Annals of Applied Probability

Approximations of the Wiener sausage and its curvature measures

Jan Rataj, Evgeny Spodarev, and Daniel Meschenmoser

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 16 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1255699545

Digital Object Identifier
doi:10.1214/09-AAP596

Mathematical Reviews number (MathSciNet)
MR2569809

Zentralblatt MATH identifier
1205.60146

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

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

Citation

Rataj, Jan; Spodarev, Evgeny; Meschenmoser, Daniel. Approximations of the Wiener sausage and its curvature measures. Ann. Appl. Probab. 19 (2009), no. 5, 1840--1859. doi:10.1214/09-AAP596. https://projecteuclid.org/euclid.aoap/1255699545


Export citation

References

  • [1] Berezhkovskiĭ, A. M., Makhnovskiĭ, Y. A. and Suris, R. A. (1989). Wiener sausage volume moments. J. Statist. Phys. 57 333–346.
  • [2] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd ed. Birkhäuser, Basel.
  • [3] Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93 418–491.
  • [4] Federer, H. (1969). Geometric Measure Theory. Springer, New York.
  • [5] Fu, J. H. G. (1985). Tubular neighborhoods in Euclidean spaces. Duke Math. J. 52 1025–1046.
  • [6] Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics 113. Springer, New York.
  • [7] Klenk, S., Schmidt, V. and Spodarev, E. (2006). A new algorithmic approach to the computation of Minkowski functionals of polyconvex sets. Comput. Geom. 34 127–148.
  • [8] Kolmogoroff, A. and Leontowitsch, M. (1933). Zur Berechnung der mittleren Brownschen Fläche. Physik. Z. Sowjetunion 4 1–13.
  • [9] Last, G. (2006). On mean curvature functions of Brownian paths. Stochastic Process. Appl. 116 1876–1891.
  • [10] Le Gall, J.-F. (1988). Fluctuation results for the Wiener sausage. Ann. Probab. 16 991–1018.
  • [11] Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.
  • [12] Rataj, J., Schmidt, V. and Spodarev, E. (2009). On the expected surface area of the Wiener sausage. Math. Nachr. 282 591–603.
  • [13] Rataj, J. and Zähle, M. (2003). Normal cycles of Lipschitz manifolds by approximation with parallel sets. Differential Geom. Appl. 19 113–126.
  • [14] Rogers, C. A. (1998). Hausdorff Measures, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [15] Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • [16] Spitzer, F. (1964). Electrostatic capacity, heat flow, and Brownian motion. Z. Wahrsch. Verw. Gebiete 3 110–121.
  • [17] Stachó, L. L. (1976). On the volume function of parallel sets. Acta Sci. Math. (Szeged) 38 365–374.
  • [18] Sznitman, A.-S. (1998). Brownian Motion, Obstacles and Random Media. Springer, Berlin.