### 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 language | English |
---|---|

Pages (from-to) | 2416-2461 |

Number of pages | 46 |

Journal | Annals of Applied Probability |

Volume | 25 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Oct 2015 |

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### Keywords

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

### Cite this

**Space-time percolation and detection by mobile nodes.** / Stauffer, Alexandre.

Research output: Contribution to journal › Article

*Annals of Applied Probability*, vol. 25, no. 5, pp. 2416-2461. https://doi.org/10.1214/14-AAP1052

}

TY - JOUR

T1 - Space-time percolation and detection by mobile nodes

AU - Stauffer, Alexandre

PY - 2015/10/1

Y1 - 2015/10/1

N2 - 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.

AB - 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.

KW - Brownian motion

KW - Fractal percolation

KW - Multi-scale analysis

KW - Poisson point process

UR - http://www.scopus.com/inward/record.url?scp=84941564478&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1214/14-AAP1052

U2 - 10.1214/14-AAP1052

DO - 10.1214/14-AAP1052

M3 - Article

VL - 25

SP - 2416

EP - 2461

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 5

ER -