Escape of a lamb to safe haven in pursuit by a lion under restarts

Abstract: We study the escape behavior of a lamb to safe haven pursued by a hungry lion. Identifying the system with a pair of vicious Brownian walkers we evaluate the probability density function for the vicious pair and from there we estimate the distribution of first passage times. The process ends in two ways: either the lamb makes it to the safe haven (success) or is captured by the lion (failure). We find that the conditional distribution for both success and failure possesses a finite mean, but no higher moments exist. This makes it interesting to study these first passage properties of this Bernoulli process under restarts, which we do via Poissonian and sharp restart protocols. We find that under both restart protocols the probability of success exhibits a monotonic dependence on the restart parameters, however, their approach to the case without restarts is completely different. The distribution of first passage times exhibits an exponential decay for the two restart protocols. In addition, the distribution under sharp resetting also exhibits a periodic behavior, following the periodicity of the sharp restart protocol itself.