LMP Seminar: Stochastic resetting prevails over sharp restart for broad target distributions

When searching for a hidden target it is often best to stop and start again from the beginning i.e. to reset. The time between two consecutive resetting events is drawn from a waiting time distribution, ψ(t), which defines the resetting protocol. But how regularly should we reset? Previously, it has been shown that deterministic resetting process with a constant time period, referred to as sharp restart, can minimize the mean first passage time to a fixed target. Here we consider the more realistic problem of a target positioned at a random distance R from the resetting site, selected from a given target distribution P_T (R). To illustrate the scenario, let’s think of the everyday example of searching for one's keys (the target). Usually when we go to retrieve our keys they are not there and are instead located at some random distance from where they should be, how far depending on how untidy we are. We show that if we are sufficiently untidy then resetting at random intervals rather than fixed intervals is best. In particular, we introduce the notion of a conjugate target distribution to a given waiting time distribution. The conjugate target distribution, P_T^* (R), is that P_T (R) for which ψ(t) extremizes the mean time to locate the target. In the case of diffusion we derive an explicit expression for P_T^* (R), conjugate to a given ψ(t) which holds in arbitrary spatial dimension. Our results show that stochastic resetting prevails over sharp restart for target distributions with exponential or heavier tails.

M. R. Evans, and S. Ray*, Phys. Rev. Lett. accepted, (2025). [https: //doi.org/10.1103/2wpq-gl2j]