Skip to main content
SpringerLink
Log in

On an Upper Bound of the Euler Characteristic of the Wiener Sausage

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

We study the asymptotic number of the connected components of the complement of a Wiener sausage in the plane. We prove the statement on the limit behaviour of the number of the connected components of the complement of a Wiener sausage with dependance on its radius. As the corollary we obtain the upper bound of the Euler characteristic of the Wiener sausage in the plane.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Borodin AN, Salminen P (2002) Handbook of Brownian motion - facts and formulae. Birkhauser, Basel

    Book  MATH  Google Scholar 

  • Berezhkovskii AM, Makhnovskii YA, Suris RA (1989) Wiener sausage volume moments. J Stat Phys 57:333–346

    Article  MathSciNet  Google Scholar 

  • Fu JHG (1985) Tubular neighborhoods in Euclidean spaces. Duke Math J 52:1025–1046

    Article  MATH  MathSciNet  Google Scholar 

  • Honzl O, Rataj J (2012) Almost sure asymptotic behaviour of the r-neighbourhood surface area of Brownian paths. Czechoslov Math J 62(1):67–75

    Article  MATH  MathSciNet  Google Scholar 

  • Kolmogoroff A, Leontowitsch M (1933) Zur Berechnung der mittleren. Brownschen Fläche Physik Z Sowjetunion 4:1–13

    Google Scholar 

  • Le Gall JF (1988) Fluctuation results for the Wiener sausage. Ann Probab 16:991–1018

    Article  MATH  MathSciNet  Google Scholar 

  • Le Gall JF (1991) On the connected components of the complement of a two-dimensional Brownian path. In: Durrett R, Kesten H (eds) Random walks, Brownian motion and interacting particle systems. Prog. Probab., vol 28, pp 323–338

  • Mountford T S (1989) On the asymptotic number of small components created by planar Brownian motion. Stochastics Stochastics Rep 28:177–188

    Article  MATH  MathSciNet  Google Scholar 

  • Rataj J, Schmidt V, Spodarev E (2009a) On the expected surface area of the Wiener sausage. Math Nachr 282:591–603

    Article  MATH  MathSciNet  Google Scholar 

  • Rataj J, Spodarev E, Meschenmoser D (2009b) Approximations of the Wiener sausage and its curvature measures. Ann Appl Probab 19(5):1840–1859

    Article  MATH  MathSciNet  Google Scholar 

  • Spitzer F (1964) Electrostatic capacity, heat flow and Brownian motion. Z Wahrscheinlichkeitstheorie verw Geb 3:110–121

    Article  MATH  MathSciNet  Google Scholar 

  • van den Berg M, Bolthausen E, den Hollander F (2001) Moderate deviations for the volume of the Wiener sausage. Ann Math 153(2):355–406

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. KPMS, Sokolovská 83, 186 75, Prague 8, Czech Republic

    Ondřej Honzl

Authors
  1. Ondřej Honzl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ondřej Honzl.

Additional information

Author was supported by grants SVV 265317/2012, GAČR 201/10/J039 and SVV 265315/2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Honzl, O. On an Upper Bound of the Euler Characteristic of the Wiener Sausage. Methodol Comput Appl Probab 16, 331–353 (2014). https://doi.org/10.1007/s11009-013-9361-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-013-9361-8

Keywords

AMS 2000 Subject Classifications

Search

Navigation