Wiener 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 was named after Norbert Wiener by 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 , and it was used by to explain results of a Bose–Einstein condensate, with proofs published by .

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
:$W_\delta(t)()$ 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''→∞).

showed that in 3 dimensions the expected value of the volume of the sausage is
:$E(, W_\delta(t), ) = 2\pi\delta t + 4\delta^2\sqrt +4\pi\delta^3/3.$
In dimension ''d'' at least 3 the volume of the Wiener sausage is asymptotic to
:$\delta^ \pi^2t/\Gamma((d-2)/2)$
as ''t'' tends to infinity. In dimensions 1 and 2 this formula gets replaced by $\sqrt$ and $2t/\log(t)$ respectively. , a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compact sets than balls.

References

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* Especially chapter 22.
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* (Reprint of 1964 edition)
* An advanced monograph covering the Wiener sausage.
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{{Stochastic processes

Mathematical physics
Statistical mechanics
Wiener process