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On the Boolean Model of Wiener Sausages

Author

Listed:
  • Rostislav Černý

    (Charles University)

  • Stefan Funken

    (University of Ulm)

  • Evgueni Spodarev

    (University of Ulm)

Abstract

The Boolean model of Wiener sausages is a random closed set that can be thought of as a random collection of parallel neighborhoods of independent Wiener paths in space. It describes e.g. the target detection area of a network of sensors moving according to the Brownian dynamics whose initial locations are chosen in the medium at random. In the paper, the capacity functional of this Boolean model is given. Moreover, the one- and two-point coverage probabilities as well as the contact distribution function and the specific surface area are studied. In $\mathbb{R}^2$ and $\mathbb{R}^3$ , the one- and two-point coverage probabilities are calculated numerically by Monte Carlo simulations and as a solution of the heat conduction problem. The corresponding approximation formulae are given and the error of approximation is analyzed.

Suggested Citation

  • Rostislav Černý & Stefan Funken & Evgueni Spodarev, 2008. "On the Boolean Model of Wiener Sausages," Methodology and Computing in Applied Probability, Springer, vol. 10(1), pages 23-37, March.
    • Handle: RePEc:spr:metcap:v:10:y:2008:i:1:d:10.1007_s11009-007-9031-9
    DOI: 10.1007/s11009-007-9031-9
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    References listed on IDEAS

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    1. Last, Günter, 2006. "On mean curvature functions of Brownian paths," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1876-1891, December.
    2. Sznitman, A. S., 1987. "Some bounds and limiting results for the measure of Wiener sausage of small radius associated with elliptic diffusions," Stochastic Processes and their Applications, Elsevier, vol. 25, pages 1-25.
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    Cited by:

    1. Dirk Erhard & Julián Martínez & Julien Poisat, 2017. "Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster," Journal of Theoretical Probability, Springer, vol. 30(3), pages 784-812, September.

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