Renewal processes with a trap under stochastic resetting

TL;DR

This paper studies how stochastic resetting affects renewal processes with a trap, deriving the mean lifetime and revealing complex behaviors depending on parameters like resetting rate, sign-keeping probability, and the distribution of intervals.

Contribution

It introduces a model combining renewal processes, stochastic resetting, and trapping, providing a closed-form expression for mean lifetime and analyzing its dependence on key parameters.

Findings

Mean lifetime is finite for all positive resetting rates.

Mean lifetime diverges as resetting rate approaches zero.

Complex extrema in mean lifetime occur depending on the sign-keeping probability.

Abstract

Renewal processes are zero-dimensional processes defined by independent intervals of time between zero crossings of a random walker. We subject renewal processes them to stochastic resetting by setting the position of the random walker to the origin at Poisson-distributed time with rate $r$. We introduce an additional parameter, the probability $\beta$ of keeping the sign state of the system at resetting time. Moreover, we introduce a trap at the origin, which absorbs the process with a fixed probability at each zero crossing. We obtain the mean lifetime of the process in closed form. For time intervals drawn from a L\'evy stable distribution of parameter $\theta$, the mean lifetime is finite for every positive value of the resetting rate, but goes to infinity when $r$ goes to zero. If the sign-keeping probability $\beta$ is higher than a critical level $\beta_c(\theta)$ (and strictly lower than $1$), the mean lifetime exhibits two extrema as a function of the resetting rate. Moreover, it goes to zero as $r^{-1}$ when $r$ goes to infinity. On the other hand, there is a single minimum if $\beta$ is set to one.