| Citation: |
Yan Xu, Hexin Zhu. DIRECTED SEARCH PROCESS DRIVEN BY LÉVY MOTION WITH STOCHASTIC RESETTING[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 137-159. doi: 10.11948/20230466
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DIRECTED SEARCH PROCESS DRIVEN BY LÉVY MOTION WITH STOCHASTIC RESETTING
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Abstract
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.
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Keywords:
- Stochastic resetting /
- search processes /
- renewal theory /
- Lévy motion
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References
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Yan Xu, Hexin Zhu. DIRECTED SEARCH PROCESS DRIVEN BY LÉVY MOTION WITH STOCHASTIC RESETTING[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 137-159. doi: 10.11948/20230466
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Figure 1.
Schematic representation of particle states: anterograde state with speed
$ v_+ $ $ v_- $ $ R $ $ \tau $ $ \psi(\tau)=\eta e^{-\eta\tau} $ $ r $ -
Figure 2.
Plots of the splitting probability
$ \pi_k^{(r)} $ $ k $ $ k=1,...,10 $ $ v_+ $ $ r=0.1 $ $ x=0 $ $ r=0.1 $ $ x=0.1 $ $ r=5 $ $ x=0 $ $ r=5 $ $ x=0.1 $ $ \lambda=0.1 $ -
Figure 3.
Plots of conditional MFPT
$ T_k^{(r)} $ $ k $ $ k=1,...,10 $ $ v_+ $ $ r=0.1 $ $ x=0 $ $ r=0.1 $ $ x=0.1 $ $ r=1 $ $ x=0 $ $ r=1 $ $ x=0.1 $ $ v_-=1 $ $ \eta=1 $ $ \lambda=0.1 $ -
Figure 4.
Plots of the splitting probability
$ \pi_k^{(r)} $ $ r $ $ k $ $ x=0 $ $ x=0.05 $ $ v_+=5 $ $ v_-=1 $ $ \eta=1 $ $ \lambda=1 $ -
Figure 5.
Plots of conditional MFPT
$ T_k^{(r)} $ $ r $ $ x=0 $ $ x=0.05 $ $ v_+=5 $ $ v_-=1 $ $ \eta=1 $ $ \lambda=1 $ -
Figure 6.
Model B. Plots of conditional MFPT
$ T_k^{(r)} $ $ r $ $ x=0 $ $ x=0.05 $ $ v_+=5 $ $ v_-=1 $ $ \eta=1 $ $ \lambda=15 $