### Abstract

Let the nodes of a Poisson point process move independently in R^d according to Brownian motions. We study the isolation time for a target particle

that is placed at the origin, namely how long it takes until there is no node of the

Poisson point process within distance r of it. In the case when the target particle

does not move, we obtain asymptotics for the tail probability which are tight up to

constants in the exponent in dimension d ≥ 3 and tight up to logarithmic factors

in the exponent for dimensions d = 1, 2. In the case when the target particle is

allowed to move independently of the Poisson point process, we show that the best

strategy for the target to avoid isolation is to stay put.

that is placed at the origin, namely how long it takes until there is no node of the

Poisson point process within distance r of it. In the case when the target particle

does not move, we obtain asymptotics for the tail probability which are tight up to

constants in the exponent in dimension d ≥ 3 and tight up to logarithmic factors

in the exponent for dimensions d = 1, 2. In the case when the target particle is

allowed to move independently of the Poisson point process, we show that the best

strategy for the target to avoid isolation is to stay put.

Original language | English |
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Pages (from-to) | 813-829 |

Number of pages | 17 |

Journal | ALEA Latin American Journal of Probability and Mathematical Statistics |

Volume | 10 |

Issue number | 2 |

Publication status | Published - 2013 |

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## Cite this

Peres, Y., Sousi, P., & Stauffer, A. (2013). The isolation time of Poisson Brownian motions.

*ALEA Latin American Journal of Probability and Mathematical Statistics*,*10*(2), 813-829.