Wiener sausage
- For the food sometimes called a Wiener sausage, see hot dog or Vienna sausage.
In the mathematical field of probability, the Wiener sausage, named after Norbert Wiener, is a neighborhood of the trace of a Brownian motion up to a time t (in layman's terms, "path traced by a ball whose center moves along a Brownian trajectory."[1] ). The Lebesgue measure of the neighborhood is called the volume of the Wiener sausage.
The Wiener sausage is one of the simplest non-Markovian functionals of Brownian motion. Its applications include stochastic phenomena including heat conduction.[2] First described in the 1960s by Harry Kesten, Frank Spitzer, and Marina von Neumann Whitman, it was more formally defined in 1974 by Mark Kac and Joaquin Mazdak Luttinger to explain results of a Bose-Einstein condensate, with proof published by M. D. Donsker and S. R. Srinivasa Varadhan.[1]
Further reading
- Jean-François Le Gall, "Fluctuation Results for the Wiener Sausage", Annals of Probability, 1988, volume 16, number 3, pages 991–1018
- M. van den Berg, E. Bolthausen, F. den Hollander, "Moderate deviations for the volume of the Wiener sausage", Annals of Mathematics, 2001, volume 153, pages 355–406
- E. Bolthausen, "On the Volume of the Wiener Sausage", Annals of Probability, 1990, volume 18, number 4, pages 1576–1582
- Uwe Schmock , "Convergence of the normalized one-dimensional wiener sausage path measures to a mixture of brownian taboo processes", Stochastics An International Journal of Probability and Stochastic Processes, Volume 29, Issue 2 February 1990 , pages 171–183
- T. Eisele and R. Lang, "Asymptotics for the wiener sausage with drift", Probability Theory and Related Fields, Volume 74, Number 1 / March, 1987, pages 125–140
- Yuji Hamana, Harry Kesten, " A large-deviation result for the range of random walk and for the Wiener sausage", Probability Theory and Related Fields, Volume 120, Number 2 / June, 2001, Pages 183–208
- A. S. Sznitman, "Some bounds and limiting results for the measure of Wiener sausage of small radius associated with elliptic diffusions", Stochastic processes and their applications, 1987, volume 25, number 1, pages 1–25
- Isaac Chavel, Edgar A. Feldman, "The Lenz shift and Wiener sausage in Riemannian manifolds", Compositio Mathematica, volume 60, number 1, (1986), pages 65–84
- M. D. Donsker and S. R. S. Varadhan, "Asymptotics for the Wiener sausage", Communications in Pure and Applied Mathematics, volume 28 (1975), pages 525–565
References
- ^ a b Natella O'Bryant. "Research of Natella O'Bryant". Abstract of Wiener Sausages and Wiener Patties (50-minute talk), probability seminar, University of California - Irvine, May 2003. Retrieved 2008-03-22.
- ^ F. den Hollander (2001) [1994], "Wiener sausage", Encyclopedia of Mathematics, EMS Press
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