To restart, or not to restart, that is the question

The mean time taken by a Brownian particle to cover a distance L is L²/2D, with D being its diffusion coefficient. We find that restarts increase this mean time. This is because restarts naturally introduce time overheads which need to be taken into account while addressing the escape properties. In addition, restarts impose a constraint on allowed overhead functions if they are to provide an advantage, that is, a reduced mean escape time. We explicitly study these constraints for Poisson and sharp restarts. Implementing an overhead function to control escape times means that the physical properties of the Brownian particle need to be appropriately modified. When restarts are non-instantaneous, an additional delay is introduced increasing the mean escape time further. Can restarts overcome the delays incurred due to finite return times? To restart, or not to restart, should be the question.