On the Boolean Model of Wiener Sausages

摘要

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.