A multidimensional first-encounter time model for random walk search based on a two-stage stochastic transport process

Mohamed Abd Allah El-Hadidy; Alaa Awad Alzulaibani; Mohamed Abd Allah El-Hadidy; Alaa Awad Alzulaibani

doi:10.3934/math.2026723

AIMS Mathematics

2026, Volume 11, Issue 6: 17722-17765. doi: 10.3934/math.2026723

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Research article

A multidimensional first-encounter time model for random walk search based on a two-stage stochastic transport process

Mohamed Abd Allah El-Hadidy ^{1
,
,},
Alaa Awad Alzulaibani ²

1.
Mathematics Department, Faculty of Science, Tanta University, Tanta 31527, Egypt
2.
Department of Mathematics and Statistics, College of Science in Yanbu, Taibah University, Yanbu Governorate, Saudi Arabia

Received: 07 April 2026 Revised: 23 May 2026 Accepted: 02 June 2026 Published: 17 June 2026
MSC : 60G50, 60J10, 60J45, 60K35, 90B40, 49J20

This paper develops a two-stage stochastic transport model for the detection and tracking of a moving target governed by a multidimensional random walk. In the first stage, the target's initial location is estimated through an optimal search strategy over independent regions, where a truncated bivariate distribution is used to describe prior uncertainty. Explicit analytical expressions for the optimal search distances minimizing the expected detection time are derived. In the second stage, the tracking problem is formulated along intersecting trajectories, where coordinated searchers aim to intercept a stochastically moving target. Closed-form expressions for the first-encounter (first-passage) time and the corresponding tracking distances are obtained. Moreover, sufficient conditions ensuring the finiteness of the expected first-encounter time are established for a bounded controlled tracking framework, where unbiased random walk motion is coupled with coordinated recurrent coverage of the admissible tracking trajectories and a uniform positive encounter probability. The proposed model provides a unified analytical framework for first-passage phenomena in multidimensional stochastic transport systems.
- stochastic transport,
- multidimensional random walks,
- first-passage time,
- target search,
- analytical modeling
Citation: Mohamed Abd Allah El-Hadidy, Alaa Awad Alzulaibani. A multidimensional first-encounter time model for random walk search based on a two-stage stochastic transport process[J]. AIMS Mathematics, 2026, 11(6): 17722-17765. doi: 10.3934/math.2026723

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Abstract

This paper develops a two-stage stochastic transport model for the detection and tracking of a moving target governed by a multidimensional random walk. In the first stage, the target's initial location is estimated through an optimal search strategy over independent regions, where a truncated bivariate distribution is used to describe prior uncertainty. Explicit analytical expressions for the optimal search distances minimizing the expected detection time are derived. In the second stage, the tracking problem is formulated along intersecting trajectories, where coordinated searchers aim to intercept a stochastically moving target. Closed-form expressions for the first-encounter (first-passage) time and the corresponding tracking distances are obtained. Moreover, sufficient conditions ensuring the finiteness of the expected first-encounter time are established for a bounded controlled tracking framework, where unbiased random walk motion is coupled with coordinated recurrent coverage of the admissible tracking trajectories and a uniform positive encounter probability. The proposed model provides a unified analytical framework for first-passage phenomena in multidimensional stochastic transport systems.

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