2D target search with boundary-induced resetting

Abstract

Most studies of stochastic search processes with resetting focus on spontaneous resetting events that occur at a random sequence of times typically generated by a Poisson process. One of the major consequences of spontaneous resetting is that it yields a finite mean first passage time (MFPT) in unbounded domains, resulting in an optimal resetting rate that minimises the MFPT. An optimal resetting rate can also occur in bounded domains provided that the reset point is not too far from the target. Recently, there has been growing interest in event-based rather than spontaneous resetting, where resetting is triggered when a specific threshold is reached. Identifying the threshold with a physical non-target boundary then leads to so-called boundary-induced resetting. In this paper, we explore the effects of boundary-induced resetting on single-particle diffusive search in a two-dimensional (2D) bounded domain $Ω$ containing one or more small interior targets. The interior target boundaries are totally absorbing, whereas the exterior boundary $\partial Ω$ is sticky. That is, whenever the particle reaches a point on $\partial Ω$ it remains attached for a random waiting time $τ$ after which it immediately resets to a fixed interior point $\x_0\in Ω$. Using renewal theory, matched asymptotic analysis and Green's functions we calculate the unconditional MFPT for absorption by any of the targets and compare this with the corresponding MFPT of a search process on the same domain with a reflecting surface $\partial Ω$ and no resetting. We show that boundary-induced resetting only reduces the MFPT relative to the no resetting case if the searcher's reset point $\x_0$ is sufficiently close to one of the targets.