Approximations of the Wiener sausage and its curvature measures

Approximations of the Wiener sausage and its curvature measures

Jan Rataj, Evgeny Spodarev, and Daniel Meschenmoser

Source: Ann. Appl. Probab. Volume 19, Number 5 (2009), 1840-1859.

Abstract

A parallel neighborhood of a path of a Brownian motion is sometimes called the Wiener sausage. We consider almost sure approximations of this random set by a sequence of random polyconvex sets and show that the convergence of the corresponding mean curvature measures holds under certain conditions in two and three dimensions. Based on these convergence results, the mean curvature measures of the Wiener sausage are calculated numerically by Monte Carlo simulations in two dimensions. The corresponding approximation formulae are given.

Primary Subjects: 60J65

Secondary Subjects: 60D05

Keywords: Brownian motion; Euler–Poincaré characteristic; intrinsic volumes; mean curvature measures; Minkowski functionals; parallel neighborhood; polyconvex approximation; tube; Wiener sausage

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Permanent link to this document: http://projecteuclid.org/euclid.aoap/1255699545
Digital Object Identifier: doi:10.1214/09-AAP596
Mathematical Reviews number (MathSciNet): MR2569809
Zentralblatt MATH identifier: 05735976

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