Wiener sausage

Last updated September 05, 2022

A short, fat Wiener sausage in 2 dimensions WienerSausage.jpg — A short, fat Wiener sausage in 2 dimensions

In the mathematical field of probability, the Wiener sausage is a neighborhood of the trace of a Brownian motion up to a time t, given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after Norbert Wiener by M. D.Donsker and S. R. Srinivasa Varadhan ( 1975 ) because of its relation to the Wiener process; the name is also a pun on Vienna sausage, as "Wiener" is German for "Viennese".

The Wiener sausage is one of the simplest non-Markovian functionals of Brownian motion. Its applications include stochastic phenomena including heat conduction. It was first described by FrankSpitzer ( 1964 ), and it was used by MarkKac and Joaquin Mazdak Luttinger ( 1973 , 1974 ) to explain results of a Bose–Einstein condensate, with proofs published by M. D.Donsker and S. R. Srinivasa Varadhan ( 1975 ).

Definitions

The Wiener sausage W_δ(t) of radius δ and length t is the set-valued random variable on Brownian paths b (in some Euclidean space) defined by

W_{\delta }(t)({b})

is the set of points within a distance δ of some point b(x) of the path b with 0≤x≤t.

Volume of the Wiener sausage

There has been a lot of work on the behavior of the volume (Lebesgue measure) |W_δ(t)| of the Wiener sausage as it becomes thin (δ→0); by rescaling, this is essentially equivalent to studying the volume as the sausage becomes long (t→∞).

Spitzer (1964) showed that in 3 dimensions the expected value of the volume of the sausage is

E(|W_{\delta }(t)|)=2\pi \delta t+4\delta ^{2}{\sqrt {2\pi t}}+4\pi \delta ^{3}/3.

In dimension d at least 3 the volume of the Wiener sausage is asymptotic to

\delta ^{d-2}\pi ^{d/2}2t/\Gamma ((d-2)/2)

as t tends to infinity. In dimensions 1 and 2 this formula gets replaced by ${\sqrt {8t/\pi }}$ and $2{\pi }t/\log(t)$ respectively. Whitman (1964), a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compact sets than balls.

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References

Donsker, M. D.; Varadhan, S. R. S. (1975), "Asymptotics for the Wiener sausage", Communications on Pure and Applied Mathematics, 28 (4): 525–565, doi:10.1002/cpa.3160280406
Hollander, F. den (2001) [1994], "Wiener sausage", Encyclopedia of Mathematics , EMS Press
Kac, M.; Luttinger, J. M. (1973), "Bose-Einstein condensation in the presence of impurities", J. Math. Phys., 14 (11): 1626–1628, Bibcode:1973JMP....14.1626K, doi:10.1063/1.1666234, MR 0342114
Kac, M.; Luttinger, J. M. (1974), "Bose-Einstein condensation in the presence of impurities. II", J. Math. Phys., 15 (2): 183–186, Bibcode:1974JMP....15..183K, doi:10.1063/1.1666617, MR 0342115
Simon, Barry (2005), Functional integration and quantum physics, Providence, RI: AMS Chelsea Publishing, ISBN 0-8218-3582-3, MR 2105995 Especially chapter 22.
Spitzer, F. (1964), "Electrostatic capacity, heat flow and Brownian motion", Probability Theory and Related Fields, 3 (2): 110–121, doi:10.1007/BF00535970, S2CID 198179345
Spitzer, Frank (1976), Principles of random walks, Graduate Texts in Mathematics, vol. 34, New York-Heidelberg: Springer-Verlag, p. 40, MR 0171290 (Reprint of 1964 edition)
Sznitman, Alain-Sol (1998), Brownian motion, obstacles and random media, Springer Monographs in Mathematics, Berlin: Springer-Verlag, doi:10.1007/978-3-662-11281-6, ISBN 3-540-64554-3, MR 1717054 An advanced monograph covering the Wiener sausage.
Whitman, Walter William (1964), Some Strong Laws for Random Walks and Brownian Motion, PhD Thesis, Cornell U.

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Wiener sausage

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Definitions

Volume of the Wiener sausage

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References