Stochastic Resetting Prevails Over Sharp Restart for Broad Target Distributions

Resetting has been shown to reduce the completion time for a stochastic process, such as the first passage time for a diffusive searcher to find a target. The time between two consecutive resetting events is drawn from a waiting time distribution ψ(t), which defines the resetting protocol. 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). 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) that holds in arbitrary spatial dimension. Our results show that stochastic resetting prevails over sharp restart for target distributions with exponential or heavier tails.