Wiener sausage explained

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)({} b}) 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{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} 2 t/\Gamma((d -2 )/ 2 ) as t tends to infinity. In dimensions 1 and 2 this formula gets replaced by \sqrt{ 8 t/\pi} and 2 {\pi}t/ 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

• • • • • Especially chapter 22.
  • (Reprint of 1964 edition)
• An advanced monograph covering the Wiener sausage.