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 $\psi (t)$, 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 $\psi (t)$ 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 $\psi (t)$ that holds in arbitrary spatial dimension. Our results show that stochastic resetting prevails over sharp restart for target distributions with exponential or heavier tails.

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• Diffusion
• Nonequilibrium statistical mechanics
• Stochastic processes
• Stochastic resetting
• Stochastic restart