Space-time percolation and detection by mobile nodes

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Abstract

Consider the model where nodes are initially distributed as a Poisson point process with intensity λ over Rd and are moving in continuous time according to independent Brownian motions. We assume that nodes are capable of detecting all points within distance r of their location and study the problem of determining the first time at which a target particle, which is initially placed at the origin of Rd, is detected by at least one node. We consider the case where the target particle can move according to any continuous function and can adapt its motion based on the location of the nodes. We show that there exists a sufficiently large value of λ so that the target will eventually be detected almost surely. This means that the target cannot evade detection even if it has full information about the past, present and future locations of the nodes. Also, this establishes a phase transition for λ since, for small enough λ, with positive probability the target can avoid detection forever. A key ingredient of our proof is to use fractal percolation and multi-scale analysis to show that cells with a small density of nodes do not percolate in space and time.

Original languageEnglish
Pages (from-to)2416-2461
Number of pages46
JournalAnnals of Applied Probability
Volume25
Issue number5
DOIs
Publication statusPublished - 1 Oct 2015

Keywords

  • Brownian motion
  • Fractal percolation
  • Multi-scale analysis
  • Poisson point process

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