On the Boolean model of Wiener sausages
type: Item
instance of: scholarly article
title: On the Boolean model of Wiener sausages
zbMATH Open document ID: 1149.60049
author: Rostislav Černý
author: Evgeny Spodarev
author: Stefan A. Funken
published in: Methodology and Computing in Applied Probability
publication date: 2008-08-20T00:00:00Z
review text: Consider a \(d\)-dimensional Brownian path \((W_t)_{t\in[0,T]}\) started at \(0\) and its corresponding Wiener sausage of radius \(r>0\) \[ S_{r,T}=\{y\in \mathbb{R}^d: \|y-W_t\|\leq r\text{ for some }t\in[0,T]\}. \] The Boolean model is the random set \(\Xi\subset\mathbb{R}^d\) obtained as the union of Wiener sausages when each point of a homogeneous Poisson point process serves as the starting point of one Brownian path. By symmetry, the two-point function \[ C(h)=\mathbf{P}[\{0,x\}\subset\Xi]\qquad\text{for \(\|x\|=h\)} \] only depends on the norm of the vector \(x\). It can be computed using the one-point function \(p=\mathbf{P}[0\in\Xi]\), the expected volume of \(S_{r,T}\) (for both quantities there are more or less explicit formulas) and the expected volume of \(S_{r,T}\cap(x+S_{r,T})\). The authors use a Monte Carlo scheme to estimate this latter volume. To this end the Brownian paths are approxiated by random walks with normally distributed increments. They present the estimated curves \(h\mapsto C(h)\) for different values of \(T\) rather than providing numerical values with errors bounds. As an alternative to computing the expected volume of \(S_{r,T}\cap (x+S_{r,T})\), it is enough to compute the expected volume of \(S_{r,T}\cup (x+S_{r,t})\). This, however, has a representation as the integral \[ \int_{\mathbb{R}^d}u(T,y)\,dy, \] where \(u(t,y)\) is the solution of the heat equation with boundary values \(u(t,y)=1\) for \(\|y\|\leq r\) or \(\|y-x\|\leq r\) and \(u(0,y)=0\) for all other \(y\). The authors propagate a finite element scheme to compute \(u\) which is more efficient than the Monte Carlo scheme considered above.
reviewed by: Achim Klenke
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 65M60
zbMATH DE Number: 5314358
zbMATH Keywords: Wiener Sausage
zbMATH Keywords: Boolean model
zbMATH Keywords: Sensor network
zbMATH Keywords: Capacity
zbMATH Keywords: Volume fraction
zbMATH Keywords: Specific surface area
zbMATH Keywords: Covariance function
zbMATH Keywords: Contact distribution function
zbMATH Keywords: Approximation
zbMATH Keywords: Heat Conduction problem
zbMATH Keywords: Finite element method
zbMATH Keywords: Monte Carlo simulation
zbMATH Keywords: Stochastic geometry
describes a project that uses: na14
MaRDI profile type: Publication
full work available at URL: s11009-007-9031-9
OpenAlex ID: W2036731934
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DOI: 10.1007/S11009-007-9031-9
description: scientific article; zbMATH DE number 5314358