Wiener sausage

A long, thin Wiener sausage in 3 dimensions

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 Template:Harvs 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 Template:Harvs, and it was used by Template:Harvs to explain results of a Bose–Einstein condensate, with proofs published by Template:Harvs.

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

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→∞).

Template:Harvtxt 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

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

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. Template:Harvtxt, 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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