DIRECTED SEARCH PROCESS DRIVEN BY LÉVY MOTION WITH STOCHASTIC RESETTING

In this paper, we demonstrate how certain active transport processes in living cells can be modeled based on a directed search process driven by Lévy motion with stochastic resetting. We focus on the motor-driven intracellular transport of vesicles to synaptic targets in the axons and dendrites of neurons, where the restart duration of the search process after reset is finite, and comprises a finite return time and a refractory period. We employ a probabilistic renewal method to calculate the splitting probabilities and conditional mean first passage times (MFPTs) for capture by a finite array of contiguous targets. We consider two different search scenarios: bounded search on the interval $ [0,L] $, where $ L $ denotes the length of the array, with a refractory boundary at $ x=0 $ and a reflecting boundary at $ x=L $ (Model A), and partially bounded search on the half-line (Model B). In the latter case, the probability that the particle cannot find a target in the absence of resetting is nonzero. We show that both models have the same splitting probability, and that increasing the resetting rate $ r $ increases the splitting probability. Furthermore, the MFPTs of Model A are monotonically increasing with respect to $ r $, whereas the MFPTs of Model B are nonmonotone with respect to $ r $, with a minimum at an optimal resetting rate.