Wiener sausage

From Wikipedia, the free encyclopedia

For the food sometimes called a Wiener (sausage), see hot dog or Vienna sausage.

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 is named after Norbert Wiener^[1] because of its relation to the Wiener process; the name is also a pun on Vienna sausage, as "Wiener" means "Viennese" in German.

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 Frank Spitzer (1964), and it was used by Mark Kac 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).

Contents 1 Definitions 2 The volume of the Wiener sausage 3 Footnotes 4 References

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

[edit] The 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

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

δ^{d − 2}π^{d / 2}2t / Γ((d − 2) / 2)

as t tends to infinity. In dimensions 1 and 2 this formula gets replaced by and 2πt / log(t) respectively.

Whitman (1964), a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compacts sets than balls.

[edit] Footnotes

[edit] References

• Donsker, M. D. & Varadhan, S. R. S. (1975), “Asymptotics for the Wiener sausage”, Communications in Pure and Applied Mathematics 28: 525–565, DOI 10.1002/cpa.3160280406 
• Hollander, F. den (2001), “Wiener sausage”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Kac, M. & Luttinger, J. M. (1973), “Bose-Einstein condensation in the presence of impurities.”, J. Mathematical Phys. 14: 1626--1628, MR0342114, DOI 10.1063/1.1666234 
• Kac, M. & Luttinger, J. M. (1974), “Bose-Einstein condensation in the presence of impurities. II.”, J. Mathematical Phys. 15: 183--186, MR0342115, DOI 10.1063/1.1666617 
• Simon, Barry (2005), Functional integration and quantum physics, Providence, RI: AMS Chelsea Publishing, MR2105995, ISBN 0-8218-3582-3  Especially chapter 22.
• Spitzer, F. (1964), “Electrostatic capacity, heat flow and Brownian motion”, Probability Theory and Related Fields 3: 110–121, DOI 10.1007/BF00535970 
• Spitzer, Frank (1976), Principles of random walks, vol. 34, Graduate Texts in Mathematics, New York-Heidelberg,: Springer-Verlag, p. 40, MR0171290  (Reprint of 1964 edition)
• Sznitman, Alain-Sol (1998), Brownian motion, obstacles and random media, Springer Monographs in Mathematics, Berlin: Springer-Verlag, MR1717054, ISBN 3-540-64554-3  An advanced monograph covering the Weiner sausage.
• Whitman, Walter William (1964), Some Strong Laws for Random Walks and Brownian Motion, PhD Thesis, Cornell U.