The isolation time of Poisson Brownian motions

Yuval Peres, Perla Sousi, A Stauffer

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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.
Original languageEnglish
Pages (from-to)813-829
Number of pages17
JournalALEA Latin American Journal of Probability and Mathematical Statistics
Issue number2
Publication statusPublished - 2013


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