### Abstract

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 |
---|---|

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

*ALEA Latin American Journal of Probability and Mathematical Statistics*,

*10*(2), 813-829.

**The isolation time of Poisson Brownian motions.** / Peres, Yuval; Sousi, Perla; Stauffer, A.

Research output: Contribution to journal › Article

*ALEA Latin American Journal of Probability and Mathematical Statistics*, vol. 10, no. 2, pp. 813-829.

}

TY - JOUR

T1 - The isolation time of Poisson Brownian motions

AU - Peres, Yuval

AU - Sousi, Perla

AU - Stauffer, A

PY - 2013

Y1 - 2013

N2 - 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 particlethat is placed at the origin, namely how long it takes until there is no node of thePoisson point process within distance r of it. In the case when the target particledoes not move, we obtain asymptotics for the tail probability which are tight up toconstants in the exponent in dimension d ≥ 3 and tight up to logarithmic factorsin the exponent for dimensions d = 1, 2. In the case when the target particle isallowed to move independently of the Poisson point process, we show that the beststrategy for the target to avoid isolation is to stay put.

AB - 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 particlethat is placed at the origin, namely how long it takes until there is no node of thePoisson point process within distance r of it. In the case when the target particledoes not move, we obtain asymptotics for the tail probability which are tight up toconstants in the exponent in dimension d ≥ 3 and tight up to logarithmic factorsin the exponent for dimensions d = 1, 2. In the case when the target particle isallowed to move independently of the Poisson point process, we show that the beststrategy for the target to avoid isolation is to stay put.

UR - http://alea.impa.br/english/index_v10.htm

M3 - Article

VL - 10

SP - 813

EP - 829

JO - ALEA Latin American Journal of Probability and Mathematical Statistics

JF - ALEA Latin American Journal of Probability and Mathematical Statistics

SN - 1980-0436

IS - 2

ER -