Encounter-based model of a run-and-tumble particle with stochastic resetting

Abstract

In this paper we analyze the effects of stochastic resetting on an encounter-based model of an unbiased run-and-tumble particle (RTP) confined to the half-line $[0,\infty)$ with a partially absorbing wall at $x=0$. The RTP tumbles at a constant rate $\alpha$ between the velocity states $\pm v$ with $v>0$. Absorption occurs when the number of collisions with the wall (discrete local time) exceeds a randomly generated threshold $\widehat{\ell}$ with probability distribution $\Psi(\ell)$. The extended RTP model has three state variables, namely, particle position $X(t)\in [0,\infty)$, the velocity direction $\sigma(t)\in\{-1, 1\}$, and the discrete local time $L(t)\in {\mathbb N}$. We initially assume that only $X(t)$ and $\sigma(t)$ reset at a Poisson rate $r$, whereas $L(t)$ is not changed. This implies that resetting is not governed by a renewal process. We use the stochastic calculus of jump processes to derive an evolution equation for the joint probability distribution of the triplet $(X(t),\sigma(t),L(t))$. This is then used to calculate the mean first passage time (MFPT) by performing a discrete Laplace transform of the evolution equation with respect to the local time. We thus find that the MFPT's only dependence on the distribution $\Psi$ is via the mean local time threshold. We also identify parameter regimes in which the MFPT is a unimodal function of both the resetting and tumbling rates. Finally, we consider conditions under which resetting is given by a renewal process and show how the MFPT in the presence of local time resetting depends on the full statistics of $\Psi$.