(a) = \frac{p^* e^{-[p^* + \alpha(\gamma + s^*)]a/(v_0^2 - \alpha^2)} \tilde{\psi}_{\pm}(s^*)}{(-s^*) \mathbb{D}'(p^*) p'(s^*)} \quad (24) $$

The prefactor $Q_{\pm}(t, a)$ comes from the integral around the branch-cut from $s_b^-$ to $s_b^+$, which we determine below.

## 2.2. Leading order decay of the survival probability

Although obtaining the exact analytical expressions for the survival probabilities in the time domain, by integrating (14), seems difficult, one can still obtain their leading order behavior at large times.
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

<figure><img src="image_5.png" /><figcaption>**Figure 2.** Graphical representation of the solution of the equation $D = 0$, where $D$ is given by (19). The trivial solution is $p=0$. There exists a non-trivial solution $p^* \in (0, \alpha\gamma)$ for $a > a_c$.</figcaption></figure>

**Case-1** ($a > a_c$): As discussed previously, after (19), when the absorbing boundary is at $a > a_c$, the decay of the survival probability at large times is dominated by the pole $s^*(a)$ in the interval $(s_b^+, 0)$, and the scaling of the decay is given by $R_\pm(a)e^{-|s^*(a)|t}$.

**Case-2** ($a = a_c$): In the case of $a = a_c$, we have the leading order for $D(p)$ as

$$D(p) = \frac{p^2 \sqrt{v_0^2 - \alpha^2}}{2\alpha\gamma} + O(p^3), \qquad (25)$$

and the pole disappears. Hence, the decay is now dominated by the integral of $\tilde{S}_\pm(s, a|0)e^{st}$ around the branch cut from $s_b^-$ to $s_b^+$, and therefore, dominated by $Q_\pm(t, a_c)e^{-|s_b^+|t}$. We now estimate the leading order dependence of $Q_\pm(t, a_c)$, on $t$, for $a = a_c$. Taking the series expansion of $\tilde{S}_\pm(s, a|0)$ near $s_b^+$ in the right hand side of (14) and integrating (see appendix D for the details) we obtain the leading order $t$-dependence of $Q_\pm(t, a_c)$ as

$$Q_\pm(t, a_c) \sim f_\pm(v_0, \alpha, \gamma, a_c) t^{-1/2}, \qquad (26)$$

where $f_\pm(v_0, \alpha, \gamma, a_c)$ are functions of $v_0, \alpha, \gamma$, and $a_c$, given by

$$f_+(v_0, \alpha, \gamma, a_c) = \frac{\alpha \sqrt{2v_0 \sqrt{v_0^2 - \alpha^2}}}{\sqrt{\pi \gamma} (v_0 + \alpha) (v_0 - \sqrt{v_0^2 - \alpha^2})} \exp \left[ - \frac{a_c \alpha \gamma}{v_0 \sqrt{v_0^2 - \alpha^2}} \right] \qquad (27)$$

$$f_-(v_0, \alpha, \gamma, a_c) = \frac{\alpha \sqrt{2v_0 \sqrt{v_0^2 - \alpha^2}}}{\sqrt{\pi \gamma} \sqrt{v_0^2 - \alpha^2} (v_0 - \sqrt{v_0^2 - \alpha^2})} \exp \left[ - \frac{a_c \alpha \gamma}{v_0 \sqrt{v_0^2 - \alpha^2}} \right] \qquad (28)$$

**Case-3** ($a < a_c$): Similarly, when $a < a_c$, the decay is dominated by the integral of $\tilde{S}_\pm(s, a|0)e^{st}$ in (14) around the branch cut from $s_b^-$ to $s_b^+$ as there is no pole (can
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

be seen from the series expansion of $\mathbb{D}(p)$ in (20)), which is given by $Q_{\pm}(t, a)e^{-|s_b^+|t}$. Taking the series expansion of $\tilde{S}_{\pm}(s, a|0)$ near $s_b^+$ in the right hand side of (14) and integrating (see appendix D for the details) we obtain the leading order $t$-dependence of $Q_{\pm}(t, a)$ as

$$Q_{\pm}(t, a) \sim g_{\pm}(v_0, \alpha, \gamma, a) t^{-3/2}, \qquad (29)$$

where

$$g_{+}(v_0, \alpha, \gamma, a) = \frac{a v_0^{3/2} (v_0^2 - \alpha^2)^{5/4} (v_0 + \sqrt{v_0^2 - \alpha^2}) \exp\left[-\frac{a \alpha \gamma}{v_0 \sqrt{v_0^2 - \alpha^2}}\right]}{\sqrt{2\pi\gamma\alpha^2} \left[(v_0 + \alpha)(v_0 - \alpha)^3 + 4a\alpha\gamma(a\alpha\gamma - (v_0 - \alpha)\sqrt{v_0^2 - \alpha^2})\right]}, \qquad (30)$$

and

$$g_{-}(v_0, \alpha, \gamma, a) = \frac{1}{\sqrt{2\pi\alpha^2}} \left(\frac{v_0}{\gamma}\right)^{3/2} \exp\left[-\frac{a\alpha\gamma}{v_0\sqrt{v_0^2-\alpha^2}}\right] \times \frac{(v_0-\alpha)(v_0^2-\alpha^2)^{1/4}(v_0+\sqrt{v_0^2-\alpha^2})(v_0^2-\alpha^2+a\gamma\sqrt{v_0^2-\alpha^2})}{\left[2v_0(\alpha^3-2a\alpha\gamma\sqrt{v_0^2-\alpha^2})+\alpha^2(4a\gamma(a\gamma+\sqrt{v_0^2-\alpha^2})-\alpha^2)+v_0^4-2\alpha v_0^3\right]}. \qquad (31)$$

Note that although in **Case-2** ($a = a_c$) and **Case-3** ($a < a_c$) the decay rates of the survival probability are dominated by the branch point $s_b^+$, the leading order time-dependence of $Q(t, a)$ are different in these two cases, as the leading singularities are different.

In summary, at large times, we have

$$S_{\pm}(t, a|0) \sim \begin{cases} R_{\pm}(a) e^{-|s^{*}(a)|t} & \text{for } a > a_c \\ f_{\pm}(v_0, \alpha, \gamma, a_c) t^{-1/2} e^{-|s_b^{+}|t} & \text{for } a = a_c \\ g_{\pm}(v_0, \alpha, \gamma, a) t^{-3/2} e^{-|s_b^{+}|t} & \text{for } 0 < a < a_c. \end{cases} \qquad (32)$$

In figures 3 and 4 we plot and compare the survival probabilities obtained from the stochastic simulations, from numerical integration of the inverse Laplace Transform in (14), and the leading order decay from (32). We set $v_0 = 5$, $\alpha = 4$, and $\gamma = 3/8$, for which we get $a_c = 1$. From figures 3(a) and 4(a) we see that, in the regime $a > a_c$, the decay rate of the survival probability decreases as the distance of the absorbing boundary from the origin increases. Figures 3(b) and 4(b) show the decay rates when the absorbing boundary is at the critical location $a_c$. Panels (c) and (d) of figures 3 and 4 plot the decay rates in the regime $a < a_c$ (see figure captions for details). Note from the figures 3(c) and 4(c) that although $S_{\pm}$ have the same decay rate $\theta(a)$ in the regime $a < a_c$, the prefactor $g_{-}$ has a smaller variation with respect to its argument $a$, compared to $g_{+}$.
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Survival probabilities for positive starting velocity $\sigma(0) = +1$. (a) Case $a > a_c$: The discrete points are simulated results and the corresponding dashed lines are the leading order obtained from (32). (b) Case $a = a_c$: The discrete points are simulated results and the corresponding dashed line is the leading order obtained from (32), and the solid line is obtained by numerically integrating (14). (c) Case $a < a_c$: The discrete points are simulated results and the corresponding solid lines are obtained by numerically integrating (14). (d) Case $a < a_c$: The discrete points are simulated results and the corresponding dashed lines are the leading order obtained from (32).</figcaption></figure>

### 3. Comparison of the decay rate $\theta(a)$ and the inverse of the mean first passage time $T_{\pm}(a)$

Since $S_{\pm}(t, a|0) \approx R_{\pm}(a)e^{-\theta(a)t}$ for $a \gg a_c$, (see (32)), the MFPT, $T_{\pm}(a)$, at the absorbing boundary $a$ is given by

$$T_{\pm}(a) = \int_{0}^{\infty} S_{\pm}(t, a|0) \, dt \approx \frac{R_{\pm}(a)}{\theta(a)}. \qquad (33)$$

Therefore, we have $1/T_{\pm}(a) \propto \theta(a)$ for $a \gg a_c$. However, for any value of $a$, using the relation $T_{\pm}(a) = \lim_{s \to 0} \tilde{S}_{\pm}(s, a|0)$, and taking $s \to 0$ limit in (6) we obtain

$$T_{+}(a) = -\frac{a}{\alpha} + \frac{v_{0}(v_{0} + \alpha)}{\alpha^{2}\gamma} \left( e^{\frac{2a\alpha\gamma}{v_{0}^{2}-\alpha^{2}}} - 1 \right) \quad \text{and} \quad T_{-}(a) = T_{+}(a) + \frac{v_{0}}{\alpha\gamma}. \qquad (34)$$

The MFPT has a non-monotonic behavior as a function of $\alpha$. It diverges as $1/\alpha$ as $\alpha \to 0$. On the other hand, $T_{\pm}(a) \sim e^{2a\alpha\gamma/(v_0^2 - \alpha^2)}$ diverges exponentially as $\alpha \to v_0$. There is an
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Survival probabilities for negative starting velocity $\sigma(0) = -1$. (a) Case $a > a_c$: The discrete points are simulated results and the corresponding dashed lines are the leading order obtained from (32). (b) Case $a = a_c$: The discrete points are simulated results and the corresponding dashed line is the leading order obtained from (32), and the solid line is obtained by numerically integrating (14). (c) Case $a < a_c$: The discrete points are simulated results and the corresponding solid lines are obtained by numerically integrating (14). (d) Case $a < a_c$: The discrete points are simulated results and the corresponding dashed lines are the leading order obtained from (32).</figcaption></figure>

optimal value of $\alpha$ where the MFPTs attain minima. We note that in the diffusive limit, $v_0 \to \infty$, $\gamma \to \infty$, keeping $v_0^2/\gamma \to 2D$ finite, the above expressions reduce to the passive case

$$T_+(a) = T_-(a) = -\frac{a}{\alpha} + \frac{2D}{\alpha^2} \left( e^{a\alpha/D} - 1 \right), \qquad (35)$$

as discussed in [47].

In figure 5 we compare $1/T_+(a)$, calculated using (34), with the dominant decay rate $\theta(a)$ in both the regimes $a > a_c$ ($|s^*|$ from (22)) and $a \le a_c$ ($|s_b^+|$ from (18)), i.e.,

$$\theta(a) = \begin{cases} \gamma - \frac{1}{v_0} \sqrt{[p^*(a)]^2 + \gamma^2(v_0^2 - \alpha^2)} & \text{if } a > a_c \\ \gamma \left(1 - \sqrt{1 - (\alpha/v_0)^2}\right) & \text{if } a \le a_c, \end{cases} \qquad (36)$$
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Comparison of the inverse of MFPT $T_{\pm}(a)$, and the decay rate $\theta(a)$. Panels (a) and (b) are for positive and negative starting velocities $\sigma(0) = \pm 1$, respectively. The red dashed lines are the values of $1/T_{\pm}(a)$ obtained from (34). The blue solid lines are the values of $\theta(a)$ both in the regimes $a > a_c$ and $a \le a_c$, obtained from (36) with numerical evaluation of $p^*(a)$ by solving $D(p) = 0$, where $D(p)$ is given in (19). The black vertical dotted lines denote the critical boundary $a = a_c$.</figcaption></figure>

where $p^*(a)$ is the root of the equation $D(p) = 0$ for $a > a_c$, where $D(p)$ is given in (19). Note that in the regime $a \le a_c$ although $\theta(a)$ freezes to a constant value, the inverse of MFPT, $1/T_{\pm}(a)$, keeps on increasing as $a \to 0$. However, for large $a$ in the regime $a \gg a_c$, $\theta(a)$ and $1/T_{\pm}(a)$ have similar values.

### 3.1. Estimation of $1/T_{\pm}(a)$ and $\theta(a)$ for $a \gg a_c$

In figure 5 we plotted the value of $\theta(a)$, for the regime $a > a_c$, from (36), with the values of $p^*$ obtained by solving $D(p) = 0$, numerically, where $D(p)$ is given by (19). In this section we derive an analytical estimate of $\theta(a)$ for the regime $a \gg a_c$ and compare its value with the corresponding value of $1/T_{\pm}(a)$. First, we note that for large $a$, the escape rates, given by (34), reduce to

$$ \frac{1}{T_{+}(a)} \approx \frac{1}{T_{-}(a)} \approx \frac{\alpha^2 \gamma}{v_0^2} \left(1 + \frac{\alpha}{v_0}\right)^{-1} e^{-2a\alpha\gamma/(v_0^2 - \alpha^2)}. \quad (37) $$

To obtain an approximate expression for the decay rate $\theta(a)$ in this large $a$ limit, we find an approximate solution for $s^*(a)$. As $a \to \infty$, from equations (19) and (22) we can approximate $p^* = \alpha\gamma - \epsilon_1$ and $s^*(a) = -\epsilon_2$ with $0 < \epsilon_1, \epsilon_2 \ll 1$. Now, substituting these values of $s^*$ and $p^*$, in terms of $\epsilon_1$ and $\epsilon_2$, in equation $D(p,s) = 0$ ((9)), for large $a$, we obtain

$$ \alpha[\alpha\gamma - \epsilon_1 - v_0\gamma + v_0\epsilon_2] = v_0[-\epsilon_1 + \alpha\epsilon_2]e^{2(\alpha\gamma - \epsilon_1)a/(v_0^2 - \alpha^2)}. \quad (38) $$

Now from (22) we obtain

$$ \epsilon_2 \approx \frac{\alpha}{v_0^2} \epsilon_1 \quad (39) $$
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

Substituting this in (38) we get

$$ \epsilon_1 \approx \alpha\gamma \left(1 + \frac{\alpha}{v_0}\right)^{-1} e^{-2\alpha\gamma a/(v_0^2 - \alpha^2)}. \quad (40) $$

Therefore, $\theta(a) = -s^*(a) = \epsilon_2$ is given by

$$ \theta(a) \approx \frac{\alpha^2 \gamma}{v_0^2} \left(1 + \frac{\alpha}{v_0}\right)^{-1} e^{-2\alpha\gamma a/(v_0^2 - \alpha^2)}. \quad (41) $$

To estimate $R_{\pm}(a)$ for large $a$, we substitute $p^* = \alpha\gamma - \epsilon_1$ and $s^*(a) = -\epsilon_2$ in (24) with $\epsilon_1, \epsilon_2$ given by (40) and (39). This yields

$$ R_{\pm}(a) = 1 - \frac{v_0}{v_0 + \alpha} e^{-2\alpha\gamma a/(v_0^2 - \alpha^2)}. \quad (42) $$

Therefore, in the limit $a \gg a_c$, equations (37), (41), and (42) together gives

$$ \frac{1}{T_{\pm}(a)} \approx \frac{\theta(a)}{R_{\pm}(a)} \approx \theta(a) \approx \frac{\alpha^2 \gamma}{v_0^2} \left(1 + \frac{\alpha}{v_0}\right)^{-1} e^{-2\alpha\gamma a/(v_0^2 - \alpha^2)}. \quad (43) $$

In the diffusive limit, $v_0 \to \infty$, $\gamma \to \infty$, keeping $v_0^2/\gamma \to 2D$ finite, the above expressions reduce to

$$ \frac{1}{T_{\pm}(a)} \approx \theta(a) \approx \frac{\alpha^2}{2D} e^{-\alpha a/D}. \quad (44) $$

Thus in the diffusive limit, when the absorbing boundary is at a large distance, or equivalently, when the height of the potential barrier is large, the escape rate $1/T_{\pm}(a)$ and the decay rate $\theta(a)$ are same, both having the Arrhenius form.

## 4. The propagator

In the previous sections we have studied the survival probabilities of the RTP in the presence of an absorbing boundary. Another relevant quantity is the position distribution of the RTP in the presence of the absorbing boundary. This position distribution is not normalized to unity but is expected to decay exponentially with time. In fact, the temporal dependence of the decay of the position distribution is exactly the same as that of the survival probability as we will see in this section. This is because the forward and backward Fokker-Planck equations possess the same eigenvalue spectrum.

In this section, we study the position distribution of the RTP at any time $t$ in the presence of an absorbing boundary at $x = a$. Let $P_{\pm}(x, t)dx$ be the probability of finding the particle between the positions $x$ and $x+dx$ at time $t$ with respectively positive and negative velocities. These two probability densities satisfy the forward Fokker-Planck equations

$$ \frac{\partial P_{+}}{\partial t} = -\frac{\partial}{\partial x} [(-\alpha \operatorname{sgn}(x) + v_0) P_{+}] - \gamma P_{+} + \gamma P_{-}, \quad (45) $$
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

$$ \frac{\partial P_{-}}{\partial t} = -\frac{\partial}{\partial x} [(-\alpha \operatorname{sgn}(x) - v_0) P_{-}] - \gamma P_{-} + \gamma P_{+}. \quad (46) $$

We consider the initial condition $P_+(x,0) = \beta\delta(x - x_0)$ and $P_-(x,0) = (1 - \beta)\delta(x - x_0)$, with $\beta \in [0,1]$. The appropriate boundary condition for the absorbing wall at $a$, is $P_-(a,t) = 0$ whereas $P_+(a,t)$ is not specified and related to $P_-(a,t)$ through the above equations. Evidently, $P_\pm(-\infty, t) = 0$. Moreover, integrating the above equation around the origin, we find that the solutions for $x < 0$ and $x > 0$ satisfy the matching conditions

$$ (v_0 - \alpha) P_+(0^+, t) = (v_0 + \alpha) P_+(0^-, t), \quad (47) $$

$$ (v_0 + \alpha) P_-(0^+, t) = (v_0 - \alpha) P_-(0^-, t). \quad (48) $$

The Laplace transforms $\tilde{P}_\pm(x,s) = \int_0^\infty e^{-st} P_\pm(x,t)] dt$ satisfy

$$ \frac{\partial}{\partial x} \left[ (v_0 - \alpha \operatorname{sgn}(x)) \tilde{P}_+ \right] + (\gamma + s) \tilde{P}_+ - \gamma \tilde{P}_- = \beta \delta(x - x_0), \quad (49) $$

$$ \frac{\partial}{\partial x} \left[ (v_0 + \alpha \operatorname{sgn}(x)) \tilde{P}_- \right] - (\gamma + s) \tilde{P}_- + \gamma \tilde{P}_+ = -(1 - \beta) \delta(x - x_0). \quad (50) $$

We have to solve these equations separately in the regions $(-\infty, \min[0, x_0])$, $(\min[0, x_0], \max[0, x_0])$, and $(\max[0, x_0], a)$ and then match the solutions at $x=0$ and $x=x_0$. Integrating the equations around $x=0$ and $x=x_0$ gives the matching conditions: (1) for $x_0 \neq 0$ (strictly)

$$ (v_0 - \alpha) \tilde{P}_+(0^+, s) = (v_0 + \alpha) \tilde{P}_+(0^-, s), \quad (51) $$

$$ (v_0 + \alpha) \tilde{P}_-(0^+, s) = (v_0 - \alpha) \tilde{P}_-(0^-, s), \quad (52) $$

and (2) for any $x_0$ (including $x_0 = 0$)

$$ [v_0 - \alpha \operatorname{sgn}(x_0^+)] \tilde{P}_+(x_0^+, s) - [v_0 - \alpha \operatorname{sgn}(x_0^-)] \tilde{P}_-(x_0^-, s) = \beta, \quad (53) $$

$$ [v_0 + \alpha \operatorname{sgn}(x_0^+)] \tilde{P}_-(x_0^+, s) - [v_0 + \alpha \operatorname{sgn}(x_0^-)] \tilde{P}_-(x_0^-, s) = -(1 - \beta). \quad (54) $$

Using the boundary conditions and the four matching conditions in equations (51)–(54) we solve for $\tilde{P}_\pm(x, s)$ (see appendix B for details). Although the solutions of $\tilde{P}_\pm(x, s)$ can be found for any $\beta \in [0,1]$ and $x_0$, here for simplicity, we now consider the case $\beta = 1/2$ and $x_0 = 0$, which corresponds to the initial condition $P_\pm(x, 0) = \frac{1}{2}\delta(x)$. For this case,

$$ \tilde{P}_{+}(x,s) = \frac{1}{4\mathbb{D}} e^{-\alpha(\gamma+s)|x|/(v_{0}^{2}-\alpha^{2})} \times \begin{cases} a_{1}^{+} e^{px/(v_{0}^{2}-\alpha^{2})} & \text{for } x < 0, \\ b_{3}^{+} e^{-px/(v_{0}^{2}-\alpha^{2})} \left[1 - \left(\frac{v_{0}(\gamma+s)-p}{v_{0}(\gamma+s)+p}\right) e^{-2p(a-x)/(v_{0}^{2}-\alpha^{2})}\right] & \text{for } 0 < x < a, \end{cases} \quad (55) $$

and
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

$$ \tilde{P}_{-}(x, s) = \frac{1}{4\mathbb{D}} e^{-\alpha(\gamma+s)|x|/(v_0^2-\alpha^2)} \times \begin{cases} a_1^- e^{px/(v_0^2-\alpha^2)} & \text{for } x < 0, \\ b_3^- e^{-px/(v_0^2-\alpha^2)} \left[1 - e^{-2p(a-x)/(v_0^2-\alpha^2)}\right] & \text{for } 0 < x < a, \end{cases} \quad (56) $$

where $p$ and $\mathbb{D}$ are given in (8) and (9), and

$$ a_1^- = [p + v_0(\gamma + s) + \gamma(v_0 + \alpha)] + [p - v_0(\gamma + s) - \gamma(v_0 + \alpha)]e^{-2ap/(v_0^2-\alpha^2)}, \quad (57) $$

$$ a_1^+ = \frac{v_0(\gamma + s) - p}{\gamma(v_0 + \alpha)} a_1^-, \quad (58) $$

$$ b_3^+ = [p + v_0(\gamma + s) + \gamma(v_0 + \alpha)], \quad (59) $$

$$ b_3^- = \frac{[v_0(\gamma + s) - p]}{\gamma(v_0 + \alpha)} b_3^+. \quad (60) $$

Similar expressions for the Laplace transform of the propagator, as in (55) and (56), have previously been presented in [58]. Let us first consider the limit $a \to \infty$, where as discussed in [54], we have

$$ \tilde{P}_{+}(x, s) = \tilde{P}_{-}(-x, s) = \frac{e^{-[\alpha(\gamma+s)+p]|x|/(v_0^2-\alpha^2)}}{4v_0[p-\alpha(\gamma+s)]} [\gamma(v_0 + \alpha \operatorname{sgn}(x)) + v_0(\gamma+s) + p \operatorname{sgn}(x)]. \quad (61) $$

Note that $\tilde{P}_{\pm}(x, s)$ has a pole at $s=0$, i.e. $p = \alpha\gamma$. This gives the steady state distribution for $t \to \infty$,

$$ P_{+}(x, t \to \infty) = P_{-}(-x, t \to \infty) = \frac{\alpha\gamma e^{-2\alpha\gamma|x|/(v_0^2-\alpha^2)}}{(v_0^2-\alpha^2)} \frac{[v_0 + \alpha \operatorname{sgn}(x)]}{2v_0}. \quad (62) $$

Evidently, the unconditional steady-state distribution of $x$

$$ P(x, t \to \infty) = P_{+}(x, t \to \infty) + P_{-}(x, t \to \infty) = \frac{\alpha\gamma e^{-2\alpha\gamma|x|/(v_0^2-\alpha^2)}}{(v_0^2-\alpha^2)}, \quad (63) $$

which matches with the result obtained in [54], and is plotted in figure 6. For any finite value of $a$, in the limit $t \to \infty$, we have,

$$ P(x, t \to \infty) = P_{+}(x, t \to \infty) + P_{-}(x, t \to \infty) = \lim_{s \to 0} s \left( \tilde{P}_{+}(x, s) + \tilde{P}_{-}(x, s) \right) = 0. \quad (64) $$

The second equality in (64) is obtained using the Tauberian theorem for the Laplace transform. Equation (64) implies that the RTP will be absorbed with probability 1 for any finite position of the absorbing boundary.
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Steady state probability density function for $a \to \infty$. The discrete points are the simulated data from the stochastic differential equation (SDE) (1), and the solid line is the theoretical result given in (63), obtained by solving the corresponding Fokker-Planck equation. We use $v_0 = 5, \alpha = 4, \gamma = 3/8$. The result matches with [54].</figcaption></figure>

To obtain the PDF of the transient state, we can take the inverse Laplace transform of the equations (55) and (56) as done for the calculation of the survival probability in (14). In this case we numerically integrate the Bromwich integral

$$P(x,t) = P_{+}(x,t) + P_{-}(x,t) = \frac{1}{2\pi i} \int_{0^{+}-i\infty}^{0^{+}+i\infty} \left[ \tilde{P}_{+}(x,s) + \tilde{P}_{-}(x,s) \right] e^{st} ds, \quad (65)$$

using the same contour given in figure D1. The analysis is similar to the one carried out in the previous section for the survival probability, and hence, we do not repeat it here. As before, by taking the contributions from the pole (for $a > a_c$) and the branch-point singularities (see figure D1), we obtain $P(x,t)$, which we compare with simulations in figure 7.

Essentially, for large $t$,

$$P_{\pm}(x,t) \sim \begin{cases} \chi_{1}^{\pm}(x) e^{-|s^{*}(a)|t} & \text{for } a > a_c \\ \chi_{2}^{\pm}(x) t^{-1/2} e^{-|s_{b}^{+}|t} & \text{for } a = a_c \\ \chi_{3}^{\pm}(x) t^{-3/2} e^{-|s_{b}^{+}|t} & \text{for } 0 < a < a_c, \end{cases} \quad (66)$$

where the functions $\chi_i^{\pm}(x)$ with $i = 1, 2, 3$, can be obtained explicitly.

We end this section by reminding the readers that $\pm$ in the expressions of $P_{\pm}(x,t)$ refer to the final orientation $\sigma(t) = \pm 1$, whereas in the expressions of the survival probabilities $S_{\pm}(t, a|x_0)$, they refer to the initial orientation $\sigma(0) = \pm 1$. Therefore, a simple integration of the expressions of $P_{\pm}(x,t)$ over $x$ does not give $S_{\pm}(t, a|x_0)$. In order to obtain the survival probabilities, one first needs to obtain the propagators with the initial orientations $\sigma = \pm$, corresponding to $\beta = 1$ and 0, respectively, as discussed in appendix C.
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

<figure><img src="image_10.png" /><figcaption>**Figure 7.** Probability density function at $t = 50$ for $v_0 = 5, \alpha = 4, \gamma = 3/8, a = 0.5$ (panel (a)) and $a = 2$ (panel (b)). The discrete points are the simulated data from the SDE (1) and the solid lines are the theoretical results.</figcaption></figure>

## 5. Conclusion

In this paper, we have studied the survival probability and the position distribution of a one-dimensional run-and-tumble particle in the presence of a confining potential $U(x) = \alpha|x|$ and an absorbing boundary at a finite distance $a$ from the potential minima. We have analytically obtained the survival probability, starting from the origin. At large times $S_{\pm}(t, a|0) \sim e^{-\theta(a)t}$, and we find a strong dependence of the decay rate $\theta(a)$ on the position of the absorbing boundary. We find that there is a critical location, $a_c$, such that the decay rate freezes to a value $\theta(a_c)$ for $a < a_c$. Above this critical value, $\theta(a)$ is monotonically decreasing with increasing value of $a$. We have derived the leading order behaviors of the survival probabilities in both of these regimes. Finally, we have compared the escape rates, defined as the inverse of MFPTs, with the corresponding decay rates. We have shown that in the regime $a \le a_c$, although the decay rates freeze at a critical value $\theta(a_c)$, the escape rates keep on increasing with decreasing $a$. We also obtained the exact expression of the MFPT $T_{\pm}(a)$. For $a \gg a_c$, the two measures of the escape rate, namely $1/T_{\pm}(a)$ and $\theta(a)$, converge to the corresponding van't Hoff-Arrhenius form. We also obtained the propagator $P_{\pm}(x, t)$ in the presence of the absorbing boundary at $x = a$. We verified our analytical results with numerical simulations.

There are many possible extensions and open questions which could be of interest. Our present formulation can be extended to study the effect of randomly switching potentials on the first passage properties of an RTP, as studied in [26, 28, 29] for Brownian particles. There have been some recent studies on stochastic resetting for a two dimensional RTP [20]. However, resetting by switching potentials would be very intriguing as they are much easier to reproduce in experiments. Another interesting question to study is the relaxation to the steady state and the first passage properties of higher dimensional RTPs under confining potentials. Although some studies have been done in this direction [59–61], systematic studies of the first passage properties are still scarce in this domain. Finally, it would be very intriguing to investigate whether similar freezing transitions exist for other types of active models such as active Brownian
particle, Active Ornstein-Uhlenbeck Particle, and direction-reversing active Brownian particles [62].

Acknowledgments

SS thanks A Kundu and SN Majumdar for useful discussions. SKN acknowledges the Leverhulme Trust (RPG-2018-137) for supporting his time and resources for research in the UK. This work was partially undertaken on ARC3, part of the High Performance Computing facilities at the University of Leeds, UK and the Computational Shared Facility at The University of Manchester, UK.

Appendix A. Details of calculations of the survival probability in section 2.1

In this appendix we give some details for the calculation of the survival probability given in section 2.1. To solve for $\tilde{S}_{\pm}(s, a|x_0)$ from (4) and (5), it is convenient to define

$$ \tilde{S}_{\pm}(s, a|x_0) = \frac{1}{s} + \tilde{q}_{\pm}(s, a|x_0), \qquad (A.1) $$

such that $\tilde{q}_{\pm}(s, a|x_0)$ satisfy a set of coupled first order homogeneous equations

$$ \frac{\partial}{\partial x_0} \begin{pmatrix} \tilde{q}_+ \\ \tilde{q}_- \end{pmatrix} = \begin{pmatrix} \frac{\gamma+s}{v_0-\alpha \operatorname{sgn}(x_0)} & \frac{-\gamma}{v_0-\alpha \operatorname{sgn}(x_0)} \\ \frac{\gamma}{v_0+\alpha \operatorname{sgn}(x_0)} & \frac{-(\gamma+s)}{v_0+\alpha \operatorname{sgn}(x_0)} \end{pmatrix} \begin{pmatrix} \tilde{q}_+ \\ \tilde{q}_- \end{pmatrix}. \qquad (A.2) $$

The general solution for $\tilde{q}_{\pm}$ can be readily written down in terms of the eigenvalues of the $2 \times 2$ matrix on the right hand side. The eigenvalues are given by

$$ \lambda_{\pm} = \frac{\alpha \operatorname{sgn}(x_0) (\gamma + s) \pm p}{(v_0^2 - \alpha^2)}, \qquad (A.3) $$

which in turn gives

$$ \tilde{q}_{\pm}(s, a|x_0) = e^{\alpha(\gamma+s)|x_0|/(v_0^2-\alpha^2)} \begin{cases} A_1^{\pm} e^{px_0/(v_0^2-\alpha^2)} & \text{for } x_0 < 0 \\ A_2^{\pm} e^{px_0/(v_0^2-\alpha^2)} + B_2^{\pm} e^{-px_0/(v_0^2-\alpha^2)} & \text{for } x_0 > 0 \end{cases} \qquad (A.4) $$

where $p$ is given by

$$ p = \sqrt{v_0^2 (\gamma + s)^2 - \gamma^2 (v_0^2 - \alpha^2)}, \qquad (A.5) $$

and we have already implemented the boundary condition $\tilde{q}_{\pm}(s, a| -\infty) = 0$. To determine the six constants $\{A_{1,2}^{\pm}, B_2^{\pm}\}$, we substitute the above solutions (obtained in (A.4)) in (4) (or (5)), and obtain

$$ A_1^{-} = A_1^{+} \frac{[v_0(\gamma + s) - p]}{\gamma(v_0 - \alpha)}, \qquad (A.6) $$
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

$$A_2^- = A_2^+ \frac{[v_0(\gamma + s) - p]}{\gamma(v_0 + \alpha)}, \qquad (A.7)$$

$$B_2^- = B_2^+ \frac{[v_0(\gamma + s) + p]}{\gamma(v_0 + \alpha)}. \qquad (A.8)$$

Integrating equations (4) and (5) around the origin gives the continuity relations $\tilde{S}_+(s, a|0^+) = \tilde{S}_+(s, a|0^-)$ and $\tilde{S}_-(s, a|0^+) = \tilde{S}_-(s, a|0^-)$. These relations together with the boundary condition $\tilde{S}_+(s, a|a) = 0$ determine all the constants. Skipping details,

$$A_1^+ = \frac{-1}{s\mathbb{D}} p(v_0 - \alpha) e^{-[p+\alpha(\gamma+s)]a/(v_0^2-\alpha^2)}, \qquad (A.9)$$

$$A_2^+ = \frac{-1}{s\mathbb{D}} v_0 [p - \alpha(\gamma + s)] e^{-[p+\alpha(\gamma+s)]a/(v_0^2-\alpha^2)}, \qquad (A.10)$$

$$B_2^+ = \frac{1}{s\mathbb{D}} \alpha [p - v_0(\gamma + s)] e^{-[p+\alpha(\gamma+s)]a/(v_0^2-\alpha^2)}, \qquad (A.11)$$

where $\mathbb{D}$ is given by

$$\mathbb{D} = v_0 [p - \alpha(\gamma + s)] - \alpha [p - v_0(\gamma + s)] e^{-2pa/(v_0^2 - \alpha^2)}. \qquad (A.12)$$

It is easily seen that substituting these constants in (A.4) and using (A.1) gives (6).

## Appendix B. Details of calculations of the propagator in section 4

In this section we provide details of the solution for the propagators $\tilde{P}_{\pm}(x, s)$ in Laplace domain using the boundary conditions described after (46) and the four matching conditions in equations (51)–(54). Away from the points $x=0$ and $x=x_0$, we get the homogeneous equations (taking Laplace transform of (45) and (46))

$$[v_0 - \alpha \operatorname{sgn}(x)] \tilde{P}'_+ + (\gamma + s) \tilde{P}_+ - \gamma \tilde{P}_- = 0, \qquad (B.1)$$

$$[v_0 + \alpha \operatorname{sgn}(x)] \tilde{P}'_- - (\gamma + s) \tilde{P}_- + \gamma \tilde{P}_+ = 0, \qquad (B.2)$$

which satisfy the general solutions (see the discussion after (A.2))

$$\tilde{P}_+(x, s) = A_i^+ e^{m_+ x} + B_i^+ e^{m_- x}, \qquad (B.3)$$

$$\tilde{P}_-(x, s) = A_i^- e^{m_+ x} + B_i^- e^{m_- x}, \qquad (B.4)$$

where the subscript $i=1,2,3$ represents the three regions $(-\infty, \min[0, x_0])$, $(\min[0, x_0], \max[0, x_0])$, and $(\max[0, x_0], a)$ respectively. Not all of the four constants $A_i^{\pm}$ and $B_i^{\pm}$, in each region, are independent of each other and relation between them can be found by substituting the above solutions in (B.1) and then setting the coefficients of $e^{m_+ x}$ and $e^{m_- x}$ to zero respectively:

$$A_i^- = \frac{A_i^+}{\gamma} [(\gamma + s) + m_+(v_0 - \alpha \operatorname{sgn}(x))], \qquad (B.5)$$

$$B_i^- = \frac{B_i^+}{\gamma} [(\gamma + s) + m_-(v_0 - \alpha \operatorname{sgn}(x))]. \qquad (B.6)$$
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

Substituting the solutions (B.3) and (B.4) in (B.2) and using equations (B.5) and (B.6) determines $m_±$ as the two solutions of

$$ (v_0^2 - \alpha^2) m_{\pm}^2 + 2\alpha(\gamma + s) \operatorname{sgn}(x) m_{\pm} - [(\gamma + s)^2 - \gamma^2] = 0. \quad (B.7) $$

Therefore,

$$ m_{\pm} = \frac{-\alpha(\gamma + s) \operatorname{sgn}(x) \pm p}{v_0^2 - \alpha^2}, \quad (B.8) $$

where

$$ p = \sqrt{v_0^2 (\gamma + s)^2 - \gamma^2 (v_0^2 - \alpha^2)}. \quad (B.9) $$

Using $m_±$, equations (B.5) and (B.6) can be simplified to

$$ A_i^- = A_i^+ \frac{v_0 (\gamma + s) + p}{\gamma [v_0 + \alpha \operatorname{sgn}(x)]}, \quad (B.10) $$

$$ B_i^- = B_i^+ \frac{v_0 (\gamma + s) - p}{\gamma [v_0 + \alpha \operatorname{sgn}(x)]}. \quad (B.11) $$

The boundary condition $\tilde{P}_-(a,s) = 0$ yields

$$ A_3^- = -B_3^- e^{-2pa/(v_0^2 - \alpha^2)}, \quad (B.12) $$

which according to equations (B.10) and (B.11) also implies

$$ A_3^+ = -B_3^+ \left[ \frac{v_0 (\gamma + s) - p}{v_0 (\gamma + s) + p} \right] e^{-2pa/(v_0^2 - \alpha^2)}. \quad (B.13) $$

On the other hand, the boundary condition $\tilde{P}_{\pm}(-\infty, s) = 0$ gives $B_1^{\pm} = 0$. Therefore, the solutions can be expressed as

$$ \tilde{P}_{+}(x, s) = e^{-\alpha(\gamma+s)|x|/(v_0^2-\alpha^2)} \times \begin{cases} A_1^+ e^{px/(v_0^2-\alpha^2)} & \text{if } x < \min(0, x_0) \\ A_2^+ e^{px/(v_0^2-\alpha^2)} + B_2^+ e^{-px/(v_0^2-\alpha^2)} & \text{if } \min(0, x_0) < x < \max(0, x_0) \\ B_3^+ \left[e^{-px/(v_0^2-\alpha^2)} - \left(\frac{v_0(\gamma+s)-p}{v_0(\gamma+s)+p}\right) e^{-p(2a-x)/(v_0^2-\alpha^2)}\right] & \text{if } \max(0, x_0) < x < a \end{cases} \quad (B.14) $$

and

$$ \tilde{P}_{-}(x, s) = e^{-\alpha(\gamma+s)|x|/(v_0^2-\alpha^2)} \times \begin{cases} A_1^- e^{px/(v_0^2-\alpha^2)} & \text{if } x < \min(0, x_0) \\ A_2^- e^{px/(v_0^2-\alpha^2)} + B_2^- e^{-px/(v_0^2-\alpha^2)} & \text{if } \min(0, x_0) < x < \max(0, x_0) \\ B_3^- \left[e^{-px/(v_0^2-\alpha^2)} - e^{-p(2a-x)/(v_0^2-\alpha^2)}\right] & \text{if } \max(0, x_0) < x < a \end{cases} \quad (B.15) $$
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

where $A_1^-, A_2^-, B_2^-$ and $B_3^-$ are related to $A_1^+, A_2^+, B_2^+$, and $B_3^+$, respectively, by

$$A_1^- = A_1^+ \frac{[v_0(\gamma + s) + p]}{\gamma(v_0 - \alpha)}, \qquad (B.16)$$

$$A_2^- = A_2^+ \frac{[v_0(\gamma + s) + p]}{\gamma[v_0 + \alpha \operatorname{sgn}(x_0)]}, \qquad (B.17)$$

$$B_2^- = B_2^+ \frac{[v_0(\gamma + s) - p]}{\gamma[v_0 + \alpha \operatorname{sgn}(x_0)]}, \qquad (B.18)$$

$$B_3^- = B_3^+ \frac{[v_0(\gamma + s) - p]}{\gamma(v_0 + \alpha)}. \qquad (B.19)$$

The remaining four independent constants $A_1^+, A_2^+, B_2^+$ and $B_3^+$ are determined by the four matching conditions in equations (51)–(54).

## Appendix C. Survival probability by integrating the propagator

Let $S_\pm(t, a|x_0)$ be the survival probability starting at position $x_0$, with positive $(\sigma(0) = 1)$, which corresponds to $\beta = 1$ in section 4) and negative $(\sigma(0) = -1)$, which corresponds to $\beta = 0$ in section 4) velocity respectively. Therefore,

$$S_+(t, a|x_0) = \int_{-\infty}^{a} [P_+(x, t) + P_-(x, t)] dx \quad \text{with } \beta = 1, \qquad (C.1)$$

$$S_-(t, a|x_0) = \int_{-\infty}^{a} [P_+(x, t) + P_-(x, t)] dx \quad \text{with } \beta = 0, \qquad (C.2)$$

where the Laplace transform of $P_\pm(x, t)$ are given in (B.14) and (B.15). We can eliminate four of the constants in terms of the others using (B.16)–(B.19). We determine the remaining four constants using the conditions (51)–(54) for arbitrary $\beta$. This determines the Laplace transforms $\tilde{P}_\pm(x, s)$ completely for arbitrary $\beta$. Setting $\beta = 1$ and $0$ respectively, and subsequently integrating $\tilde{P}_+(x, s) + \tilde{P}_-(x, s)$ over $x$, yields the expressions for the Laplace transforms $\tilde{S}_\pm(s, a|x_0) = \int_0^\infty S_\pm(t, a|x_0) e^{-st} dt$, as

$$\tilde{S}_{\pm}(s, a|x_0) = \frac{1}{s} \left[ 1 - \tilde{F}_{\pm}(s, a|x_0) \right], \qquad (C.3)$$

with

$$\tilde{F}_{\pm}(s, a|x_0) = e^{-[\alpha(\gamma+s)(a-x_0)+pa]/(v_0^2-\alpha^2)} \tilde{\psi}_{\pm}(s|x_0). \qquad (C.4)$$

The functions $\tilde{\psi}_\pm(s|x_0)$ in the above expression are given separately for the two cases $0 < x_0 \le a$ and $x_0 < 0$.

For $0 < x_0 \le a$, we have

$$\tilde{\psi}_{+}(s|x_0) = \frac{1}{\mathbb{D}} \left\{ v_0 \left[ p - \alpha(\gamma + s) \right] e^{px_0/(v_0^2 - \alpha^2)} - \alpha \left[ p - v_0(\gamma + s) \right] e^{-px_0/(v_0^2 - \alpha^2)} \right\}, \qquad (C.5)$$
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

$$ \tilde{\psi}_{-}(s|x_0) = \frac{1}{\mathbb{D}} \left[ \frac{v_0(\gamma + s) - p}{\gamma(v_0 + \alpha)} \right] \left\{ v_0[p - \alpha(\gamma + s)] e^{px_0/(v_0^2 - \alpha^2)} + \alpha[p + v_0(\gamma + s)] e^{-px_0/(v_0^2 - \alpha^2)} \right\}, \quad (C.6) $$

where $\mathbb{D}$ is given by (9). It is easily checked that $\tilde{F}_{+}(s,a|a) = 1$, and consequently, $\tilde{S}_{+}(s,a|a) = 0$.

For $x_0 < 0$ we have

$$ \tilde{\psi}_{+}(s|x_0) = \frac{p}{\mathbb{D}}(v_0 - \alpha) e^{px_0/(v_0^2 - \alpha^2)} \quad (C.7) $$

$$ \tilde{\psi}_{-}(s|x_0) = \frac{p}{\gamma \mathbb{D}} [v_0(\gamma + s) - p] e^{px_0/(v_0^2 - \alpha^2)}. \quad (C.8) $$

It is easily checked that $\tilde{F}_{\pm}(s,a|x_0) \to 0$ for both $x_0 \to -\infty$ (for a given $a$) and $a \to \infty$ (for a given $x_0$), and consequently, $\tilde{S}_{\pm}(s,a|x_0) \to 1/s$ i.e. $S_{\pm}(t,a|x_0) \to 1$. On the other hand, for any finite $a$, we have $\tilde{F}_{\pm}(s \to 0, a|x_0) = 1 + O(s)$, and hence, $s=0$ is no longer a pole. We also note the continuity of the $\tilde{F}_{\pm}$ across $x_0=0$, i.e. $\tilde{F}_{+}(s,a|0^{+}) = \tilde{F}_{-}(s,a|0^{-})$ and $\tilde{F}_{-}(s,a|0^{+}) = \tilde{F}_{+}(s,a|0^{-})$.

Thus, we have verified that the Laplace transform $\tilde{S}_{\pm}(s,a|x_0)$ of the survival probabilities, in (C.3)–(C.8) obtained here by integrating the propagator are exactly the same as (6)–(13) obtained earlier in section 2.1 directly from the Backward Fokker–Planck.

## Appendix D. Contour integration for evaluating the survival probability in $(x, t)$ space

The survival probability $S_{\pm}(t,a|x_0)$, for $x_0 = 0$, is given by the Bromwich integral

$$ S_{\pm}(t, a|0) = \frac{1}{2\pi i} \int_{0^{+}-i\infty}^{0^{+}+i\infty} S_{\pm}(s, a|0) e^{st} ds. \quad (D.1) $$

Considering the integral along the contour, as drawn in figure D1. Using Cauchy–Goursat theorem, we obtain

$$ \begin{aligned} & \frac{1}{2\pi i} \int_{0^{+}-i\infty}^{0^{+}+i\infty} S_{\pm}(s,a|0) e^{st} ds \\ = & -\frac{1}{2\pi i} \sum_{j=1}^{11} \int_{C_j} S_{\pm}(s,a|0) e^{st} ds - \frac{1}{2\pi i} \int_{C_R} S_{\pm}(s,a|0) e^{st} ds. \end{aligned} \quad (D.2) $$

It can be shown that the integrals on the contours $C_1$ and $C_3$ are canceled by the integrals along the contours $C_{11}$ and $C_9$, respectively. The integral around the pole $s^*$ (on the contours $C_2$ and $C_{10}$) is given by the residue of $S_{\pm}(s,a|0)$ at $s^*$. The integrals around the branch points $s_b^+$ (on the contours $C_4$ and $C_8$) and $s_b^-$ (on the contour $C_6$)
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

<figure><img src="image_11.png" /><figcaption>**Figure D1.** Construction of the Bromwich contour. Red, green, and blue points are the schematic positions of the branch points $s_b^-$, $s_b^+$, and the pole $s^*$, respectively.</figcaption></figure>

and the integral along the circular arc $C_R$ tend to zero. Therefore, as there is no pole for $a \le a_c$, we obtain

$$ \begin{aligned} & \frac{1}{2\pi i} \int_{-i\infty}^{+i\infty} S_{\pm}(s, a|0) e^{st} ds \\ &= -\frac{1}{2\pi i} \int_{C_5} S_{\pm}(s, a|0) e^{st} ds - \frac{1}{2\pi i} \int_{C_7} S_{\pm}(s, a|0) e^{st} ds \\ &= -\frac{1}{2\pi i} \int_{s_b^+}^{s_b^-} S_{\pm 5}(s, a|0) e^{st} ds - \frac{1}{2\pi i} \int_{s_b^-}^{s_b^+} S_{\pm 7}(s, a|0) e^{st} ds, \end{aligned} \quad (D.3) $$

where $S_{\pm 5}(s, a|0)$ and $S_{\pm 7}(s, a|0)$ are the values of $S_{\pm}(s, a|0)$ on the contours $C_5$ and $C_7$, respectively. For $a > a_c$, as there is a pole $s^*$ of the integrand in the interval $(s_b^+, 0)$, we get

$$ \begin{aligned} & \frac{1}{2\pi i} \int_{0^+ - i\infty}^{0^+ + i\infty} S_{\pm}(s, a|0) e^{st} ds = -\frac{1}{2\pi i} \int_{C_5} S_{\pm}(s, a|0) e^{st} ds - \frac{1}{2\pi i} \int_{C_7} S_{\pm}(s, a|0) e^{st} ds \\ &= -\frac{1}{2\pi i} \int_{s_b^+}^{s_b^-} S_{\pm 5}(s, a|0) e^{st} ds - \frac{1}{2\pi i} \int_{s_b^-}^{s_b^+} S_{\pm 7}(s, a|0) e^{st} ds + R_{\pm} e^{-|s^*|t}. \end{aligned} \quad (D.4) $$
Survival probability and position distribution of a run and tumble particle in $U(x) = \alpha|x|$ potential

When the particle starts with a positive initial velocity, for $j \in \{5, 7\}$, we obtain

$$S_{+j}(s, a|0) = \frac{1}{s} \left( 1 + \frac{q_j(\alpha - v_0) e^{a(q_j - \alpha(\gamma+s)) / (v_0^2 - \alpha^2)}}{v_0 e^{2aq_j / (v_0^2 - \alpha^2)} (q_j - \alpha |s+\gamma|) + \alpha v_0 |s+\gamma| - \alpha q_j} \right), \quad (D.5)$$

and for negative initial velocity we obtain

$$S_{-j}(s, a|0) = \frac{1}{s} \left( 1 + \frac{\frac{q_j}{\gamma} (q_j - (v_0\gamma + s)) e^{a(q_j - \alpha(\gamma+s)) / (v_0^2 - \alpha^2)}}{v_0 e^{2aq_j / (v_0^2 - \alpha^2)} (q_j - \alpha |s+\gamma|) + \alpha v_0 |s+\gamma| - \alpha q_j} \right), \quad (D.6)$$

where $q_5 = -i\sqrt{-\alpha^2\gamma^2 - s^2v_0^2 - 2\gamma sv_0^2}$, and $q_7 = i\sqrt{-\alpha^2\gamma^2 - s^2v_0^2 - 2\gamma sv_0^2}$.

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