Renewal processes with a trap under stochastic resetting

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.