Journal of the Mathematical Society of Japan

On the expected volume of the Wiener sausage

Yuji HAMANA

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Abstract

We consider the expected volume of the Wiener sausage on the time interval [0,t] associated with a closed ball. Let L(t) be the expected volume minus the volume of the ball. We obtain that L(t) is asymptotically equal to a constant multiple of t1/2 as t tends to 0 and that it is represented as an absolutely convergent power series of t1/2 for any t > 0 in the odd dimensional cases. Moreover, the explicit form of L(t) can be given in five and seven dimensional cases.

Article information

Source
J. Math. Soc. Japan, Volume 62, Number 4 (2010), 1113-1136.

Dates
First available in Project Euclid: 2 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1288703099

Digital Object Identifier
doi:10.2969/jmsj/06241113

Mathematical Reviews number (MathSciNet)
MR2761916

Zentralblatt MATH identifier
1223.60066

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 44A10: Laplace transform 30B10: Power series (including lacunary series)

Keywords
Wiener sausage power series Laplace transform Bessel process

Citation

HAMANA, Yuji. On the expected volume of the Wiener sausage. J. Math. Soc. Japan 62 (2010), no. 4, 1113--1136. doi:10.2969/jmsj/06241113. https://projecteuclid.org/euclid.jmsj/1288703099


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References

  • A. M. Berezhkovskii, Yu. A. Makhnovskii and R. A. Suris, Wiener sausage volume moments, J. Math. Phys., 57 (1989), 333–346.
  • M. van den Berg, E. Bolthausen and F. den Hollander, Moderate deviations for the volume of the Wiener sausage, Ann. of Math. (2), 153 (2001), 355–406.
  • A. N. Borodin and P. Salminen, Handbook of Brownian Motion, Birkhäuser, Basel, 1996.
  • R. K. Getoor, Some asymptotic formulas involving capacity, Z. Wahr. Verw. Gebiete, 4 (1965), 248–252.
  • Y. Hamana and H. Kesten, A large deviation result for the range of random walks and for the Wiener sausage, Probab. Theory Related Fields, 120 (2001), 183–208.
  • G. A. Hunt, Some theorems concerning Brownian motion, Trans. Amer. Math. Soc., 81 (1956), 294–319.
  • K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths, Springer-Verlag, Berlin, 1974.
  • J.-F. Le Gall, Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan, Séminaire de Probabilitiés X\!I\!X, Lecture Notes in Math., 1123, Springer-Verlag, Berlin, 1985, pp.,314–331.
  • J.-F. Le Gall, Fluctuation results for the Wiener sausage, Ann. Probab., 16 (1988), 991–1018.
  • J.-F. Le Gall, Sur une conjecture de M. Kac, Probab. Theory Related Fields, 78 (1988), 389–402.
  • W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed., Springer-Verlag, Berlin, 1966.
  • A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, 4: Direct Laplace Transforms, Gordon Breach Science Publishers, New York, 1992.
  • A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, 5: Inverse Laplace Transforms, Gordon Breach Science Publishers, New York, 1992.
  • F. Spitzer, Electrostatic capacity, heat flow and Brownian motion, Z. Wahr. Verw. Gebiete, 3 (1964), 110–121.