Mathematics – Probability
Scientific paper
2011-08-31
Mathematics
Probability
Improved tail bounds and proofs reorganized
Scientific paper
Consider the model where nodes are initially distributed as a Poisson point process with intensity \lambda over R^d and are moving in continuous time according to independent Brownian motions. We assume that nodes are capable of detecting all points within distance r to their location and study the problem of determining the first time at which a target particle, which is initially placed at the origin of R^d, 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 \lambda 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 \lambda since, for small enough \lambda, with positive probability the target can avoid detection forever. Our proof combines ideas from fractal percolation and multi-scale analysis to show that cells with a small density of nodes do not percolate in space and time. We believe this technical result to have wide applicability, since it establishes space-time percolation of many increasing events.
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