Wiener sausage

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A long, thin Wiener sausage in 3 dimensions Wiener process 3d.png
A long, thin Wiener sausage in 3 dimensions
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".

Contents

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

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

Volume

There has been a lot of work on the behavior of the volume (Lebesgue measure) |Wδ(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

as t tends to infinity. In dimensions 1 and 2 this formula gets replaced by and 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.

References