The mobile Boolean model: an overview and further results

TL;DR

This paper reviews the mobile Boolean stochastic geometric model, focusing on detection times of fixed sets, providing formulas, special case solutions, and asymptotic results for different dimensions, and suggesting future research directions.

Contribution

It introduces new formulas and solutions for the mobile Boolean model, including detection time distributions and asymptotics, expanding understanding of this stochastic geometric process.

Findings

Derived formulas for detection times in mobile Boolean models

Obtained explicit distributions for detection times of a ball in odd dimensions

Provided asymptotic behavior of detection times in even dimensions

Abstract

This paper offers an overview of the mobile Boolean stochastic geometric model which is a time-dependent version of the ordinary Boolean model in a Euclidean space of dimension $d$. The main question asked is that of obtaining the law of the detection time of a fixed set. We give various ways of thinking about this which result into some general formulas. The formulas are solvable in some special cases, such the inertial and Brownian mobile Boolean models. In the latter case, we obtain some expressions for the distribution of the detection time of a ball, when the dimension $d$ is odd and asymptotics when $d$ is even. Finally, we pose some questions for future research.