Search processes with stochastic resetting and multiple targets

Search processes with stochastic resetting provide a general theoretical framework for understanding a wide range of naturally occurring phenomena. Most current models focus on the first-passage-time problem of finding a single target in a given search domain. Here we use a renewal method to derive general expressions for the splitting probabilities and conditional mean first passage times (MFPTs) in the case of multiple targets. Our analysis also incorporates the effects of delays arising from finite return times and refractory periods. Carrying out a small-r expansion, where r is the mean resetting rate, we obtain general conditions for when resetting increases the splitting probability or reduces the conditional MFPT to a particular target. This also depends on whether π_{tot}=1 or π_{tot}<1, where π_{tot} is the probability that the particle is eventually absorbed by one of the targets in the absence of resetting. We illustrate the theory by considering two distinct examples. The first consists of an actin-rich cell filament (cytoneme) searching along a one-dimensional array of target cells, a problem for which the splitting probabilities and MFPTs can be calculated explicitly. In particular, we highlight how the resetting rate plays an important role in shaping the distribution of splitting probabilities along the array. The second example involves a search process in a three-dimensional bounded domain containing a set of N small interior targets. We use matched asymptotics and Green's functions to determine the behavior of the splitting probabilities and MFPTs in the small-r regime. In particular, we show that the splitting probabilities and MFPTs depend on the "shape capacitance" of the targets.